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https://mathoverflow.net/questions/225254 | 1 | Let's say that a set $X\subseteq [0,1]$ has Property Q if the following holds: For every continuous $f:[0,1]\to\mathbb{R}$ with $f(0)=0$ and derivative existing and bounded by 1 on $[0,1]\setminus X$, we have $f(1)\le 1$.
All countable sets have Property Q since then $f'(x)$ is integrable and $\int\_0^x f'(u)\,du=f(x... | https://mathoverflow.net/users/9025 | A boundary of the second fundamental theorem of calculus | Since $\{ x: f'(x) \textrm{ does not exist}\}$ is a Borel set, it seems reasonable to restrict ourselves to Borel sets. Then only countable sets work. An uncountable Borel set $X$ has a perfect subset $A$; see [here.](https://en.wikipedia.org/wiki/Perfect_set_property) So $A$ is compact, uncountable, and has no isolate... | 2 | https://mathoverflow.net/users/48839 | 225306 | 105,315 |
https://mathoverflow.net/questions/225324 | 2 | Without applying Fermat's Last Theorem, how can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution $(x,y) = (0, \frac{1}{2})$ for ever prime $p \geq 5$ ?
| https://mathoverflow.net/users/nan | How can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution for every prime $p>3$? | It is equivalent to FLT. Indeed, if $a^p+1=b^p$ for positive rational $a,b$, we have $x^p:=(ab)^p=a^p(a^p+1)=(a^p+1/2)^2-1/4:=y^2-1/4$.
Opposite implication (moving from the comments): if $x^p=y^2-1/4=(y-1/2)(y+1/2)$, $x\ne 0$, denote $y-1/2=a/m$ for coprime non-zero integers $a,m$. Then also $a+m\ne 0$, $y+1/2=(a+m)... | 12 | https://mathoverflow.net/users/4312 | 225325 | 105,320 |
https://mathoverflow.net/questions/225333 | 2 | Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}\_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = \text{diag}(\zeta, \zeta^{-1})$. The group $G$ acts naturally on $\mathbb{C}[x, y]$. Let $\mathbb{C}[x, y]^G$ be the algebra of... | https://mathoverflow.net/users/83694 | Description of the algebra of $G$-invariant polynomials by generators and relations | **EDIT.** This is a new version of the answer. Since at some point there was confusion between the $(\zeta, \, \zeta)$-action and the $(\zeta^{-1}, \, \zeta)$-action, for the sake of clarity let me discuss both.
**The $(\zeta^{-1}, \, \zeta)$-action (the OP case).**
---------------------------------------------------... | 6 | https://mathoverflow.net/users/7460 | 225335 | 105,321 |
https://mathoverflow.net/questions/225329 | 7 | I accept that my question seems so vague and broad, and I already looked into some similar questions in MO. But I would like to learn specifically about some open problems and conjectures regarding elliptic curves in finite fields. Also if there is something about their isogeny in particular it is highly welcomed. If y... | https://mathoverflow.net/users/24541 | What are some open problems regarding elliptic curves in finite fields? | We know that, given an elliptic curve over a finite field $\mathbb{F}\_q$, there exists integers $m,n$ with $m|n$ such that the group of rational points is a product of a cyclic group of order $m$ and a cyclic group of order $n$. I believe it is still open to deterministically, in polynomial time, compute $m,n$ (a big ... | 6 | https://mathoverflow.net/users/2290 | 225336 | 105,322 |
https://mathoverflow.net/questions/225337 | 9 | In his paper *Higher Algebraic K-Theory I* (see [here][1]), Quillen introduces a topological space $BC$, called the classifying space of $C$, and tries to relate its topology to the categorical structure of $C$. I am curious as to what is the importance of 1. adjoint functors inducing homotopy equivalences, and 2. comp... | https://mathoverflow.net/users/nan | Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I | N.B.: I have reread your question and it occured to me that you a probably asking something entirely different. However since I'm unclear what exactly is your question and since I don't want to delete this wall of text, here it is.
Firstly, both categories and (homotopy types of) topological spaces admit a descriptio... | 11 | https://mathoverflow.net/users/10605 | 225347 | 105,328 |
https://mathoverflow.net/questions/225360 | 8 | Is there an way to estimate (if not calculate) the number of possible non-isomorphic graphs of 50 vertices and 150 edges?
| https://mathoverflow.net/users/49739 | How many non-isomorphic graphs of 50 vertices and 150 edges | The simplest guess one could make is $\frac{1}{50!} { {50 \choose 2} \choose 150}$. That is, we first count the number of labeled such graphs, then assume that most of them have trivial automorphism group so we can approximately divide out by $50!$ when removing the labels. You can estimate how big this is using Stirli... | 9 | https://mathoverflow.net/users/290 | 225363 | 105,332 |
https://mathoverflow.net/questions/225361 | -1 | Let’s have the equation $a^k+\frac{k \cdot a^k}{x \cdot y^n}=m^k$ where $k≥2$ and $x≠y$ and $a, k, x, y, n, m$ positive integers greater than zero. If $x \cdot y^n = f \cdot a^g$ where $f$ and $g$ any positive integer greater than zero then the equation $ a^k + \frac{k \cdot a^k}{x \cdot y^n}=L$ has infinite solutions ... | https://mathoverflow.net/users/18229 | Fractional equations | It's elementary: let $d=(a,m)$, $a=a\_1d$ and $m=m\_1d$. Then dividing both sides by $d^k$ gives $a\_1^k+\frac{ka\_1^k}{xy^n}=m\_1^k$. Let $p$ be any prime divisor of $a\_1$. Since $(a\_1,m\_1)=1$, $p\nmid m\_1$, so $p\nmid \frac{ka\_1^k}{xy^n}$. In particular, the number of $p$ factors in $a\_1^k$ is no greater than t... | -1 | https://mathoverflow.net/users/37103 | 225364 | 105,333 |
https://mathoverflow.net/questions/225362 | 1 | Let $G$ be a finite group, $R = \mathbb{C}G$ the regular representation of $G$, and $\rho : G \to \text{GL}(V)$ a finite dimensional representation of $G$. Write $\sigma: G \to \text{GL}(V \otimes R)$ for the tensor product of $\rho$ and the regular representation. How do I construct a vector space isomorphism $f: V \o... | https://mathoverflow.net/users/83694 | Construction of vector space isomorphism where $f(v \otimes gh) = \sigma(g)(f(v \otimes h)),\text{ }\forall v \in V,\text{ }g,\,h \in G$ | The map $\phi: v\otimes g\mapsto g^{-1}v\otimes g$ gives an isomorphism from $V\otimes R$ to $V\_0\otimes R$, where $G$ acts trivially on $V\_0$. Taking a basis of $V\_0$ gives you an isomorphism $V\_0\otimes R\simeq R^{\dim V}$. I don't think there is a canonical isomorphism, though.
| 2 | https://mathoverflow.net/users/37103 | 225365 | 105,334 |
https://mathoverflow.net/questions/224461 | 6 | even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.
Let us consider a smooth projective algebraic variety $X$ over $\mathbb{C}$.
Whenever I have a coherent sheaf $F$ over $X$, I can get a resolution of $F$ by a (bounded) co... | https://mathoverflow.net/users/67157 | Information and intuition packed in the Chern character for coherent sheaves | I'm expanding on my comment, not really giving a complete answer.
Say that $L$ is a polarization on your smooth projective $X$. Then HRR applied to $F\otimes L^n$ gives
$$
P\_L(F)(n)=\chi(F\otimes L^{\otimes n})=\int\_X ch(F\otimes L^{\otimes n})\cdot td(X)=\int\_X ch(F)\cdot e^{c\_1(L)}\cdot td(X)
$$
where $P\_L... | 5 | https://mathoverflow.net/users/5516 | 225367 | 105,335 |
https://mathoverflow.net/questions/225381 | 4 | In [Diophantine equations over the twentieth century: a (very) brief overview
, p. 5](http://www-irma.u-strasbg.fr/~bugeaud/travaux/prockyoto.ps)
>
> **Problem** Let $f(x) \in \mathbf{Z}[x]$ be an irreducible polynomial of degree at least 2. Do the Diophantine equation $$f(x)=y^2z^3$$ have only finitely many soluti... | https://mathoverflow.net/users/12481 | Is this problem of Schinzel and Tijdeman misquoted? It appears easy with Pell equations | The problem is indeed misquoted. Here is the correct statement:
>
> If a polynomial $P(x)$ with rational coefficients has at least three simple zeros then the equation $y^2z^3=P(x)$ has only finitely many solutions in integers $x,y,z$ with $yz\neq 0$.
>
>
>
The paper can be found [here](http://matwbn.icm.edu.p... | 5 | https://mathoverflow.net/users/30186 | 225382 | 105,337 |
https://mathoverflow.net/questions/225387 | 2 | Is the tangent bundle of $S^2 \times S^1$ trivial or not?
| https://mathoverflow.net/users/nan | Tangent bundle of $S^2 \times S^1$ trivial or not | It is trivial.
More generally, by a result of Kervaire a product of any number $n \geq 2$ of spheres is a parallelizable manifold if one of them has odd dimension.
For a short proof see
*E. B. Staples*, [**A short and elementary proof that a product of spheres is parallelizable if one of them is odd**](http://ww... | 11 | https://mathoverflow.net/users/7460 | 225389 | 105,338 |
https://mathoverflow.net/questions/225391 | 29 | This question is probably too vague for experts, but I really don't know how to avoid it.
I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the base-change functor it induces is monadic.
Now, I don't really know anything about descent and I'm trying to probe aro... | https://mathoverflow.net/users/69037 | Why are monadicity and descent related? | Probably the reason "monadicity" gets connected with descent (and the associated terminology of descent theory) is because of its relevance to the question of descent for rings.
If you're talking about a morphism of rings $\phi:A\to B$ there is a functor $-\otimes\_AB:Mod\_A\to Mod\_B$. Then you can ask the question... | 31 | https://mathoverflow.net/users/11546 | 225394 | 105,340 |
https://mathoverflow.net/questions/225398 | 5 | What feature(s) must a (non 1st-order) language with proper-class-many formulas have in order to guarantee that:
There is a proper class P of formulas such that both
(a) every set-sized sub-collection of P is satisfiable, and
(b) no proper-class-sized sub-collection of P is satisifable?
| https://mathoverflow.net/users/83742 | A proper class of formulas with every set-sized (but no proper-class-sized) subcollection satisfiable | Let me give a few examples.
**Example 1.** Let us work in Gödel-Bernays set theory, and
assume that $T\subset {}^{<\text{Ord}}2$ is a proper class tree of
height Ord, but there is no cofinal branch.
(This theory is consistent relative to an inaccessible cardinal,
because if $\kappa$ is inaccessible and not weakly c... | 5 | https://mathoverflow.net/users/1946 | 225399 | 105,341 |
https://mathoverflow.net/questions/225327 | 12 | This was [asked on MSE](https://math.stackexchange.com/q/1546777/52694) and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line?
I'd love to say that it's a weighted sum of del... | https://mathoverflow.net/users/24611 | Fourier transform of the critical line of zeta? | If $\varphi$ is in the class of Schwartz
we have
$$\int\_{-\infty}^{+\infty}\varphi(t)\zeta(\frac12+it)\,dt=
\sum\_{n=0}^\infty\Bigl\{
\frac{1}{\sqrt{n+1}}\widehat{\varphi}\Bigl(\frac{1}{2\pi}\log
(n+1)\Bigr)- 2\pi\int\_{x\_n}^{x\_{n+1}}e^{\pi
y}\widehat{\varphi}(y)\,dy\Bigr\}$$
where $x\_0=-\infty$ and $x\_n=\frac{1}{... | 8 | https://mathoverflow.net/users/7402 | 225409 | 105,346 |
https://mathoverflow.net/questions/225404 | 9 | I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U\_q(\mathfrak{sl}\_2)$ and I know that when $q$ is a primitive root of unity (say, not equal to 1 or -1) then there are two irreducible representation... | https://mathoverflow.net/users/83561 | $U_q(\mathfrak{sl}_2)$ representations of "quantum dimension" zero | Yes, you are interpreting it right. For $U\_q(\mathfrak{sl}\_2)$, there's a very concrete interpretation of the quantum dimension: it's the trace (in the usual sense) of the element $K$. So, on the Weyl module of highest weight $n$ (the representation generated by $v$ with the relation $Kv=q^nv$ and $F^{(n+k)}v=E^{(k)}... | 7 | https://mathoverflow.net/users/66 | 225414 | 105,347 |
https://mathoverflow.net/questions/225370 | 18 | Recall that a monoidal category $\mathcal C$ is *rigid* if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes X^l$ and $X \otimes X^r \to \mathbf 1 \to X^r \otimes X$ satisfying certain equations. It is a fundamental fact about... | https://mathoverflow.net/users/78 | What categorical property of monoidal categories picks out the ones with duals? | The notion of rigidity can be defined for any pseudomonoid in a monoidal proarrow equipment. See for instance *Dualizations and Antipodes* by Day, McCrudden, and Street (although they work only with monoidal bicategories and don't make the equipments explicit).
| 8 | https://mathoverflow.net/users/49 | 225420 | 105,349 |
https://mathoverflow.net/questions/225415 | 6 | Given list of symmetric matrices $\{A\_i,B\_i\}\_{i=1}^r\in\Bbb R^{n\times n}$ where $r\in\Bbb N$ is arbitrary what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$A\_iX=XB\_i$$ holds?
Note: $X'$ means the transpose of $X$.
**(1)** Is there a test to see if there is no such $X$... | https://mathoverflow.net/users/10035 | On $XX'=I$ such that $AX=XB$ is true | (1) Robert Israel's reduction to testing if $det(X(\vec{t}))$ is identically zero is valid, however, doing this test deterministically is a long-standing open problem in computational complexity, that would imply fairly strong computational lower bounds that currently seem out of reach.
Nonetheless, there is an effic... | 6 | https://mathoverflow.net/users/38434 | 225429 | 105,351 |
https://mathoverflow.net/questions/225397 | 3 | Let $\Gamma$ be the semidirect product of $\mathbb{Z}$ and $\mathbb{Z}/4$,
where the action of $\mathbb{Z}/4$ on $\mathbb{Z}$ is defined by $\bar{k} \cdot x = (-1)^k x$. Clearly $\Gamma$ has virtual cohomological dimension (*vcd*) one.
Is it possible to construct a one-dimensional contractible CW complex $X$
such th... | https://mathoverflow.net/users/64639 | How to construct a proper action of a group of finite virtual cohomological dimension? | You can let it act on the real line with ℤ acting by integral translations and the generator of ℤ/4 acting by multiplication by (−1).
| 2 | https://mathoverflow.net/users/nan | 225441 | 105,355 |
https://mathoverflow.net/questions/225459 | 24 | **Definitions:**
Lagrange's theorem implies that for each prime $p$, the factors of $(p − 1)!$ can be arranged in unequal pairs, with the exception of $±1$, where the product of each pair $≡ 1 \pmod p$. *[See Wiki article on Wilson's theorem.](https://en.wikipedia.org/wiki/Wilson's_theorem#Prime_modulus)*
From the ... | https://mathoverflow.net/users/45057 | A conjecture based on Wilson's theorem | For an integer $n$ with $1\leq n\leq p-1$, let $n^{-1}$ be the inverse of $n$ modulo $p$. It follows from Weil's bound on Kloosterman sums that for every $\epsilon>0$ the set $\{n: xp\leq n\leq (x+\epsilon) p, yp\leq n^{-1}<(y+\epsilon) p\}$ has cardinality $\epsilon^2p+\mathcal{O}(\sqrt{p}\log^2 p)$. Hence up to a rel... | 35 | https://mathoverflow.net/users/37555 | 225460 | 105,364 |
https://mathoverflow.net/questions/225448 | 7 | This is inspired by a [recent question about complete multipartite integral graphs](https://mathoverflow.net/questions/225251/can-a-block-matrix-with-at-least-3-zero-blocks-of-different-size-on-the-diagonal/225289#225289). I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It ... | https://mathoverflow.net/users/29783 | Why are some solutions of these diophantine equations off the usual patterns? | I take this from my comments above and add something.
The question is about rational points on the surface $S$ given by
$$ \Delta(a,b,c) := (ab+bc+ca)^3 - 27(abc)^2 = z^2 $$
in the weighted projective space ${\mathbb P}\_{1,1,1,3}$. Along the line
$b+c = 4a$ the left hand side factors as a square times $b^2 + 18bc + ... | 10 | https://mathoverflow.net/users/21146 | 225465 | 105,368 |
https://mathoverflow.net/questions/225466 | 3 | Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}\_p$ which is complementably $\ell\_p$-saturated but admits no unconditional basis. I was wondering if such an example has been proved for the case $p=1$.
I have managed to constru... | https://mathoverflow.net/users/73784 | example of an $\ell_1$-saturated Banach space without an unconditional basis | Szankowski (Subspaces without the approximation property. Israel J. Math. 30 (1978), no. 1-2, 123–129) for each $1\le p<2$ constructed a subspace $X$ of $\ell\_p$ without the approximation property and thus without an unconditional basis. On the other hand, as a subspace of $\ell\_p$, $1\le p<\infty$, the space $X$ is ... | 5 | https://mathoverflow.net/users/37822 | 225473 | 105,371 |
https://mathoverflow.net/questions/225471 | 5 | I have noticed that many authors tend to use "let" instead of "for all". For example, they write something like this:
>
> Let $n$ be an even natural number. Then also $n^2$ is even.
>
>
>
I wonder, why they use "let" instead of "for all", also in cases where the "for all"-version sounds quite good:
>
> For... | https://mathoverflow.net/users/83783 | "Let" versus "for all" | Language, written and spoken, is a flexible beast. English is particularly flexible. There are often many grammatically correct ways to say the same thing. Good, careful writing requires the ability to use this flexibility and to rein it in as appropriate, but this takes a lot of time and requires lots and lots of self... | 1 | https://mathoverflow.net/users/20787 | 225480 | 105,375 |
https://mathoverflow.net/questions/225477 | 2 | Forgive me to ask an elementary question, because I really need the answer to this today (I already asked this in Stackexchange.)
Let $S$ be the rational quartic scroll in $\mathbb{P}^5$ ($S$ is the image of the embedding $\mathbb{P}^1\times\mathbb{P}^1\rightarrow\mathbb{P}^5$ via $|\mathcal{O}\_{\mathbb{P}^1\times\m... | https://mathoverflow.net/users/83786 | Elementary question: Cubic 4-fold and rational quartic scroll | The question about secant lines reduces to a Chern class computation on the blowing up $\nu:\widetilde{\Sigma \times \Sigma}\to \Sigma \times \Sigma$ along the diagonal, where $\Sigma$ is the ~~cubic~~ quartic scroll. Your claim is equivalent to the assertion that in $A^4(\widetilde{\Sigma \times \Sigma})$, the followi... | 0 | https://mathoverflow.net/users/13265 | 225486 | 105,378 |
https://mathoverflow.net/questions/222567 | 7 | **Let a finite cyclic group $G = \mathbb Z/n$ act continuously on an open $d$-ball $B^d$. Suppose further that this action extends to the closed ball $\overline{B^d}$. Is there necessarily a fixed point in the interior?**
Note that by Brouwer's fixed point theorem, there has to be some fixed point in the closed ball.... | https://mathoverflow.net/users/14233 | Cyclic groups acting on balls, and interior fixed points | There need not be a fixed point in the interior. As in Bredon's book, p. 61, there exist smooth actions of a cyclic group $C\_{r}$, of order $r$, without fixed points on $\mathbb{R}^{n}$ for large enough $n$ (e.g. $n\ge 8$) whenever $r$ is not a prime power. By one-point compactification one obtains a continuous, one-f... | 5 | https://mathoverflow.net/users/1822 | 225495 | 105,380 |
https://mathoverflow.net/questions/225427 | 4 | I'm trying to make sense of a definition appearing in Hua Luogeng's book ["Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains"](http://www.ams.org/bookstore-getitem/item=MMONO-6 "Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains").
Consider the Herm... | https://mathoverflow.net/users/75321 | Hua Luogeng's definition of automorphism group for Hermitian symmetric space | I was able to prove that, indeed, it must necessarily be $C=-\bar B$ and $D=\bar A$.
First consider [Woodbury identity](https://en.wikipedia.org/wiki/Woodbury_matrix_identity)
$$
(A-BD^{-1}C)^{-1}=A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1};
$$
using the constraints on $A,B,C,D$, it reduces to
$$
D^\text{t}=A^{-1}(\mathbb... | 2 | https://mathoverflow.net/users/75321 | 225508 | 105,386 |
https://mathoverflow.net/questions/225386 | 9 | Let $\pi \colon \mathfrak{X} \rightarrow B$ be a deformation of complex compact manifolds and $E$ be a holomorphic vector bundle on $\mathfrak{X}$ (or a coherent sheaf on $\mathfrak{X}$ that is flat over $B$).
The function $b \rightarrow \mathrm{h}^0(X\_b, E\_{|X\_b})$ is known to be upper semi-continuous, but I wonder... | https://mathoverflow.net/users/13503 | Cohomology of vector bundles in families | The constructibility result you ask for is Satz 7.7(1) in [Ein Kriterium für die Offenheit der Versalität (Flenner, 1981)](http://dx.doi.org/10.1007/BF01174768).
The appropriate generalization of the result in Voisin's book that you mention is the topic of [Eine Bemerkung über relative Ext-Garben (Flenner, 1981)](htt... | 5 | https://mathoverflow.net/users/70322 | 225517 | 105,390 |
https://mathoverflow.net/questions/225493 | 3 | Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy
$-\Delta u+a(x)u=f, \ \ u|\_{\partial \Omega}=0$,
where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following estimate holds
$||a(x)u||\_{H^1(\Omega)} \leq C ||a||\ \||f||\_{H^1(\Omega)}$,
where $C$ is independent of $u$ and $a... | https://mathoverflow.net/users/42326 | A Global Estimates for Linear Elliptic PDE | I disagree with the accepted answer. You do not need a Lipschitz bound on $a$. I assume for simplicity that $a\in L^\infty(\Omega)$.
Step 1. The solution $u$ exists and is unique in $H^1\_0(\Omega)$, by Lax-Milgram for example.
Step 2. Write $g:=f-au$. Then $u$ satisfies $-\Delta u = g$ in $\Omega$, so provided $\O... | 2 | https://mathoverflow.net/users/40120 | 225518 | 105,391 |
https://mathoverflow.net/questions/225487 | 3 | In dimension 2, the euler poincare formula restricts the incidence properties of edges in a triangulation of a surface.
Are there analogous generalizations for higher dimensions, like elaborations on the Schenzel formula for the specific case pf spheres?
| https://mathoverflow.net/users/21985 | f vectors of simplicial complexes homeomorphic to n dimensional spheres | The $g$-conjecture (or $g$-theorem in the polytopal case) aims the characterize what possible $f$-vectors can arise from simplicial spheres. [Here](https://gilkalai.wordpress.com/2009/04/04/how-the-g-conjecture-came-about/) is some nice history and introduction to the $g$-conjecture by Gil Kalai. In brief the $g$-conje... | 4 | https://mathoverflow.net/users/51668 | 225520 | 105,393 |
https://mathoverflow.net/questions/225537 | 2 | Let $R$ -- be an irreducible plane real algebraic curve (without isolated points).
Suppose that $(x,y)\in R\Leftrightarrow (x,-y)\in R.$
**Question:** could one find a polynomial $f(x,y)$ with zero set $R$ such that
$$\forall (x,y)\in{\mathbb R^2}\quad f(x,y)=f(x,-y)$$
| https://mathoverflow.net/users/16044 | Could a real curve symmetric across the line be defined only by polynomial that is not reflection-invariant? | Let $f$ be any polynomial whose zero set is $R$. Then $F(x,y)=f(x,y)f(x,-y)$ works.
| 1 | https://mathoverflow.net/users/37103 | 225539 | 105,403 |
https://mathoverflow.net/questions/225546 | 2 | How is the knot equivalence problem represented?
By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G\_1$ and $G\_2$ with adjacency matrices $A$ and $B$ respectively and we seek if there is a permutation matrix $P$ with transpose $P'$ such that ... | https://mathoverflow.net/users/10035 | On Knot Equivalence problem statement | By knot, I assume you mean a knot in $S^3$. Typically a knot is encoded as a diagram, which you might think of as a tetravalent planar graph, with some kind of description of undercrossings and overcrossings, say by some kind of labelling of the half edges incident to every vertex. Abstractly, the notion of equivalence... | 8 | https://mathoverflow.net/users/68910 | 225547 | 105,405 |
https://mathoverflow.net/questions/225551 | 5 | Let $\ p\_1\ < p\_2 < \ldots\ $ be the sequence of all primes $\ (2\ 3\ 5\ \ldots)$.
Let $\ x\_1 < x\_2 < \ldots\ $ be an arbitrary increasing sequence of positive integers such that $\ x\_n\le p\_n\ $ for every $\ n=1\ 2\ldots\,$.
**QUESTION:** Does sequence $\ (x\_1 < x\_2 < \ldots)\ $ contain a 3-term arithmetic... | https://mathoverflow.net/users/8385 | 3-term arithmetic progressions of terms as frequent as primes | It is open. Best current results in the quantitative version of Roth's theorem belong to Sanders and allow to find 3-term arithmetic progression between something like $O(n/\log^{1-\varepsilon} n)$ numbers not exceeding $n$, for any given $\varepsilon>0$.
UPD: already not to Sanders, but to Bloom (see quid's comment)... | 4 | https://mathoverflow.net/users/4312 | 225552 | 105,407 |
https://mathoverflow.net/questions/225550 | 4 | The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix
$$
\left\|\frac{\partial^2u}{\partial x^i\partial x^j}\right\|\, .
$$
For instance, with $n=2$, one has
$$
A+Bu\_{xx}+Cu\_{xt}+Du\_{tt}+E(u\_{xx}u\_{tt}-u\_{xt}^2)=0\... | https://mathoverflow.net/users/22606 | Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type? | Two points:
First, the equation ${u\_{xx}}^2+{u\_{yy}}^2-1 =0$ is not of Monge-Ampère type.
Second, the class of Monge-Ampère equations is preserved under contact transformations on the $1$-jet space, so if an equation is of Monge-Ampère type up to contact transformation, then it *is* a Monge-Ampère equation alrea... | 7 | https://mathoverflow.net/users/13972 | 225554 | 105,408 |
https://mathoverflow.net/questions/225545 | 1 | I need to do some numerical computation on special values of a Hecke L-function $L(s,\chi)$. To do this, I want to construct a Hecke character in MAGMA, given that I know its infinity type.
In other words, suppose we are working on a totally real field $K$. My Hecke character $\chi$ is first defined over the principa... | https://mathoverflow.net/users/32631 | construct a Hecke character in MAGMA with given infinity type | It sounds like Magma does not currently include the exact functionality that you want to use. An alternative that you could try is manually building the $L$-function you wish to evaluate numerically (see the section of the Magma documentation "Arithmetic Geometry - $L$-functions - Constructing a general L-series"). Thi... | 1 | https://mathoverflow.net/users/48142 | 225569 | 105,412 |
https://mathoverflow.net/questions/225567 | 3 | **Weak Version**:
Is there a 1st order language $L$ (with only countably-many formulas) such that for each recursive coding $C$ of the formulas of $L$, there is a theory $T$ of $L$ where
1. $T$ is not satisfiable,
2. every $C$-recursive proper subtheory of $T$ (meaning the set of $C$-codes of the sentences in the sub... | https://mathoverflow.net/users/83742 | When can we have "each subtheory is satisfiable iff it is recursively axiomatizable"? | Perhaps this is the kind of example for which you are searching.
Let's use the logic $L\_{\omega\_1,\omega}$, which allows for
countable conjunctions and disjunctions. Let $A\subset\mathbb{N}$
be any infinite set with no infinite computably enumerable
subset. In the language with a constant symbol $c$ and infinitely
... | 3 | https://mathoverflow.net/users/1946 | 225574 | 105,414 |
https://mathoverflow.net/questions/225556 | 2 | If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s\_0, \dots, s\_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil generated by $s\_i$ and $s\_j$ does not have a curve in its base locus?
I know that for generic choices for the basis it ... | https://mathoverflow.net/users/82672 | Pencils in very ample linear systems without curve in its base locus | Here is a more precise formulation of my suggestion above. First of all, I do not believe that you can find such a surface in $\mathbb{P}^3$. The base locus curves must be coordinate lines. In order to get the behavior you describe, you will need the entire coordinate tetrahedron to be contained in the surface. But the... | 1 | https://mathoverflow.net/users/13265 | 225575 | 105,415 |
https://mathoverflow.net/questions/225571 | 2 | Given real symmetric matrices $A,B\in\{0,1\}^{n\times n}$ is it true that $$AX=XB$$ has a solution of form $X$ a permutation matrix iff a solution with $XX'=I$ exists? We are over reals.
It is clear if there is a solution $X$ of permutation matrix form then $XX'=I$ solution exists. Is there any truth in converse stat... | https://mathoverflow.net/users/10035 | On $XX'=I$ such that $AX=XB$ is true when $A,B\in\{0,1\}^{n\times n}$ | The answer is no for $n\ge4$. The matrices
\begin{equation}
A=\left(\begin{array}{rrrr}
1 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)\text{ and }
B=\left(\begin{array}{rrrr}
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{array}\right)
\end{equation}
have the ... | 6 | https://mathoverflow.net/users/18739 | 225580 | 105,418 |
https://mathoverflow.net/questions/225512 | 8 | In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A \bigoplus A^\*)$. There are also computations by Grossman and Snyder for examples that come from the Asaeda-Haagerup subfactor. ... | https://mathoverflow.net/users/69850 | Brauer-Picard for a fusion category coming from a quantum group | As far as I know, no one has written this up, but I think you should be able to find the Brauer-Picard groupoid for quantum groups at roots of unity by the following techniques. Now that I've written it down, there are a lot of gaps which would need to be filled in, several of which are not straightforward. This might ... | 8 | https://mathoverflow.net/users/22 | 225584 | 105,421 |
https://mathoverflow.net/questions/225534 | 10 | Suppose $f: X \rightarrow Y$ is a flat projective morphism of finite type schemes over an algebraically closed field so that the fibers over the closed points of Y are (geometrically) reduced. Why is it true (or do I need some additional assumptions?) that the number of irreducible components of the geometric fibers is... | https://mathoverflow.net/users/75970 | Why is the number of irreducible components upper semicontinuous in nice situations? | Here is an alternative to Count Dracula's (correct) argument that emphasizes instead the constancy of the Hilbert polynomial for a flat family of projective schemes.
As above, assume that $Y$ is a DVR. For one fixed irreducible component $Z\_{\eta}$ of $X\_\eta$ of minimal dimension $d$, denote by $Z$ the Zariski cl... | 4 | https://mathoverflow.net/users/13265 | 225585 | 105,422 |
https://mathoverflow.net/questions/225559 | 5 | Does anyone know a good reference to understand the historical background of Castelnuovo-Mumford regularity?
I know the backgound for the modern commutative-algebra approach (using free graded resolutions and Betti numbers) but I'd like to know the geometric motivations that led to the sheaf cohomology definition of... | https://mathoverflow.net/users/83833 | Historical refererences for Castelnuovo-Mumford regularity | You might like looking at Eisenbud, Green and Harris "[Cayley-Bacharach theorems and conjectures](http://www.ams.org/journals/bull/1996-33-03/S0273-0979-96-00666-0/home.html)". They start with classical theorems of Euclidean geometry, such as Pappus and Pascal's theorem, and relate them to questions in commutative alge... | 3 | https://mathoverflow.net/users/297 | 225588 | 105,424 |
https://mathoverflow.net/questions/225572 | 13 | **I.** The Diophantine equation,
$$x^3+y^3+z^3 = 3w^3\tag1$$
with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. *Are there larger ones?* As Noam Elkies points out in [this post](https://mathoverflow.net/a/66740/12905), Cassels in a [1985 paper](http://www.ams.org/journals/mco... | https://mathoverflow.net/users/12905 | On cubic reciprocity for $x^3+y^3+z^3 = 996$? | Problems of this type are studied in the article
Colliot-Thélène, Wittenberg - Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines.
Here, in remark 5.7, they give an explanation of Cassels' result in terms of a Brauer-Manin obstruction to strong approximation. Other examples of Brauer-M... | 14 | https://mathoverflow.net/users/5101 | 225593 | 105,428 |
https://mathoverflow.net/questions/225592 | 3 | It is known that non-isotrivial fibrations of genus $g>0$ curves over the projective line have a bunch of singular fibers. There are at least three of them.
It is not difficult to prove that an elliptic fibration $X \longrightarrow \mathbb{P}^1$ have at least three singular fibers and at least two of them are not mul... | https://mathoverflow.net/users/82672 | Elliptic fibrations with few singular fibers | Consider first an elliptic fibration with a section over $\mathbb{P}^1$. (In this case none of the singular fibers are multiples of smooth curves.)
Assume that the minimal discriminant has degree $12n$.
Then the number of components of singular fibers which do not intersect the zero section equals $12n-2a-m$, where $a... | 5 | https://mathoverflow.net/users/8621 | 225595 | 105,430 |
https://mathoverflow.net/questions/225532 | 10 | Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of $\frac{D}{A}$ and what arrangement achieves it for specific values of $n$?
| https://mathoverflow.net/users/75293 | Optimization of points on a plane | I conjecture that the minimum is $2n$, for all $n\geq 4$.
This is obtained by putting four points in a square and the remaining points in the center.
There are then $(n-4)$ vertices in the center, and thus the total sum of distances squared is $4 \cdot \frac{1}{2}(n-4) + 4 + 2\cdot 2= 2n$,
assuming that the square ha... | 6 | https://mathoverflow.net/users/1056 | 225598 | 105,431 |
https://mathoverflow.net/questions/225596 | 7 | I have often seen the assertion that for a smooth plane curve $C$ of degree $d$ the gonality of $C$ is $d-1$ and each gonality pencil is obtained by projection from a point of $C$ onto a line.
(let me recall that the gonality of $C$ is by definition the minimal degree $d$ of divisors $D$ on $C$ with $r(D)=1$, those d... | https://mathoverflow.net/users/40038 | The gonality of smooth plane curves | Let $X \subset \mathbb{P}^2$ be a smooth planar curve of degree $d$. It is fairly well known that $\Omega^1(X) \cong \mathcal{O}(d-3)|\_{X}$, and $H^0(\mathbb{P}^2, \mathcal{O}(d-3)) \to H^0(X, \Omega^1)$ is an isomorphism. So the canonical embedding of $X$ is isomorphic to the composition $X \to \mathbb{P}^2 \stackrel... | 9 | https://mathoverflow.net/users/297 | 225603 | 105,433 |
https://mathoverflow.net/questions/225600 | -4 | Following [this](https://math.stackexchange.com/questions/1566087/can-the-existence-of-infinitely-many-even-perfect-numbers-be-settled-by-a-diagon/1566168?noredirect=1#comment3188832_1566168), let's define the notion of perfect sequence as follows:
$(u\_{i})\_{i}$ is a perfect sequence if and only if it is the sequen... | https://mathoverflow.net/users/13625 | Diagonal argument for even perfect numbers | You are asking if a sum of $i$ numbers equals $2$.
The partial sum
$S\_k=\frac{1}{u\_{1,1}}+...+\frac{1}{u\_{k,k}}$ does not change as $i$ grows for every $k<i$ so, if you want to reach $2$ there are these possibilities :
**1.** $i=\infty$
**2.** $i<\infty$
We don't know if there are infinitely or finite... | 2 | https://mathoverflow.net/users/38851 | 225610 | 105,436 |
https://mathoverflow.net/questions/225544 | 7 | Let $M\_n^{sa}$ be the space of $n\times n$ complex Hermitian matrices, let $r < n$, and let $E$ and $F$ be (real) linear subspaces of $M\_n$ with ${\rm codim}(E) < r^2$ and ${\rm codim}(F) = 1$. Let $V$ be the set of matrices in $E$ whose rank is at most $r$ and suppose $V$ is not contained in $F$. Is $V \cap (M\_n\se... | https://mathoverflow.net/users/23141 | Geometry of Hermitian rank $\leq r$ matrices | Let $n=4$, $r=3$. Let $E = \mathbb{R} \oplus M\_3^{sa}(\mathbb{C})$ (viewed as $4\times 4$ matrices which are $0$ in the first row and column except for the $(1,1)$ entry). Then the (real) codimension of $E$ in $M\_4^{sa}(\mathbb C)$ is $6<9=r^2$. Let $F\subset M\_4^{sa}(\mathbb{C})$ be the space of all matrices with $... | 6 | https://mathoverflow.net/users/13360 | 225616 | 105,439 |
https://mathoverflow.net/questions/225626 | 5 | Consider a monoidal 2-category (or bicategory) B. For example, B could by the 2-category (finite sets, finite correspondences, isomorphisms of correspondences) with monoidal structure given by product of sets (etc). By a *category C enriched over B* I mean:
* A collection of objects ob(C)
* For each pair of objects x... | https://mathoverflow.net/users/83857 | Category enriched over a monoidal 2-category | Yes, here: R. Garner, M. Shulman, *Enriched categories as a free cocompletion*. [arxiv](http://arxiv.org/abs/1301.3191).
| 4 | https://mathoverflow.net/users/4262 | 225629 | 105,443 |
https://mathoverflow.net/questions/225620 | 2 | This question is a reformulation of a special case of the question
[An $\ell$-adic local system which is trivial on every fiber of a morphism](https://mathoverflow.net/questions/225468/an-ell-adic-local-system-which-is-trivial-on-every-fiber-of-a-morphism)
(this special case did not receive an answer on MathOverflow).... | https://mathoverflow.net/users/12395 | Is an $\ell$-adic local system on $\mathbb A^2$ that restricts to a trivial local system on every vertical line a pullback via the first projection? | First translate to the case of a continuous map $\rho : \pi\_1(\mathbf{A}^2) \to G$ with $G$ finite. Using a limit argument reduce to the case where $\rho$ is defined on $\pi\_1(\mathbf{A}^2\_{\mathbf{F}\_q})$ for some $q$. Let $\sigma : \mathbf{A}^1\_{\mathbf{F}\_q} \to \mathbf{A}^2\_{\mathbf{F}\_q}$, $x \mapsto (x, 0... | 3 | https://mathoverflow.net/users/80000 | 225630 | 105,444 |
https://mathoverflow.net/questions/225455 | 12 | Concerning the article "Bernstein, Allen R.; Robinson, Abraham.
Solution of an invariant subspace problem of K. T. Smith and
P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in
finding out the following : (1) who was the editor of the Pacific
Journal of Mathematics at the time? as well as (2) who was the... | https://mathoverflow.net/users/28128 | Bibliographic request concerning an article by Bernstein and Robinson | I can identify one individual who scrutinized the Bernstein-Robinson manuscript and established its validity before it was published in PJM:
Paul Halmos, [I Want to be a Mathematician: An Automathography](https://rads.stackoverflow.com/amzn/click/com/0387960783) (19... | 9 | https://mathoverflow.net/users/11260 | 225658 | 105,450 |
https://mathoverflow.net/questions/225653 | 0 | From [HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY](http://alpha.math.uga.edu/~pollack/impossibility.pdf).
On p. 13
>
> BUNYAKOVSKY’S CONJECTURE ( WEAK FORM ). Let $f$ be a polynomial with
> integer coefficients and positive leading coefficients which
> is irreducible over $\mathbf{Q}$. Let $d := \gc... | https://mathoverflow.net/users/12481 | What is wrong with this counterexample to the Weak Bunyakovsky's conjecture and reformulation of Bunyakovsky's conjecture? | $f(x)/6 + N\mathbb Z$ is *not* periodic modulo $N$. It is periodic modulo $6N$, so you have to check a larger range. And indeed, $\gcd(f(637)/6,N)=1$.
| 11 | https://mathoverflow.net/users/18739 | 225660 | 105,451 |
https://mathoverflow.net/questions/225635 | 6 | Is there a good reference that explains mod p K-theory and p-adic or p-complete K- theory? All I know about K-theory is the topological K-theory of "vector bundles and k-theory" in Switzer's book (and similar expositions found in a book by Hatcher, and a chapter in May's Concise course). There seems to me be a jump fro... | https://mathoverflow.net/users/80739 | reference request for mod p and p-adic K-theory | You have learned about topological K-theory (of topological spaces).
The Quillen result from
"On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field", Daniel Quillen, Ann. Math., Vol. 96, No. 3 (Nov., 1972), pp. 552-586,
is about algebraic K-theory (of rings).
| 5 | https://mathoverflow.net/users/83633 | 225662 | 105,453 |
https://mathoverflow.net/questions/225663 | 4 | Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of them might actually be reflecting limits in disguise - that some types of limits in localizations must arise from limits ... | https://mathoverflow.net/users/69037 | What kinds of limits does localization of commutative rings reflect? | One of the fundamental results in commutative algebra is the following:
>
> Let $M$ be an $A$-module. Then $M = 0$ if and only if $M\_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m} \triangleleft A$.
>
>
>
You can think of it as a local-to-global principle if you like. Regardless, it follows that:
> ... | 8 | https://mathoverflow.net/users/11640 | 225671 | 105,457 |
https://mathoverflow.net/questions/225674 | 11 | Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in $\mathrm{GL}(T\_{2}(E))$ of the $2$-adic Galois representation (and I would be interested in analogous descriptions for $\ell$-adic Galois ... | https://mathoverflow.net/users/24757 | Galois representations for the curve $y^{2} = x^{3} - x$ | I don't know that this is written down anywhere, but it's possible. It is known in general that $GL(T\_{\ell}(E))$ is contained in the normalizer of $R\_{\ell}^{\times}$, where $R\_{\ell} = \mathbb{Z}[i] \otimes \mathbb{Z}\_{\ell}$. (This follows for example from Corollary 2 on page 502 of Serre and Tate's ''Good reduc... | 14 | https://mathoverflow.net/users/48142 | 225680 | 105,458 |
https://mathoverflow.net/questions/225683 | 3 | For each positive integer $n$, write $S\_n$ for the symmetric on $n$-letters. Suppose that $m | n$ is a proper divisor of $n$, and write $n = km$. Consider the element
$$\displaystyle u(m,n) = \underbrace{\begin{pmatrix} 1 & 2 & \cdots & k\end{pmatrix} \cdots \begin{pmatrix} n-k+1 & n-k+2 & \cdots & n \end{pmatrix}}\... | https://mathoverflow.net/users/10898 | What do we know about these subgroups of $S_n$? | If I understand your notation correctly, these subgroups are wreath products $C\_{k} \wr S\_{m}$, where $C\_{k}$ is a cyclic group of order $k$ (acting as a single $k$-cycle). These groups are a semidirect product of a "base group" which is a direct product of $m$ cyclic subgroups of order $k$, and a group $S\_{m}$ whi... | 10 | https://mathoverflow.net/users/14450 | 225684 | 105,460 |
https://mathoverflow.net/questions/225677 | 7 | A result commonly, and [probably erroneously](https://mathoverflow.net/a/223245/16537), attributed to W. Sierpiński is that every non-atomic, countably additive, nonnegative measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on a set $S$, has the weak, and hence the strong, Darboux property, which means... | https://mathoverflow.net/users/16537 | Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC? | This theorem follows from Dependent Choice, and thus is strictly weaker than the Axiom of Choice. Here is a proof using only DC. Fix $X\in\Sigma$ such that $\mu(X)>0$ and let $a\in(0,\mu(X))$. We will use DC to inductively construct a sequence of disjoint measurable subsets $Y\_n$ of $X$ such that $\mu(\bigcup Y\_n)=a$... | 6 | https://mathoverflow.net/users/75 | 225689 | 105,462 |
https://mathoverflow.net/questions/225659 | 14 | In what follows, $\mathsf{ZCKP}$ refers to the subset of $\mathsf{ZFC}$ consisting of the axioms of [Zermelo set theory](https://en.wikipedia.org/wiki/Zermelo_set_theory) with choice and foundation ($\mathsf{ZC}$) plus those of [Kripke-Platek set theory](https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory) (... | https://mathoverflow.net/users/17064 | "Largish" cardinals | Your cardinals are known as the [$\Sigma\_n$-correct cardinals](http://cantorsattic.info/Reflecting#Reflection_and_correctness), and they arise in diverse set-theoretic contexts. For example, we use them extensively in our paper:
* J. Bagaria, J. D. Hamkins, K. Tsaprounis, T. Usuba, [Superstrong
and other large cardi... | 16 | https://mathoverflow.net/users/1946 | 225697 | 105,464 |
https://mathoverflow.net/questions/225652 | 0 | Good morning everybody. My question is inspired from the following fact:
Consider $\mathbb R^3$ endowed with coordinates $(x,y,z)$. Of course if we were to solve the second order pde $\partial\_x^2 g(x,y,z)=0$ we would find $g(x,y,z)=x p\_0(y,z)+p\_1(y,z)$. Then it will be possible, playing with $p\_0$ and $p\_1$, to... | https://mathoverflow.net/users/57571 | Boundary behaviour of a second order pde with characteristics | Let us look at the local problem: taking $X$ a non-zero smooth vector field in a neighborhood of 0 in $\mathbb R^3$, you may choose local coordinates such that $X=\partial\_z$. If $π\_1, π\_2$ are smooth hypersurfaces such that $X$ is transverse to both of them, you may assume that they are given locally by
$$
π\_j=\{(... | 2 | https://mathoverflow.net/users/21907 | 225698 | 105,465 |
https://mathoverflow.net/questions/225599 | 12 | Let $K\_0(Var\_k)$ be the abelian group generated by the isomorphism classes of varieties over the field $k$ with the relations
$$[X]=[U]+[X\setminus U]$$
for every variety $X$ and open subvariety $U$.
Is this abelian group torsion free?
(Any positive or negative result for any particular field $k$ is welcome.)
| https://mathoverflow.net/users/37059 | Does the Grothendieck ring of varieties contain torsion? | As per Theo Johnson-Freyd's request, I'm converting my comment to an answer.
Larsen-Lunts [show](http://arxiv.org/pdf/math/0110255.pdf) that if $k$ is algebraically closed of characteristic zero, then there is a natural isomorphisms $$K\_0(\text{Var}\_k)/(\mathbb{L})\overset{\sim}{\longrightarrow} \mathbb{Z}[SB],$$
w... | 12 | https://mathoverflow.net/users/6950 | 225718 | 105,471 |
https://mathoverflow.net/questions/225722 | 6 | Let $f\in \mathbb{Q} [x]$ be a polynomial, and $a\_0 = a$ be an arbitrary integer. Let us define a sequence $\{a\_n \} $ by the recurrence relationship : $$a\_n = f(a\_{n-1} ). $$ I want to show that $a\_n $ cannot always be a prime number, with $\{a\_n \}$ being pairwise distinct. I am pretty sure that this is a very ... | https://mathoverflow.net/users/68663 | Non-existence of a prime generating polynomial recurrence relation | Let $a\_0 = 2^{2^k} + 1$ for $k$ sufficiently large, and let
$$a\_n = (a\_{n-1} - 1)^2 + 1.$$
Then $a\_n = 2^{2^{k+n}} + 1$, so this sequence can't always be prime regardless of the value of $k$ iff there are infinitely many composite [Fermat numbers](https://en.wikipedia.org/wiki/Fermat_number), and as far as I kn... | 13 | https://mathoverflow.net/users/290 | 225726 | 105,475 |
https://mathoverflow.net/questions/225678 | 3 | I have a question regarding counting permutations of a multiset's elements. The problem is the following:
Given a multi-set $M=\{0^{m}, 1^{n-m}\}$ the number of all possible permutations of its elements is given by the formula $\frac{n!}{m!(n-m)!}$. How can I come up with a formula that gives the number of permutatio... | https://mathoverflow.net/users/83898 | Counting Specific Permutations of Elements in a Multiset | **UPDATE**. I've simplified the exposition of my approach below and added an example of $k=2$.
Construct a [de Bruijn graph](https://en.wikipedia.org/wiki/De_Bruijn_graph) $G$ with the vertices representing the $(k+1)$-mers that have at least $k/2$ ones, which we denote $u\_1, u\_2, \dots, u\_d$. Then every restricte... | 2 | https://mathoverflow.net/users/7076 | 225768 | 105,485 |
https://mathoverflow.net/questions/225764 | 4 | Let $A$ be a finite dimensional algebra over an algebraically closed field $K$.
If $A$ is semisimple, then $A$ is Morita equivalence with a commutative algebra, that is $A \backsim K^n$ where $n$ is the number of isomorphism classes of simple representations of $A$.
For a general, non-semisimple algebra, we cannot expe... | https://mathoverflow.net/users/41644 | A canonical representative in Morita equivalence class | I don't know what $J(B)$ means, but perhaps the result you are looking for is one from Morita's original paper (Kiiti Morita. Duality for modules and its applications to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, 6:83–142, 1958. Google has a link to a PDF, but I'm having trouble... | 4 | https://mathoverflow.net/users/78 | 225770 | 105,486 |
https://mathoverflow.net/questions/225773 | 5 | Haw can we prove that an arbitrary set $A$ of $n$ positive integers is 2-Freiman isomorphic to a subset of {$ 1,2,...,4^{n}$} and $4^{n}$ cannot be improved to $2^{n}$?
| https://mathoverflow.net/users/83948 | Freiman-isomorphic sets | It is an open conjecture from [a paper of Konyagin and myself](http://math.haifa.ac.il/seva/Papers/colifr.dvi) that every $n$-element set of integers is Freiman-isomorphic to a subset of $[0,2^{n-2}]$. There are some counterexamples for small values of $n$, but it is believed that the conjecture is "essentially true"; ... | 11 | https://mathoverflow.net/users/9924 | 225776 | 105,488 |
https://mathoverflow.net/questions/225745 | 9 | Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières modularisation/deequivariantisation exists.
*Is the resulting modular fusion category unitary? Is the modularisation functor u... | https://mathoverflow.net/users/13767 | Is the modularisation of a unitary fusion category always unitary? | The answer is yes to both questions (see [Müger's](http://arxiv.org/abs/math/9812040) paper). Müger's version of modularization is done in the unitary setting. The only new information that you need for full generality is that the unitary Drinfel'd center of a unitary fusion category is equal to the usual Drinfel'd cen... | 6 | https://mathoverflow.net/users/6517 | 225779 | 105,490 |
https://mathoverflow.net/questions/225785 | 2 | $\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\scrT{\mathscr T}\def\ssp{\kern.4mm}
$More specifically, **I ask** whether $S$ *be a Borel set in the topological space* $(\Omega,\scrT)$ in the following situation.
Let $\Omega$ be the set of all real analytic functions $x:\bbR\to\bbR$, and let $X$ be the abstract real vect... | https://mathoverflow.net/users/12643 | Is the set of entire functions Borel in the space of analytic functions? | $f$ has an entire extension iff the radius of convergence of the Maclaurin series of $f$ is $+\infty$, i.e. iff $\limsup\_{n \to \infty} |a\_n|^{1/n} = 0$ where $a\_n$ are the coefficients of that series. The coefficients can
be defined as limits of expressions defined in terms of point evaluations
(finite-difference... | 7 | https://mathoverflow.net/users/13650 | 225792 | 105,495 |
https://mathoverflow.net/questions/225793 | 3 | **Question 1** Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology that's sufficient.
So, I'm coming into this with the perspective of Polish spaces(i.e. completely metrizable sepa... | https://mathoverflow.net/users/81420 | Under what conditions can we put a complete norm on a linear subspace of a separable Banach space? | The main restriction on putting a complete norm is that Hamel bases in infinite-dimensional separable Banach spaces have cardinality continuum (see Exercise 6.4 on page 191 in Fabian-Habala-Hajek-Montesinos-Pelant-Zizler, Functional Analysis and Infinite-Dimensional Geometry). This gives an answer to your first questio... | 5 | https://mathoverflow.net/users/37822 | 225796 | 105,498 |
https://mathoverflow.net/questions/225798 | -2 | Do you know anything special about that kind of planar graphs? An article that covers these graphs might be helpful.
| https://mathoverflow.net/users/83962 | Planar Graphs with #Vertices = #Faces | Unlike some other planar graphs, these ones always contain at least one triangular face and at least one vertex of degree $\le 3$. The reason for the first property is that, by Euler's formula, the number of edges in a planar graph with no triangles is at most $2n-4$, but your graphs always have exactly $2n-2$ edges. T... | 4 | https://mathoverflow.net/users/440 | 225804 | 105,502 |
https://mathoverflow.net/questions/225784 | 4 | What I'm looking for is a non-asymptotic bound on the probability that the smallest gap between eigenvalues of a GUE matrix does not exceed a certain value.
I'm aware of the bounds in
<http://imrn.oxfordjournals.org/content/2010/3/436.full.pdf>
and <https://people.math.osu.edu/nguyen.1261/cikk/gap.pdf> , but was wond... | https://mathoverflow.net/users/83939 | Tail bounds on eigenvalue gaps for GUE | This is studied in
*Gérard Ben Arous and Paul Bourgade*, [**Extreme gaps between eigenvalues of random matrices**](http://dx.doi.org/10.1214/11-AOP710), *Ann. Probab.* **41** (2013), no. 4, 2648--2681.
(Ah, so that's how the "insert citation" button works!) In particular, for GUE, the smallest gap has size about $... | 3 | https://mathoverflow.net/users/766 | 225806 | 105,503 |
https://mathoverflow.net/questions/225820 | 13 | All pure mathematicians know that the goal is to produce *insight*, rather than to simply obtain results. However, it might sometimes be of value to disseminate largely empirical work. In the same spirit as [this question](https://mathoverflow.net/questions/15366/which-journals-publish-expository-work), which journals ... | https://mathoverflow.net/users/30147 | Which journals publish experimental results in pure maths? | I think one suitable journal is "Mathematics of Computation", an AMS journal.
<http://www.ams.org/publications/journals/journalsframework/mcom>
Also, specific to algebra, the Journal of Algebra has a computational section: <http://www.journals.elsevier.com/journal-of-algebra/>
| 9 | https://mathoverflow.net/users/35394 | 225821 | 105,510 |
https://mathoverflow.net/questions/225842 | 6 | *I had [asked something very similar before on math.se](https://math.stackexchange.com/q/1560243) (deleted now) but unfortunately it hadn't received a lot of attention. I decided to re-ask here.*
Let $S$ be a fixed scheme. Is the following true?
>
> **Theorem(?):** The category of affine schemes over $S$ is contr... | https://mathoverflow.net/users/22810 | An apparent equivalence of the category of affine schemes over $S$ and the category of quasi-coherent $\mathcal{O}_S$-algebras | No, it's not true in general (EGA 2, (1.2.3)).
The following example is taken from EGA 2, (1.3.3). Over a field $K$, let $S$ be the affine plane with a doubled origin. Then $S$ is the union of two affine open subsets $Y\_1$ and $Y\_2$, each of them is isomorphic to the affine plane, glued along the complementary sub... | 16 | https://mathoverflow.net/users/10696 | 225843 | 105,518 |
https://mathoverflow.net/questions/225826 | 0 | Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^\*\mathbf{M})^{-1}\mathbf{M}^\*$ be the pseudo-inverse of $\mathbf{M}$, where $^\*$ denotes the conjugate transpose. Let $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$, i.e. $\mathbf{G}$ is the pseudo-inverse of $a\mathbf{H}+\mathbf{M}$, for some mat... | https://mathoverflow.net/users/51469 | Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion | to test if $G=(aH+M)^+=M^+(I-aHM^+ + {\cal O}(a^2))$ you can multiply from the right with $aH+M$ (using $M^+M=I$) and see if you obtain the identity to first order in $a$:
$$M^+(I-aHM^+ + {\cal O}(a^2))(aH+M)=M^+(aH+M-aHM^+M+{\cal O}(a^2))=I+{\cal O}(a^2)$$
| 2 | https://mathoverflow.net/users/11260 | 225854 | 105,526 |
https://mathoverflow.net/questions/225868 | 3 | Several threads (e.g. [Integration of the product of pdf & cdf of normal distribution](https://mathoverflow.net/questions/101469/integration-of-the-product-of-pdf-cdf-of-normal-distribution) ) have shown that
$E[\Phi(x)]=\Phi(\mu/\sqrt{\sigma^2+1})$ when $x\sim N(\mu,\sigma^2)$.
I'd like to compute $Var(\Phi(x))$ ... | https://mathoverflow.net/users/83997 | Variance of the normal CDF | I think I figured it out, actually. Just as for the mean we can reinterpret $E[\Phi(x)]$ as $E[\mathbf 1[y<x]]$ for $y\sim N(0,1)$, we can reinterpret $E[\Phi(x)^2]$ as $E[\mathbf 1[y<x,y^\prime<x]]$ for independent standard normals $y,y^\prime$. Therefore,
$Var(\Phi(x)) = E[\Phi(x)^2]-E[\Phi(x)]^2=\Phi(\frac{\mu}{\s... | 3 | https://mathoverflow.net/users/83997 | 225877 | 105,532 |
https://mathoverflow.net/questions/225870 | 12 | Suppose:
* I am a 'problem-solver' rather than a 'theory-builder'
* I am an undergraduate student
* I have a passion for solving mathematical problems
* The homework I get is not satisfying (in the sense that the problems are computing-problems rather than problems that require creative thinking), and I get far too l... | https://mathoverflow.net/users/84000 | Open Problems for Undergraduates | Richard Guy compiled a list of open problems in combinatorial game theory, available at <http://library.msri.org/books/Book29/files/unsolved.pdf> . His book "Unsolved problems in number theory" also contains parts which are more combinatorial in nature. In the realm of Davenport's constant there are many open problems,... | 4 | https://mathoverflow.net/users/37555 | 225881 | 105,533 |
https://mathoverflow.net/questions/225786 | 4 | Let $M$ be a compact, orientable and irreducible 3-manifold with with boundary consisting of two incompressible components $N\_0,N\_1$, with $N\_i \stackrel{f\_i}{\cong} S\_g$ for some diffeomorphism $f\_i: S\_g \to N\_i$, where $S\_g$ is the closed orientable surface of genus $g $. I wonder if the following is true:
... | https://mathoverflow.net/users/78554 | Is the following 3-manifold always a trivial I-bundle over a surface? | This follows, fairly easily, from the hypothesis of irreducibility and from the "annulus theorem" (see page 130 of Jaco-Shalen's book "Seifert Fibered Spaces in 3-Manifolds"). You can remove the hypothesis of irreducibility if you are willing to use the Poincaré conjecture.
| 2 | https://mathoverflow.net/users/1650 | 225884 | 105,536 |
https://mathoverflow.net/questions/179493 | 4 | We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact ($1\leq p<\infty$) if there exists a $p$-summable sequence $(x\_n)\_{n=1}^{\infty}$ in $X$ such that
$$ K\subseteq \left\{\sum\_{n=1}^{\infty}\alpha\_{n}x\_{n}:(\alpha\_{n})\_{n}\in B\_{l\_{q}}\right\}\quad\quad(\frac{1}{p}+\frac{1}{q}=1).$$
A B... | https://mathoverflow.net/users/41619 | A question on $p$-approximation property | The answer for both questions are positive. The first is an easy consequence of a version of Grothendieck's Theorem that characterizes the p-approximation property in terms of the density of finite-rank operators w.r.t uniform convergence on p-compact sets (1). Since p-compact sets are q-compact for p
It follows the n... | 2 | https://mathoverflow.net/users/84004 | 225888 | 105,538 |
https://mathoverflow.net/questions/220079 | 6 | Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?
| https://mathoverflow.net/users/50629 | Heegard genus of hyperbolic Haken 3-manifolds | Even better, there are hyperbolic surface bundles with Heegaard genus two. These are all described in Jesse Johnson's paper, titled [Surface bundles with genus two Heegaard splittings](http://arxiv.org/abs/math/0607513). You will need to use some criterion to recognize pseudo-Anosov maps, however.
| 5 | https://mathoverflow.net/users/1650 | 225899 | 105,543 |
https://mathoverflow.net/questions/225814 | 34 | I was curious about a physics question which I thought might be suitable for mathoverflow. I looked at [the answer to this question](https://mathoverflow.net/questions/102313/classical-limit-of-quantum-mechanics), but it's not what I'm looking for.
Basically, classical mechanics and the $\hbar \to 0$ limit of quantu... | https://mathoverflow.net/users/7108 | Does quantum mechanics ever really quantize classical mechanics? | It is perhaps helpful to distinguish between four types of mechanics here:
1. *Pure-state classical mechanics*. Here, the mechanics are classical, and the system is described by a single point $(q,p)$ in phase space. This point evolves via Hamilton's equations of motion $\partial\_t q = \frac{\partial H}{\partial p};... | 44 | https://mathoverflow.net/users/766 | 225901 | 105,545 |
https://mathoverflow.net/questions/225880 | 2 | As we know, a [*tree-decomposition*](https://en.wikipedia.org/wiki/Tree_decomposition) of a graph must have following features:
1. All vertices are covered
2. All edges are covered
3. The connectivity condition
I think using repeated nodes in *tree-decomposition* must be allowed by definition. Is it true?
| https://mathoverflow.net/users/83880 | Repeated nodes in tree-decomposition of a graph is allowed or not? | The basic definition of tree decompositioning allows it. If we are trying to minimize the graph diameter or node count, then obviously they are redundant, and for algorithmic calculations they can be eliminated.
From the Wikipedia definition in your link is the definition of **smooth**:
>
> A tree decomposition $... | 2 | https://mathoverflow.net/users/70355 | 225904 | 105,547 |
https://mathoverflow.net/questions/225895 | 4 | Let $(X,\mu)$ be a probability space and $f\colon (X,\mu)\to (X,\mu)$ be an ergodic automorphism. Let $\phi\in L^\infty(X,\mu)$ be such that $\int\phi d\mu=0$.
Suppose that for $\mu$-a.e. $x\in X$, it holds
$$
\sup\_{n\geq 1} \left|\Sigma\_{j=0}^{n-1}\phi(f^j(x))\right|=\infty.
$$
Then, is it possible that
$$\... | https://mathoverflow.net/users/889 | Two-side deviations for ergodic sums | Sure. Here is a recipe to construct such systems. Start from a dynamical systems which preserves a non-atomic probability measure and which is ergodic $(Y, \nu, T)$. Given an integrable random variable $r : Y \to \mathbb{N}^\*$, you can construct a tower over $Y$ of height $r$. Define a space:
$$X := \{(y,n) : \ y \i... | 4 | https://mathoverflow.net/users/75670 | 225906 | 105,549 |
https://mathoverflow.net/questions/225903 | 47 | Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled [Undecidability of the Spectral Gap](http://www.nature.com/nature/journal/v528/n7581/full/nature16059.html), there is an [extended version](http://arxiv.org/pdf/1502.04573v2.pdf) on arxiv which is 146 pages long. Here is from the abstract:"*Many chal... | https://mathoverflow.net/users/51484 | How undecidable is the spectral gap? | I haven't read the paper carefully, but this appears to be a standard undecidability result, of the sort of which there are dozens if not hundreds in the literature, of the same ilk as the undecidability of Wang tilings, the undecidability of the existence of solutions to Diophantine equations, the word problem for gro... | 37 | https://mathoverflow.net/users/8991 | 225918 | 105,554 |
https://mathoverflow.net/questions/225855 | 5 | In Barry Mazur's paper introducing Galois deformations, he hints at having a general theory for representations which are not residually Schur, but with more complicated statements. Does anyone know where/if this got written up with a similar level of detail as in Mazur's paper? I'm particularly interested in applicati... | https://mathoverflow.net/users/6084 | Mazur's Galois Deformations paper for non-residually irreducible case | I don't know where this got written, but it's certainly well-known. Here's how it works. There are usually problems with deforming objects that have automorphisms because in many cases the corresponding functors are "obviously" not representable (Gabber produces a fine counterexample to an overoptimistic attempt to mak... | 9 | https://mathoverflow.net/users/43076 | 225936 | 105,563 |
https://mathoverflow.net/questions/225737 | 3 | By the Brill-Noether Theorem, a general curve $C$ of genus $g\geq2$ has maximal Clifford index $\lfloor \frac{g-1}{2}\rfloor$. Hence a very naive question is:
(Q1) Is a curve with maximal Clifford index a *general* curve (in the sense of Brill-Noether theory) ? And if not, what can be said about such a curve (when no... | https://mathoverflow.net/users/40038 | Are curves with maximal Clifford index Brill-Noether general? | Q2) as stated is obviously true, because if the curve had a pencil of degree less than or equal to c+1, then the Clifford index would be at most c-1 (the clifford index of this pencil). So you probably meant to state the converse statement, which asks if a curve of maximal gonality has maximal Clifford index.
At leas... | 1 | https://mathoverflow.net/users/36622 | 225937 | 105,564 |
https://mathoverflow.net/questions/225953 | 6 | I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum number of terms is 9:
$$x\_1^3 + x\_1^2 x\_2 + x\_1 x\_2^2 + x\_2^3+ x\_1^2 +x\_1 x\_2 + x\_2^2 + x\_1 + x\_2 + 1$$
How ... | https://mathoverflow.net/users/84033 | Number of polynomial terms for certain degree and certain number of variables | If you have $k$ variables and want degrees $1,...,n$, you get the sum $$\sum\_{i=1}^n h\_i(1,...,1)=\sum\_{i=1}^n\binom{k+i-1}{i}=\binom{k+n}{n}-1.$$ Here the $h\_i$ are the [complete homogeneous symmetric polynomials](http://en.wikipedia.org/wiki/Complete_homogeneous_symmetric_polynomial).
| 7 | https://mathoverflow.net/users/29783 | 225956 | 105,569 |
https://mathoverflow.net/questions/225828 | 3 | Is there any general theory to find the numbers represented by ternary forms of the type
$q(x,y,z)=ax^2+bx^2-abz^2,$
when $a,b$ are prime?
By doing an internet search, the closest I found was the paper [1], that deals with quadratic forms when the discriminant is not a square.
For instance, consider the indefi... | https://mathoverflow.net/users/44293 | Indefinite Ternary Forms with Square Discriminant | For your form
$$q(x,y,z)=ax^2+bx^2-abz^2,$$
everything is completely predictable by congruences as long as
$\gcd(a,b) = 1$ and both $a,b$ are squarefree.
Oh, your original form $2 x^2 + 5 y^2 - 10 z^2$ integrally represents all numbers that are neither $4^k (8n+1)$ nor $25^k (5n \pm 1).$ For the negative of the form,... | 6 | https://mathoverflow.net/users/3324 | 225971 | 105,576 |
https://mathoverflow.net/questions/225861 | 1 | Let $G = (E,V)$ be an undirected graph (which can have multiple edges or loops).
Let $k,l,m\colon E\to \mathbb{R}\_{\geq 0}$ be three edge-weight functions that satisfy $2k(e) + l(e) + m(e) = 1$ for all edges $e \in E$. Let $D\_G(k,l,m) = (V,A)$ be the random digraph obtained from $G$ in the following manner: indepe... | https://mathoverflow.net/users/25028 | $q$-connectedness of random digraphs obtained from a fixed graph | Instead of thinking of the initial graph as undirected, replace each edge with directed edges in each direction. The meaning of $k(e)+l(e)$ is the probability that you remove the directed edge $uv$. (You also want to assume the probability of removing $vu$ stays the same.)
Choose any edge $e$. Condition on the rest o... | 1 | https://mathoverflow.net/users/2954 | 225976 | 105,577 |
https://mathoverflow.net/questions/225823 | 11 | Sample $n^2$ integers $a\_{11},\dots,a\_{nn}$ in $\{-d,\dots,-1,0,1\dots,d\}$ uniformly.
What is the probability that the resulting matrix $[a\_{ij}]$ has rank $r$?
Is there a nice parametrization of such matrices that helps us generate such a rank $r$ matrix quickly deterministically?
In general what is a good ... | https://mathoverflow.net/users/10035 | On Sampling rank $r$ matrices | The case $r=n$ is considered in this [paper](http://www.math.ubc.ca/~gerg/papers/downloads/AAIMHNIE.pdf) by Martin and Wong. They prove that for every $n \geq 2$ and every $\epsilon >0$, the probability that a random $n \times n$ matrix with entries from $\{-k, \dots, 0, \dots, k\}$ is singular is $\ll \frac{1}{k^{2-\e... | 4 | https://mathoverflow.net/users/2233 | 225977 | 105,578 |
https://mathoverflow.net/questions/225972 | 25 | In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F\_1$, I was just reading through Lorscheid's paper [The geometry of blueprints](http://arxiv.org/abs/1103.1745). I certainly like the idea a lot and I get the feeling of it being the 'right' approach to $\mathbb F\_1$.
A blue... | https://mathoverflow.net/users/76299 | A geometric theory of Blueprints? (Algebras over the field with one element) | $\newcommand{\N}{\mathbb N}\newcommand{\paren}[1]{\left(#1\right)}\newcommand{\T}{\mathbb{T}}\newcommand{\m}{\mathfrak{m}}\newcommand{\E}{\mathbf{E}}$I can answer your first set of questions:
There is a geometric theory of blueprints. It's easiest\* to work from the definition of a blueprint as a pair $B = (A,R)$ con... | 15 | https://mathoverflow.net/users/62519 | 225979 | 105,579 |
https://mathoverflow.net/questions/222952 | 9 | I have a matrix $A \in \mathbb{R}^{p \times q}$ of rank $r$ and its SVD decomposition, i.e,
$$
A = U S V^\top,
$$
where $U \in \mathbb{R}^{p \times r}$ and $V \in \mathbb{R}^{q \times r}$ are orthonormal, $S = \mathrm{diag}\left(s\_1, \dots, s\_r\right)$, and $s\_i \in \mathbb{R}$. I would like to compute SVD for $B = ... | https://mathoverflow.net/users/76501 | Efficient SVD of a matrix without some of the columns | After some research, I managed to find a reasonable answer. The operation is called updating (in case of adding new columns to the original matrix) or donwdating (in case of removing) of the SVD. Full update/downdate of SVD with a single column can be done in $\mathcal{O}\left(r^2(1 + p + q)\right)$ time [[1](http://ww... | 5 | https://mathoverflow.net/users/76501 | 225981 | 105,580 |
https://mathoverflow.net/questions/225984 | 6 | Chebotarev's theorem on roots of unity says that all the minors of a prime-length DFT matrix over the complex numbers are nonzero. I was wondering if there was an analogue for finite fields.
More precisely, let $p$ be prime and $\omega=e^{2\pi i/p}$, the complex $p$th root of unity, and let $\Omega$ be the matrix giv... | https://mathoverflow.net/users/41530 | Finite field analogue of Chebotaryov theorem on roots of unity? | Not for $GF(p)$ of prime cardinality. Consider a nontrivial factorization $p-1=uv$ which always exists. Let $w$ be primitive in $GF(p)^{\ast}$ The "regular" DFT submatrix which is made up of $u^{th}$ row and every $v^{th}$ column entry is an all 1 matrix and singular. More generally, when we have $GF(p^m)$ a primitive ... | 3 | https://mathoverflow.net/users/17773 | 225989 | 105,582 |
https://mathoverflow.net/questions/226005 | 2 | Let $bw(G)$ be the *branchwidth* of graph $G$ and $\omega(G)$ be the size of *maximum clique* in $G$. I think the following inequality holds:
$$
\omega(G)\leq bw(G)
$$
**Intuition:** Assume (in reverse of process of building a [branch-decomposition](https://en.wikipedia.org/w/index.php?title=Branch-decomposition&oldi... | https://mathoverflow.net/users/83880 | What is the relation between size of maximum clique and branchwidth? | No, your inequality does not hold. You are off by a constant factor. Probably the easiest way to see this is to consider the dual notion of a *tangle*, which I will define now.
A *separation* in a graph $G$ is a pair $(A,B)$ where $A$ and $B$ are edge-disjoint subgraphs of $G$ whose union is $G$. The *order* of $(A,B... | 6 | https://mathoverflow.net/users/2233 | 226012 | 105,588 |
https://mathoverflow.net/questions/226019 | 5 | Call a square matrix **Galois-irreducible** if all its eigenvalues are Galois conjugates of each other.
>
> Let $M$ be an integer $n\times n$ matrix which is not Galois-irreducible. Is it always possible to find an integer matrix $S$ such that $S^{-1}MS=diag(A\_1,...,A\_k)$ is a block diagonal matrix with Galois-i... | https://mathoverflow.net/users/29783 | Is it always possible to "separate" the eigenvalues of an integer matrix? | Think of $M$ first as a linear operator acting on $V = \mathbb{Q}^n$. Pass to a splitting field $K$ and consider the induced action on $V \otimes K$. This splits up into a direct sum of generalized eigenspaces of $M$, which are also then permuted by the Galois action of $G = \text{Gal}(K/\mathbb{Q})$ into orbits. By Ga... | 5 | https://mathoverflow.net/users/290 | 226022 | 105,591 |
https://mathoverflow.net/questions/226021 | 5 | $G = (V, E)$ is a 3-connected plane triangulation. Let $S \subset V$ such that $G(V - S)$ is disconnected. Is it true that $G(S)$ must contains a separating cycle?
My intuition is leading me to believe it is true, but I can not prove it.
| https://mathoverflow.net/users/23850 | does every vertex-cut set in a maximal planar graph contain a cycle? | Take one component $H$ of $G(V-S)$ and let $S'$ be the set of vertices in $S$ that are adjacent to a vertex in $H$. Then $S'$ is separating: it separates $H$ from the rest of $G$. Draw $G$ in the plane. There is a simple closed (Jordan) curve $C$ in the plane which intersects the drawing exactly at $S'$ (elsewhere lyin... | 9 | https://mathoverflow.net/users/9025 | 226027 | 105,594 |
https://mathoverflow.net/questions/226042 | 4 | A Hopf-Galois extension of commutative rings, as defined by Montgomery [here](http://msp.org/gtm/2009/16/gtm-2009-16-012s.pdf), is a morphism of commutative rings $\phi:A\to B$ with a Hopf-algebra $H$ coacting on $B$ by a ring map $c:B\to B\otimes H$ such that the following two maps are *bijections*:
1. The canonical... | https://mathoverflow.net/users/11546 | Hopf-Galois Structure Maps | Unless I am very mistaken:
1. If the coaction of $H$ on $B$ is not $A$-linear, then there is not a canonical map $A \to B^H$, and so condition 1 would not make sense.
2. Commutativity of $B$ together with the fact that the coaction is a ring map together imply that the map $B\otimes B \to B\otimes H$ IS a ring homomo... | 3 | https://mathoverflow.net/users/78 | 226047 | 105,598 |
https://mathoverflow.net/questions/226040 | 5 | I'm looking for a characterization of graphs that are prime under the Cartesian product, with prime defined as in [this question](https://mathoverflow.net/questions/225094/connected-and-vertex-transitive-prime-graphs-with-respect-to-cartesian-product). Does such a characterization exist, either in general or after rest... | https://mathoverflow.net/users/84093 | Which graphs are prime under the Cartesian product? | For connected graphs, they are the graphs in which every two edges are connected by a sequence of pairwise relations using one or both of the following two types of relation:
* Edge $xy$ and $uv$ are related if $d(x,u)+d(y,v)\ne d(x,v)+d(y,u)$
* Edge $xy$ and $yv$ are related if $y$ is the only common neighbor of $x$... | 5 | https://mathoverflow.net/users/440 | 226048 | 105,599 |
https://mathoverflow.net/questions/226050 | 7 | It is well-known that the semicircle law holds for a wide class of matrices with independent and identically distributed (mean zero) entries.
My question is: is there any study about the more general case, in which we drop the "identically distributed" condition? That is, the matrix still have independent entries, bu... | https://mathoverflow.net/users/34309 | Extension of Wigner's semicircle law? | Yes, there is a central limit theorem that guarantees convergence in probability to the Wigner semicircle law, see for example [Central limit theorem for linear eigenvalue statistics of random matrices with independent entries,](http://arxiv.org/abs/0809.4698) or Chapter 2 of Tao's [Topics in random matrix theory.](htt... | 5 | https://mathoverflow.net/users/11260 | 226052 | 105,600 |
https://mathoverflow.net/questions/226049 | 6 | It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in <https://en.wikipedia.org/wiki/Real_tree> admits an isometric embedding into the Banach space $\ell\_1(\Gamma)$ for a set $\Gamma$ of sufficiently large cardinality (the space $\ell\_1(\Gamma)$ is defined as the space of real... | https://mathoverflow.net/users/37822 | Embedding of real trees into $\ell_1(\Gamma)$ | It's true: there's an isometric embedding of every real tree $T$ on some $\ell^1$-space, i.e., a set $A$ and a map $f:T\to \ell^1(A)$ that is an isometric embedding, that is, satisfies $\|f(x)-f(x')\|=d(x,x')$ for all $x\in T$.
I don't know a reference; here's a proof. Let $(x\_t)\_{t<\alpha}$ be an enumeration of po... | 10 | https://mathoverflow.net/users/14094 | 226062 | 105,603 |
https://mathoverflow.net/questions/226033 | 1 | Does anybody know whether there is an analysis of when the monotone decreasing chain has the [Feller property](https://en.wikipedia.org/wiki/Feller_process)?
The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down $n \mapsto n-1$ in each step is $q\_n>0,$ the one of staying at level $n$... | https://mathoverflow.net/users/84089 | Markov chain with Feller property | Probably Feller unless $\sum \frac 1 {q\_i} < \infty $. If the sum is finite, you reach 0 in bounded expected time starting from anywhere, and the Feller condition is not satisfied with the state 0 being a counterexample. If the sum is infinite, assume wlog that the $q\_i$ are bounded below by $1$ for large $i$. If not... | 1 | https://mathoverflow.net/users/nan | 226070 | 105,607 |
https://mathoverflow.net/questions/226056 | 1 | In his answer to the following mathoverflow question, [The (un)decidability of Robinson Arithmetic without multiplication](https://mathoverflow.net/q/177047/1946), Emil Jerabek proved that the following fragment:
1. $\forall$x(Sx$\neq$0)
2. $\forall$x$\forall$y(Sx=Sy $\Rightarrow$ x=y)
3. $\forall$x(x$\neq$0 $\Righta... | https://mathoverflow.net/users/20597 | A question regarding a fragment of Robinson Arithmetic | Just turning what SJR wrote to an answer: the true but undecidable sentence is:
$$ \forall x((\exists y(x=y+y))\lor(\exists y(x=S(y+y)))). $$
| 1 | https://mathoverflow.net/users/37103 | 226072 | 105,608 |
https://mathoverflow.net/questions/226089 | 3 | Is there any general relation between [*Hadwiger number*](https://en.wikipedia.org/wiki/Hadwiger_number) and [*Treewidth*](https://en.wikipedia.org/wiki/Treewidth) of a graph? Intuitively I think Hadwiger number is greater than or equal to Treewidth, but I couldn't prove it.
| https://mathoverflow.net/users/83880 | What is the relation between Hadwiger number and Treewidth? | Planar graphs have Hadwiger number at most 4, but can have arbitrarily high tree width (as evidenced by the $n \times n$ grid).
| 6 | https://mathoverflow.net/users/1492 | 226091 | 105,614 |
https://mathoverflow.net/questions/226093 | 20 | As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$).
Contrary to the case of Fermat, the case where $n=3$ has infinitely many solutions, because the surface is rational. For $n=4$, we get a K3 surface and ... | https://mathoverflow.net/users/23758 | Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$ | It has been conjectured by Euler that this equation has no solutions in positive integers when $n\geq 4$.
When $n=4$, this was disproved by Elkies in the paper [Elkies, On A4+B4+C4=D4] in a very strong way: he proves that the rational points of this K3 surface are dense in the real points for the euclidean topology.
... | 22 | https://mathoverflow.net/users/2868 | 226094 | 105,615 |
https://mathoverflow.net/questions/226086 | 24 | Let $M^3$ be an oriented 3-manifold, and let $f:M^3\looparrowright \mathbb R^4$ be a codimension one immersion. Is it possible to find a small deformation of the composite map
$$
M^3 \to \mathbb R^4 \to \mathbb R^6
$$
which is an embedding?
(I expect the answer to be "no", and so I'm mostly interested in the method o... | https://mathoverflow.net/users/5690 | Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$? | Quoting Theorem F of [this paper](http://www.maths.ed.ac.uk/~aar/papers/koschork.pdf) by Ulrich Koschorke:
>
> For any self-transverse immersion $j$ of a closed 3-manifold $M$ into $\mathbb{R}^4$ the following integers are equal modulo 2:
>
>
> * the Euler number of the surface of double points of $j$;
> * the nu... | 27 | https://mathoverflow.net/users/8103 | 226103 | 105,617 |
https://mathoverflow.net/questions/226085 | 6 | Let the kernel be $f(\mathbf{x},\mathbf{y}) = \arccos(\mathbf{x}^T \mathbf{y})$, where $\mathbf{x}$ and $\mathbf{y}$ are $\ell\_2$ normalized vectors of the same dimensionality, and $\arccos(\cdot): [-1,1] \to [0,\pi]$ is the inverse cosine function.
Question: Is $f$ conditionally negative definite? If yes, how can I... | https://mathoverflow.net/users/84114 | On proof of the conditionally negative definiteness of a kernel | Here a direct approach. Recall the power-series
\begin{equation\*}
\arccos(z) = \frac\pi2 - \sum\_{k\ge0}\binom{2k}{k}\frac{z^{2k+1}}{4^k(2k+1)}.
\end{equation\*}
From this series it is clear that $\arccos(x^Ty)$ is conditionally negative definite (because it is of the form "const $-$ positive definite").
---
*E... | 5 | https://mathoverflow.net/users/8430 | 226104 | 105,618 |
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