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/429464 | 5 | $\DeclareMathOperator{\graph}{\operatorname{graph}}$
I would like to know if, given $f\in W^{1,2}(\mathbb{R}^n,\mathbb{R})$, it is true that we can always cover $\graph(f)\subset\mathbb{R}^{n+1}$ with the images of countably many Lipschitz maps $g\_k:\mathbb{R}^n\to\mathbb{R}^{n+1}$ and a set $E$ with $\mathscr{H}^n(E)... | https://mathoverflow.net/users/351083 | Is the support of a Sobolev function a varifold? | Yes if you choose a suitable representative of a Sobolev function.
**Lemma.** *Let $f\in W^{1,p}(\mathbb{R}^n)$, $1\leq p<\infty$. Then for every $\epsilon>0$, there is a Lipschitz function $g:\mathbb{R}^n\to\mathbb{R}$ such that
$$
|\{ x\in\mathbb{R}^n:\, f(x)\neq g(x)\}|<\epsilon.
$$*
The proof follows from the poi... | 7 | https://mathoverflow.net/users/121665 | 429467 | 174,031 |
https://mathoverflow.net/questions/429452 | 6 | Say that $\{a\_n\}\_{n\geq 1}$, $|a\_n|\leq 1$, are such that $$\left|\sum\_{n\leq x} a\_n \log \frac{x}{n}\right|\leq \epsilon x\quad\text{for all $x\geq x\_0$.}$$ What sort of bound can we deduce on $S(x)=\sum\_{n\leq x} a\_n$?
---
*Naïve answer.*
It is easy to give a simple bound:
since, for $y>1$, $$\begin{al... | https://mathoverflow.net/users/398 | (Explicit) Tauberian theorems: removing $(\log x/n)$ | $\newcommand\ep\epsilon$You cannot improve the upper bound $c\sqrt\ep\,x$ on $|S(x)|$ (where $c>0$ is a universal real constant factor) by more than a universal positive real constant factor.
Indeed, take any $\ep\in(0,1/4)$ and let
$$a\_n:=\sum\_{j\ge1}(-1)^j1(c^{j-1}<n<c^j),\quad c:=e^{\sqrt\ep}.$$
Let
\begin{equat... | 8 | https://mathoverflow.net/users/36721 | 429469 | 174,032 |
https://mathoverflow.net/questions/427693 | 4 | This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple, but keeps eluding each of my attempts when we get into the finer details.
Let's start by calling:
$$
f(z) = e^{z}-... | https://mathoverflow.net/users/133882 | Borel summation and the Abel function of $e^z-1$ | So, there's been some developments on the tetration forum, and I came across two papers:
*Introduction to 1-summability and the resurgence theory*, David Sauzin
<https://hal.archives-ouvertes.fr/hal-00860032v1/document#cite.LY>
And:
*The Fixed Point of the Parabolic Renormalization Operator*,
Oscar Lanford III,... | 2 | https://mathoverflow.net/users/133882 | 429477 | 174,034 |
https://mathoverflow.net/questions/429471 | 4 | **Question.** Let $\mathcal{C}$ be a small abelian category. Does the category $\mathrm{Ind}(\mathcal{C})$ of ind-objects of $\mathcal{C}$ have enough injectives?
I have seen many times that $\mathrm{Ind}(\mathcal{C})$ automatically has enough injectives. For example, Proposition 5 of [Akhil Mathew's note](https://am... | https://mathoverflow.net/users/490435 | Does $\mathrm{Ind}(\mathcal{C})$ have enough injectives, if $\mathcal{C}$ is an abelian category? | As Dan Petersen said in the comments, $Mod(k)$ isn't small. Note that even in this case, $Ind(Mod(k))$ only has *small* direct sums, and in particular you cannot take the direct sum that appears in point 2. of your proof sketch (I'm assuming you meant direct sum, and not tensor product); and so you don't have that gene... | 7 | https://mathoverflow.net/users/102343 | 429492 | 174,036 |
https://mathoverflow.net/questions/429497 | 18 | For $n\geq 4$, let $V\_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the problem of computing $V\_n$ has been extensively studied and has a standard name: what is this name?
For which values of $... | https://mathoverflow.net/users/17064 | Known configurations maximizing the volume of the convex hull of n points on the unit sphere | The problem is elementary for $n=5$.
We may regard that case as a combination of two triangular pyramids sharing a base $\triangle ABC$. Then the volume is always bounded by one-third the area of the common base times the distance between the two remaining points $D,E$ (the latter is always greater than or equal to t... | 11 | https://mathoverflow.net/users/86625 | 429498 | 174,037 |
https://mathoverflow.net/questions/429506 | 3 | Let $G=GL\_n(\mathbb{F}\_q)$ be the (finite) group of all linear invertible transformations of the vector space $(\mathbb{F}\_q)^n$ over the finite field $\mathbb{F}\_q$.
$G$ acts naturally on the Grassmannian $Gr\_{k,n}$ of $k$-dimensional linear subspaces of $(\mathbb{F}\_q)^n$.
**What is the length of the corres... | https://mathoverflow.net/users/16183 | Length of representation of $GL_n(\mathbb{F}_q)$ in functions on Grassmannian | I believe that this is answered in Proposition 5.1 of [this](https://arxiv.org/abs/1811.08675) paper, which says that it is $\text{min}(k,n-k) + 1$.
| 5 | https://mathoverflow.net/users/317 | 429510 | 174,039 |
https://mathoverflow.net/questions/429473 | 0 | Computation model is defined as Hartmanis and Stearns [4](https://www.ams.org/journals/tran/1965-117-00/S0002-9947-1965-0170805-7/S0002-9947-1965-0170805-7.pdf), it is well known that Liouvilles constant
$$C\_L=\sum\_{i=1}^{\infty} 10^{-i!}$$ is computable in real time or linear time [1](https://www.ams.org/journals/tr... | https://mathoverflow.net/users/14024 | Examples of real-time transcendental number and superlinear-time trancsendental number | Too long for a comment, though not a complete answer, sorry.
I don't understand the exact influence of allowing multiple tapes on complexity for Turing machines, but let me offer the following anyway.
Any number whose binary expansion is a so-called Sturmian or Beatty sequence (definition below) is transcendental; ... | 2 | https://mathoverflow.net/users/116357 | 429516 | 174,040 |
https://mathoverflow.net/questions/429518 | 3 | By a theorem of Hagemann and Mitschke, a condition (A) that a variety $\mathcal{V}$ is congruence $n$-permutable, is equivalent to a condition (B) that there exist ternary terms $p\_1,\dots,p\_{n-1}$ such that $\mathcal{V}$ satisfies identities
$$x = p\_1(x,y,y),\ \ p\_i(x,x,y) = p\_{i+1}(x,y,y),\ \ p\_{n-1}(x,x,y) = y... | https://mathoverflow.net/users/490477 | Reference request for a proof of the Mal'cev condition for congruence $n$-permutability | The original paper is in English:
Hagemann, Joachim; Mitschke, A.
On $n$-permutable congruences.
Algebra Universalis 3 (1973), 8-12.
The proof given there is only a partial proof which depends
on an earlier paper (in German) by Hagemann.
I did not find the argument in any of the standard textbooks by Burris ... | 1 | https://mathoverflow.net/users/75735 | 429525 | 174,041 |
https://mathoverflow.net/questions/429528 | 1 | Let $\mathcal{H}$ be a separable Hilbert space and let $x\_1,...,x\_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures
$$
\mu := \frac1{n}\,\sum\_{i=1}^n\, \delta\_{x\_i} \mbox{ and }
\mu^{\varepsilon} := \frac1{n}\,\sum\_{i=1}^n\, G(x\_i,\varepsilon\, T).
$$
Here $T$ is any trace-... | https://mathoverflow.net/users/36886 | Distance between empirical measures and thickened version | The claim follows by a synchronous coupling: $X = x\_I$ and $X^{\epsilon} = x\_I + \sqrt{\epsilon} \sum\_{j=1}^{\infty} \sqrt{\lambda\_j} \rho\_j e\_j$ where $I \sim \operatorname{Uniform}(\{1, \dots, n \})$; $e\_j$ are the eigenfunctions of $T$; $\lambda\_j$ are the corresponding eigenvalues; and $\{ \rho\_j \} \overs... | 1 | https://mathoverflow.net/users/64449 | 429532 | 174,043 |
https://mathoverflow.net/questions/428777 | 6 | $\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called *algebraic* if $A=\{x\in X: a\_0xa\_1x\dotsm xa\_n=1\}$ for some elements $a\_0,a\_1,\dotsc,a\_n\in X$.
Let $\mathcal A\_X$ be the family of all algebraic sets in $X$.
**Definition.** The *Steinhaus number* $\Sn(X)$ of an infinite group $X$ is the l... | https://mathoverflow.net/users/61536 | Steinhaus number of a group | The answer to this problem is negative: *For the compact Polish group $X=S\_3^\omega$ we have $Sn(X)\le\mathfrak r$ where $$\mathfrak r=\min\{|\mathcal R|:\mathcal R\subseteq [\omega]^\omega\;\wedge\;\forall f\in 2^\omega\;\;\exists R\in\mathcal R\;\;|f[R]|=1\}.$$*
By induction it can be shown $\mathfrak r=\min\{|\math... | 1 | https://mathoverflow.net/users/61536 | 429535 | 174,045 |
https://mathoverflow.net/questions/429079 | 7 | For $f: \mathbb R^n \to \mathbb R$ a locally integrable function, $\varepsilon \in (0, \infty)$, and $x \in \mathbb R^n$, define $I(f, \varepsilon, x)$ to be the averaged integral of $f$ over $B\_{\varepsilon} (x)$, the ball of radius $\varepsilon$ around $x$. That is,
$$I(f, \varepsilon, x) := \frac{1}{\mu(B\_\varep... | https://mathoverflow.net/users/173490 | Does this "local time" type limit exist a.e. for $C^2$ functions? | Even $C^\infty$ isn't enough and even on $\mathbb R$. Take any nowhere dense compact set $A\subset\mathbb R$ of positive measure and set $f=0$ on $A$. Now let $I\_k$ be the complementary intervals to $A$ (ignore the two rays). Draw some positive bumps on the middle halves of $I\_1, I\_2,\dots I\_n$. If $n$ is large eno... | 3 | https://mathoverflow.net/users/1131 | 429544 | 174,050 |
https://mathoverflow.net/questions/429561 | 1 | I am looking for an example of a function $f:[0,1]\to\mathbb{R}$ which is in $L^p$ for some $p$ and whose graph is not a $1$-dimensional varifold in $\mathbb{R}^2$, that is such that it is not possible to write
$$
\operatorname{graph}(f)\subset \bigcup\_{n\in\mathbb{N}} g\_k(\mathbb{R}) \cup E
$$
for Lipschitz function... | https://mathoverflow.net/users/351083 | $L^p$ function whose graph is not a varifold | While writing this question down I realized that its answer is actually pretty trivial, but I am going to keep it here for the benefit of other users. It suffices to take any measurable function $\tilde{f}:[0,1]\to\mathbb{R}$ whose graph is not a varifold and compose it with a diffeomorphism $\phi$ that sends $\mathbb{... | 2 | https://mathoverflow.net/users/351083 | 429562 | 174,056 |
https://mathoverflow.net/questions/429547 | 2 | $(X, \tau\_X) $ and $(Y, \tau\_Y) $ be two topological spaces.
$\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $.
Question : Does this implies $(Y, \tau\_Y) $ is compact?
---
Notation:
$Y^X$: Set of all functions from $X$ to $Y$.
$C(X, Y) =\{f\in Y^X: f \text{ is continuous }\}$
... | https://mathoverflow.net/users/483536 | (Dis)prove : if every function with closed graph are continuous then the target space is compact | Let me try to answer these questions under the assumption that $Y$ is $T\_1$.
We start with a set $E$ along with a filter $\mathcal F\in\mathcal P(E)$. We can then cook up a topological space $X$ with underlying set $\{x\}\sqcup E$ with topology given by the discrete topology $E^\delta$ on $E$, and $\{x\}\sqcup U$ fo... | 5 | https://mathoverflow.net/users/176381 | 429564 | 174,057 |
https://mathoverflow.net/questions/429579 | 8 | Let $M$ be a compact contractible manifold, $X\subset\partial M$ and $C\_X$ the cone over $X$.
>
> **Question:** Is it true that $C\_X$ embeds in $M$ with its boundary $\partial C\_X$ mapped to $X\subset \partial M$?
>
>
>
I am mostly interested in the piecewise linear case, that is, $M$ is a PL manifold, $X$ ... | https://mathoverflow.net/users/108884 | If $M$ is contractible manifold and $X\subset \partial M$, does the cone over $X$ embed in $M$? | Not in the PL case - this follows from the results of ["Knot concordance in homology cobordisms"](https://arxiv.org/abs/1801.07770) by Hom, Levine, and Lidman.
They prove that for many pairs of a 3-manifold $Y$ and knot $K \subset Y$, any contractible 4-manifold with boundary $Y$ does not contain a PL embedded disc w... | 10 | https://mathoverflow.net/users/33041 | 429581 | 174,064 |
https://mathoverflow.net/questions/429199 | 1 | This might be related to [counting hamiltonian cycles](https://mathoverflow.net/questions/429025/counting-hamiltonian-cycles-in-graph-and-finding-a-coefficient-of-polynomial).
@**Peter Taylor** gave [negative result](https://mathoverflow.net/a/428266/12481) about the one dimensional case, but we believe his attack is... | https://mathoverflow.net/users/12481 | Only trivial solutions to system of linear diophantine equations possibly related to hamiltonian cycles in graphs | Yes, we can get $\exp(\omicron(n))$.
Assume for a moment that $n$ is a perfect square and $m=\sqrt{n}$.
The general case is essentially the same, just a bit more complicated.
The idea is to partition those $n=m^2$ variables $y\_1,\ldots\, y\_n$ in $m$ blocks of $m$ digits in base $b=n+1$. Define
$$a\_{ij} = \left\{... | 2 | https://mathoverflow.net/users/155625 | 429592 | 174,066 |
https://mathoverflow.net/questions/429396 | 2 | The suggested intuition behind mixed Hodge structures - developed
in particular to generalize Hodge decomposition of cohomology
groups from complex smooth complete varieties to more general algebraic varieties - is that one should think the cohomology groups $H^k(X)$ to be endowed with increasing filtrations whose succ... | https://mathoverflow.net/users/108274 | Example motivating mixed Hodge structures | We think of the cycle $\alpha\_i$ as coming from a point, specifically, the point we need to add to compactify the puncture $p\_i$.
Here "come from" refers to the excision exact sequence in compactly supported cohomology
When we have a variety $X$ obtained as the open subset of a variety $\overline{X}$ whose closed... | 3 | https://mathoverflow.net/users/18060 | 429606 | 174,067 |
https://mathoverflow.net/questions/429262 | 2 | Let $\overline{X}$ be a smooth proper curve over $\mathbb{F}\_q$, for some $q$, $S$ a collection of $\mathbb{F}\_q$ points of $\overline{X}$, and set $X=\overline{X}-S$.
For a rank $n$ $\overline{\mathbb{Q}}\_{\ell}$-local system on $X$, it is known that the coefficients of the characteristic polynomials of Frobenii ... | https://mathoverflow.net/users/484855 | Generation of trace fields of Frobenii on local systems | Your proof in the case of four points assumes that the local monodromies at those four points are unipotent.
In general, one needs a bound not just on the set of ramification points but on the breaks/slopes of the sheaf at those points.
To see this is necessary, one can work already in the case of sheaves lisse of ... | 2 | https://mathoverflow.net/users/18060 | 429607 | 174,068 |
https://mathoverflow.net/questions/429604 | 5 | Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}\_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L\_{\varepsilon}$ be a line bundle on $V\_{\varepsilon} = V \times Spec[\mathbb{C}[\varepsilon]/(\varepsilon ^2)$. If we denote by $j\_{\varepsilon... | https://mathoverflow.net/users/45597 | Extension of first order deformations of a line bundle | Under some conditions on $X,V$, your line bundle can be extended to $X\_{\varepsilon}$. Indeed, let $\imath\_X:X\hookrightarrow X\_{\varepsilon}$ and $\imath\_V:V\hookrightarrow V\_{\varepsilon}$ be two closed immersions and $\mathcal{I}\_X,\mathcal{I}\_V$ be the ideal sheaves respectively. By the following exact seque... | 4 | https://mathoverflow.net/users/490443 | 429616 | 174,074 |
https://mathoverflow.net/questions/429588 | 9 | On a scheme, the coherent sheaves that are invertible objects for the tensor product (monoid) operation are precisely the coherent sheaves that are (Zariski) locally free of rank one. Is the same true for algebraic spaces? (I believe that this follows from a theorem of Nisnevich since there is simultaneously an etale l... | https://mathoverflow.net/users/13265 | Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one? | There are counterexamples by Stefan Schröer. One of them is not locally separated (a *bug-eyed cover*, as Kollár calls it), another is a (non-normal) proper surface. See the paper [here](https://doi.org/10.1016/S0022-4049(02)00012-9).
About the link: some characters do not display properly, but one can click "View PD... | 8 | https://mathoverflow.net/users/7666 | 429622 | 174,075 |
https://mathoverflow.net/questions/429614 | 3 | Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$.
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$ ?
I have explained my thoughts [here](https://math.stackexchange.com/q/4470490/977780) on MSE.
| https://mathoverflow.net/users/483536 | Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$? | Yes, there exists such a function: Consider the real line as a linear space over the field $\mathbb Q$ and find a linearly independent Cantor set $C\subseteq \mathbb R$
(using the Kuratowski-Mycielski Theorem 19.1 in Kechris' "Classical Descriptive Set Theory"). Identifying $C$ with $C\times C$, we can write $C$ and th... | 5 | https://mathoverflow.net/users/61536 | 429623 | 174,076 |
https://mathoverflow.net/questions/429619 | 25 | A function $f:X\to X$ on a group $X$ is called a *polynomial* if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a\_0,a\_1,\dots,a\_n\in X$ such that $f(x)=a\_0xa\_1x\cdots xa\_n$ for all $x\in X$. The smallest possible number $n$ in this representation is called the *degree* of the polynomial $f$ and is deno... | https://mathoverflow.net/users/61536 | The number of polynomials on a finite group | $\DeclareMathOperator\Poly{Poly}$**Proposition.** If $G$ is a simple non-abelian finite group, then $\Poly(G)=G^G$.
(Edit: this observation appears as the main therorem in [this paper](https://www.ams.org/journals/proc/1965-016-03/S0002-9939-1965-0175971-0/) by Maurer and Rhodes, Proc. AMS 1965. See also Theorem 2 [h... | 22 | https://mathoverflow.net/users/14094 | 429635 | 174,078 |
https://mathoverflow.net/questions/429633 | 3 | We are given a convex shape $S$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume $V(S)$ of $S$ be $\tfrac12$ (*I guess nothing changes for any other fixed constant in $(0,1)$*).
---
**Question:** How can we prove or disprove that, for all $S$ and all $d\in\mathbb{N}$, th... | https://mathoverflow.net/users/115803 | Bounding the number of facets of a polytope to approximate a given convex shape in higher dimensions | I think ***if we assume the facets are simplexes***, the number of facets of such a polytope must grow more than exponentially, even in the easiest case where $S=[0,1]^d$. Fix a constant $\epsilon>0$.
Choose a finite subset of $\partial( [0,1]^d)$, spanning a polytope $C$ with $\phi\_C$ facets, and volume $V(C)\ge\epsi... | 3 | https://mathoverflow.net/users/6101 | 429639 | 174,079 |
https://mathoverflow.net/questions/429638 | 3 | The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian subvariety translated by a torsion element.
Both Raynaud's proof and the equidistribution proof prove this by reduction t... | https://mathoverflow.net/users/157701 | Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0? | What you want to do is, by induction on the theorem in the number field $K$ case, prove that all torsion points in $\mathcal Z\_K$ lie in a finite union of torsion translates of abelian varieties (contained in $Z\_K$).
It follows that all torsion points in $\mathcal Z\_{\eta}$ specialize to points in a finite union o... | 5 | https://mathoverflow.net/users/18060 | 429640 | 174,080 |
https://mathoverflow.net/questions/429545 | 10 | Suppose $\langle L,\leq\rangle$ is a lattice with join $\sqcup$. Let $F\_1$ and $F\_2$ be principal filters on $L$. Thus, for $i\in I=\{1,2\}$ there are $x\_i\in L$ so that $F\_i=\{y\in L:x\_i\leq y\}$.
In this situation, $F\_1\cap F\_2$ is also a principal filter, because $F\_1\cap F\_2=\{y\in L:x\_1\sqcup x\_2\leq ... | https://mathoverflow.net/users/25415 | Are arbitrary nonempty intersections of principal filters principal? | Nope. For a silly counterexample consider the lattice of nonzero real numbers with the usual order. Consider the principal filters $F\_i = \left\{y \ne 0: -\frac{1}{i} \le y \right\}$. Then $F=\bigcap\_i F\_i $ should be the principal filter generated by zero. But since zero is not in the lattice $F$ is the filter of p... | 6 | https://mathoverflow.net/users/58082 | 429647 | 174,083 |
https://mathoverflow.net/questions/429620 | 0 | Let $K$ be a imaginary quadratic field, $R\_K$ be ring of integers of $K$, and $E/K$ be elliptic curve which has CM over $K$.
Let $\psi\_E$ be Hecke (Grössencharakter) character of $E/K$.
Let fix prime ideal $I＝(\pi)$ of $K$.
Then, why does $[I](P)＝0$ ($P\in E$) imply $[\psi(I)](P)＝0$?
If I could write $\psi(I)$ like... | https://mathoverflow.net/users/144623 | Why does $[I](P)＝0$ ($P\in E$) imply $[\psi(I)](P)＝0$ ? ($\psi$ is Hecke character of elliptic curve) | This is essentially the same as your other recent CM-theory question, in a mild disguise; for both questions the point is that ***$\psi(I)$ is a generator of $I$.*** This follows easily from the fact that $\psi$ takes values in $K$, and for principal ideals $(\lambda)$ with generators sufficiently congruent to 1 we hav... | 2 | https://mathoverflow.net/users/2481 | 429649 | 174,084 |
https://mathoverflow.net/questions/429598 | 3 | Let $V$ be a Riemann surface, $x\in V$, and $B:=B(x,r)$ some small ball (in a local chart). It is well known that there is a meromorphic function $f$ on $V$ with the only pole at $x$. What I’d like to ask of is if there is a meromorphic f on V that has a pole at x and additionally such that $|f|<1$ outside $B$?
| https://mathoverflow.net/users/138007 | Meromorphic function on the Riemann surfaces | For open surfaces, there are counterexamples. The first one was constructed by P. Myrberg:
Ueber die analytische Fortsetzung von beschrankten Funktionen, Ann. Acad. Sci. Fenn., Ser. A. I N:o 58 (1949)
Since this paper is difficult to obtain (and written in German), I refer to another paper
Heins, Maurice,
Riemann... | 7 | https://mathoverflow.net/users/25510 | 429653 | 174,085 |
https://mathoverflow.net/questions/422885 | 1 | Let $\overline L= (L, h)$ be a hermitian $C^
\infty$ line bundle on an arithmetic variety $X\to\operatorname{Spec }\mathbb Z$ (I am reasoning in terms of higher Arakelov geometry, like in Gillet & Soule' papers).
$\overline L$ is said to be arithmetically ample if:
1. $L$ is relatively ample on $X$
2. $L\_{\mathbb ... | https://mathoverflow.net/users/65980 | Arithmetic ampleness and scalings of the metric | Let $\overline{L}$ be any arithmetically ample line bundle. In the way you have written it down, $\overline{L}\_{\alpha}$ is not arithmetically ample for $\alpha$ sufficiently large, more precisely for $\alpha\ge\alpha\_0=\exp\left(\frac{2\overline{L}^{\dim X}}{L\_{\mathbb{C}}^{\dim X-1}}\right)$:
Let $s\in H^0(X,L^{... | 1 | https://mathoverflow.net/users/61532 | 429661 | 174,087 |
https://mathoverflow.net/questions/429648 | 6 | Let $ t>0 $, and we look at the random walk $S\_{n}=\sum\_{i=1}^{n}X\_{n}$ on $\mathbb{Z}$ with $S\_0=0$ where $$ \mathbb{P}\left(X\_{n}=1\right) =\frac{1}{2}\left(1+\frac{1}{n^{t}}\right)
$$ $$ \mathbb{P}\left(X\_{n}=-1\right) =1-\mathbb{P}\left(X\_{n}=1\right)=\frac{1}{2}\left(1-\frac{1}{n^{t}}\right)$$We would like ... | https://mathoverflow.net/users/490586 | Parameterized simple asymmetric random walk | Recall that a random walk (or a Markov chain in general) is called recurrent if it almost surely (a.s.) returns to the initial state infinitely often.
We will show that in our case the walk is recurrent iff $t\ge1/2$.
The key here is the law of the iterated logarithm. Indeed, according to (say) [Theorem 1 of Chapte... | 5 | https://mathoverflow.net/users/36721 | 429665 | 174,090 |
https://mathoverflow.net/questions/428994 | 8 | This is a soft question, I guess. $\Gamma$-convergence is a notion of convergence of functionals so that if $F\_n$ $\Gamma$-converges to $F$, then cluster points of $\arg\inf F\_n$ are minimizers of $F$. This is especially helpful if you want to minimize $F$ but find it easier to minimize $F\_n$.
However, if you look... | https://mathoverflow.net/users/479223 | How do people prove $\Gamma$-convergence in more complicated settings? | I am mostly familiar with the simpler definition ("Definition in first-countable spaces" from the Wikipedia link):
>
> Given the functionals $F\_\varepsilon, F: X \to \overline{\Bbb{R}}$
> (indexed for $\varepsilon>0$, we say that $F\_\varepsilon$
> $\Gamma$-converges to $F$ if the two properties hold:
>
>
> (LI)... | 4 | https://mathoverflow.net/users/13093 | 429666 | 174,091 |
https://mathoverflow.net/questions/429671 | 1 | Recall that a [preadditive category](https://en.wikipedia.org/wiki/Preadditive_category) is just a category $\mathcal{C}$ enriched in the category of abelian groups such that composition is linear with respect to the various group operations, so $$f\circ(g+h)=f\circ g+f\circ h,$$ $$(g+h)\circ f=g\circ f+h\circ f,$$ and... | https://mathoverflow.net/users/92164 | One object preadditive groupoids as a categorification of skew fields | There aren't any nontrivial preadditive groupoids; a preadditive category always has zero morphisms, and if zero morphisms are invertible then every object is a zero object.
If you think of rings as one-object preadditive categories, then commutative rings can be thought of as one-object preadditive monoidal categori... | 3 | https://mathoverflow.net/users/290 | 429673 | 174,093 |
https://mathoverflow.net/questions/429374 | 2 | While looking at an analogue of Pontryagin duality for compact Discrete Valuation Rings (DVRs), I came about the observation that generally one *should* have an isomorphism of $A$-modules
$$\hom\_{\mathbb{Z}}\left(A,\mathbb{R}/\mathbb{Z}\right)\xrightarrow{\sim} K/A,$$
where $A$ is a DVR that is compact with respec... | https://mathoverflow.net/users/159298 | Why is the natural map $\hom(A,\mathbb{R}/\mathbb{Z})\to K/A$ an isomorphism, $K/\mathbb{Q}_p$ unramified, $A=\mathcal{O}_K$? | This answer does not prove that the map described in the question is an isomorphism, but it does prove that an isomorphism between $\hom\_{\mathbb{Z}}(A,\mathbb{R}/\mathbb{Z})$ and and $K/A$ exists.
To begin, we note that $M \mapsto \hom\_{\mathbb{Z}}(A,M)$ is a right adjoint to the forgetful functor from the categor... | 0 | https://mathoverflow.net/users/159298 | 429675 | 174,094 |
https://mathoverflow.net/questions/291293 | 7 | Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim\_n \frac{d(e,g^n)}{n}$.
This number can actually be defined in a more general context: if $G$ acts by isometries on a set $X$, define ... | https://mathoverflow.net/users/111917 | Rational stable translation length | As already mentioned in the comments, there exist finitely generated groups having non-discrete translation spectra. (There seems to be only a few examples though, it would be interesting to have more.) About relatively hyperbolic groups, we have the following statement:
>
> **Proposition.** *Let $G$ be a relativel... | 1 | https://mathoverflow.net/users/122026 | 429696 | 174,099 |
https://mathoverflow.net/questions/429642 | 5 | I've only recently learned about Girard's theory of Dilators and Ptykes, and I find this theory very elegant, but it is not clear at all to me whether/how it can be used to produce ordinal notations for all the large recursive ordinals used in proof theory and ordinal analysis. The introduction of several papers on the... | https://mathoverflow.net/users/22131 | How to build large recursive ordinals using Dillator and/or Ptykes? | There are at least two fundamentally different ways how one could reach Bachmann-Howard ordinal using dilator and ptykes.
One way is to allow recursion on ordinals for ptykes for all finite types. Then the supremum of all ordinals describable in this way will be exactly B-H ordinal. See outline of ptyx interpretation... | 4 | https://mathoverflow.net/users/36385 | 429708 | 174,105 |
https://mathoverflow.net/questions/429711 | 2 | Let $A>0$ be fixed and consider $X\_1,\ldots$ i.i.d. nonnegative random variables such that $E[1/X\_1]<\infty$.
Is is true that $$\sup\_{a\in \big (0,\frac A{\sqrt n} \big]} \sum\_{i=1}^n 1\_{X\_i>a} \frac{a^3}{X\_i^2}$$
converges in probability to $0$ ?
With the crude bound $\sum\_{i=1}^n 1\_{X\_i>a} \frac{a^3}{X\... | https://mathoverflow.net/users/490637 | Convergence in probability of a supremum | As suggested by Anthony Quas, the supremum in question can be rewritten as
$$s\_n:=\sup\_{b\ge\sqrt n/A}S\_n(b),$$
where
$$S\_n(b):=\frac1{b^3}\sum\_{i=1}^n Z\_i^2\,1(Z\_i<b)$$
and $Z\_i:=1/X\_i$, so that the $Z\_i$'s are iid positive random variables with $EZ\_i<\infty$.
Take now any real $c>0$. Then for all large e... | 3 | https://mathoverflow.net/users/36721 | 429734 | 174,112 |
https://mathoverflow.net/questions/429688 | 2 | Fix a continuously differentiable but nowhere twice differentiable function $f$ on $\mathbb{R}$ supported on $[0,1]$. Is it true that for all $x\in[0,1]$ and all $\delta$ sufficiently small
\begin{align\*} & \sup\_{0<h\leq \delta}|f'(x+2h)-2f'(x+h)+f'(x)|\leq \\ & \quad C \delta^{-1}\sup\_{0<h\leq \delta}|f(x+2h)-2f(x+... | https://mathoverflow.net/users/152618 | An inequality about the second-order difference | $\newcommand\de\delta\newcommand\lhs{\text{lhs}}\newcommand\rhs{\text{rhs}}$No. E.g., let
$$f(x):=x^3\sin\frac1x$$
for $x\in(0,1/2]$, with $f(0):=0$. Then $f$ can be obviously extended to a continuously differentiable function $f$ on $\mathbb R$ supported on $[0,1]$.
Moreover,
$$\rhs(\de):=\de^{-1}\sup\_{0<h\le\de... | 1 | https://mathoverflow.net/users/36721 | 429735 | 174,113 |
https://mathoverflow.net/questions/429744 | 3 | Denote by $f(n)$ the maximal number of distinct divisors of $k$ integer numbers $1\leq a\_1<a\_2<\ldots<a\_k\leq n$, where $k$ is not fixed and $a\_1+\ldots+a\_k\leq n$. I'm interested in the asymptotics of $f(n)$.
For example, $f(4)=3$ since 4 has 3 divisors, $f(10)=5$ since 10=6+4 and 1,2,3,4,6 divide 4 or 6.
Not... | https://mathoverflow.net/users/103116 | Maximal number of divisors of numbers whose sum does not exceed $n$ | No, if you do not care on multiplicative factor: namely, $f(n)\leqslant 2\sqrt{n}$. Use the following
**Lemma.** For any real $x>0$ and any positive integer $a$, $a$ has at most $a/x$ divisors which are not less than $x$.
**Proof.** Let $d\_1>d\_2>\ldots>d\_m\geqslant x$ be these divisors. Then $a/d\_1<a/d\_2<\ldot... | 6 | https://mathoverflow.net/users/4312 | 429746 | 174,116 |
https://mathoverflow.net/questions/429752 | 0 | Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$.
Let $R\_K$ be ring of integers of $K$.
Let $ \hat{E}$ be its formal group of $E$.
Take $a \in{R\_K}$ and $[a] \in \operatorname{End}E$ . Then there is unique corresponding homomorphism of formal group, $[a]... | https://mathoverflow.net/users/144623 | Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$? | Let $\omega\_E$ be an invaraint differential on $E$. Then $[a]$ satisfies $[a]^\*\omega\_E=a\omega\_E$. That's over $K$, so the same formula holds on the formal group, i.e., $\widehat{[a]}\omega\_{\hat E}=a\omega\_{\hat E}$. On the other hand, if you write $\widehat{[a]}(T)=cT+\text{h.o.t.}$, then $\widehat{[a]}\omega\... | 5 | https://mathoverflow.net/users/11926 | 429755 | 174,117 |
https://mathoverflow.net/questions/429764 | 0 | I sent yesterday a paper to a journal of publisher Springer for consideration, in the same time I have got a new free distribution service and an open-access called [Research Square](https://www.researchsquare.com/). It looks like Arxiv; it is not peer-reviewed and makes research available in a fast way. Now I want to ... | https://mathoverflow.net/users/51189 | Is it ethical to submit a paper to journal then to Research Square? And what is the difference between that research square and ArXiv? |
>
> I want to know the difference between that Research Square and Arxiv.
>
>
>
One of the differences is that Arxiv is a not-for-profit service and is not actively trying to sell you stuff.
| 7 | https://mathoverflow.net/users/1898 | 429771 | 174,121 |
https://mathoverflow.net/questions/429758 | 7 | $\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\HH}{{\mathbb H}}$Let $A$ be an abelian group. We denote by $A\_\Tors$ the torsion subgroup of $A$.
We set $A\_\tf=A... | https://mathoverflow.net/users/4149 | Is this exact sequence known? | $\newcommand{\bQ}{\mathbb{Q}}\newcommand{\bZ}{\mathbb{Z}}\DeclareMathOperator{\Tor}{Tor}\newcommand{\Tors}{\mathrm{Tors}}$I tried to write up the computation with some level of details, please let me know if anything looks dubious.
Consider the long exact sequence $$\ldots\to H\_1(\Gamma, M\_2)\to H\_1(\Gamma,M\_3)\t... | 5 | https://mathoverflow.net/users/39304 | 429772 | 174,122 |
https://mathoverflow.net/questions/429778 | 4 | Let $X$ be a finite $p$-local spectrum. For each $h \in \mathbb{N} \cup \{\infty\}$, let $K(h)$ be [Morava $K$-theory](https://en.wikipedia.org/wiki/Morava_K-theory) of height $h$. Recall that the coefficients $K(h)\_\ast$ are a graded field, and $K(h)\_\ast(X)$ is a finite-dimensional vector space over this graded fie... | https://mathoverflow.net/users/2362 | Is $\operatorname{dim}_{K(h)_\ast} K(h)_\ast X$ increasing in $h$? | Yes, this is true, and appears in the early literature, although I do not immediately remember exactly where; I'd guess work of Wilson and/or Ravenel. I'll assume that $h>0$ for notational simplicity. There is a homology theory $E$ with $E\_\*=\mathbb{F}\_p[v\_h,v\_{h+1}^{\pm 1}]$, and this is a discrete valuation ring... | 8 | https://mathoverflow.net/users/10366 | 429779 | 174,124 |
https://mathoverflow.net/questions/429776 | 4 | Consider a pointed compact Hausdorff space $(X,x\_0)$ and a closed pointed subspace $i:A\subset X$ such that there exists a continuous map $r:X\rightarrow A$ such that $r|\_A=\text{Id}\_A$. Set
$$q:(X,x\_0)\rightarrow(X/A,A/A)$$
be the collapsing map and let $\tilde{K}^0$ denote the reduced (complex) topological $K$-gr... | https://mathoverflow.net/users/nan | Pullback of complex vector bundles along a retraction of compact Hausdorff spaces: a direct proof instead? | You may add $\mathbb C^n$ to $E$ so that in fact $q^\ast E$ has a trivialization $\phi$. Say that $E$ has rank $m$. The restriction of $q^\ast E$ to $A$ has also a trivialization $\psi$, simply because the restriction of $q$ to $A$ is a constant map. Comparing $\psi $ with the restriction of $\phi$ we get a map $u:A\to... | 9 | https://mathoverflow.net/users/6666 | 429781 | 174,125 |
https://mathoverflow.net/questions/429780 | 4 | **Short version:** If I have a map $f:Y \to I$, and $\mu$ an ultrafilter on $Y$, under what condition can $\mu$ be written as a limit/sum/integral of ultrafilters on the fibers of $f$ along the ultrafilter $\eta = f\_\*(\mu)$ on $I$ ? Is it always possible? If not is there an explicit condition on $\mu$ and $f$ to dete... | https://mathoverflow.net/users/22131 | Decomposition of an ultrafilter on the fibers of a map | First, let me dispose of the trivial cases where $f$ is constant or one-to-one on a set in $\mu$. In the case of constant $f$, say with value $i$, you can take $\eta$ principal at $i$ and let $\mu\_i$ be a suitable copy of $\mu$. In the one-to-one case, $\eta$ is a copy of $\mu$, and all the $\mu\_i$ can be principal.
... | 5 | https://mathoverflow.net/users/6794 | 429785 | 174,126 |
https://mathoverflow.net/questions/429796 | 5 | This question is follow up of [this MO-post](https://mathoverflow.net/q/429619/61536).
First let us recall the necessary definitions.
A function $f:X\to X$ on a group $X$ is called a *polynomial* if there exists $n\in\mathbb N$ and elements $a\_0,\dots,a\_n\in X$ such that $f(x)=a\_0xa\_1x\cdots xa\_n$ for all $x\i... | https://mathoverflow.net/users/61536 | The number of polynomials on a finite group, II | This is an answer to problem 1 and problem 3:
As noted in the comments, $\operatorname{Poly}(G)$ with pointwise multiplication is a group for finite $G$. Consider the homomorphism $\operatorname{Poly}(G)\to G$ given by evaluating at $1$. It is surjective (as seen by constant polynomials), and the kernel contains an e... | 10 | https://mathoverflow.net/users/39747 | 429798 | 174,132 |
https://mathoverflow.net/questions/429757 | 1 | How to derive that after stereo-graphical projection, $\Delta u$ in $\mathbb{R}^n$ is transformed to
$$
\Delta\_{\mathbb{S}^n}u - \frac{n(n-2)}{4}u\ \text{in}\ \mathbb{S}^n.
$$
To be more precise, in this [paper](https://pdf.sciencedirectassets.com/272601/1-s2.0-S0022123610X00052/1-s2.0-S0022123609005114/main.pdf?X-Amz... | https://mathoverflow.net/users/480661 | Laplacian on sphere after stereographic projection | the relation between the two Laplacians is a bit more complicated. I follow the survey [paper](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-17/issue-1/The-Yamabe-problem/bams/1183553962.full) of Lee and Parker on the Yamabe problem.
The conformal Laplacian of a Ri... | 3 | https://mathoverflow.net/users/127247 | 429805 | 174,137 |
https://mathoverflow.net/questions/429808 | 8 | Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that
$$
\lim\_{r\to 0}\frac{\displaystyle\int\_{B\_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ?
$$
This is obvious if $0$ is a Lebesgue point of $f$ or if $n=1$, but I would like to know if it's true in general.
| https://mathoverflow.net/users/351083 | How badly can the Lebesgue differentiation theorem fail? | Metafune has given an example of the limit failing to be $0$ at a particular point - namely for $n > 1$, the function $|x|^{-\alpha}$, with $1 \leq \alpha < n$ has that limit equal to $\infty$ at $0$.
However, you can still get some kind of affirmative result.
In general the limit in question is zero $\mathcal H^{n... | 17 | https://mathoverflow.net/users/173490 | 429810 | 174,138 |
https://mathoverflow.net/questions/429789 | 4 | Suppose that $\pi:X\to S$ is a *smooth projective* morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3\_S$.
**Question**: Can a similar statement be made for more general $S$?
I am primarily interested in Dedekind schemes $S$... | https://mathoverflow.net/users/94086 | Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$? | This answer addresses the second question: "If we assume the fibers of $\pi$ are curves of genus $0$, can we embed $X$ into $\mathbb{P}^2\_S$?"
The answer to this is also **no** (providing there is a singular fibre).
Let $\pi: X \to \mathbb{P}^1$ be a conic bundle surface over an algebraically closed field $k$, i.e... | 4 | https://mathoverflow.net/users/5101 | 429813 | 174,139 |
https://mathoverflow.net/questions/429236 | 2 | I am interested in the following question:
Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known invariants that completely characterize congruency via unimodular integer matrices?
More precisely: Are there invariants ... | https://mathoverflow.net/users/157865 | Classification of congruent integer matrices | [My Ph.D. thesis](http://www-personal.umich.edu/%7Ealimil/phdthesis.pdf) studied this and related questions from the perspective of arithmetic invariant theory. (There's also various other literature out there addressing related questions from a different number of directions; my contribution was mainly putting this ma... | 7 | https://mathoverflow.net/users/422 | 429820 | 174,141 |
https://mathoverflow.net/questions/429827 | 5 | Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}\_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}\_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram.
Given the functor $F$, we can also construct $F^{\prime}: C \to Cat\_\infty$, $c \mapsto Mod\_{F(c)}$, where $Mod\_{F(c)}$ is ... | https://mathoverflow.net/users/70889 | Explicit description of the right adjoint | See Theorem B in [On conjugates and adjoint descent](https://arxiv.org/abs/1705.04933), Horev-Yanovski, which states exactly this. The statement doesn't include this, but the body of the paper describes explicitly in what way the right adjoint is a limit of the right adjoints.
Note that your example is not a special ... | 6 | https://mathoverflow.net/users/102343 | 429831 | 174,145 |
https://mathoverflow.net/questions/429809 | 3 | I have a little technical question on Peter Scholze's [lectures on condensed mathematics](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf).
On page 12, right above the Proof of Theorem 2.2, he says that for extremally disconnected sets the condition (ii) on page 7 is automatic. I can't see why. I understand... | https://mathoverflow.net/users/473423 | Condensed mathematics | The key point is to use the fact that extremally disconnected sets are projective (in the category of compact Hausdorff spaces) to note the coequalizer diagram involved in the analogue of condition (ii) is a [*split* coequalizer.](https://ncatlab.org/nlab/show/split+coequalizer)
I'll explain some additional details i... | 8 | https://mathoverflow.net/users/490721 | 429839 | 174,147 |
https://mathoverflow.net/questions/416510 | 4 | I'm reading the Atanas Atanasov's course notes of Joe Harris' course [Geometry of Algebraic Curves](https://staff.math.su.se/shapiro/UIUC/curvesHarris.pdf)
and have a question about a suggested modification of an dimension
countinging argument applying methods from deformation theory.
On page 22 one consideres a vers... | https://mathoverflow.net/users/108274 | Deformation theoretic argument on dimension counting of naive Hurwitz scheme | You can get a lower bound on the dimension of $V\_{d,g}$ using deformation theory as follows. The deformation obstruction theory of a map $f : X \to Y$ between smooth varieties (where $f$ and $X$ are allowed to deform by $Y$ is held fixed) is governed by the complex
$$
\mathbb{L}\_f = \left[f^\*\Omega\_Y \to \Omega\_X\... | 4 | https://mathoverflow.net/users/12402 | 429841 | 174,148 |
https://mathoverflow.net/questions/429739 | 15 | **Qeustion:**
Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}\_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$,
such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$.
Now given a faithful representation
$$\varphi:\mathfrak{g}\hookrightarrow \mathfrak{gl}(V)$$
such that the restrictions o... | https://mathoverflow.net/users/486528 | A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$ | WLOG, we may assume that $\phi$ is injective and identify $\mathfrak{g}$ with its image in $\mathrm{End}(V)$. Our goal is to construct a reductive algebraic subgroup of $\mathrm{GL}(V)$, whose Lie algebra coincides with $\mathfrak{g}$.
We may assume, thanks to Deligne's results, that $\mathfrak{g}^0$ (which comes fro... | 11 | https://mathoverflow.net/users/9658 | 429849 | 174,149 |
https://mathoverflow.net/questions/429855 | 2 | Let $G$ be a (connected ?) algebraic group and $X$ a smooth, projective, and connected algebraic curve, both over an algebraically closed field $k$ of characteristic $0$.
My questions are then as follows:
1. By "local systems" in this context, *do we mean lisse $\ell$-adic étale sheaves* on $X$ (with coefficients i... | https://mathoverflow.net/users/143390 | The stack of equivariant local system is quasi-smooth | These are both answered in section 10 of the linked paper but since this is done at a very high level I will try to provide a down-to-earth explanation.
1 No, absolutely not. We mean local systems in the D-module sense, i.e. $G$-torsors with flat connection. So for $G =G L\_n$ these are $D$-modules that, restricted t... | 6 | https://mathoverflow.net/users/18060 | 429856 | 174,155 |
https://mathoverflow.net/questions/47009 | 19 | Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m\_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded by a representation over the field $K(\chi)$. The most interesting case is $K=\mathbb{Q}$. Given the character table, or... | https://mathoverflow.net/users/10266 | What can be said about Schur indices, given only the character table? | The following is a theorem of K. Kronstein:
>
> **Theorem:** for $k$ a number field or a nonarchimedean completion of a number field, if it is possible to detect the Schur index $m\_k$ of all finite groups from their character table and power maps, then $m\_k(\chi) \leq 2$ for all characters $\chi$ of finite groups... | 2 | https://mathoverflow.net/users/125523 | 429861 | 174,156 |
https://mathoverflow.net/questions/429868 | 6 | In other words, does there exist a metric space $(E,\rho)$ with finite Hausdorff dimension but infinite packing dimension?
Here are my thoughts:
* I know that it is generally hard to relate Hausdorff and packing measures and dimensions (other than the fact that Hausdorff dimension is always less than or equal to pa... | https://mathoverflow.net/users/322634 | Does finite Hausdorff dimension imply finite packing dimension? | A construction used (repeatedy) in the paper
*Edgar, G. A.*, [**Centered densities and fractal measures**](http://nyjm.albany.edu:8000/j/2007/13-4.html), New York J. Math. 13, 33-87 (2007). [ZBL1112.28004](https://zbmath.org/?q=an:1112.28004).
For more information, see that paper.
---
We construct a compact m... | 7 | https://mathoverflow.net/users/454 | 429872 | 174,158 |
https://mathoverflow.net/questions/429862 | 5 | Let $G$ be a finite group and let $p$ be a prime number such that $p\mid |G|$.
Let $\text{IBr}(G)$ denote the set of irreducible Brauer characters of $G$ for the prime $p$.
Assume $\mathbb{F}\_{q}$ is a splitting field for $G$ where $q=p^f$ for some positive integer $f$.
Set $r:=|\text{IBr}(G)|$.
Let $\{\rho\_1... | https://mathoverflow.net/users/91107 | Do $F$-traces of simple modules at $p'$-classes uniquely determine the module? | It is still the case that the $\mathbb{F}\_{q}$-valued trace functions of the (say) $\ell$ non-isomorphic simple $\mathbb{F}\_{q}$-modules $V\_{1},V\_{2}, \ldots V\_{\ell}$, are linearly independent, where $G$ has $\ell$ $p$-regular classes.
This was known to Brauer and Nesbitt, and a proof may be found (for example)... | 4 | https://mathoverflow.net/users/14450 | 429879 | 174,159 |
https://mathoverflow.net/questions/429886 | 1 | Let us define
$$
\mathbb{H}^{1} = H^{1}(-L,0) \times H^{1}(0,L) \ \ \text{and} \ \ \mathbb{L}^{2} = L^{2}(-L,0)\times L^{2}(0,L),
$$
where $H^{1}(I) = \big\lbrace u \in L^{2}(I) \ \text{and} \ u\_{x} \in L^{2}(I); I = (a,b) \big\rbrace$.
Besides these,
$$
\mathbb{M} = \big\lbrace (u,v) \in \mathbb{H}^{1}; u(-L) = v(L... | https://mathoverflow.net/users/481556 | Inequality involving Sobolev spaces | Just take a derivative, and you get
$$ i \lambda u\_x = f\_x - w\_x $$
So taking the $L^2(-L,0)$ norms on both sides, you get
$$ \lambda^2 \int |u\_x|^2 \leq \int |f\_x - w\_x|^2 $$
The RHS can be expanded and estimated using AM-GM to be
$$ \lambda^2 \int |u\_x|^2 \leq 2 \int |f\_x|^2 + |w\_x|^2 $$
The firs... | 2 | https://mathoverflow.net/users/3948 | 429890 | 174,163 |
https://mathoverflow.net/questions/426434 | 3 | *This question was originally posted last week in Math Stack Exchange (see [here](https://math.stackexchange.com/questions/4483240/compatibility-of-pullbacks-with-an-equivalence-relation)).*
I'm currently working on the proof of the existence of the sheafification in Angelo Vistoli’s 2007 [*Notes on Grothendieck topo... | https://mathoverflow.net/users/485069 | Compatibility of pullbacks with an equivalence relation | After I saw my mistake I got the proof really quick. We need the fibre product of $f$ and $\phi$, where $f$ is an arbitrarily morphism and $\phi$ the covering from $a\sim b$.
So we get the following commutative diagram:
$\require{AMScd}$
\begin{CD}
S\times\_UT @>{pr\_2}>> T\\
@V{pr\_1}VV @VV{\phi}V\\
S @>{f}>> U
\end{C... | 0 | https://mathoverflow.net/users/485069 | 429901 | 174,166 |
https://mathoverflow.net/questions/429896 | 0 | How can we represent F(x,m) in the infinte polynominal of x,m?
(Note that F(x,m) is the incomplete elliptical integral of the first kind, and I used its representation in the wikipedia)
More specifically, what is the value of [F(x/2,(cos m)^-0.5)-F(m,(cos m)^-0.5)]\*(cos m)^-0.5 when terms of O(m^3),O(x^3) are ignored?... | https://mathoverflow.net/users/489097 | Approximation of Incomplete elliptic integral of first kind | The series expansion in powers of $k$ of the incomplete elliptic integral of the first kind
$$F(\varphi,k)= \int\_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}$$
can be simply obtained by expanding the integrand,
$$F(\varphi,k)=\varphi+\frac{1}{4} k^2 (\varphi-\sin \varphi \cos \varphi)+\frac{3}{256} k^4 (12 ... | 1 | https://mathoverflow.net/users/11260 | 429905 | 174,168 |
https://mathoverflow.net/questions/429893 | 3 | Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator
between the spaces of Sobolev sections $D: W^{k+1,2}(X;E)\to W^{k,2}(X;F) $.
Therefore we can compute its index which is a topological ... | https://mathoverflow.net/users/99042 | Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$) | Local boundary conditions such as the Dirichlet condition you mention were considered in Atiyah-Bott, The index problem for manifolds with boundary. 1964 Differential Analysis, Bombay Colloq., 1964 pp. 175–186 Oxford Univ. Press, London. (There is also a chapter by Atiyah in Palais's Seminar on the Atiyah-Singer Index ... | 4 | https://mathoverflow.net/users/3460 | 429916 | 174,171 |
https://mathoverflow.net/questions/429915 | 2 | Assume we have an irreducible algebraic cycle $Z$ on $X\times Y$ where $X$ and $Y$ are projective varieties ($X$ is smooth) such that restriction of $Z$ to $U\times Y$ where $U\subset X$ is a Zariski open is a graph of a regular map from $U$ to $Y$. Is any cycle close enough to $Z$ on the Chow variety also graph of a m... | https://mathoverflow.net/users/127776 | Cycles that are graphs of morphisms | Not as stated. Let $X$ be $\mathbb P^2$, $Y$ be the blowup of $\mathbb P^2$ on one point $P$, $Z$ the graph of the blowup map $Y \to X$, which is the graph of a regular map on the open set $U$ defined as the complement of $P$.
We can deform $Z$ in a family that moves the point $P$. The cycles in that family will each... | 2 | https://mathoverflow.net/users/18060 | 429917 | 174,172 |
https://mathoverflow.net/questions/429914 | 0 | I am looking for an example of the following:
Find a bijective, differentiable function $f$ and continuous probability density functions $q\_1\ne q\_2$ such that $f\_\*q\_1=p=f\_\*q\_2$, where $f\_\*$ is the pushforward density and $p$ is continuous as well. What if continuity is strengthened to differentiability?
Ed... | https://mathoverflow.net/users/486206 | Transforming two smooth densities to the same density | This is impossible if $f$ is injective, without further assumptions such as bijective, differentiable, etc. Let $Q\_1,Q\_2$ be probability measures on a measurable space $(\Omega, \mathcal{F})$, and assume $f\_\* Q\_1 = f\_\* Q\_2$ for some injective (bimeasurable) $f : (\Omega,\mathcal{F}) \to (\Xi,\mathcal{G})$. For ... | 3 | https://mathoverflow.net/users/99418 | 429919 | 174,173 |
https://mathoverflow.net/questions/429725 | 2 | There's been [some debate at the nLab](https://nforum.ncatlab.org/discussion/3306/identity-type/#Item_0) recently over the names of "identity type" and "path type" in certain dependent type theories.
One user wrote that
>
> Many cubical type theorists make the distinction between identity types like Martin-Löf’s ... | https://mathoverflow.net/users/483446 | Path types and identity types in dependent type theory | Prior to about a decade ago, no one used "path" terminology for identity types. The identification of the semantics of identity types with path objects dates to Awodey and Warren's [Homotopy theoretic models of identity types](https://arxiv.org/abs/0709.0248) and Voevodsky's [simplicial model of univalent foundations](... | 7 | https://mathoverflow.net/users/49 | 429931 | 174,177 |
https://mathoverflow.net/questions/429925 | 4 | I have this problem at the moment which the strong topology $\beta (E;E^\* )$ is defined, when $E$ is a locally convex space. This topology is generated by the basic open sets:
$$U=\{x \in E : \sup\_{f \in B} |\langle f,x \rangle|<\varepsilon\},$$
where $B\subset E^\* $ is bounded. In this way, we say that $B$ is bou... | https://mathoverflow.net/users/489810 | Weak* bounded and strong bounded are the same? | In general, $\sigma(E^\*,E)$-bounded sets need not be $\beta(E^\*,E)$-bounded. For an example, let $E$ be the set of scalar sequences with only finitely many non-zero terms endowed with the norm $\|x\|\_\infty=\sup\{|x\_n|:n\in\mathbb N\}$. For the evaluations $\delta\_n(x)=x\_n$, the set $B=\{n\delta\_n:n\in\mathbb N\... | 8 | https://mathoverflow.net/users/21051 | 429940 | 174,180 |
https://mathoverflow.net/questions/427437 | 2 | Let $n\geq 1$. Let $[n]=\{0<1\}^n$ equipped with the product order. Let $f:[n]\to [n]$ be a strictly increasing map. When $f$ is bijective, there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that $f(\epsilon\_1,\dots,\epsilon\_n)=(\epsilon\_{\sigma(1)},\dots,\epsilon\_{\sigma(n)})$.
>
> Is there such a rep... | https://mathoverflow.net/users/24563 | How to describe this set of maps of posets? | I have found a way published in a recent preprint (<https://doi.org/10.48550/arXiv.2209.02667>).
>
> **Theorem**: Let $n\geq 1$. Let $f=(f\_1,\dots,f\_n):[n]\to [n]$ be a stricly increasing map. Then there is the equality $
> f\_i(x\_1,\dots,x\_n) = \max\_{(\epsilon\_1,\dots,\epsilon\_n)\in
> f\_i^{-1}(1)} \min \{x... | 1 | https://mathoverflow.net/users/24563 | 429941 | 174,181 |
https://mathoverflow.net/questions/365603 | 0 | Let $f\_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F\_n:= f\_n\circ \dots\circ f\_1$. Then $F\_n$ is continuous. However, the pointwise limit need not be (consider Mateusz's example of
$$
f\_n = \frac{\sqrt{2} x}{\sqrt{1 + 4 x^2}} \qquad F\_n \to \operatorname{sign}(x)
$$
the sign funct... | https://mathoverflow.net/users/36886 | Infinite composition of continuous functions | Since you’re searching for names: there is a rich literature on *discrete non-autonomous dynamical systems* generated by an infinite family of continuous maps. Here "non-autonomous" means of course that the map you apply depends on the order of iteration, which is exactly your subject.
For instance:
1. Cánovas, J. ... | 1 | https://mathoverflow.net/users/167834 | 429955 | 174,185 |
https://mathoverflow.net/questions/429967 | 5 | Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book *Algebraic Geometry and Arithmetic Curves*, a regular but non-smooth curve is given. But that curve is not geometrically reduced.
| https://mathoverflow.net/users/11599 | A regular, geometrically reduced but non-smooth curve | I believe a classic example is the curve define in $\mathbb P^2\_{\mathbb F\_p(t)}$, with coordinates $(x:y:z)$, by the equation
$$ t x^p + z^{p-1} y + y^p=0$$
for $p>2$.
Differentiating with respect to $y$, one can see that the curve is smooth wherever $z \neq 0$, and substituting in $z=0$, one can see the curve... | 14 | https://mathoverflow.net/users/18060 | 429970 | 174,189 |
https://mathoverflow.net/questions/425948 | 10 | Let $\{b\_n\}\_{n\geq0}$ be a sequence such that $b\_nb\_{n+1}=0$ and define
$$a\_n:=\sum\_{k=0}^n(-1)^{n-k}\binom{n}{k}b\_k.$$
If $\lim\_{n\to\infty}a\_n=0$, can we conclude that $b\_n=0$ for all $n$?
More generally, if $\{b\_n\}\_{n\geq0}$ is a sequence with infinitely many zeros and $\lim\_{n\to\infty}a\_n=0$, can... | https://mathoverflow.net/users/168342 | A number sequence problem involving binomial transform | Unfortunately the argument that I originally posted contained a gap at the end.
The gap is explained at the end of the proof, where I also state 3 partial results.
Let
$$f(z)=\sum\_{n=0}^\infty b\_nz\_n/n!,\quad g(z)=\sum\_{n=0}^\infty a\_nz^n/n!.$$
Then your relation between $a\_n$ and $b\_n$ means
$$g(z)=e^{-z}f(z)... | 6 | https://mathoverflow.net/users/25510 | 429979 | 174,192 |
https://mathoverflow.net/questions/429977 | 11 | David Roberts wrote in the comment section of the blog post "[Convergence of an infinite sum in the rationals](https://thehighergeometer.wordpress.com/2022/07/29/convergence-of-an-infinite-sun-in-the-rationals/)" the following paragraph:
>
> Someone mentioned (I think on Twitter) that the Taylor series of rational ... | https://mathoverflow.net/users/483446 | In the rational numbers, is every convergent power series a Taylor series for a rational function? | No. Enumerate the rational numbers $a\_1,a\_2,\dots$. Then for every sequence $c\_1, c\_2,\dots$ of rational numbers decreasing rapidly enough, the series
$$ \sum\_{n=1}^{\infty} c\_n x^n \prod\_{i=1}^{n-1} (x-a\_i ) $$
converges on each rational number. On $a\_m$ it takes the value $$ \sum\_{n=1}^{m} c\_n a\_m^n \... | 21 | https://mathoverflow.net/users/18060 | 429981 | 174,193 |
https://mathoverflow.net/questions/429889 | 3 | Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "[G.Laumon - Fonctions zeta des variétés de Siegel](http://www.numdam.org/article/AST_2005__302__1_0.pdf)" (Lemma 4.1) that to an algebraic representation $W$ of $\mathrm{GSp}\_{4}(\mathbb{Q})$ we c... | https://mathoverflow.net/users/169282 | $l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$ | This is a special case of Pink's "canonical construction" functor, which associates various kinds of coefficient sheaves on a Shimura variety (etale $\ell$-adic sheaves, vector bundles with connection, variations of Hodge structures, etc) to algebraic representations of the underlying group.
For more information see ... | 2 | https://mathoverflow.net/users/2481 | 429982 | 174,194 |
https://mathoverflow.net/questions/429845 | 4 | I am looking into the practicalities of doing Math in FOL + PA with the FOL extended with equality and functions.
For a predicate you can easily extend the language such that a predicate is defined as a logical expression, which then can be comprehended or expanded in the proofs and theorems that follow.
However, i... | https://mathoverflow.net/users/5917 | Defining functions in FOL + PA | You might avoid the issue in first-order arithmetic by assuming that a function always selects the least option, or zero if there is no option. So a definition like
$$R(x,y,z)=z<y \wedge \exists v:x = vy+z$$
would actually be transformed into
$$R'(x,y,z)=(R(x,y,z) \wedge \forall w:R(x,y,w)\implies z\le w) \vee (z=0 \we... | 3 | https://mathoverflow.net/users/nan | 429983 | 174,195 |
https://mathoverflow.net/questions/429933 | 2 |
>
> Is there a discrete space Markov chain, starting from a fixed state, whose stationary distribution is a multimodal distribution and that mixes in polynomial time?
>
>
>
For example, Ising model on say a complete graph has a multimodal stationary distribution at low temperature. Critical $\beta$ (i.e. inverse... | https://mathoverflow.net/users/479350 | Polynomial time mixing Markov chain for multimodal distribution | The OP seems specifically interested in ergodic Markov chains with a unique stationary distribution $\pi$. The following example is admittedly a bit contrived, but is helpful to illustrate some general principles discussed more below.
>
> **Claim:** There exists a Markov chain with a "bimodal" stationary distributi... | 2 | https://mathoverflow.net/users/64449 | 429990 | 174,197 |
https://mathoverflow.net/questions/425726 | 8 | Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G\_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}\_p(y)$ and $G\_p$ be the Galois group of the same polynomial over $\mathbb{F}\_p(y)$. It is known (see B. Birch and H. Swinnerton-Dyer, "Note on a problem of Chowla... | https://mathoverflow.net/users/101078 | The distribution of certain Galois groups | Denote by $\Omega$ the splitting field of $f(x)-y$ over $\mathbb{Q}$, by $k/\mathbb{Q}$ the maximal constant extension inside $\Omega$, and set $G:=Gal(\Omega/\mathbb{Q}(y))$, $G^+ =Gal(\Omega/k(y))$. Up to avoiding finitely many primes $p$, one has "good constant reduction" and can obtain the picture over $\mathbb{F}\... | 3 | https://mathoverflow.net/users/127660 | 430001 | 174,203 |
https://mathoverflow.net/questions/429999 | 1 | Geometry and combinatorics are two different branches of mathematics. Does there exist any connection between them? In many cases, mathematicians solve some geometric problems by reducing them to a combinatorial language. What are the general techniques to convert a geometrical problem to a combinatorial one? What are ... | https://mathoverflow.net/users/490039 | Bridges between geometry and combinatorics | I would recommend the work of Adiprasito, Huh and Katz:
>
> K. Adiprasito, J. Huh, E. Katz, Hodge Theory for Combinatorial
> Geometries, Annals of Mathematics 188 (2018), 381–452. [[arXiv](https://arxiv.org/abs/1511.02888)]. [[Journal](https://web.math.princeton.edu/%7Ehuh/MatroidHodge.pdf)]
>
>
>
They actuall... | 0 | https://mathoverflow.net/users/7031 | 430002 | 174,204 |
https://mathoverflow.net/questions/430003 | 18 | I am a Masters student of math interested in physics. When I was an undergraduate, I took the introductory course of physics, but it is just slightly harder than high school physics course. To be precise, it just taught us how to use calculus in physics, without involving the higher knowledge of math such as manifold, ... | https://mathoverflow.net/users/490869 | How does a Masters student of math learn physics by self? | I can recommend Leonard Susskind's [Theoretical Minimum](https://theoreticalminimum.com):
>
> A number of years ago I became aware of the large number of physics
> enthusiasts out there who have no venue to learn modern physics and
> cosmology. Fat advanced textbooks are not suitable to people who have
> no teacher... | 13 | https://mathoverflow.net/users/11260 | 430004 | 174,205 |
https://mathoverflow.net/questions/429980 | 1 | Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product
$$X \otimes\_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \iota)(z) \in M \text{ and } (\iota \otimes \tau)(z)\in X \text{ for all }\sigma\in B(H)\_\*, \tau \in B(K)\_\*\}.$$
We ha... | https://mathoverflow.net/users/470427 | Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$? | I follow the book of Effros+Ruan (which is a book, so not viewable online, but really is the nicest source I think). For any operator spaces $X,Y$ we can consider the operator space projective tensor product $\newcommand{\proten}{\widehat\otimes}X\proten Y$ whose dual satisfies
$$ (X\proten Y)^\* = CB(X,Y^\*). $$
(To b... | 2 | https://mathoverflow.net/users/406 | 430008 | 174,207 |
https://mathoverflow.net/questions/430016 | 5 | Let $G=Gl\_n(\mathbb{C})$ and $\mathcal{N}$ be the nilpotent cone associated to it i.e nilpotent matrices inside $\mathfrak{g}=\mathfrak{gl}\_n(\mathbb{C})$.
We have the variety $\tilde{\mathcal{N}}$ with the Springer resolution $p:\tilde{\mathcal{N}} \to \mathcal{N}$ with the Springer sheaf $p\_{!}(\mathbb{Q}[\dim \... | https://mathoverflow.net/users/146464 | Invariants of cohomology of Springer sheaf | You want to look at the partial Grothendieck-Springer resolution, i.e. the variety of pairs $ (g \in G/ P\_\mu, v \in g \mathfrak p\_\mu g^{-1})$.
The partial Grothendieck-Springer resolution is smooth, so the shifted constant sheaf on it is perverse, and its pushforward to $\mathfrak g$ is pure.
The partial Grothe... | 9 | https://mathoverflow.net/users/18060 | 430029 | 174,214 |
https://mathoverflow.net/questions/429512 | 3 | $\DeclareMathOperator\ht{ht}$All rings are commutative Noetherian with identity.
Exercise 9.8 of Matsumura's book *Commutative ring theory*: Let $A$ be a ring and $A\subset B$ an integral extension. If $P$ is a prime ideal of $B $ with $\mathfrak p = P\cap A$ then $\ht P \leq \ht \mathfrak p$.
There are many exam... | https://mathoverflow.net/users/47763 | Examples of integral ring extensions that $\operatorname{ht}P \lt \operatorname{ht}P\cap A$ | Take $A=\mathbb{Z}$ and $B=\mathbb{Z}[X]/(X^2+X,2X)$ $=$ $A[x]$, where $x$ denotes the residue class of $X$. Clearly, $x$ is integral over $A$. Every prime ideal of $B$ either contains $x$ or both $x+1$ and $2$. Hence $P$ = $(2,x+1)$ is a minimal prime ideal, so of height $0$. (Note that $P$ is also a maximal ideal of ... | 5 | https://mathoverflow.net/users/31923 | 430039 | 174,216 |
https://mathoverflow.net/questions/430035 | 6 | For $n\geq 1$, $f\_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f\_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and
$$\lim\limits\_{n\to+\infty}\sup\_{x\in[0,1]}\big|f\_n(x)-\sqrt{x}\big|= 0.$$
Let $y\_n$ be the unique solution of
$$\begin{cases}
y\_n(0)=0 \\
y\_n'=f\_n(y\_n) \text{ on [0,1]}.
\end{cases}$$
**Question... | https://mathoverflow.net/users/159940 | Forcing the uniqueness of a solution of an ODE | $\newcommand\ep\varepsilon$First, the conditions that $f\_n\in\mathcal{C}^1([0,1],\mathbb{R})$ and $f\_n(x)\ge\sqrt{x}$ for $x\in[0,1]$ imply $f\_n(0)>0$. Since
\begin{equation\*}
\begin{cases}
y\_n(0)=0, \\
y\_n'=f\_n(y\_n) \text{ on [0,1]},
\end{cases}
\tag{2}\label{2}
\end{equation\*}
we see that $y\_n>0$ in a rig... | 8 | https://mathoverflow.net/users/36721 | 430048 | 174,219 |
https://mathoverflow.net/questions/430068 | 0 | Consider $f \in L^{2}(0,1)$ and $g \in L^{\infty}(0,1)$ such that
1. $ \text{lim} ~g(x) = 0 \ \ \text{when} \ \ x \to 0^{+};$
2. $g(x) > 0 \ \forall x \in (0,1)$;
3. $\text{lim}~\dfrac{g(x)}{x^{\alpha}} = N > 0,$ when $x \to 0^{+}$, $0 < \alpha < 1$
Moreover, suppose
$$
\int\_{0}^{1}g(x)|f(x)|^{2} = M < \infty
$$
*... | https://mathoverflow.net/users/490936 | Get an estimate on $L^{2}(0,1)$ | No. Think of a sequence of $f\_n$ such that $\| f\_n\|\_2=1$ while $\|g f\|\_2\to 0$. And you might as well assume $g(x)= x^{\alpha}$.
It is equivalent to asking the same question with $f\_n\geq0$, $\| f\_n\|\_1 =1$, $\| x^{2\alpha}f\_n\|\_1\to0$ (just take $g\_n=\sqrt{f\_n}$ for the $L^2$ case).
Let us try $f\_n=\... | 1 | https://mathoverflow.net/users/40120 | 430084 | 174,228 |
https://mathoverflow.net/questions/427590 | 0 | There is a great introduction by May, "[The Geometry of Iterated Loop Spaces](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.170.2840&rep=rep1&type=pdf)". I really enjoy reading it, but it was written 50 years ago and contains outdated technical details related to the language of topological spaces. Now, as ... | https://mathoverflow.net/users/148161 | How now to study operads in homotopy theory? | One of the most comprehensive references today is certainly:
>
> B. Fresse, *Homotopy of Operads and Grothendieck–Teichmüller Groups*, Mathematical Surveys and Monographs 217. <https://bookstore.ams.org/surv-217/>
>
>
>
However, the idea that topological spaces are obsolete and that we should only use simplici... | 8 | https://mathoverflow.net/users/36146 | 430092 | 174,231 |
https://mathoverflow.net/questions/430081 | 6 | I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is appreciated.
But let's consider a percolation model on $\mathbb{Z}^d$ (or even $\mathbb{Z}^2$ if that's easier), with each ... | https://mathoverflow.net/users/121745 | Infinite clusters for loopless percolation | These fascinating questions have been studied recently, e.g. by [Bauerschmidt, Crawford, Helmuth and Swan](https://arxiv.org/abs/1912.04854) (no percolation on $\mathbb{Z}^2$) and by [Bauerschmidt, Crawford and Helmuth](https://arxiv.org/abs/2107.01878) (percolation phase transition on $\mathbb{Z}^d$ for $d\geq 3$).
| 7 | https://mathoverflow.net/users/5784 | 430093 | 174,232 |
https://mathoverflow.net/questions/425711 | 3 | The [diamond lemma](https://www.cip.ifi.lmu.de/%7Egrinberg/algebra/diamond-talk.pdf) has recently come up in my teaching, and as always I've been looking for nice and simple applications. This has reminded me of the thesis
Kimmo Eriksson, *Strongly convergent games and Coxeter groups*, KTH Stockholm 1993,
which I h... | https://mathoverflow.net/users/2530 | Eriksson's thesis "Strongly convergent games and Coxeter groups" | The thesis can now be found at <https://archive.org/details/eriksson-strongly-convergent-games-thesis> .
Thanks to Kimmo Eriksson for sending me a hard copy and allowing it to be shared!
| 2 | https://mathoverflow.net/users/2530 | 430106 | 174,235 |
https://mathoverflow.net/questions/430071 | 1 | Suppose you have Banach spaces $\mathcal B\_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu\_\alpha$ be Gaussian measures on $\mathcal B\_\alpha$ with Cameron-Martin spaces $\mathcal H\_{\mu\_\alpha}$.
Is it then true that the product space of all the Banach spaces with the product measure (independent comp... | https://mathoverflow.net/users/479223 | Cameron-Martin space of product space | At least if $I$ is countable this should be true, but I do not have a reference:
Let $H$ be the CM space of the Frechet space $\mathcal{B} := \prod\_{\alpha \in I} \mathcal{B}\_{\alpha}$ with the product topology and the product measure. Recall that the Hilbert space reproducing kernel of $(\mathcal{B}, \mu)$ is isom... | 1 | https://mathoverflow.net/users/117692 | 430114 | 174,236 |
https://mathoverflow.net/questions/429812 | 15 | Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. We say $x \in \mathbb R^n$ is a *strong Lebesgue point* of $f$ if
$$\lim\_{r \to 0} \frac{\int\_{B\_r (x)} |f(y) - f(x)| \, dy}{r^{n+1+\varepsilon}} = 0$$
for some $\varepsilon > 0$, potentially depending on $x$.
**Question:** Suppose every poin... | https://mathoverflow.net/users/173490 | If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function? | I realise I'm bumping into you again and already gave you an answer elsewhere after you posted this, but I thought I'd post my answer here for others to see. The answer is yes, $f$ has to be constant even if $\varepsilon = 0$. Here's the proof (where $\varepsilon = 0$):
---
Fix $x \in \mathbb{R}^n$. Then by Marko... | 12 | https://mathoverflow.net/users/120665 | 430115 | 174,237 |
https://mathoverflow.net/questions/430089 | 2 | I have started reading about subgroup growth and, to my surprise, I haven't found a reference to whether direct products preserve subgroup growth.
Recall that, given a finitely generated group $G$, the function $s\_n(G)$ is given by
$$s\_n(G)=\#\{\text{subgroups of } G\text{ of index }\leq n\}.$$
We say that $G$ has ... | https://mathoverflow.net/users/44172 | Subgroup growth of direct product | Subgroup growth of direct product is quite difficult in general. However, it is easier for pro-$p$ groups. You might like to see a recent paper: Y. Barnea and J.-C. Schlage-Puchta, [Branch groups, orbit growth, and subgroup growth types for pro-p group](https://www.cambridge.org/core/services/aop-cambridge-core/content... | 2 | https://mathoverflow.net/users/5034 | 430122 | 174,240 |
https://mathoverflow.net/questions/430100 | 1 | I try to understand the Proof of Theorem 4.21 in [Carmona Delarue (2018)](https://link.springer.com/content/pdf/10.1007/978-3-319-58920-6.pdf). In the following, what I don't understand:
Processes are assumed to be defined on a complete filtered probability space $(\Omega, \mathcal F, \mathbb F=(\mathcal F\_t)\_{t \i... | https://mathoverflow.net/users/490964 | Finding an existence and uniqueness result of a strong solution of Lipschitz SDEs | Following the proof of [Theorem 4.21](https://link.springer.com/content/pdf/10.1007/978-3-319-58920-6.pdf), fix the environment $\mu$, and additionally suppress the dependence of the SDE coefficients on $\mu$, so that the nonlinear SDE reduces to a classical one.
>
> **Claim:** In the classical existence/uniqueness... | 2 | https://mathoverflow.net/users/64449 | 430124 | 174,242 |
https://mathoverflow.net/questions/430121 | 1 | I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces:
Let $X$ be a complex space(i.e. analytic variety) with normal singularity.
>
> Lemma 4.6.1. Any pluriharmonic distribution on $X$ is locally the real part of a... | https://mathoverflow.net/users/167083 | Bott-Chern cohomology for singular complex spaces |
>
> closed (1,1)-forms and currents on X
> are not necessary locally $dd^c$-exact in general
> What makes it different when X is singular?
>
>
>
The obstruction to local $dd^c$-lemma
is $R^1\pi\_\*(O\_{X'})$, where
$\pi:\; X' \to X$ is the resolution of
singularities. When $X$ is smooth
(more generally, when it
... | 3 | https://mathoverflow.net/users/3377 | 430128 | 174,244 |
https://mathoverflow.net/questions/430126 | 4 | I've asked two years ago a post on Mathematics Stack Exchange, were provided two excellent answers. I'm asking on MathOverflow in the hope that some professor can to expand/improve (if it is possible) these results answering my question. The post has the same title and identifier [*3757149*](https://math.stackexchange.... | https://mathoverflow.net/users/142929 | On the diophantine equation $x^{m-1}(x+1)=y^{n-1}(y+1)$ with $x>y$, over integers greater or equal than two | You can make a lot of progress if you're willing to assume a deep conjecutre. The $N$-variable generalization of the $abc$-conjecture (<https://en.wikipedia.org/wiki/N_conjecture>) applied to your equation (with $N=4$) says that if $\gcd(x,y)=1$ and if no subsum of the $4$-term sum
$$ x^n + x^{n-1} - y^m - y^{m-1} $$
v... | 10 | https://mathoverflow.net/users/11926 | 430130 | 174,245 |
https://mathoverflow.net/questions/430082 | 4 | Let $n$ be a positive integer and $0 \leq i < n$. Define
$$
N(i) = \# \left\{ (x\_1,\dots, x\_s) \in [1, n]^s: x\_1^2 +\dots + x\_s^2 \equiv i \mod n \right\}.
$$
I am looking for a reference for the following result: if $s$ is large enough,
$$
n^{s-1} \ll N(i) \ll n^{s-1}
$$
is true for all $i$ and $n$. Could anyone p... | https://mathoverflow.net/users/48408 | Sum of many squares modulo $n$ | The estimate is false for $s\leq 4$, but it is true for $s\geq 5$. For example, if $s=3$ and $8\mid n$ and $i=7$, then $N(i)=0$. Or if $s=4$ and $n=4^k$ and $i=0$, then $N(i)\leq n^2$. For $s\geq 5$ the result follows easily from the fact that the number of ways to write a positive integer $m$ as a sum of $s$ squares i... | 11 | https://mathoverflow.net/users/11919 | 430133 | 174,246 |
https://mathoverflow.net/questions/430129 | 7 | I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example.
Given a faithfully flat map of commutative rings $A \to B$, the usual rules of faithfully flat descent let us identify $A$-modules ... | https://mathoverflow.net/users/490054 | Basic example of derived descent | $\newcommand{\Z}{\mathbf{Z}}\newcommand{\FF}{\mathbf{F}}\newcommand{\H}{\mathrm{H}}$Let me do the universal case of your situation, which is understanding descent along the map $\Z[t] \to \Z$ sending $t\mapsto 0$. [To get your case, note that $\FF\_2 = \Z \otimes\_{\Z[t]} \Z$, where one of the maps $\Z[t] \to \Z$ sends... | 8 | https://mathoverflow.net/users/102390 | 430134 | 174,247 |
https://mathoverflow.net/questions/430136 | 21 | A well known equivalent of the Axiom of Choice is Krull's Maximal Ideal Theorem (1929): if $I$ is a proper ideal of a ring $R$ (with unity), then $R$ has a maximal ideal containing $I$. The proof is easy with Zorn's Lemma. The converse, Krull implies Zorn, is due to Hodges (1979).
A ring with a unique maximal ideal i... | https://mathoverflow.net/users/159728 | A Krull-like Theorem and its possible equivalence to AC | Nice question ! I believe the homework exercise implies AC.
Indeed, assume its conclusion holds, and let $R$ be a ring with no maximal ideal. I'm going to prove that $R$ is zero, thus proving Krull's theorem (apply this to $R/I$ for a proper ideal $I$).
Let $k$ be your favourite field. Then $k\times R$ is local : $... | 33 | https://mathoverflow.net/users/102343 | 430138 | 174,249 |
https://mathoverflow.net/questions/429844 | 2 | *The sets are defined in $\mathbb{R}\_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ is denoted $\mathring C$.*
Let $S$ be the $n$-simplex:
$$S=\left\{x\in\mathbb{R}\_+^n,\,\sum\_{i=1}^n x\_i=1\right\}$$
and $E$ be a linear subspace such that $E\cap\mathring S\neq\varnothing$ (hence, $E$ contains a v... | https://mathoverflow.net/users/159940 | Intersection of the simplex with a linear subspace of codimension $2$ | The answer is *Yes*.
By your assumption there exists $x\in\mathring F\subset \mathring G$.
We also have $x\in F\subset G'$, and the only way a face $G'$ can contain an interior point of a face $G$ is if $G\subseteq G'$.
Now, consider a small neighborhood of $x$. In this neighborhood, $F$, $G$ and $E$ look like line... | 1 | https://mathoverflow.net/users/108884 | 430140 | 174,250 |
https://mathoverflow.net/questions/430158 | 2 | Under suitable large-cardinal assumptions, in the inner model $L(\mathbb R)$ one can have $\omega\_1$ and $\omega\_2$ measurable (this follows from determinacy).
I was wondering if it is possible to reconcile measurability of $\omega\_1$ (or other cardinals) with the model $L( \mathbb R^{\omega\_1} )$ or objects alik... | https://mathoverflow.net/users/491011 | Large cardinals and measurability in $L(A)$ | For each $\alpha<\omega\_1$, choose a real $r\_\alpha$ that codes the ordertype $\alpha$. This sequence $\langle r\_\alpha : \alpha < \omega\_1 \rangle$ then codes a sequence of surjections from $\omega$ to each countable ordinal. From this, you can then run the Ulam matrix construction and show that there is no counta... | 6 | https://mathoverflow.net/users/11145 | 430161 | 174,254 |
https://mathoverflow.net/questions/430143 | 4 | I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below:
"The function $\frac{\xi^\beta}{|\xi|^s}\hat\phi(\xi)$ is the Fourier transform of an integrable function." Here $\hat\phi(\xi)\in C^\infty\_0(\mathbb{R}^n)$ is a bump f... | https://mathoverflow.net/users/490996 | A proof of Bernstein's inequality | Well, I find that there could be another answer. We still only need prove the integrablity at inifity. Considering the homogeneous function $\frac{\xi^\beta}{|\xi|^s}$, by the proposition 2.4.8 in "Classical Fourier Analysis", its original function, denoted by $g(x)$, is smooth on $\mathbb{R}^n\backslash\{0\}$ and homo... | 1 | https://mathoverflow.net/users/490996 | 430174 | 174,262 |
https://mathoverflow.net/questions/430176 | 8 | Let $E$ be a topos and let $\Omega\_E$ be its subobject classifier. We can consider $\Omega$ as an internal poset in $E$, and thus we may form the $E$-topos of internal presheaves $\hat{\Omega}\_E := [\Omega\_E^{\text{op}},E]$. This feels like a "fatter" version of the relative Sierpinski $E$-topos $\mathbb{S}\_E := [\... | https://mathoverflow.net/users/51336 | Internal presheaves on the subobject classifier | I'm not sure this constitute an answer to your question, but from what you are telling of your motivation it very well might be, and in any case it was too long to be a comment.
First - I don't think there is a good answer to your questions 1 and 2. The problem is that considering $\Omega$ as just a poset is a very "... | 10 | https://mathoverflow.net/users/22131 | 430179 | 174,263 |
https://mathoverflow.net/questions/430131 | 6 | $\newcommand{\GL}{\mathrm{GL}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\Spin}{\mathrm{Spin}}\renewcommand{\O}{\mathrm
O}\newcommand{\R}{\mathbb
R}\newcommand\Z{\mathbb Z}$If $\rho\colon \SU\_8\to\GL\_n(\R)$ is a representation, then we can lift $\rho$ to land in the spin group
$\Spin\_n$ in... | https://mathoverflow.net/users/97265 | Is there a representation of $\mathrm{SU}_8/\{\pm 1\}$ that doesn't lift to a spin group? | $\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\CC\mathbb{C}\def\SU{\text{SU}}\def\SO{\text{SO}}\def\Spin{\text{Spin}}\def\diag{\text{diag}}\def\Id{\text{Id}}$Such a lift is always possible. We will show:
**Theorem** Let $n \equiv 0 \bmod 8$ be a positive integer and let $\rho : \SU\_n/\{ \pm \Id\_n \} \to \SO\_m$ be a (... | 6 | https://mathoverflow.net/users/297 | 430180 | 174,264 |
https://mathoverflow.net/questions/430185 | 3 | I am trying to understand the the following paper <https://arxiv.org/pdf/1810.10971.pdf>, in particular Example 2:
If $ Y \sim N(0,1)$, the standard normal on $\mathbb{R}$, then
$ \begin{align\*} \Big( \mathbb{E} \Big[ \frac{1}{m!} Y^{\otimes m }\Big]\Big)\_{m \geq 0 } = \exp\Big(\frac{1}{2} e\_1 \otimes e\_1 ... | https://mathoverflow.net/users/490927 | What is a tensor product of random variables? | In this context, I believe the tensor product on random variables is nothing other than the tensor product over the values of the RVs. (In other words, if $\Omega$ is a sample space and $X : \Omega \rightarrow V$ and $Y : \Omega \rightarrow W$ are RVs, then $X \otimes Y : \Omega \rightarrow V \otimes W$ is defined by $... | 2 | https://mathoverflow.net/users/7961 | 430192 | 174,268 |
https://mathoverflow.net/questions/413965 | 8 | Let $Cat\_n$ denote the category of $n$-categories. Then [Joyal's category $\Theta\_n$](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/Theta+category) is (1) a full [dense subcategory](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/dense+subcategory) of $Cat\_n$ which (2) is also a [Reedy category](http... | https://mathoverflow.net/users/2362 | Is Joyal's category $\Theta_n$ the "only" Reedy category which is dense in $n$-categories? | I now believe what's really going on here isn't so much a question of Reedyness. Rather, we have the following:
**Claim:**
Let $\mathcal A \subseteq Cat\_n$ be a dense subcategory which is idempotent-complete. Then $\Theta\_n \subseteq \mathcal A$.
Here $Cat\_n$ means weak $(\infty,n)$-categories.
**Remarks:**
... | 2 | https://mathoverflow.net/users/2362 | 430196 | 174,269 |
https://mathoverflow.net/questions/430168 | 1 | (I asked this question [on MSE](https://math.stackexchange.com/questions/4521591) 10 days ago, but I got no answer.)
Let $X$ and $Y$ be two independent identically distributed binomial random variables with parameters $n \in \mathbb{N}$ and $p \in (0,1)$. Let $Z := XY$ be their product.
Is it true or false that $\t... | https://mathoverflow.net/users/491015 | Does the (normalized) product of two independent binomial variables converge in distribution to a normal variable? | $\newcommand{\R}{\mathbb R}\newcommand\ep\epsilon\newcommand\tsi{\tilde\sigma}$Yes, of course. This follows by the multivariate (here, bivariate) so-called delta method.
Indeed, we may assume that
\begin{equation\*}
X=\sum\_{i=1}^n X\_i,\quad Y=\sum\_{i=1}^n Y\_i,
\end{equation\*}
where $X\_1,Y\_1,\dots,X\_n,Y\_n$ a... | 1 | https://mathoverflow.net/users/36721 | 430198 | 174,271 |
https://mathoverflow.net/questions/430193 | 4 | Question:
---------
Did Kolmogorov develop a set of axioms for probability theory motivated by Algorithmic Information Theory in the 1960s?
Context:
--------
In 1965, Andrey Kolmogorov considered three approaches to information theory(combinatorial, probabilistic, and algorithmic) [1] where in the algorithmic app... | https://mathoverflow.net/users/56328 | Kolmogorov's approach to probability theory | In 1970, Kolmogorov developed the 'Combinatorial foundations of information theory and the calculus of probabilities' in relation to a presentation at the International Congress of Mathematicians in Nice(1970). This text was eventually published in 1983:
* A.N. Kolmogorov. Combinatorial foundations of information the... | 4 | https://mathoverflow.net/users/56328 | 430209 | 174,276 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.