parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/292457 | 9 | Let $\lambda\geq \omega\_2$ be a regular cardinal and $S\subset[\lambda]^\omega$ be a stationary set. I'm looking for a property of $S$, say "shootable", such that there exists a forcing extension preserving $\lambda, \omega\_1$ as cardinals that shoots a club into $S$. I've encountered ad hoc examples, but I'd really ... | https://mathoverflow.net/users/119731 | What are the typical forcings to shoot a club through a stationary subset of $[\lambda]^\omega$ | There is a notion of fat stationary subset of $P\_\kappa(\lambda)$ and a natural forcing for shooting a club through it, that you can find in the thesis [Fat subsets of $P\_{\kappa}(\lambda)$](https://open.bu.edu/bitstream/handle/2144/14099/Zaigralin_bu_0017E_10097.pdf?sequence=1) by Ivan Zaigralin. In particular look ... | 7 | https://mathoverflow.net/users/11115 | 292459 | 128,787 |
https://mathoverflow.net/questions/292469 | 1 | Let $\Omega$ be a Polish space and $\mathcal{B}(\Omega)$ be its Borel $\sigma$-algebra. Let $\{\mu\_n\}$ be a sequence of probability measures on $\mathcal{B}(\Omega)$ such that $\mu\_n$ weak-converges to $\mu$. Then in general, $\mu\_n(E) \nrightarrow \mu(E)$ for $E \in \mathcal{B}(\Omega)$.
But if for some $F \in \... | https://mathoverflow.net/users/120390 | Weak-convergence of probability measures implies the convergence of the measure of a continuity set | A trivial counterexample is $\Omega=\mathbb R$, $\mu\_n=\delta\_{1/n}, \mu=\delta\_0$, and $F=\{0\}$. A standard result is $\mu\_n(F)\to \mu(F)$ if $\mu(\partial F)=0$. This looks somehow opposite to what you want.
| 2 | https://mathoverflow.net/users/21051 | 292475 | 128,794 |
https://mathoverflow.net/questions/292499 | 3 | As we know every normal Noetherian domain $R$ can be written as $$R=\bigcap\_{\mathsf{ht}(\mathfrak p)=1}R\_\mathfrak p.$$ I'm asking myself the following question:
**Question:** If the normalization of $\widetilde{R}$ is given by $$\widetilde{R}=\bigcap\_{\mathsf{ht}(\mathfrak p)=1}R\_\mathfrak p,$$ then is $R$ norm... | https://mathoverflow.net/users/80084 | $\widetilde{R}=\bigcap_{\mathsf{ht}(\mathfrak p)=1}R_\mathfrak p$ | I am just posting my comments as an answer. First, in the positive direction, for a Noetherian, integral domain $R$ with fraction field $K$, if $R$ satisfies Serre's condition $S\_2$, then the natural inclusion of subrings of $K$, $$R\to \bigcap\_{\text{ht}(\mathfrak{p}) = 1} R\_{\mathfrak{p}}, $$ is an isomorphism. Th... | 4 | https://mathoverflow.net/users/13265 | 292502 | 128,800 |
https://mathoverflow.net/questions/290926 | 2 | **[INTRODUCTION]**
Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of $G$ on a Hilbert space, then the restriction $\pi|\_{G'}$ of $\pi$ to $G'$ is **(analytic) discretely decomposable... | https://mathoverflow.net/users/56989 | Discrete decomposability of unitary representation | Check the home page of Toshiyuki Kobayashi ([link](https://www.ms.u-tokyo.ac.jp/~toshi/)), download the earlier papers of his,
over there you fill an answer.
Edit: in a 2017 note of Duflo-Galina-Vargas ([link behind paywall](http://www.heldermann.de/JLT/JLT27/JLT274/jlt27051.htm)), you will find a proof that the ans... | 2 | https://mathoverflow.net/users/67162 | 292504 | 128,801 |
https://mathoverflow.net/questions/292498 | 2 | Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow.
>
> *Definition:* Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative if $\forall X,Y,Z \in \mathfrak{X}(M)$: $\nabla\_{\nabla\_X Y}Z = \nabla\_X\nabla\_Y Z$.
>
>
>
Using the de... | https://mathoverflow.net/users/1849 | Do "associative" connections exist / arise naturally in some context? | No, such a connection cannot exist: Consider a function which vanish to first order at a point $p\in M$, i.e., $f(p)=0$ and $d\_pf=0$, but assume that there are vector fields $X,Y$ with $(X\cdot (Y\cdot f))(p)\neq0$.Such a function clearly exist. Moreover, let $\tilde Z$ be a vector field which does not vanish at $p.$ ... | 4 | https://mathoverflow.net/users/4572 | 292505 | 128,802 |
https://mathoverflow.net/questions/292508 | 9 | Let $g,n$ be positive integers, is there a reference that $\mathrm{Sp}(2g,\mathbb{Z})\to\mathrm{Sp}(2g,\mathbb{Z}/n\mathbb{Z})$ is surjection?
The only reference I could find is lemma 5.16 in Deligne–Mumford, but to be honest, I don't quite understand the argument: Why is the reduction map surjective on unipotent ele... | https://mathoverflow.net/users/nan | Reduction mod $n$ of symplectic group | This was originally proved in
M. Newman, J. R. Smart, [Symplectic modulary groups](https://eudml.org/doc/207465), Acta Arith 9 (1964), 83-89.
| 10 | https://mathoverflow.net/users/317 | 292513 | 128,806 |
https://mathoverflow.net/questions/292514 | 12 | Let $R$ be a domain and $\tilde R$ its integral closure in its fraction field: $R\subset \tilde R\subset Frac(R)$.
Is it true that a prime ideal $ \tilde {\mathfrak p} \subset \tilde R$ and its trace $\mathfrak p= \tilde {\mathfrak p}\cap R\subset R$ are related by the equality of heights $$ ht(\tilde {\mathfrak p})... | https://mathoverflow.net/users/450 | Is height preserved in a normalization? | The answer for a general commutative Noetherian domain is "no". The following is adapted from a paper I wrote with Jay Shapiro a few years back.
Consider Nagata's book *Local Rings*, Appendix, Example 2 (see also Stacks Project, tag 02JE). I'm changing notation and specializing a bit (by setting his parameter $m$ to ... | 13 | https://mathoverflow.net/users/19045 | 292519 | 128,809 |
https://mathoverflow.net/questions/291888 | 40 | The *formal group law* associated with a generating function $f(x) = x + \sum\_{n=2}^\infty a\_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my [thesis](https://digital.lib.washington.edu/researchworks/handle/1773/36757), I found a large number of examples of formal group laws that have combinatorial interpreta... | https://mathoverflow.net/users/103524 | Characterizing positivity of formal group laws | Given $\phi(x)\in\mathbb{R}[[x]]$, with $\phi(0)=1$, we have defined $g(x):=\int^x\_0{dt\over \phi(t)}$, $f:=g^{-1}$ and $$F(x,y)=f\big(g(x)+g(y)\big)=\sum\_{n=0}^\infty \psi\_n(x) {y^n\over n!}\in\mathbb{R}[[x,y]].$$ Let's write a recursion for the coefficient sequence $\psi\_n=\partial\_y^nF(x,0)\in\mathbb{R}[[x]]$, ... | 16 | https://mathoverflow.net/users/6101 | 292537 | 128,817 |
https://mathoverflow.net/questions/292531 | 3 | I'm trying to compute $H^3(point group,\mathbb{Z})$ for all the 32 point groups in 3D which has some applications in physics. Unfortunately, I could not find literature discussing this problem. So I tried to use GAP program to compute it.
I used the following code to do the computation:
```
gap> GroupCohomology(P... | https://mathoverflow.net/users/95652 | Calculating cohomology group $H^3(point group,\mathbb{Z})$ using GAP program | The Kunneth formula for group cohomology is:
$$H^n(G\_1 \times G\_2; \Bbb Z) \cong
\bigoplus\_{i= 0}^n H^i(G\_1;\Bbb Z) \otimes\_{\Bbb Z} H^{n-i}(G\_2;\Bbb Z)
\oplus\bigoplus\_{p =0}^{n+1} \text{Tor}^{\Bbb Z}(H^p(G\_1;\Bbb Z),H^{n+1-p}(G\_2;\Bbb Z))
$$
Most of these terms will be $0$ due to the fact that there are ... | 3 | https://mathoverflow.net/users/54401 | 292541 | 128,819 |
https://mathoverflow.net/questions/292516 | 3 | The following Question has a yes answer when $K$ is algebraically closed. I am looking for an elementary proof of it, as well as an answer for arbitrary $K$ of characteristic $0$.
>
>
> >
> > **Question.** Is a connected subgroup of $(K^\*,\times)^n$ defined by polynomial equations and having Zariski dimension 1... | https://mathoverflow.net/users/18583 | Connected subgroup of $\mathbb G^n_m$ of Zariski dimension 1 | Over any field, every smooth connected algebraic subgroup of $\mathbb{G}\_m^n$ of dimension one is isomorphic to $\mathbb{G}\_m$. First note that such a subgroup is diagonalizable because it has a faithful diagonalizable representation. Then use that there is a contravariant category equivalence between diagonalizable ... | 5 | https://mathoverflow.net/users/120592 | 292543 | 128,821 |
https://mathoverflow.net/questions/292461 | 0 | Suppose $B\_r\subset \mathbb{R}^2$ is a hemidisc, i.e., $x^2+y^2 \leq r^2, y\geq 0$. Is there a regularity result of the type $\Vert \psi \Vert\_{W^{2,p}(B\_{1/2})} \leq C (\Vert \psi \Vert\_{L^p(B\_{1})} + \Vert \Delta \psi \Vert\_{L^p(B\_1)}) $?
What about similar Schauder estimates ?
| https://mathoverflow.net/users/3709 | $L^p$ regularity for semidisc | No, you need some information on what $\psi$ does on the real line. A counterexample for your estimate is given by the bounded harmonic function $$ \psi(x,y)=\arctan\frac xy, $$ which does not even belong to $W^{1,p}(B\_{1/2})$ for $p\ge 2$.
To see that Schauder estimates fail, you can similarly use the counterexamp... | 2 | https://mathoverflow.net/users/90407 | 292558 | 128,824 |
https://mathoverflow.net/questions/292566 | 9 | For trying to understand how general a certain theorem is, I'm looking for an example of an essentially small abelian category which has enough projectives and enough injectives, but whose category of projectives is not equivalent to the category of injectives.
By a result of Auslander, each such category can be writ... | https://mathoverflow.net/users/15887 | Example of an abelian category with enough projectives and injectives which are not dual | The category of countable abelian groups is an essentially small abelian category, and has enough projectives and injectives (the countable free abelian groups and the countable divisible groups respectively). However, there is an injective with endomorphism ring $\mathbb{Q}$, but no such projective, so the categories ... | 22 | https://mathoverflow.net/users/22989 | 292572 | 128,830 |
https://mathoverflow.net/questions/292562 | 7 | The integral $$\int\_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$$ is related to the convolution of two [half-normal distribution](https://en.wikipedia.org/wiki/Half-normal_distribution)s. This can be inferred from this [question](https://math.stackexchange.com/questions/2539306/sum-of-independent-half-normal-distribut... | https://mathoverflow.net/users/93724 | What is $\int_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$? | This indefinite integral is a special function, called [Owen's T](https://en.wikipedia.org/wiki/Owen%27s_T_function):
$$\int\_0^z e^{-a^2 x^2}{\rm erf}\,(bx)\,dx=\frac{\arctan(b/a)}{a\sqrt\pi}-\frac{2\sqrt\pi}{a} T\left(\sqrt{2} az,b/a\right)$$
---
Here is the requested derivation:
$$\int\_0^z e^{-a^2x^2}{\rm... | 10 | https://mathoverflow.net/users/11260 | 292574 | 128,832 |
https://mathoverflow.net/questions/292547 | 7 | We call a map $f:{\mathbb Z}\times {\mathbb Z} \to {\mathbb Z}$ an *[additive grid](https://mathoverflow.net/questions/292383/fibonacci-grids)* if for all $x,y \in {\mathbb Z}$ we have that $f(x,y)$ is the sum of the neighboring values, that is, $$f(x,y) = f(x-1,y)+f(x+1,y) + f(x,y-1) + f(x,y+1).$$
To me it feels like ... | https://mathoverflow.net/users/8628 | Injective additive grids | Yes, such a grid can be constructed easily by induction.
As can be seen from the [answer given to your previous question](https://mathoverflow.net/a/292400/955), the adjacent rows can be chosen arbitrarily.
Denote the numbers in these rows by $\ldots,x\_{-1},x\_0,x\_1,\ldots$ and $\ldots,y\_{-1},y\_0,y\_1,\ldots$.
How ... | 2 | https://mathoverflow.net/users/955 | 292579 | 128,834 |
https://mathoverflow.net/questions/292554 | 4 | Let $X$ be a smooth, compact, orbifold of dimension $4$, where the stabilisers are only allowed to be cyclic groups. Let $p \in X$ be an isolated orbifold point (i.e. the orbifold chart about $p$ consists of a finite (cyclic) group acting linearly on $\mathbb{R^{4}}$, and the action is free on $\mathbb{R}^{4} \setminus... | https://mathoverflow.net/users/99732 | Can we perturb a surface away from an orbifold point? | The answer to the question is no. In general the intersection $[f(S)] \cdot [f(S)]$ can lie in $\mathbb{Q} \setminus \mathbb{Z}$, then of course in this case there is no such two cycle.
| 4 | https://mathoverflow.net/users/99732 | 292581 | 128,835 |
https://mathoverflow.net/questions/292580 | -1 | in the book "etale cohomolgy" milne is using two notions: coprime ideals and strictly coprime ideals. It seems to me that both the notions are same.
because (f(t))+(g(t))=(f(t),g(t)).
What am i doing wrong?
| https://mathoverflow.net/users/nan | coprime and strictly coprime ideals | I believe Milne's definitions are as follows. Let $R$ be a ring, and $f, g\in R[x]$ two polynomials. $f$ and $g$ are *coprime* if they share no factors in $R$; they are *strictly coprime* if $(f,g) = R[x]$ as ideals. Strictly coprime implies coprime, and if $R$ is a field then they're equivalent, but in general coprime... | 3 | https://mathoverflow.net/users/119745 | 292587 | 128,837 |
https://mathoverflow.net/questions/292596 | 1 | The Hamiltonian of the p-spherical spin glass model is
$$H\_{N,p}(\sigma)=\frac{1}{N^{\frac{p-1}{2}}} \sum\_{i\_1,...,i\_p=1}^N X\_{i\_1,...,i\_p} \sigma\_{i\_1}\cdot...\cdot \sigma\_{i\_p}$$
where $\sigma \in \mathbb S^{N-1}(\sqrt{N}).$ The $X\_{i\_1,...,i\_p}$ are random i.i.d. standard centred Gaussians.
In th... | https://mathoverflow.net/users/119875 | Random matrix and spherical spin-glass | The spherical two-spin model has Hamiltonian
$$H=N^{-1/2}\sum\_{i\neq j}X\_{ij}\sigma\_i\sigma\_j$$
where the $\sigma\_i$, $i=1,2,\ldots N$, are continuous spin variables subject to the constraint $\sum\_i\sigma\_i^2=N$. The matrix $X$ is a symmetric $N\times N$ matrix of independently distributed Gaussian random varia... | 1 | https://mathoverflow.net/users/11260 | 292597 | 128,840 |
https://mathoverflow.net/questions/292582 | 47 | A few years ago, I came up with this proof of Perron's theorem for a class presentation:
[https://pi.math.cornell.edu/~web6720/Perron-Frobenius\_Hannah%20Cairns.pdf](https://pi.math.cornell.edu/%7Eweb6720/Perron-Frobenius_Hannah%20Cairns.pdf)
I've written an outline of it below so that you don't have to read a link.
... | https://mathoverflow.net/users/120600 | Is this proof of Perron's theorem correct, and if so is it original? | (1) Correctness: I read all arguments in detail and couldn't find anything wrong with them. Of course, this doesn't mean too much...
(2) Orginality: I think in a topic which has such an extensive historical record as Perron-Frobenius theory does, the question of "originality" or "novelty" of any particular proof is a... | 56 | https://mathoverflow.net/users/102946 | 292611 | 128,844 |
https://mathoverflow.net/questions/292594 | 5 | Let $f(z)=\sum\_{n\ge 1}a(n)e(nz)$, be a newform of CM-type, and let $\psi\_f$ be the associated Hecke character, so that,
$$
f(z)=\sum\_{\mathfrak{a}}\psi\_f(\mathfrak{a})e(N(\mathfrak{a})z),
$$
and let $\rho\_{\lambda,f}$ be the associated Galois representation.
Let $\frak{p}$ be a prime ideal of the field by which ... | https://mathoverflow.net/users/44319 | Galois representation associated to CM-newforms | Let me abbreviate $\rho\_{\lambda,f}$ as $\rho$, and $\psi\_f$ as $\psi$.
By definition, $L(s,\rho)=L(s,f)=L(s,\psi)$. The equality of the Euler factors of $L(s,\rho)$ and $L(s,\psi)$ at the split prime $p=\mathfrak{p}\mathfrak{p}'$ means that
$$\det(1-\rho(\mathrm{Frob}\_{\mathfrak{p}})p^{-s})=(1-\psi(\mathfrak{p})p^... | 8 | https://mathoverflow.net/users/11919 | 292612 | 128,845 |
https://mathoverflow.net/questions/292614 | 3 | If $X$ is a proper smooth complex analytic space, one can define Chow groups of **analytic** cycles on $X$ the usual way.
We have a cycle map
$$c^p\_X: \text{CH}^p(X) \to \text{H}^{2p}\_{D}(X,\mathbf{Z}(p))$$
to Deligne cohomology of $X$.
Is $c^p\_X$ an isomorphism? Is $c^p\_X\otimes\mathbf{Q}$ an isomorphism?
... | https://mathoverflow.net/users/nan | Analytic cycles on complex-analytic spaces | Unless I misunderstand your definition, wouldn't $CH^p(X)$ coincide with the usual Chow group when $X$ is smooth projective, by GAGA? And of course Deligne cohomology would be the same. So the answers should be no and no,
for a general variety $X$.
| 6 | https://mathoverflow.net/users/4144 | 292616 | 128,846 |
https://mathoverflow.net/questions/292625 | 7 | One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes.
**A refresher (you may skip to the question at the bottom)**
One defines
(1) $z\_n(X,d) :=$ free abelian group generated by all reduced, irreducible closed $k$-subschemes $W\subset X\times(\mathbf{P}^1\... | https://mathoverflow.net/users/nan | How to think about $\mathbf{Z}(n)_{\mathcal{M}}$ | [All cohomology will be reduced cohomology for ease of notation].
There is no analog for classical homotopy theory. This is related to the fact that the Picard group of the category of spectra is $\mathbb{Z}$ (so the only twists are shifts in degree).
But not all is lost.
Let us enter the more exotic, but still q... | 11 | https://mathoverflow.net/users/43054 | 292628 | 128,848 |
https://mathoverflow.net/questions/292606 | 11 | Let us say that two sets $A$ and $B$ are comparable if there is an injection from $A$ to $B$ or there is an injection from $B$ to $A$. Obviously, in a model of ${\rm ZFC}$ any two sets are comparable by comparing their cardinalities. But this is not necessarily the case in a model of ${\rm ZF}$. For example in Cohen's ... | https://mathoverflow.net/users/5984 | A model of ZF without a well-ordering of the reals in which any two sets of reals are comparable | Yes. The perfect set property will ensure every set of reals is countable or has size continuum.
>
> *Solovay, R.M.*, [**A model of set-theory in which every set of reals is Lebesgue measurable**](http://dx.doi.org/10.2307/1970696), Ann. Math. (2) 92, 1-56 (1970). [ZBL0207.00905](https://zbmath.org/?q=an:0207.00905... | 8 | https://mathoverflow.net/users/7206 | 292630 | 128,849 |
https://mathoverflow.net/questions/289740 | 3 | If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$.
Is there a space $(X,\tau)$ with $|X| > 1$ and with the following properties?
1. The identity map $\text{id}\_X$ is the only continous sur... | https://mathoverflow.net/users/8628 | Maximal elements in the partially ordered set of image spaces | A positive answer to this problem is given by the known answers to the following problem posed by de Groot in [New Scottish book](http://www.wmi.uni.wroc.pl/sites/default/files/upload_attach/ksiega_szkocka_2.pdf).
**Problem 393** (de Groot; 28 May, 1958). *Does there exist a (plane) continuum which does not admit any... | 2 | https://mathoverflow.net/users/61536 | 292637 | 128,852 |
https://mathoverflow.net/questions/292591 | 5 | Say that a partial order $P$ is *forcing-rigid* in a model $V$ if whenever $G \subseteq P$ is generic over $V$, then in $V[G]$, $G$ is the only filter which is $P$-generic over $V$. This implies there are no nontrivial automorphisms of $P$.
If $P$ is forcing-rigid and $P$ forces "$\dot Q$ is forcing-rigid," then is $... | https://mathoverflow.net/users/11145 | preservation of forcing rigidity in iterations | No, rigidity is not preserved in iterations. In particular,
**Proposition:** If $T$ is a rigid Souslin-tree with the property that
$$1 \Vdash\_T T\_{s} =\{ t \in T : t\le\_T s \}\text{ is rigid and Souslin for every }s \not\in \dot{G}$$
(rigid and Souslin off-the-generic-branch in the terminology of [FuchsHam2008]... | 6 | https://mathoverflow.net/users/8843 | 292638 | 128,853 |
https://mathoverflow.net/questions/292631 | 3 | Note:
=====
In this question, a complex number is counted as a vector initiated from the origin.
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_-
>
> Is there a holomorphic function $B:\mathbb{C}^2 \to \mathbb{C}$ such that for every t... | https://mathoverflow.net/users/36688 | Can the "Bisector" be represented by a holomorphic function? | Here is a better answer than the other answer I gave, which is currently accepted. It also answers some of your questions in the comments on that answer. Pick a branch of log, then $B$ and $ (zw)^{1/2}$ are holomorphic and their arguments differ by a multiple of $\pi$ wherever they are both defined. So their quotient i... | 6 | https://mathoverflow.net/users/116794 | 292640 | 128,854 |
https://mathoverflow.net/questions/292669 | 4 | Let $X$ be a smooth, connected, complex analytic variety, and $Y\subset X$ a closed, analytic subvariety of codimension at least 2. Now let $V\subset X\backslash Y$ be a closed, analytic subvariety. Is the closure $\bar{V}$ of $V$ in $X$ also analytic?
**Motivation:** In the case where $X$ is a surface, this seems to... | https://mathoverflow.net/users/4181 | Can an analytic variety extend along a codimension 2 subvariety? | Yes, by Remmert-Stein's extension theorem. See e.g. Fritsche-Grauert, *From holomorphic functions to complex manifolds*, Theorem 6.9.
| 5 | https://mathoverflow.net/users/4721 | 292670 | 128,859 |
https://mathoverflow.net/questions/223909 | 6 | Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is *compatible with* ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\to (X,\tau)$. We denote the collection of $T\_2$-topologies compatible with ${\cal F}$ by $T\_2({\cal F})$.
Note th... | https://mathoverflow.net/users/8628 | Minimal Hausdorff topologies compatible with a bunch of functions | The answer to this question is affirmative:
**Theorem.** There exists a countable set $X$ and an uncountable family $\mathcal F$ of self-functions of $X$ such that the poset $T\_2(\mathcal F)$ has no minimal elements.
Here $T\_2(\mathcal F)$ is the poset of all Hausdorff topologies on $X$ making all functions $f\... | 3 | https://mathoverflow.net/users/61536 | 292673 | 128,861 |
https://mathoverflow.net/questions/292678 | 4 | Let $S$ be an affine scheme, $X$ be a projective $S$-scheme, $W,Z\to X$ two reduced, irreducible closed $S$-subschemes, flat over $S$. Let $S'\to S$ be a faithfully flat map, with $S'$ affine.
Assume there exists an $S'$-isomorphism $g: W\_{S'}\xrightarrow{\simeq} Z\_{S'}$, commuting with the closed immersions into $... | https://mathoverflow.net/users/nan | Descent of isomorphisms between irreducible closed subschemes | $W,Z\to X$ are classified by two points $s,s'\in\text{Hilb}\_{X/S}(S)$. Here I mean the Hilbert functor, ie. I am not using its representability.
Since the Hilbert functor is an fpqc sheaf (ie. with no need of a sheafification) and $S'\to S$ is an fpqc cover, the map
$$\text{Hilb}\_{X/S}(S)\to (S'\to S)^{-1}\_{\rm ... | 3 | https://mathoverflow.net/users/nan | 292680 | 128,863 |
https://mathoverflow.net/questions/292681 | 2 | I'm looking for a survey or a source for Cutting Lemma. I looked at Matusek's Discrete Geometry textbook, but it only proved Cutting Lemma for lines in $\mathbb{R}^2.$ I need to know the proof in $\mathbb{R}^d$ and other variations of Cutting Lemma.
Thanks in advance for any help!
| https://mathoverflow.net/users/118765 | Cutting Lemma in Discrete Geometry |
>
> Chazelle B, "Cuttings." *Handbook of Data Structures and Applications* (D. Mehta and S. Sahni, editors), chap. 25. 2005. [PDF download](https://www.cs.princeton.edu/~chazelle/pubs/cuttings.pdf).
>
>
>
**Theorem 1.1**. Given a set $H$ of $n$ hyperplanes in $\mathbb{R}^d$,
for any $0 < \epsilon < 1$, there ex... | 3 | https://mathoverflow.net/users/6094 | 292684 | 128,864 |
https://mathoverflow.net/questions/292677 | 2 | Fix $a>0$ and $b>0$. Does the following ODE
\begin{equation}
G(x)^2+2axG(x)G'(x)+2aG'(x)(x-b)=0 \tag{\*}
\end{equation}
have a solution, say, $F(x)$, that satisfies $F(x)>0$ and $F'(x)<0$ on $(b,\infty)$?
I tried to solve it by Mathematica, and it gives $G$ as solution to
\begin{equation} x=e^{-2a\left(\text{log}\,G... | https://mathoverflow.net/users/116983 | A first order ODE problem | The answer is yes: indeed, for any $a>0$ and $b>0$, your equation $(\*\*)$ with any $C\_0>0$ determines
a solution $G$ to your ODE $(\*)$ such that $G>0$ and $G'<0$ on $(b,\infty)$ -- and even on $(0,\infty)$.
Indeed, rewrite $(\*\*)$ as
\begin{equation}
x=X(G(x)),\quad X(g):=
e^{-2a\left(\text{log}\,g-\frac{1... | 3 | https://mathoverflow.net/users/36721 | 292688 | 128,867 |
https://mathoverflow.net/questions/292682 | 5 | The Giry monad consists of an endofunctor, $P$, on the category of measureable spaces $\mathcal{M}$, as well as two natural transformations $\mu, \eta$ known as the product and unit respectively. $P$ maps a measurable space, $X$, to the measurable space $P(X)$ of all probability measures on $X$. The unit takes a point ... | https://mathoverflow.net/users/10007 | Is the Giry Monad also a Comonad and if not, is there a probability measures (Co)monad? | The Giry monad is not a comonad because it doesn't admit a counit.
---
Let $P$ be the Giry endofunctor, which assigns to a space $X$ the (suitably topologized) space of probability measures on the Borel subsets of $X$.
For $X$ a single point $\{a\}$, the space $P(X)$ is a single point (with element the trivial... | 4 | https://mathoverflow.net/users/3075 | 292692 | 128,868 |
https://mathoverflow.net/questions/208051 | 7 | Let $n$ be a positive integer and let $K\subseteq \mathbb{R}^n$ be compact. Pick $x^\* \in \mathbb{R}^n\setminus K$.
Let $E$ be the connected component of $\mathbb{R}^n\setminus K$ that contains $x^\*$. Let ${\cal C}$ be the collection of connected components of $K$. For each $C\in {\cal C}$ let $E\_C$ be the connect... | https://mathoverflow.net/users/8628 | Intersection of connected components in $\mathbb{R}^n$ | The answer to this problem is Yes.
Indeed, the inclusion $E\subset \bigcap\_{C\in\mathcal C}E\_C$ is trivial, so it remains to prove that for any point $x\in\mathbb R^n\setminus E$ there exists a connected component $C\in\mathcal C$ of $K$ such that $x\notin E\_C$. By Zorn's Lemma, the compact set $K$ contains a mini... | 5 | https://mathoverflow.net/users/61536 | 292693 | 128,869 |
https://mathoverflow.net/questions/292646 | 4 | Let $M$ be a manifold, $P$ a closed subset and sub-manifold of $M$, and $j : (M,\emptyset) \to (M,P)$ the injection (I note $f : (M,P) \to (N,Q)$ a smooth map $M \to N$ such that $f(P) \subset Q$). The pullback $j^\* : \Omega\_{dR,c}(M,P) \to \Omega\_{dR,c}(M)$ is just $\alpha \mapsto \alpha$ for a form $\alpha$ null o... | https://mathoverflow.net/users/74372 | Relative de rham cohomology with compact support | Not necessarily. For simplicity, let's assume a closed manifold $M$ and a closed submanifold $P$. Then all forms automatically have closed support and the de Rham cohomology is isomorphic to singular cohomology with real coefficients (my preferred way of thinking about cohomology). Then your map $\bar j^\*$ is just the... | 2 | https://mathoverflow.net/users/6646 | 292696 | 128,870 |
https://mathoverflow.net/questions/292708 | 6 | Let $C,\Gamma\subset\mathbb{P}^n$ be degree $n$ rational normal curves in $\mathbb{P}^n$, such that for any $p\in C$ the tangent line $T\_pC$ of $C$ at $p$ is tangent to $\Gamma$ as well. This means that there exists a point $q\in\Gamma$ such that $T\_q\Gamma = T\_pC$.
Clearly, if $n = 2$ this means that $C$ and $\Ga... | https://mathoverflow.net/users/14514 | Rational normal curves and tangent lines | **Edit.** This answer has been edited to address positive characteristic, as alerted by @FelipeVoloch. The result holds **except** in characteristic $2$. In characteristic $2$, the result fails.
Let $k$ be an algebraically closed field. Let $C$ be a smooth, connected $k$-curve. Let $V$ be a $k$-vector space of dimens... | 6 | https://mathoverflow.net/users/13265 | 292711 | 128,873 |
https://mathoverflow.net/questions/292709 | 4 | EDIT: the original $\ge$ is now $>$ (sorry for the typo!)
Let $B\_1(\cdot)$ and $B\_2(\cdot)$ denote independent, standard Brownian bridges, i.e., they are mean-zero Gaussian processes on $[0,1]$ with $B(0)=B(1)=0$ and covariance function $t\_1(1-t\_2)$ for $t\_1 \le t\_2$.
Let $F\_1(\cdot)$ and $F\_2(\cdot)$ each b... | https://mathoverflow.net/users/120669 | Supremum of difference of Brownian bridges: strictly positive wp 1? | The answer is $1$, even if $B\_1$ and $B\_2$ are not independent. Indeed, take any $u>0$. Then
\begin{align\*}
P( \sup\_{x\in\mathbb{R}} [B\_1(F\_1(x)) - B\_2(F\_2(x))] \ge -u )
&\ge \sup\_{x\in\mathbb{R}}P(B\_1(F\_1(x)) - B\_2(F\_2(x)) \ge -u ) \\
&\ge \lim\_{x\to-\infty}P(B\_1(F\_1(x)) - B\_2(F\_2(x)) \ge -u ) \... | 4 | https://mathoverflow.net/users/36721 | 292716 | 128,874 |
https://mathoverflow.net/questions/292694 | 3 | I am reading the lecture notes [INTRODUCTION TO DONALDSON–THOMAS INVARIANTS](https://www.maths.tcd.ie/~mozgovoy/data/dt_lecture.pdf). I have a question in the end of page 1 about the proof of a map is an automorphism.
Let $m>0$ be an integer. Let $\overline{A} = Q[x\_1, x\_2]$ be an algebra with multiplication:
\begi... | https://mathoverflow.net/users/11877 | How to show that a map which relates to Donaldson–Thomas invariants is an automorphism? | The fact that the algebra is commutative has been discussed in the comments. So, I will address the bijectivity of $T\_{a,b}$. I will assume we believe that we have a well defined algebra homomorphism since it is only asked why $T\_{a,b}$ is a bijection. Here is a sketch of bijectivity.
Our multiplication respects th... | 3 | https://mathoverflow.net/users/51668 | 292730 | 128,877 |
https://mathoverflow.net/questions/292548 | 4 | Let $f\_1,f\_2$ be two smooth quasiconcave functions defined on a convex subset of $\mathbb{R}^d$.
It is known that $f\_1+f\_2$ is not necessarily quasiconcave.
**Does there always exist monotonically increasing functions $g\_1,g\_2$ such that the sum $g\_1\circ f\_1 + g\_2\circ f\_2$ is quasiconcave?**
The motia... | https://mathoverflow.net/users/34461 | Can the sum of quasiconcave functions always be made quasiconcave? | There is an easy counterexample. Let
$$
f\_1(x)=\begin{cases}
x-n,&\mbox{when $2n\le x\le 2n+1$,}\\
n+1,&\mbox{when $2n+1\le x\le 2n+2$,}
\end{cases}
$$
and let $f\_2(x)=f\_1(-x)$.
Now whenever $g\_1$ and $g\_2$ are strictly increasing, the function $g\_1\circ f\_1+g\_2\circ f\_2$ is strictly increasing on interval... | 3 | https://mathoverflow.net/users/120173 | 292733 | 128,878 |
https://mathoverflow.net/questions/292718 | 0 | Thanks to sound remarks [here](https://mathoverflow.net/questions/271936/closed-form-for-int-0t-e-x-fraci-n-alpha-xxdx?rq=1) and [here](https://mathoverflow.net/questions/289163/what-can-we-know-about-the-half-of-the-generating-series-of-bessel-function), and looking again at these equations, I noticed that my puzzle b... | https://mathoverflow.net/users/111000 | Literature about the integral of Bessel $\int_0^x I_{0,1}(u) e^{-a u}du$? | The integral is studied in [On Certain Indefinite Integrals Involving Bessel Functions](http://onlinelibrary.wiley.com/doi/10.1002/sapm1958371157/abstract) (1958).
(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)
The paper is ... | 4 | https://mathoverflow.net/users/11260 | 292741 | 128,881 |
https://mathoverflow.net/questions/280027 | 16 | In the article "Hodge theory for combinatorial geometries" by Adiprasito, Huh and Katz, it it claimed in the proof of theorem 5.12 that there is a Chow equivalence between the de Concini-Processi wonderful model $Y$ of an arrangement, and a certain toric variety $X$.
However, the source they cite only proves that
$... | https://mathoverflow.net/users/64302 | Is the Chow ring of a wonderful model for a hyperplane arrangement isomorphic to the singular cohomology ring? | The isomorphism $H^\cdot(Y)\cong Ch^\cdot(X)$ is shown by
Eva Feichtner and Sergey Yuzvinsky in Feichtner, E. & Yuzvinsky, S. Invent. Math. (2004) 155: 515. <https://doi.org/10.1007/s00222-003-0327-2>
It's worth emphasizing that the inclusion $Y\subset X$ is not a homotopy equivalence, and generally $H^\cdot(X)\no... | 6 | https://mathoverflow.net/users/5045 | 292745 | 128,882 |
https://mathoverflow.net/questions/289951 | 14 | Let $X$ be a smooth projective complex analytic space.
We can cook up a complex analytic version of Bloch's cycle complex by declaring
$z^n(X^{\rm an}, m)$
is the free abelian group on all codimension $m$ analytic cycles on $X\times\Delta^n$ ($\Delta^n$ being the usual standard $n$ simplex in complex analytic space... | https://mathoverflow.net/users/nan | Is Deligne cohomology the motivic cohomology of analytic spaces? | The answer is no.
So far, you have checked only very special cases, such as weight $n$ and degree $2n$, for $n = 0,1$.
Consider the analytic hypercohomology of your $\mathbf{Z}(n)\_{\mathcal{M}}$ in degree $2n$, denoted $H^{2n}\_{\rm an}(X,\mathbf{Z}(n))$. The same (elementary) argument as in several references (e... | 7 | https://mathoverflow.net/users/nan | 292756 | 128,885 |
https://mathoverflow.net/questions/292749 | 11 | In this [question](https://mathoverflow.net/questions/292625/how-to-think-about-mathbfzn-mathcalm/292628?noredirect=1#comment725975_292628) I previously asked how to think about the motivic complex $\mathbf{Z}(1)\_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^\*((-)\wedge ... | https://mathoverflow.net/users/nan | How to think about infinite generatedness of motivic cohomology | While waiting for someone more competent than me to answer, let me turn the question right back to you. Why *should* motivic cohomology be finitely generated?
The answer is, of course, that there's no reason for it. And it is not. Let us take a look at a special example
**The Picard group**
Let us fix the ground ... | 14 | https://mathoverflow.net/users/43054 | 292762 | 128,889 |
https://mathoverflow.net/questions/292738 | 5 | If a Hausdorff space $\ X\ $ admits a dense subset $ A \hookrightarrow X\ $ such that
$$|X|^{|A|}\ =\ |X|$$
then indeed $$|X| \leq |\text{End}\_{\text{Top}}(X)| \leq |X|^{|A|}\ = \ |X|.$$
It is the case of $\mathbb{Q} \hookrightarrow \mathbb{R}$. Thus, if there is a small enough dense subspace, there are not so m... | https://mathoverflow.net/users/104432 | Some T2 spaces must have a small dense? | **Is the converse true?**
No.
The paper *[Constructions and Applications of Rigid Spaces, I](http://dx.doi.org/10.1016/0001-8708(78)90006-3)*, Advances in Mathematics 29 (1978), 89-130, by Kannan and Rajagopalan, describes a countaby infinite Hausdorff space $X$ such that the only continuous maps $f\colon X\to X$ a... | 9 | https://mathoverflow.net/users/75735 | 292763 | 128,890 |
https://mathoverflow.net/questions/292728 | 7 | From several lecture notes and some posts, people claim that while schemes are constructed by gluing affine schemes over the Zariski topology, algebraic spaces are constructed by gluing affine schemes over the étale topology, which I do not really understand. Could someone explain this point carefully? Examples?
| https://mathoverflow.net/users/87910 | Clarifying an interpretation of algebraic spaces | If I remember correctly, this goes roughly as follows. Consider the category $\mathcal C=\operatorname{Rings}^{op}$, first endowed with the Zariski topology. You can consider sheaves on this site that are locally covered by representable sheaves. Such sheaves form a category equivalent to the category of schemes.
As... | 3 | https://mathoverflow.net/users/11682 | 292768 | 128,891 |
https://mathoverflow.net/questions/292607 | 4 | Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function and $n>1$. I define the function $g(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]).
Can we say that the Hessian of $g$ is invertible for all $x$ with probability... | https://mathoverflow.net/users/nan | Question on Hessian of a function (probability question) | Here is my counterexample:
Take $f(x)=\|x\|^4$, then for any $x\neq 0$, by spherical symetry, $x$ is a eigenvector of $Hf|\_x$. we have then
$$Hf|\_x(x)=a\|x\|^2 x $$
We can choose $x=v$ an eigenvector on $\frac{1}{2}(A+A^T)$. ie $\frac{1}{2}(A+A^T)(v)=\lambda v$ then with $t\in \mathbb{R}$
$$ (Hg)|\_{tv}=(Hf+A)|\_{tv... | 1 | https://mathoverflow.net/users/99045 | 292769 | 128,892 |
https://mathoverflow.net/questions/292729 | -1 | I'm doing research in Optimization and I have found this obstacle in the way.
>
> If we have set of half planes like $c\_ix\leq b\_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better if it is polynomial time) such that we can certify $\{\cap \_{i\neq p}\{c\_ix\leq b\_i\}\}\subset \{c\_px\leq b\... | https://mathoverflow.net/users/120677 | Does a half plane contain intersection of some other half planes? | Find the maximum of $c\_{p}x$ subject to the constraints $c\_{i}x\le b\_i$ for $i\ne p$. If the maximum is at most $b\_p$, the subset relation holds.
| 2 | https://mathoverflow.net/users/120173 | 292771 | 128,893 |
https://mathoverflow.net/questions/292767 | 3 | I am trying to understand the CM method for elliptic curves. Suppose we fix a discriminant $D<0$ and a prime $p$. In the CM method, we look for integer solutions $(t,y)$ to the norm equation $4p = t^2 -Dy^2$. If these solutions exist, then we can construct an elliptic curve over $\mathbb{F}\_p$ with $p+1 \pm t$ rationa... | https://mathoverflow.net/users/37982 | Trace of elliptic curve in CM method | The solution to the norm equation is unique, under the additional assumptions we make on $D$. Recall that we require either that $D \equiv 1 \mod 4$ or that $D \equiv 0 \mod 4$.
Moreover, your claim about recovering such a curve is only true, if in addition, $D \notin \{-3,-4\}$.
The reason for this uniqueness res... | 3 | https://mathoverflow.net/users/74819 | 292781 | 128,896 |
https://mathoverflow.net/questions/292777 | 3 | I am looking for the tightest known bound for the sum
$$\sum\_{\substack{1\leq k\leq j^\alpha \\ k\mid j}}k^\lambda$$
where $j$ is a large positive integer, $\alpha\in(0,1)$ and $\lambda\geq 1$.
I am also interested whether it is possible to give any better estimate on the sum
$$\sum\_{j=n}^{n^2} \frac{1}{j^\l... | https://mathoverflow.net/users/16040 | Sum of small divisors with powers | The second sum, can be rewritted as
$$\sum\_{k=1}^{n^{2\alpha}} k^\lambda \sum\_{j=\max\{n,k^{1/\alpha}\}\atop k\mid j}^{n^2} \frac{1}{j^\lambda} = \sum\_{k=1}^{n^{2\alpha}} k^\lambda \sum\_{\ell=\lceil \max\{n/k,k^{1/\alpha-1}\}\rceil}^{\lfloor n^2/k\rfloor} \frac{1}{(k\ell)^\lambda} = \sum\_{k=1}^{n^{2\alpha}} \sum\_... | 3 | https://mathoverflow.net/users/7076 | 292798 | 128,901 |
https://mathoverflow.net/questions/292801 | 12 | Let $M$ be a (possibly simply connected) compact manifold $M$. Are there always non-zero classes in the homotopy or homology of $\mathrm{Diff}(M)$ that directly arise from the topology of $M$ itself?
As an example of the type of answers I am looking for I construct non-zero classes in the homotopy and homology of the... | https://mathoverflow.net/users/12156 | Non-zero homotopy/homology in diffeomorphism groups | Here is a very naive approach: choose a basepoint in the manifold (call it $M$). Then evaluation at the basepoint gives a map
$$
\text{Diff}(M) \to M
$$
and so cohomology classes on $M$ pull back to ones on $\text{Diff}(M)$.
If for example, ~~$M$ admits a nowhere zero vector field, then using the associated flow one... | 10 | https://mathoverflow.net/users/8032 | 292803 | 128,903 |
https://mathoverflow.net/questions/292800 | 8 | Let $A$ be a nonempty set. Then we call a nonempty set $p(A)$ to be a *partition set* of $A$ if and only if all the following are true :
1. $p(A)$ is a subset of the power set of $A$.
2. The elements of $p(A)$ are pairwise disjoint.
3. Every element of $A$ is present in some element of $p(A)$.
Now, for a set $A$ an... | https://mathoverflow.net/users/109471 | Cardinalities of which there exists partitions of a set containing elements of the same size | This is equivalent to the axiom of choice.
If the axiom of choice holds, then given $A$ and $B$ which are infinite, then $|A\times B|=\max\{|A|,|B|\}$. So let's say $|A|$ is the maximal one, then this means there is a bijection between $A$ and $A\times B$, so we can partition $A$ to sets of size $|B|$ by considering ... | 16 | https://mathoverflow.net/users/7206 | 292807 | 128,905 |
https://mathoverflow.net/questions/292818 | 4 | Regular categories may equivalently defined as those with:
* finite limits
* coequalizers of kernel pairs
* pulback stable regular epis
or
* finite limits
* pullback stable regular epi/mono factorization
When carefully proving the equivalence, the only limits required are pullbacks i.e. in a category with pullb... | https://mathoverflow.net/users/31420 | Why are Regular Categories assumed to be finitely complete? | A category with pullbacks and equalizers that satisfies the rest of the definition of a regular category is called [locally regular](https://ncatlab.org/nlab/show/locally+regular+category), since this is equivalent to saying that all of its slice categories (which of course have terminal objects) are regular in the usu... | 6 | https://mathoverflow.net/users/49 | 292830 | 128,915 |
https://mathoverflow.net/questions/292833 | 19 | Two simple remarks:
1. The polynomial $x^k-1$ can be factorised over the integers as a product of (irreducible) cyclotomic polynomials: $$x^k-1 = \prod\_{d|k}\Phi\_d(x).$$
If we choose $k$ to be a number that has a lot of divisors, then $x^k-1$ will have a lot of factors. For example, if $k$ is a product of $b$ disti... | https://mathoverflow.net/users/8217 | Why can’t you use cyclotomic polynomials to factor big numbers really quickly? | Lets suppose you're in the worst case scenario: $N=pq$ where $p=2p′+1$ and $q=2q′+1$ with $p$, $q$, $p′$ and $q′$ all prime and $p$ and $q$ roughly the same size. Then the order of $x$ modulo $p$ will be either $p′$ or $2p′$ for any $x \neq \pm 1 \mod p$, and similarly modulo $q$. So we expect $GCD(N,x^k−1)$ to be nont... | 32 | https://mathoverflow.net/users/297 | 292834 | 128,916 |
https://mathoverflow.net/questions/292829 | 10 | This is a very soft question, and I'm not sure what I expect as an answer.
In SGA6, Expose XIII, Theoreme 5.1 it is proven that, if $X$ is a proper scheme over a field $k$, then $NS(X)$ is finitely generated. Here $NS(X) := \mathrm{Pic}\_{X/k}(k)/\mathrm{Pic}\_{X/k}^0(k)$.
However, on wikipedia's page for the "the... | https://mathoverflow.net/users/120713 | Why is the theorem of the base mostly cited only for smooth proper varieties | The comment by nfdc23 answers the question:
"There's no good reason, and in particular nothing pathological for the non-smooth case. Perhaps some paper working with smooth varieties stated the result in the relevant context and someone getting it from there copied the SGA6 reference without reading it and carried ove... | 5 | https://mathoverflow.net/users/120713 | 292847 | 128,919 |
https://mathoverflow.net/questions/272864 | 3 | It is [well documented](https://en.wikipedia.org/wiki/String_art) that certain string-art patterns generate quadratic Bezier curves: let $x, y\_1, y\_2$ be three points in $\mathbb{E}^2$, consider the family of line segments joining $x + (1-t) (y\_1 - x)$ to $x + t (y\_2 - x)$ for $t\in [0,1]$, its envelope forms a qua... | https://mathoverflow.net/users/3948 | Conics, string art, and Bezier-like curves | You could do the usual string-art construction in 3D (giving a 3D parabola), and then do a central projection $(x,y,z) \mapsto (x/z,y/z,1)$ down onto the plane $z=1$. This will give you any conic section curve you like.
Suppose the three 3D points are $\mathbf{P}\_i = (x\_i, y\_i,z\_i)$ for $i=1,2,3$. We can assume ... | 2 | https://mathoverflow.net/users/50265 | 292848 | 128,920 |
https://mathoverflow.net/questions/292665 | 8 | Let $G = \operatorname{GL}\_n(F)$ for a $p$-adic field $F$, and let $G\_D = \operatorname{SL}\_n(F)$. I am wondering if there is a connection between irreducible, admissible representations of $G$ and of $G\_D$.
If $(\pi,V)$ is one for $G$, then I am not sure whether the restriction of $\pi$ to $G\_D$ remains irreduc... | https://mathoverflow.net/users/38145 | Connections between representations of $\operatorname{SL}_n$ and $\operatorname{GL}_n$ | The answer to your questions (with proofs) may be found in
C.J. Bushnell, P.C. Kutzko, The admissible dual of SL(N). I
Annales scientifiques de l'École Normale Supérieure, Série 4 : Volume 26 (1993) no. 2 , p. 261--280
First if $\pi$ is an irreducible smooth (complex) representation of $G$, then $\pi\_{\mid G\_{D... | 5 | https://mathoverflow.net/users/4767 | 292869 | 128,926 |
https://mathoverflow.net/questions/292868 | 4 | Let $k$ be an algebraically closed field of characteristic zero. Let $G$ be a group algebraic space over $k$ such that $G\to $ Spec $k$ is locally of finite type.
Suppose that $G(k)$ is finite.
Does it follow that $G\to$ Spec $k$ is finite (and thus $G$ is a scheme)?
If we assume $G$ is quasi-separated over $k$,... | https://mathoverflow.net/users/120713 | Group algebraic spaces that are locally of finite type and have only finitely many points | The comment of nfdc23 answers the question:
No: for any algebraically closed field $k$, let $C$ be any commutative $k$-group scheme of finite type with positive dimension and $G=C/H$ where $H\to C$ is the $k$-subgroup functor given by the (etale) constant group on $C(k)$. In other words, $G$ is the quotient of $C$ mo... | 3 | https://mathoverflow.net/users/120713 | 292873 | 128,927 |
https://mathoverflow.net/questions/292559 | 18 | [Saito (1988)](https://projecteuclid.org/euclid.dmj/1077306852) gives a proof that
$$\textrm{Art}(M/R) = \nu(\Delta)$$
Here, $M$ is the minimal regular projective model of a projective smooth and geometrically connected curve $C$ of positive genus over the field of fractions of $R$, a ring with perfect residue field. T... | https://mathoverflow.net/users/104436 | How does Saito's treatment of the conductor and discriminant reconcile with an elliptic curve? | This is a great question, but I don't think there is an easy answer.
[Saito](https://projecteuclid.org/euclid.dmj/1077306852) himself proves on p.156 (Cor. 2) that his results imply Ogg's formula, including the missing
case of 2-adic fields. However, the proofs are quite condensed and the underlying technology very a... | 15 | https://mathoverflow.net/users/3132 | 292877 | 128,928 |
https://mathoverflow.net/questions/292650 | 9 | There is a general principle that, for finite simply-connected CW complexes, things that are true rationally are usually true once you localize away from a finite list of primes.
I'm interested in the possibility that a wedge $S^a \vee S^b$ might be a retract of a space, and I'll be satisfied if there is such a retr... | https://mathoverflow.net/users/3634 | Retracting a wedge of spheres off a homotopy fiber | Take the homotopy fibre of the inclusion $f:\mathbb{C}P^2\vee\mathbb{C}P^2\hookrightarrow \mathbb{C}P^2\times\mathbb{C}P^2$. Both of these spaces are simple connected and neither has a wedge of spheres as a retract, even after inverting any finite collection of primes (unless I've overlooked something). The homotopy fi... | 7 | https://mathoverflow.net/users/54788 | 292881 | 128,931 |
https://mathoverflow.net/questions/292878 | 11 | Let $(u\_1, u\_2, u\_3, u\_4)$ and $(v\_1, v\_2, v\_3, v\_4)$ be vectors in $\mathbb R\_+^4$. Is the following inequality true?
\begin{align\*}
\left(\sum\_{{[4] \choose 3}} \sqrt{u\_i u\_j u\_k}\right)^{2/3} + \left(\sum\_{{[4] \choose 3}} \sqrt{v\_i v\_j v\_k}\right)^{2/3} \leq \left(\sum\_{{[4] \choose 3}} \sqrt{... | https://mathoverflow.net/users/20062 | A (reverse)-Minkowski type inequality for symmetric sums | Rewrite the inequality in question as
\begin{equation\*}
f(u+v)\le f(u)+f(v)
\end{equation\*}
for $u,v$ in $\mathbb R\_+^4$,
where
\begin{equation\*}
f(u):=-\left(\left(\frac{1}{\sqrt{u\_1}}+\frac{1}{\sqrt{u\_2}}+\frac{1}{\sqrt{u\_3}}
+\frac{1}{\sqrt{u\_4}}\right) \sqrt{u\_1 u\_2 u\_3 u\_4}\right)^{2/3}.
\end{eq... | 5 | https://mathoverflow.net/users/36721 | 292884 | 128,933 |
https://mathoverflow.net/questions/292828 | 0 | Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K\*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $\*$ is the free product. Then we can construct a group algebra $$\mathbb C(D)=\left\{h\_{0}g\_{0}+\Sigma\_{i=1}^{n}h\_{k}g\_{k}\mid h\_{i}\in \mathbb C... | https://mathoverflow.net/users/110479 | Find the trace for some elements in group algebra | You essentially want to count the number of 'words' of length 2k in the 'alphabet' $a$, $(ab)$, $(ac)$ and $(ad)$ that evaluate to 1. A quick computational check for $k=0,1, ..., 6$ suggests that this number corresponds to [sequence A194724 on the OEIS](http://oeis.org/A194724). The reason for this eludes me at the mom... | 0 | https://mathoverflow.net/users/108905 | 292887 | 128,934 |
https://mathoverflow.net/questions/292897 | 0 | Let $R(m,n)$ be defined on all the integers such that $R(m,0)=m, R(0,n)=n, R(m,n)=R(n,m)$ and $R(R(m,n),p)=R(m,R(n,p))$ for all integers $p$. Thus $R$ satisfies the *Abel associativity equation*. Let $f$ be defined for all integers, and $f(0)=0$. Then the Forsyth/Abel functional equation is
>
> $$f(m)+f(n)=f(R(m,n... | https://mathoverflow.net/users/62343 | Find the general solution to the Forsyth/Abel functional equation | This is wrong. Let $f$ be any permutation of the integers such that $f(0)=0$, and $R(m,n) = f^{-1}(f(m)+f(n))$.
| 1 | https://mathoverflow.net/users/13650 | 292899 | 128,935 |
https://mathoverflow.net/questions/292893 | 14 | Let $G$ be a hyperbolic group. I know that it is an open problem whether $G$ has a torsion-free subgroup of finite index. But if we let $N$ be the subgroup of $G$ generated by its non-torsion elements, then is $G/N$ necessarily finite?
If not, $G/N$ would be a finitely generated infinite torsion group with finitely m... | https://mathoverflow.net/users/35840 | Subgroup of hyperbolic group generated by non-torsion elements | Yes. The following answer is inspired by Andy Putman's comment. Let $N\_\infty(G)$ be the subgroup generated by elements of infinite order, in a group $G$.
Every non-elementary hyperbolic group $G$ with trivial finite radical has a useful property, which bears the ridiculous name "Pnaive", namely that for every finit... | 19 | https://mathoverflow.net/users/14094 | 292900 | 128,936 |
https://mathoverflow.net/questions/292905 | 7 |
>
> Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements?
>
>
>
>
>
I have to check positive semi-definiteness of many symmetric matrices with integer elements. First I used eigenvalues, but floating point round error happens :... | https://mathoverflow.net/users/120767 | Checking positive semi-definiteness of integer matrix | For small symmetric matrices, you could look at the characteristic polynomial.
The real symmetric matrix $A$ is positive semidefinite iff the coefficients of the characteristic polynomial are alternating in sign. For $n \times n$ matrices this gives you $n$ integer expressions to check.
| 5 | https://mathoverflow.net/users/13650 | 292921 | 128,945 |
https://mathoverflow.net/questions/292907 | 1 | My goal is to generate an irreducible polynomial over $GF(2^{12})$ with degree $t$, which can get fairly big, let's say up to $t=200$ or so. I've found this [very helpful paper](http://www.math.clemson.edu/~sgao/papers/GP97a.pdf) that walks me through the Ben-Or irreducibility test. I've implemented it, and it works pe... | https://mathoverflow.net/users/120765 | Efficient algorithm for $x^n-x \mod P(x)$ over $GF(2^{12})$ | OP here. Achim Krause gave the answer in his comment on my question; I'm putting it here to show the question is resolved.
>
> You can compute $x^{2^n}$ mod $f$ by starting with $x$, and then repeatedly squaring and reducing modulo $f$ in each step. Takes you $n$ steps, and the degree of none of the intermediate va... | 1 | https://mathoverflow.net/users/120765 | 292924 | 128,947 |
https://mathoverflow.net/questions/292913 | 3 | I am looking for a standard name (if it exists) for the following property of a Schauder basis $(e\_i)\_{i=1}^\infty$ in a Banach space $X$:
$$\|\sum\_{i\in F}x\_ie\_i\|\le\|x\|$$for any $x=\sum\_{i=1}^\infty x\_ie\_i\in X$ and any finite subset $F\subset\mathbb N$.
This condiion implies that the Schauder basis is ... | https://mathoverflow.net/users/61536 | What is a standard name for this kind of unconditional bases in Banach spaces? | The terminology I have seen in the literature refers to such a sequence as being a *$1$-suppression unconditional basis*. More generally, if for $K\geq1$ we have \begin{equation}\Vert \sum\_{i\in F}x\_ie\_i\Vert \leq K\Vert x\Vert\end{equation} for every $x=\sum\_{i=1}^\infty x\_i\in X$ and finite $F\subset\mathbb{N}$,... | 8 | https://mathoverflow.net/users/848 | 292925 | 128,948 |
https://mathoverflow.net/questions/243865 | 5 | Let $R$ be an integral $\bar{\mathbb{F}}\_p$-algebra of finite type, let $V$ be an $R$-algebra. Consider a morphism $f \colon \mathrm{Spec}(V) \rightarrow \mathbb{A}^n\_R$ that has the following properties:
* every fibre of $f$ is a closed immersion $f\_K\colon \mathrm{Spec}(V)\_K \hookrightarrow \mathbb{A}^n\_K$
* ... | https://mathoverflow.net/users/33573 | A sufficient condition for a morphism to be a closed immersion? | Yes, it is true. More precisely any universally closed monomorphism of schemes (your second and third condition) with locally Noetherian target is a closed immersion. This is proven in Proposition 3.8 of Ferrand: Monomorphismes de schémas noethérian (<http://www.numdam.org/article/SAC_1967-1968__2__A7_0.pdf>).
| 5 | https://mathoverflow.net/users/120529 | 292935 | 128,951 |
https://mathoverflow.net/questions/292938 | 4 | Let $(\Omega,\mu)$ be a measure space, say $\sigma$-finite for the sake of simplicity, and let $L^1 := L^1(\Omega,\mu)$ denote the real-valued $L^1$-space over $(\Omega,\mu)$.
For all $f,h \in L^1$ we call the set
\begin{align\*}
[f,h] := \{g \in L^1: \; f \le g \le h\}
\end{align\*}
the **order interval** between $... | https://mathoverflow.net/users/102946 | Weak compactness of order intervals in $L^1$ | As Jochen Wengenroth mentioned in the comments, the weak compactness of order intervals follows from their uniform integrability. I think the proof of the Dunford-Pettis theorem is reasonably elementary; anyway, here is a sketch tailored for this special situation.
Let $\mathcal{F}$ be the weak-$\ast$ closure of $[f,... | 3 | https://mathoverflow.net/users/95776 | 292947 | 128,955 |
https://mathoverflow.net/questions/292942 | 1 | I have asked this question on [stats.se.com](https://stats.stackexchange.com/questions/325194/reinforcement-learning-definition-construction-of-state-and-action-random-var) but I did not receive an answer. Given is the description of a probabilistic finite state machine and I want to 'translate this' into a Markov proc... | https://mathoverflow.net/users/39310 | Markov processes: Construction of the state variables | The Markov chain construction you want you should able to find in any text on Markov chains/processes or even in texts on probability/stochastic processes in general. Yet, it is simpler to give the construction than to look for it in the literature.
Indeed, let $p\_0:=I^d$ and $p:=\Delta^p$, the "initial distributio... | 0 | https://mathoverflow.net/users/36721 | 292957 | 128,960 |
https://mathoverflow.net/questions/292934 | 5 | Suppose that $\langle X,Y\rangle$ is a dual pair of Banach spaces satisfying $|\langle x,y\rangle|\leq \Vert x\Vert\Vert y\Vert$ for all $x\in X$, $y\in Y$.
Is it true that the unit ball of $X$ is $\sigma(X,Y)$-closed?
| https://mathoverflow.net/users/70540 | weak closedness of the unit ball for a dual pair of Banach space | Okay, even assuming that $Y$ separates points the answer is still no. Take $X = l^1$ and let $Y$ be the set of elements $(a\_n)$ of $l^\infty$ which satisfy $\lim a\_n = 2a\_1$. It's easy to see that $Y$ separates the points of $X$, but $e\_n \to 2e\_1$ weakly where $(e\_n)$ is the standard basis of $l^1$. So $2e\_1$ i... | 10 | https://mathoverflow.net/users/23141 | 292958 | 128,961 |
https://mathoverflow.net/questions/292968 | 0 | In finitely presented groups, we can define equivalence classes simply by writing equations in the generators : $abc=d$. In this equivalence class we find elements like this $a(aa^{-1})bc$. We can interpret these equations on a finite category, where the generators are just the arrows. Then we are basically saying this... | https://mathoverflow.net/users/10007 | The Abstraction of Equality | I'm not quite sure what you mean, but one possibility for what you may be driving at is the following. It may seem like an "arrow of diagrams" should be a functor, but usually the categorification of "imposing an equality" is not acting by a functor but rather "adding a morphism". We can thus categorify a group present... | 5 | https://mathoverflow.net/users/49 | 292973 | 128,964 |
https://mathoverflow.net/questions/292955 | 3 | Suppose $f \colon I \to \mathbb{R}$ is a function in, say, $L^\infty$, and $I \subset \mathbb{R}$ is a bounded interval. We may assume further regularity on $f$, such as Lipschitz continuity or strict positivity, in case it matters.
Consider all polynomials of $f$, such as $3f^4 - f^2 +2$. These are in $L^2(I)$, sinc... | https://mathoverflow.net/users/1445 | Closure of polynomials of a function in $L^2$ | Let $\sigma(f)$ be the smallest $\sigma$-algebra on $[0,1]$ which makes $f$ measurable. I claim that $\overline{P\_f} = L^2(I, \sigma(f), m)$, the space of all square-integrable $\sigma(f)$-measurable functions. In particular, $P\_f$ is dense iff the completion of $\sigma(f)$ under $m$ contains all the Borel sets (equi... | 5 | https://mathoverflow.net/users/4832 | 292978 | 128,966 |
https://mathoverflow.net/questions/292979 | 3 | If $G$ is a finite solvable group, it is known (for example, Murty & Raghuram 2000, lemma 2.4) that $Ind\_H^G 1\_H-1\_G$ can be expressed directly (with all coefficients $=1$) as a sum of monomial characters.
What about $G$ non-solvable?
I guess it is one of two options: 1) it is known to be false, 2) the same is e... | https://mathoverflow.net/users/120797 | $Ind_H^G 1_H-1_G$ as direct sum of monomials | It does not hold in general: For instance take $G=S\_5$, the symmetric group on $5$ letters, and $H=S\_4$. Then $\text{Ind}\_H^G1\_H-1\_G$ is an irreducible character of degree $4$ (by $2$-transitivity of $S\_5$). Thus if it were a sum of monomial characters, it would have to be monomial, implying that $S\_5$ has a sub... | 11 | https://mathoverflow.net/users/18739 | 292985 | 128,969 |
https://mathoverflow.net/questions/292824 | 6 | In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P\_1$ to be locally free:
If for all closed points $t:x \hookrightarrow X$ we have $Ext^i(P\_1, t\_\* \mathscr{O}\_x)=0$ for all $i>0$, then $P\_1$ is a locally free sheaf.
... | https://mathoverflow.net/users/119460 | Question on condition for a sheaf to be locally free in Orlov 2004 | The question is local, so it is enough to show that if $A$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $M$ is a finitely generated module such that $Ext^i(M,A/\mathfrak{m}) = 0$ for $i > 0$ then $M$ is free. Let $n = \dim(M/\mathfrak{m}M)$ and let $f:A^{\oplus n} \to M$ be a homomorphism that induc... | 7 | https://mathoverflow.net/users/4428 | 292989 | 128,970 |
https://mathoverflow.net/questions/292962 | 11 | I would like to know if there are sources on the history of the classification of mathematical subjects.
Gérard Lang
| https://mathoverflow.net/users/30395 | History of the classification of mathematical subjects | Here is one such historical overview:
[Mathematics in library subject classification systems](https://books.google.ca/books?id=fL5DDwAAQBAJ&pg=PA181), by Craig Fraser (2016). ([Springer link](https://link.springer.com/chapter/10.1007/978-3-319-64551-3_12))
>
> Insofar as library science is concerned, modern class... | 6 | https://mathoverflow.net/users/11260 | 292991 | 128,971 |
https://mathoverflow.net/questions/292967 | 3 | Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane. Consider two positive reals $T\_1,T\_2>0$ such that $T\_1\le T\_2$.
>
> **My question.** Does the following inequa... | https://mathoverflow.net/users/62673 | A matrix monotonicity question | The answer is no, in general. Here is a counterexample:
Let
\begin{align\*}
X =
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix},
\quad \text{and} \quad
A =
\begin{pmatrix}
-1 & -1 \\
1 & -1
\end{pmatrix}
=
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
- I,
\end{align\*}
where $I \in \mathbb{R}^{2 \time... | 4 | https://mathoverflow.net/users/102946 | 292995 | 128,972 |
https://mathoverflow.net/questions/292993 | 2 | Let $X$ be a quasi-projective integral variety over $\mathbb{C}$. If $X$ is projective, then $\mathrm{H}^2(X,\mathbb{Z})$ contains "ample" classes. These "ample" classes are defined as being the image of an ample line bundle on $X$ via $\mathrm{Pic}(X) \to \mathrm{H}^2(X,\mathbb{Z})$.
If $X$ is not projective, I woul... | https://mathoverflow.net/users/120713 | Does cohomology with compact support contain "ample" elements | The answer is "not always": for example, if $X=\mathbb A^2$ and $\overline X=\mathbb P^2$, then $H^2\_c(X)=0$.
| 5 | https://mathoverflow.net/users/5690 | 292996 | 128,973 |
https://mathoverflow.net/questions/293001 | 3 | Let $X$ is a smooth projective variety *defined over a finite extension $K/\mathbf{Q}$*, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{\rm an}$ the complex-analytic space associated to $$(X\times\_{K,\sigma}\mathbf{C})(\mathbf{C}).$$
>
> Is it know... | https://mathoverflow.net/users/nan | Absolute Hodge cycles | A couple of comments. First of all, the question of absoluteness of Hodge cycles is only interesting if there is more than one embedding if your field of definition into $\mathbb{C}$. So you really want to formulate your question for a field different from $\mathbb{Q}$, otherwise it's vacuously true. [Added: This was i... | 4 | https://mathoverflow.net/users/4144 | 293003 | 128,977 |
https://mathoverflow.net/questions/292923 | 9 | I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to correct me in the probably various places I am wrong)
I am trying to make some basic computations, like for example c... | https://mathoverflow.net/users/119805 | Hecke algebra of GL(2,F) | Let $G$ be a reductive $p$-adic group. First fixing a Haar measure on $G$, you can identify the algebra of distributions of $G$ with the ("big") Hecke algebra $H(G)$ of locally constant complex functions with compact support equipped with convolution $\star$. If $K$ is any compact open subgroup, the bi-$K$-invariant fu... | 5 | https://mathoverflow.net/users/4767 | 293023 | 128,982 |
https://mathoverflow.net/questions/293021 | 2 | Let $M$ be a smooth, compact manifold and $\xi: \mathcal B \to M$ a smooth complex [Banach bundle](https://en.wikipedia.org/wiki/Banach_bundle) over $M$. Here, smooth is understood to be in the [Fréchet-sense](https://de.wikipedia.org/wiki/Fr%C3%A9chet-Ableitung). Further, let $p: V \to M$ be an ordinary smooth finite-... | https://mathoverflow.net/users/78554 | On sections into Banach bundles over a compact manifold | if you go back to the proof you linked to you realize that you only need to embed one of the vector bundles into a trivial bundle.
In detail: For $V$ a trivial bundle your map is an isomorphism. In general, let $V\oplus V^{\perp}$ be a trivial bundle. Then you have $\Gamma((V\oplus V^{\perp})\otimes \mathcal B)\cong ... | 3 | https://mathoverflow.net/users/105652 | 293027 | 128,983 |
https://mathoverflow.net/questions/293013 | 7 | Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing notion $\mathbb{P}\_G$: for $A, B \in \mathcal{B}$ let $A \sim B \iff m(A \bigtriangleup B) = 0$, and let $\mathbb{P}\_G$ co... | https://mathoverflow.net/users/11115 | Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give | If $G$ is a compact group with infinite weight, then the Maharam type of the Haar measure on $G$ is equal to its weight - See Theorem 2.4 in S. Grekas, On products of topological measure spaces, Handbook of measure theory, Vol. 1, edited by E. Pap, Elsevier 2002.
As the measure algebra of a compact group is Maharam h... | 4 | https://mathoverflow.net/users/120832 | 293032 | 128,986 |
https://mathoverflow.net/questions/293017 | 1 | Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does $\mathrm{Aut}^0\_{X/k}(k)$ act trivially on $NS(X)$?
| https://mathoverflow.net/users/120713 | does $Aut^0$ act trivially on the Neron-Severi group? | Yes, as abx already said. The morphism $\mathrm{Aut}(X) \to \mathrm{Aut}(\mathrm{NS}(X))$ is algebraic. Thus, it maps the connected component of the algebraic group $\mathrm{Aut}(X)$ to the **trivial** connected component of $\mathrm{Aut}(\mathrm{NS}(X))$. QED
| 2 | https://mathoverflow.net/users/4333 | 293036 | 128,987 |
https://mathoverflow.net/questions/292988 | 4 | In [pg. 24](https://books.google.com/books?id=5lv_CAAAQBAJ&pg=PA24) of his book on Galois cohomology, Serre gives the following exercise:
"Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties:
(i) $P$ is a pro p-group
(ii) $G$ is finite.
(iii) A Sylow... | https://mathoverflow.net/users/nan | No lifts in an exact sequence of profinite groups? | Here is the problem with my conjectural argument.
My hope that one could reduce this to the case where $P$ is finite abelian, but this does not work. This reduction can however be made if one further assumes that $H^1(G,P) \to H^1(S,P)$ is an isomorphism for any abelian $p$-group. To be able to reduce the general ca... | 0 | https://mathoverflow.net/users/nan | 293046 | 128,990 |
https://mathoverflow.net/questions/293045 | 1 | This is an exercise from Serre's book on Galois cohomology.
Let $p>5$ and consider the groups $SL\_2(\mathbb{F}\_p)$ and $SL\_2(\mathbb{Z}\_p[w])$ where $w$ is a primitive $p$th root of unity.
Is there an elementary reason for why there can be no embedding $SL\_2(\mathbb{F}\_p) \rightarrow SL\_2(\mathbb{Z}\_p[w])... | https://mathoverflow.net/users/nan | Is there an elementary reason for why $SL_2(\mathbb{F}_p)$ for $p>5$ does not embed into $SL_2(\mathbb{Z}_p[w])?$ | Notice that $\mathrm{SL}\_2(\mathbb F\_p)$ has a cyclic subgroup $U$ of order $p$ (the upper triangular unipotent matrices) such that some conjugacy class in $\mathrm{SL}\_2(\mathbb F\_p)$ contains at least $(p - 1)/2$ elements of $U$. (This can be seen already if you conjugate only by diagonal matrices. I don't know i... | 3 | https://mathoverflow.net/users/2383 | 293049 | 128,991 |
https://mathoverflow.net/questions/293047 | 17 | When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the Constructible Universe. For instance, in Jech's 'Set Theory' pg 176, in order to prove the consistency of the Axiom of Choic... | https://mathoverflow.net/users/120841 | Taking a proper class as a model for Set Theory | What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the *scheme* of statements that the model satisfies every individual axiom of ZFC, as a separate statement for each axiom.
The difference is between asserting "$L$ is a model of ZFC" and the scheme ... | 19 | https://mathoverflow.net/users/1946 | 293050 | 128,992 |
https://mathoverflow.net/questions/293037 | 9 | I'm mainly concerned with countable support iterations of proper forcings that add reals of some large cardinal length. It is known that countable support iteration of Sacks forcing/Cohen forcing of weakly compact ($\kappa$) length forces $\kappa=\omega\_2$ has the tree property. Is there a general theorem, like: for a... | https://mathoverflow.net/users/23835 | Countable support iteration of proper forcings and the tree property | If $\kappa$ is huge, then any countable support iteration $\mathbb{P}$ of proper forcing up to $\kappa$ (where each component is of size $<\kappa$) that forces $2^\omega = \kappa = \omega\_2$, must also force the tree property at $\omega\_2$.
To see this, suppose $j: V \to N$ is a huge embedding with critical point ... | 8 | https://mathoverflow.net/users/26319 | 293054 | 128,994 |
https://mathoverflow.net/questions/293025 | 8 | Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$.
>
> Let $M\in \mathcal{B}(F)$ be a bounded operator. Suppose
... | https://mathoverflow.net/users/116483 | Are the following subsets of a Hilbert space always homeomorphic? | The topological equivalence of the set $S\_M:=\{x\in F:\langle Mx,x\rangle=1\}$ and the unit sphere $S:=\{x\in F:\|x\|=1\}$ can be proved as follows.
The assumptions on $M$ and the [spectral theorem](https://en.wikipedia.org/wiki/Spectral_theorem#Multiplication_operator_version) (or just the equality $\langle Mx,x\ra... | 7 | https://mathoverflow.net/users/61536 | 293057 | 128,996 |
https://mathoverflow.net/questions/290694 | 1 | If $\left( p\_n \right)\_{n=0}^{\infty}$ is a family of orthogonal polynoamials with respect to a measure $\mu$ on $[-1,1]$, and $\left( x\_j, w\_j \right)$ are the quadrature points and weights for the respective Gaussian quadrature rule, we can easily prove that
$$ f\_N (x) : \, = \sum\limits\_{n=0}^{N-1} \sum\limit... | https://mathoverflow.net/users/42864 | Proof Reference - Polynomial interpolation at quadrature points | Here, three possible references for the formula:
>
> P. J. Davis and P. Rabinowitz, Methods of Numerical Integration,
> Computer Science and Applied Mathematics. Academic Press, New York,
> 1984 (see p.88)
>
>
> J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, CRC Press, New
> York, 2003 (see section 8.... | 1 | https://mathoverflow.net/users/89429 | 293065 | 129,000 |
https://mathoverflow.net/questions/293026 | 15 | In the summer I will be teaching a course in (plane) Euclidean geometry to future high school teachers and I am looking for a suitable axiom system (unlike [College (Euclidean) geometry textbook recommendations](https://mathoverflow.net/questions/101883/college-euclidean-geometry-textbook-recommendations) where books a... | https://mathoverflow.net/users/5339 | Axioms for constructive Euclidean geometry | Have a look at Hartshorne's *Geometry: Euclid and Beyond*. He uses [Hilbert's axioms](https://en.wikipedia.org/wiki/Hilbert%27s_axioms) for geometry and discusses (section 11) the following "circle-circle intersection axiom (E)":
>
> Given two circles $\Gamma,\Delta$, if $\Delta$ contains at least one point inside ... | 17 | https://mathoverflow.net/users/49 | 293068 | 129,001 |
https://mathoverflow.net/questions/293063 | 13 | Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in L(X,Y) \text{ are equiv., if there is an isometry P on Y with }PT=S \right\}.$$
I ask: Does $$\sup\_{x \in X} \left\l... | https://mathoverflow.net/users/119875 | The geometry of $\mathbb{R}^n$ | The answer is no, in general.
In order to construct a counterexample, let $X = Y = \mathbb{R}^n$ for any $n \ge 2$ and endow this space with the $p$-norm for your favourite $p \in [1,\infty] \setminus \{2\}$. The point about this choice of the norm is that the (linear) isometries on $\mathbb{R}^n$ with respect to thi... | 12 | https://mathoverflow.net/users/102946 | 293070 | 129,003 |
https://mathoverflow.net/questions/293005 | 17 | As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem?
In particular, what is known about the arithmetic systems $PA + \text{Fermat's Last Theorem}$ and $PA + \lnot \text{Fermat's Last Theorem}$?
Note in particular that if $PA \vdash \te... | https://mathoverflow.net/users/65915 | What is known about the relationship between Fermat's last theorem and Peano Arithmetic? | The main reference for this topic is Angus Macintyre's appendix to Chapter 1 ("The Impact of Gödel's Incompleteness Theorems on Mathematics") of *Kurt Gödel and the Foundations of Mathematics: Horizons of Truth* (Cambridge University Press, 2011).
There are a a couple of reasons why one might wonder whether the proof... | 27 | https://mathoverflow.net/users/3106 | 293081 | 129,006 |
https://mathoverflow.net/questions/293083 | 2 | Given $F \subseteq C\_C(\mathbb{R}^d, \mathbb{R}^p)$, $F$ is dense in $C\_C(\mathbb{R}^d, \mathbb{R}^p)$ in the supremum norm $\|\cdot\|\_\infty$. Also given $G \subseteq C\_C(\mathbb{R}^p, \mathbb{R}^s)$, $G$ is dense in $C\_C(\mathbb{R}^p, \mathbb{R}^s)$ in $\|\cdot\|\_\infty$. Is the set $G \circ F := \{g \circ f: g... | https://mathoverflow.net/users/120854 | Is the "composition" of two dense subsets of functions dense? | This is hopeless in general: if $p=1$ and $d=s=2$, then the function $h(x,y)=(x,y)$ (or a compactly supported version of it) is very far from being a composition of a function in $C(\mathbb R,\mathbb R^2)$ with a function in $C(\mathbb R^2,\mathbb R)$. To see this, suppose that $f$ and $g$ belong to $C(\mathbb R^2,\mat... | 6 | https://mathoverflow.net/users/11054 | 293095 | 129,010 |
https://mathoverflow.net/questions/293031 | 11 | [Gabriel-Ulmer duality](https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality) states that 2-categories $\mathrm{Lex}$ (of small finitely complete categories and functors preserving finite limits) and $\mathrm{LFP}$ (of locally finitely presentable categories and finitary right adjoints) are dual. This duality should be ... | https://mathoverflow.net/users/62782 | Gabriel-Ulmer duality for $\infty$-categories | I'm not aware of anyone writing the proof down, but I think we can patch it together as an easy consequence of several facts in Lurie's *Higher Topos Theory* (henceforth HTT).
The statement, as I understand it, is that the functor
$$\mathrm{Fun}^{lex}(-,\mathrm{Space}):(\mathrm{Cat}\_\infty^{lex,\natural})^{op}\to\ma... | 11 | https://mathoverflow.net/users/43054 | 293105 | 129,013 |
https://mathoverflow.net/questions/293133 | 3 | Let $\mathcal{K}$ be a $\lambda$-accessible category and $\hat{\mathcal{K}}$ its free completion under connected limits.
* Is $\hat{\mathcal{K}}$ still accessible?
* $\mathcal{K}$ can be identified with a subcategory of $\text{Set}^{\text{Pres}\_{\lambda}(\mathcal{K}) ^{\text{op}}}$, is it true that $\hat{\mathcal{K... | https://mathoverflow.net/users/104432 | Freely adding connected limits preserves accessibility? | An easy way to see the failure of (2) is that if $\mathcal{K}$ already has connected limits, then its restricted Yoneda embedding into $\mathrm{Set}^{\mathrm{Pres}\_\lambda(\mathcal{K})^{\mathrm{op}}}$ preserves them, so that its closure under such limits therein would be just itself. But the *free* completion under co... | 4 | https://mathoverflow.net/users/49 | 293135 | 129,021 |
https://mathoverflow.net/questions/293124 | 6 | While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.
**Question** : Suppose $K$ is continuous non-negative function on $[0,T]^2$ such that $K(t,s)= 0 $ if and only if $s \ge t$. Prove or disprove the fol... | https://mathoverflow.net/users/115814 | Injectivity of a Fredholm operator | Surprisingly (to me), the statement is false. My counterexample is a little messy, but the idea is fairly simple.
Take $T = 1$ and set $a\_n = \frac{1}{n}$ and $b\_n = 1 - \frac{1}{n}$ for $n \in \mathbb{N}$. Define $f$ by setting $f(t) = 1$ on the intervals $[a\_{2n+2}, a\_{2n+1})$ and $f(t) = -1$ on the intervals $... | 7 | https://mathoverflow.net/users/23141 | 293138 | 129,023 |
https://mathoverflow.net/questions/293129 | 3 | Let $\rho$ be a group action by a compact group $G$
\begin{equation}
\rho:G\times M \rightarrow M \\
\rho:(g,p) \rightarrow \rho\_g(p)
\end{equation}
Denote the orbit of $p\in M$ by $\mathcal{O}\_p$ and the isotropy group of $p$ by $G\_p$. We have a natural representation $G\_p$ on the vector space $T\_p(M)/T\_p... | https://mathoverflow.net/users/86065 | Is there an easy example of group action where the slice theorem produces a non-trivial principal bundle? | Consider the usual $G = S^1$ action on $S^2$ given by rotations. This action respects the antipodal map, so descends to a $G$ action on $M = \mathbb{R}P^2$. Let $p\in M$ be any point on the "equator", where the isotropy group is $G\_p = \mathbb{Z}/2\mathbb{Z}\subseteq S^1$.
Then the $G\_p$ action on the slice at $p$ ... | 3 | https://mathoverflow.net/users/1708 | 293144 | 129,026 |
https://mathoverflow.net/questions/292896 | 10 | This question is a follow-up to [this question](https://mathoverflow.net/questions/215802/vanishing-natural-transformation-and-strong-generator) I asked some time ago. Let $X$ be a **smooth** projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K\_X)$, $\omega \neq 0$. Let
$$A \longrightarrow... | https://mathoverflow.net/users/37214 | Vanishing natural transformation exact triangle | So I think I have an argument which shows that if $\mathrm{id}\_{A} \otimes \omega : A \longrightarrow A \otimes K\_X[n]$ is zero and $\mathrm{id}\_{B} \otimes \omega : B \longrightarrow B \otimes K\_X[n]$ is zero then $\mathrm{id}\_{C} \otimes \omega : C \longrightarrow C \otimes K\_X[n]$ is not necessarily zero. This... | 4 | https://mathoverflow.net/users/37214 | 293145 | 129,027 |
https://mathoverflow.net/questions/293024 | 7 | By a Zero Line-Sum (ZLS) matrix I mean matrices with the property, that each row sum and each column sum equals zero:
$$A\in\mathbb{R}^{m\times n}:\ \sum\_{i=1}^{n}a\_{ij}=\sum\_{j=1}^{m}a\_{ij}=0$$
These can be thought of as being the difference of two "ordinary" doubly stochastic matrices.
ZLS matrices obvious... | https://mathoverflow.net/users/31310 | Properties of Zero Line-Sum Matrices | Below find six suggestions.
**Case of general ZLS matrices.**
Examples of properties are:
1. All cofactors of a *square* ZLS are equal.
2. Each *square* ZLS matrix of dimension $n$ has eigenvector $1^{n\times 1}$ with eigenvalue $0$.
3. If an $n\times n$ ZLS $A$ is moreover *symmetric*, and if $\mathrm{Sp}(\cdot... | 4 | https://mathoverflow.net/users/108556 | 293146 | 129,028 |
https://mathoverflow.net/questions/292965 | 1 | Let $E$ be a complex Hilbert space. We recall that an operator $T\in\mathcal{L}(E)$ is said to be hyponormal if $[T^\*, T]\geq 0$ (i.e. $\langle (T^\*T-TT^\*)x,x \rangle\geq 0$ for all $x\in E$). Let $E\overline{\otimes}E$ denotes the completion,
endowed with a reasonable uniform cross-norm, of the algebraic tensor pro... | https://mathoverflow.net/users/113054 | Proving the hyponormality of $A\otimes B$ | I think the ``if part'' is correct and it follows from the following fact: if the operator-matrices
$$T = \begin{bmatrix}
T\_{11} & T\_{12}\\
T\_{21} & T\_{22}
\end{bmatrix},\quad
S = \begin{bmatrix}
S\_{11} & S\_{12}\\
S\_{21} & S\_{22}
\end{bmatrix}$$
are both positive, then the matrix obtained by entry-wise tensor
$... | 1 | https://mathoverflow.net/users/85652 | 293149 | 129,030 |
https://mathoverflow.net/questions/293099 | 3 | Let $P$ be a probability distribution and let $A$ and $B$ be some events, and suppose that we want to minimise an $f$-divergence between $P$ and the set of all distributions $Q$ that satisfy that constraint that $Q(B|A) = q$ for some fixed $q \geq P(B|A)$. Let $P\_{f}$ denote the result of minimising a given $f$-diverg... | https://mathoverflow.net/users/45570 | Minimising the f-divergence to a conditional probability constraint | After having some trouble understanding this question, I have come to interpret it as follows: Let $P$ be a probability measure on a measurable space $(S,\Sigma)$. Suppose that $A$ and $B$ are in $\Sigma$, and $P(A)>0$. Fix any $q\in[P(B|A),1]$. Let $f\colon[0,\infty]\to(-\infty,\infty]$ be a function that is convex an... | 1 | https://mathoverflow.net/users/36721 | 293153 | 129,032 |
https://mathoverflow.net/questions/293156 | 5 | I was trained in reaction-diffusion (parabolic/elliptic) PDEs, and my research now focuses on applied optimal tranport. I'd like to learn probability and stochastic processes, mostly their connection with PDEs (Feynman-Kac formulas, Itô's calculus, etc.)
Could anyone recommend a nice PDE-oriented textbook, anything alo... | https://mathoverflow.net/users/33741 | PDE-oriented textbook on probability and random processes? | Stochastic processes and application by Pavliotis is a good one.
| 3 | https://mathoverflow.net/users/30684 | 293158 | 129,034 |
https://mathoverflow.net/questions/293150 | 0 | Let $A$ be a $n\times n$ real matrix. Is there a vector $\vec x \in \mathbb{R}^n$ with exactly $0 \le k < n$ zero entries such that $A \vec x = 0$?
Is there an efficient algorithm to tackle this question?
| https://mathoverflow.net/users/16615 | Kernel vectors with given number of non-zero entries? | A first step would be to find a (nice) basis of the null space of $A$. That is easy if the matrix is in RREF. Once that is done (which would take about $n^3$ steps with a lazy algorithm), answering your question would seem easy in practice for reasonable size $n.$ However the problem is NP-complete: As $n$ grows to be ... | 2 | https://mathoverflow.net/users/8008 | 293161 | 129,036 |
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