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https://mathoverflow.net/questions/238380 | 9 | I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}\_m$ action. Here's what I have so far:
Let $X=Spec\,A$ be an affine scheme (after this case is setteled I imagine it would be relatively easy to generalize to relatively affine schemes and get global Proj). Let $\mathbb{G}\_m = Spec\... | https://mathoverflow.net/users/22810 | Geometric construcion of Proj as a quotient by a $\mathbb{G}_m$ action | First of all, the ideal $I$ corresponding to the fixed point set is generated by all $A\_d$ with $d\ne0$. Thus $I=\bigoplus\_d I\_d$ with $I\_d=A\_d$ for $d\ne0$ and $I\_0=\sum\_{d\ne0}A\_dA\_{-d}\subseteq A\_0$.
Secondly, as already stated, the answer to Q1 in this generality is "no" but, under the given circumstanc... | 8 | https://mathoverflow.net/users/89948 | 238399 | 110,129 |
https://mathoverflow.net/questions/238394 | 10 | Over at <http://www.scottaaronson.com/blog/?p=2725#comment-1089004> we had a discussion of intermediate Turing degrees.
The following function came up:
>
> Take Chaitin’s constant, and rearrange its binary digits as follows: for each of the sets {1st digit} {2-3rd digits} {4-7th digits} {2^n-(2^n+1)-1}, order the... | https://mathoverflow.net/users/44423 | What Turing degree is this function? | I think it's strictly below $0'$. Namely let's call your number $\Gamma(\Omega)$ where $\Gamma$ is a Turing functional. Let $\Phi$ be any other Turing functional. Then show that the set
$$
S = \{X: X = \Phi(\Gamma(X))\}
$$
has measure 0 (which is easy since $\Gamma$ erases a lot of information about $X$)
and moreover s... | 6 | https://mathoverflow.net/users/4600 | 238400 | 110,130 |
https://mathoverflow.net/questions/238397 | 2 | in a class gave me the concept of initial and terminal object. I started to look for this objects in different categories. I already proved that $\mathbb{Z}$ is initial and the zero-ring is terminal in the category of rings, but I can't find the initial and terminal objects (if they exists) in the category of local rin... | https://mathoverflow.net/users/91417 | initial and terminal objects in local rings | A correct but somewhat uninteresting answer (as pointed out by Johannes Hahn) is that there is no initial local ring, in the vanilla category of local rings and local homomorphisms. As Johannes said, this is more or less for the same reason that the category of fields has no initial object and no terminal object. In mo... | 5 | https://mathoverflow.net/users/2926 | 238404 | 110,132 |
https://mathoverflow.net/questions/238377 | 12 | Denote the commuting probability (the probability that two randomly chosen elements commute) of a finite group $G$ by $\operatorname{cp}(G)$. By a result of Gustafson [2], $\operatorname{cp}(G)=\operatorname{k}(G)/|G|$, where $\operatorname{k}(G)$ is the number of conjugacy classes of $G$. My question is the following:... | https://mathoverflow.net/users/57975 | Finite groups with lots of conjugacy classes, but only small abelian normal subgroups? | No: there exists a sequence of finite groups with commuting probability bounded away from 0 but with no abelian (normal) subgroup of bounded index.
Fix a prime power $q$. Consider the "higher Heisenberg" group $G\_n$ of order $q^{2n+1}$ consisting of those square matrices of size $n+2$ over $\mathbf{F}\_q$ of the for... | 14 | https://mathoverflow.net/users/14094 | 238407 | 110,133 |
https://mathoverflow.net/questions/180223 | 5 | Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$.
Then we know that the eigenvalues of $-\Delta$ form an increasing sequence of eigenvalues $0<\lambda\_1<\lambda\_2\leq \lambda\_3\leq....$ and we have a corresponding orthonormal b... | https://mathoverflow.net/users/21870 | Boundary regularity of Dirichlet Eigenfunction on bounded domains | In case this helps, the answer is yes for a Lipschitz domain.
From McLean's book *Strongly Elliptic Systems and Boundary Integral Equations*:
If $\Omega$ is a Lipschitz domain, there is a bounded trace operator $\gamma\colon H^s(\Omega) \to H^{s-1/2}(\partial\Omega)$ for any $1/2 < s < 3/2$. Moreover, if $1/2 < s \... | 2 | https://mathoverflow.net/users/91324 | 238412 | 110,135 |
https://mathoverflow.net/questions/238112 | 5 | Let $K$ be a field, $\partial\_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial\_1\ldots \partial\_n]$ (it is implicitly assumed that the derivations do not commute with elements of $K$ and the commutation relation is the obvious $\partial\_i k - k\partial\_i=\partial\_i(k)$).
For example, $R$ ... | https://mathoverflow.net/users/32985 | Groebner bases for differential operators with field coefficients (reference request) | Elaborating on Michael's comments and your concerns about Saito, Sturmfels, Takayama, "Grobner Deformations of Hypergeometric Differential Equations", the case this book addresses is the case of the Weyl algebra over a characteristic 0 field $k$. I.e., the noncommutative polynomial ring $$D=k[x\_1,\ldots, x\_n, \partia... | 2 | https://mathoverflow.net/users/36720 | 238419 | 110,136 |
https://mathoverflow.net/questions/238413 | 3 | My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of integers in cosets coprime to a fixed ideal (in some reasonable subset of the canonical embedding).
In particular, for t... | https://mathoverflow.net/users/50426 | Primes in arithmetic progressions in number fields | Theorem 6 in Chapter XV, $\S$5 of Lang *Algebraic Number Theory* is a result of the sort you want. Lang formulates it using ideles, but he gives an application in more classical language in Example 3 on the following page. I first quote what he shows in Example 3, then I'll try to rewrite it to sound a bit more like wh... | 10 | https://mathoverflow.net/users/297 | 238427 | 110,138 |
https://mathoverflow.net/questions/238422 | 3 | First of all, sorry for my bad english. I tried to find out whether the following statement is true or not:
Let $X$ be a operator system, $B$ a $C^\*$-algebra and $f:X\to B$ linear such that $f(1)\ge 0$ and $\|f(1)\|\_B=\|f\|\_{op}$. Then $f$ is positive.
I already know that the statement is true if $B=\mathbb{C}$ ... | https://mathoverflow.net/users/nan | is a linear map on an operator system into a $C^*$-algebra (+ extra conditions) positive? | It's false. Take $X = B = \mathbb{C}^2 \cong l^\infty(\{0,1\})$, and define $f: X \to B$ by $f(a,b) = (a,.5(a-b))$. Then $f(1,1) = (1,0) \geq (0,0)$ and $\|f\| = \|f(1,1)\|\_\infty = 1$, but $f(0,1) = (0, -.5) \not\geq 0$.
Edit: but maybe it is worth pointing out that if $f(1\_X) = 1\_B$ and $\|f\| = 1$ then $f$ must... | 6 | https://mathoverflow.net/users/23141 | 238428 | 110,139 |
https://mathoverflow.net/questions/238431 | 6 | **For 5 months!** I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):
>
> **Main equation:**
>
>
> $$
> \biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\mathbf{x}}{\mathbf{x^{\text{T}}}\mathbf{x}}E\biggr)\mathbf{x}=\mathbf{1}
> $$
>
>
> **Constraint equati... | https://mathoverflow.net/users/91441 | Explicit solution to a Rayleigh quotient equation | Since $M^2$ is symmetric, it is diagonalizable with orthogonal eigenvectors. Also the eigenvalues are positive. Let $(v\_i)$ be an orthonormal basis of eigenvectors for $M^2$ with eigenvalues $\lambda\_i\ge 0$. Now express $\mathbf 1$ in terms of the eigenvectors as $\sum b\_iv\_i$ and write $\mathbf x=\sum a\_iv\_i$. ... | 4 | https://mathoverflow.net/users/11054 | 238437 | 110,142 |
https://mathoverflow.net/questions/238405 | 9 | Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations.
Generally $Vect(BG) \ne Rep(G)$ since by Atiyah Segal completion the grothendiek ring of one is the completion of the other. Unfortunately I d... | https://mathoverflow.net/users/22810 | Simplest explicit counterexample for $Vect(BG) \ne Rep(G)$ as monoids | Recall that $H^\*(BS^3)=\mathbb{Z}[y]$ with $|y|=4$. For any odd integer $k>0$ there is a map $\psi^k\colon BS^3\to BS^3$ with $(\psi^k)^\*(y)=k^2y$. (I am not actually sure if $k$ needs to be odd.) There are evident inclusions $S^1\xrightarrow{i}S^3\xrightarrow{j}SU$. The composite $\psi^k\circ Bi$ just comes from the... | 3 | https://mathoverflow.net/users/10366 | 238447 | 110,145 |
https://mathoverflow.net/questions/238443 | 4 | For given $g$, consider the family of graphs which may be embedded to the compact orientable surface of genus $g$. For this family, consider maximal clique $\alpha(g)$, maximal chromatic number $\chi(g)$ and maximal choice number $cn(g)$. Famous results are $\alpha(g)=\chi(g)=[(7+\sqrt{1+48g})/2]$ for all $g$ (Heawood ... | https://mathoverflow.net/users/4312 | Choice number of embedded graphs | I will use the *Euler genus*, so that I can also talk about non-orientable surfaces. The Euler genus of a sphere with $g$ handles is $2g$ and the Euler genus of a sphere with $g$ crosscaps is $g$.
The maximal choice number for a surface of Euler genus $\epsilon$ turns out to be almost always equal to the Heawood Boun... | 3 | https://mathoverflow.net/users/2233 | 238451 | 110,147 |
https://mathoverflow.net/questions/238441 | 4 | Is the following statement consistent in $\mathsf{ZFC}$?
>
>
> >
> > For every ordinal $\beta$ there is an ordinal $\lambda\_0$ such that for all ordinals $\lambda\geq\lambda\_0$ we have $2^{\aleph\_{\lambda}}\geq \aleph\_{\lambda+\beta}.$
> >
> >
> >
>
>
>
| https://mathoverflow.net/users/8628 | Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ "get larger and larger"? | Yes, in the Foreman-Woodin model for the global failure of $GCH$ your statement is true.
See [The generalized continuum hypothesis can fail everywhere.](http://www.jstor.org/stable/2944324?origin=crossref&seq=1#page_scan_tab_contents)
| 5 | https://mathoverflow.net/users/11115 | 238455 | 110,148 |
https://mathoverflow.net/questions/166249 | 3 | Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ such that the adjoint map of $\phi$
$$ \phi' \colon P \to P^\ast = Hom(P,A) $$
restricts to $0$ on $L$ and $L = L^\perp$... | https://mathoverflow.net/users/1123 | Metabolic vs stably metabolic | I think there is no general answer to this question, and partial answers can only be obtained in sufficiently low dimensions. Very similar to the question when a stably free projective module is actually free (which doesn't have a complete answer). I explain below my favourite way of thinking about this kind of problem... | 4 | https://mathoverflow.net/users/50846 | 238467 | 110,151 |
https://mathoverflow.net/questions/238468 | 11 | The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor
$$
X\mapsto \Omega^\infty (W\_j \wedge\_{h\Sigma\_j} X^{[j]})
$$
in which $W\_j$ is a spectrum with (naive) $\Sigma\_j$-action, $X^{[j]}$ is the $j$-fold smash power and ${}\_{h\Sigma}$ d... | https://mathoverflow.net/users/8032 | Reference request: Goodwillie tower of the identity | This is done by identifying the partition complex of the set with j elements with the space of fully grown trees with j+1 leaves. Reference: "Partition complexes, duality and integral tree representations" by Alan Robinson, Algebr. Geom. Topol. 4. See specifically proposition 2.7 and corollary 2.8. <http://msp.org/agt/... | 13 | https://mathoverflow.net/users/6668 | 238479 | 110,156 |
https://mathoverflow.net/questions/238424 | 7 | (This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\subseteq^\*$, where $a\subseteq^\* b$ means $a\setminus b$ is finite.
Let $\mathfrak{ht}$ (sometimes called $\operator... | https://mathoverflow.net/users/2415 | What is the height (or depth) of $[\mathbb{N}]^\infty$? | In the Hechler model, $\aleph\_1 = \mathfrak{t} < \mathfrak{b} = \mathfrak{c} = \aleph\_2 < \mathfrak{ht}$.
(By "the Hechler model" I mean the result of a length-$\omega\_2$ finite support iteration of the forcing to adjoin a dominating real.)
That $\aleph\_1 = \mathfrak{t} < \mathfrak{b} = \mathfrak{c} = \aleph\_2... | 3 | https://mathoverflow.net/users/70618 | 238481 | 110,157 |
https://mathoverflow.net/questions/238497 | 1 | Let $f(n)$ be
$$
f(n)=\begin{cases}
1,&\small{\text{if there are digits 1 in the constant $\textit{e}$ $\textit{n}$ times in a row}}\\
0,&\small{\text{otherwise.}}\\
\end{cases}
$$
Is it true that $f(n)$ is computable?
I strongly believe that is not, but I can't even understand how to prove it. What to do?
| https://mathoverflow.net/users/91470 | Computable function | The function is computable. It is either the constant $1$ function, if there are arbitrarily long sequences of consecutive $1$s in the expansion of the number $e$ (and I guess you mean the decimal expansion or the binary expansion), or if there is not, then the function is a simple step function, with value $1$ up to s... | 4 | https://mathoverflow.net/users/1946 | 238498 | 110,164 |
https://mathoverflow.net/questions/238066 | 10 | Is there a family $\{A\_\alpha:\alpha<2^{\omega\_1}\}\subset [\omega\_1]^{\omega\_1}$ in ZFC such that for each countable set $I\subset 2^{\omega\_1}$ and $\alpha\in 2^{\omega\_1}\setminus I$ we have
$$A\_\alpha\not\subset\bigcup\{A\_\beta:\beta\in I\}.$$
Of course, we can find such a family of size $\omega\_2$ inste... | https://mathoverflow.net/users/71011 | A question on subsets of $\omega_1$ | Today I could prove that the existence of such family is not provable from ZFC.
**Theorem.**
If GCH holds in $V$, and $\mu\ge \omega\_3$ then in $V^{Fn(\mu,2)}$ the following holds:
*If $\{A\_i:i<{\omega}\_3\}\subset [{\omega}\_1]^{{\omega}\_1}$, then there is
$I\in [\omega\_3]^\omega$ and $\alpha\in \omega\_3\setm... | 6 | https://mathoverflow.net/users/71011 | 238502 | 110,166 |
https://mathoverflow.net/questions/238490 | 4 | The [Fisher information](https://en.wikipedia.org/wiki/Fisher_information) of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}\_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^2} \,\log f\left(Y;\theta\right)\right | \theta \right]\enspace ,... | https://mathoverflow.net/users/2051 | Can samples be compressed? | No, the inequality is not guaranteed. One example with $k=2$ is for $g(x\_1,x\_2) = OR(x\_1,x\_2)$. Then $f\_g(1;\theta) = 1-(1-\theta)^2$, and $\mathcal{I}\_g(\theta) = 4/(2\theta-\theta^2)$. We have $\mathcal{I}\_g(\theta)>\mathcal{I}\_{x\_1}(\theta)$ when $\theta<\tfrac23$ and the reverse inequality when $\theta>\tf... | 6 | https://mathoverflow.net/users/20186 | 238504 | 110,168 |
https://mathoverflow.net/questions/238172 | 18 | The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred to the other completions of $Q$, i. e. the $p$-adic fields. Of course, since the topology of $\mathbb Q^p$ behaves rath... | https://mathoverflow.net/users/37059 | $p$-adic Bott periodicity? | The $p$-completed algebraic $K$-theory of the algebraic closure of $\mathbb{Q}\_p$, i.e., $K(\bar{\mathbb{Q}}\_p; \mathbb{Z}\_p)$, is equivalent to its second loop space, up to an issue about path components. This is due to Suslin. The descent to $\mathbb{Q}\_p$ is more subtle than the descent from $\mathbb{C}$ to $\ma... | 18 | https://mathoverflow.net/users/9684 | 238507 | 110,169 |
https://mathoverflow.net/questions/238505 | 21 | Suppose that $S$ and $T$ are sets of Fibonacci numbers greater than $1$. Let $S^\*$ be the product of numbers in $S$, and likewise for $T^\*$. If $S^\*=T^\*$, must $S=T$?
| https://mathoverflow.net/users/61426 | Distinctness of products of Fibonacci numbers | (**Finite** sets, of course, if you don't want to allow $S^\* = T^\* = \infty$)
Yes. We may assume wlog $S$ and $T$ are disjoint, and $\max S < \max T$. By [Carmichael's theorem](https://en.wikipedia.org/wiki/Carmichael's_theorem), every Fibonacci number except $1$, $8$ and $144$ has a prime factor that does not divi... | 43 | https://mathoverflow.net/users/13650 | 238508 | 110,170 |
https://mathoverflow.net/questions/238482 | 11 | Is there a simple smooth closed curve $\gamma$ in $\mathbb{R}^{3}$ such that for all $x,y\in \gamma$ with $x \neq y $, $l\_{x}$ and $l\_{y}$ are [skew lines](https://en.wikipedia.org/wiki/Skew_lines), where $l\_{x} $ and $l\_{y}$ are straight lines tangent to $\gamma$ at $x,y$, respectively? What about real analytic ca... | https://mathoverflow.net/users/36688 | A very torsioned closed curve in space | If by "skew" we mean *nonparallel and nonintersecting*, then the answer is NO. Every smooth closed curve in $\mathbf{R^3}$ has uncountably many pairs of intersecting tangent lines. This follows from [Poincare-Hopf index theorem](https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem); see
[M. Ghomi, Tangent... | 21 | https://mathoverflow.net/users/68969 | 238515 | 110,173 |
https://mathoverflow.net/questions/238510 | 18 | What is the expected value of an N-dimensional vector of uniformly distributed random numbers which sum to 1 and have been sorted in descending order?
Here is the algorithm for drawing a sample from the distribution of this N-dimensional vector:
* Create an N-1 dimensional vector of uniformly distributed random num... | https://mathoverflow.net/users/91482 | What is the expected value of an N-dim vector of uniform randoms that sum to 1 which have been sorted into descending order? | You pick $N-1$ points $X\_1,\dotsc, X\_{N-1}$ independently and uniformly at random in $[0,1]$. They divide the segment $[0,1]$ into $N$ segments of lengths $S\_1,\dotsc, S\_N$ usually referred to as *spacings*.
To use your description, rearrange the $X\_i$ in increasing order $X\_{(1)}\leq \cdots \leq X\_{(N-1)}$ a... | 17 | https://mathoverflow.net/users/20302 | 238524 | 110,176 |
https://mathoverflow.net/questions/238525 | 6 | Let $\mathbb F\_q$ be a finite field and $n$ an integer.
>
> What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F\_q[X\_1,\dots,X\_n]$ such that the set $Z(f)$ of zeros of $f$ in the affine space $\mathbb F\_q^n$ has cardinality $q^n-1$, that is misses exactly one point?
>
>
>
It is clear t... | https://mathoverflow.net/users/9317 | Hypersurface missing just one point | It's $n(q-1)$. We must have
$$\sum\_{x\_1,\dots,x\_n \in \mathbb F\_q}f(x\_1,\dots,x\_n)=f(0,\dots,0) \neq 0$$
but
$$\sum\_{x\_1,\dots,x\_n \in \mathbb F\_q} \prod\_i x\_i^{e\_i}$$
vanishes unless each $e\_i$ is a positive multiple of $q-1$, so vanishes for all monomials of degree $<n(q-1)$, so unless the degre... | 12 | https://mathoverflow.net/users/18060 | 238527 | 110,177 |
https://mathoverflow.net/questions/238531 | 6 | Let $D$ be a degree $3$ division algebra over a field $k$ of char not 2 and 3.
Any such division algebra is cyclic. I am interested in knowing the cases when the reduced norm map $Nrd : D^\* \rightarrow k^\*$ is surjective. Of course, this happens over $\bar k$ and finite field etc. Here is my explicit question.
I wa... | https://mathoverflow.net/users/69977 | reduced norm from degree 3 division algebra | The Merkurjev-Suslin Theorem says that an element $x \in k^\*$ is a norm from $D$ if and only if $[D] \cup (x)$ is zero in the Galois cohomology group $H^3(k, \mathbb{Z}/3)$. Therefore, for the reduced norm to be surjective for every division $k$-algebra of degree 3, it suffices to assume that $H^3(k, \mathbb{Z}/3) = 0... | 7 | https://mathoverflow.net/users/6486 | 238534 | 110,179 |
https://mathoverflow.net/questions/238532 | 0 | Following a problem I found on [mathstack](https://math.stackexchange.com/questions/1762753/proof-of-an-inequality-in-c-2), with no solution, and no comment, so I think this inequality is not easy, so I post it here (because I think there are more some good math job, maybe someone can solve it).
Let $n\ge 2$ be an in... | https://mathoverflow.net/users/38620 | Absolute value inequality with complex numbers | If you have it for $n=2$, just sum up over all pairs $(z\_i,z\_j)$ with $i<j$ and divide by $n-1$.
As for the proof for $n=2$, yours is quite ok for me, and the proof by math110 is especially elegant, but well, here is another approach. We need two easy lemmata:
Lemma 1. For real $t$ and non-negative real $R$ we h... | 9 | https://mathoverflow.net/users/4312 | 238540 | 110,181 |
https://mathoverflow.net/questions/238318 | 5 | Let Rel be the category whose objects are sets and whose morphisms are binary relations, with composition defined by $x (S \circ R) z \Leftrightarrow (\exists y : x R y \wedge y S z)$, and identity the equality relation.
If we have relations $R \in Rel(X, X')$ and $S \in Rel(Y, Y')$, we can define a relation $(R \to ... | https://mathoverflow.net/users/66017 | Set of functions is not a bifunctor on Rel | The operation you are describing is a cartesian closed structure, not on the category $Rel$ of sets (as objects) and relations (as morphisms), but rather on a category whose *objects* are relations and whose morphisms are pairs of functions mapping related arguments to related results. This is a bit easier to see/expla... | 3 | https://mathoverflow.net/users/1015 | 238547 | 110,183 |
https://mathoverflow.net/questions/238514 | 7 | Let $X$ be a del Pezzo surface over a number field $k$. (A *del Pezzo surface* over $k$ is a smooth, projective, geometrically connected surface whose anti-canonical class $K\_X$ is ample.) Let $d := K\_X^2$ be the *degree* of $X$. It is well-known that $d$ satisfies $1 \leq d \leq 9$.
If $d \geq 5$, and $X(k) \neq \... | https://mathoverflow.net/users/17907 | Obstruction to rationality of del Pezzo surfaces of degree 4 | Interesting question.
But, alas, the answer is *no*.
The issue is that you have missed an extra non-rationality criterion. Namely, it is possible that such a surface $X$ has $\mathrm{Br}(X) = \mathrm{Br}(k)$, yet we have $\mathrm{Br}(X\_K) \neq \mathrm{Br}(K)$ for some finite extension $K/k$.
Problems of this typ... | 7 | https://mathoverflow.net/users/5101 | 238553 | 110,185 |
https://mathoverflow.net/questions/238538 | 8 | Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$.
I am interested the status in a much weaker form of the conjecture:
>
> Is it true that for all even numbers $K$, there exist primes $p,q$
> with $p-q=K$?
>
>
>
... | https://mathoverflow.net/users/74918 | Do arbitrary $K$-twin primes exist? | I think this is open just as much as the original conjecture. In fact, in analytic number theory, we usually prove the existence of certain objects by showing that there are many of them (certainly infinitely many, which can be further defined by density etc.).
| 7 | https://mathoverflow.net/users/11919 | 238556 | 110,186 |
https://mathoverflow.net/questions/238565 | 5 | Suppose we have a simple connected graph $G=(V,E)$. Then let $A$ be its $|E|\times |V|$ incidence matrix. Here I am considering the unoriented incidence matrix. I want to know when the row span of $A$ contains the all ones vector in $\mathbb{R}^{|V|}$. I believe this happens if and only if $G$ has a spanning regular su... | https://mathoverflow.net/users/41283 | Characterizing graphs whose incidence matrix has the all ones vector in its row span | The rank of the incidence matrix is $|V|$ minus the number of bipartite components. I assume the graph is connected. If it is not bipartite, it follows that the row space is $\mathbb{R}^{V}$ and hence it contains the all-ones vector. If the graph is bipartite, the vector that is 1 on the vertices in the first colour cl... | 4 | https://mathoverflow.net/users/1266 | 238571 | 110,192 |
https://mathoverflow.net/questions/238578 | 1 | Let $\mathcal L$ be an *infinite* signature and $\mathcal A$, $\mathcal B$ two *finite* $\mathcal L$-structures such that
for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\iff\mathcal B\models\varphi.$$
>
> Does it follow that $\mathcal A$ and $\mathcal B$ are isomorphic?
>
>
>
C... | https://mathoverflow.net/users/91516 | $\mathcal A\equiv\mathcal B\implies \mathcal A\cong\mathcal B$ for finite $\mathcal L$-structures where $\mathcal L$ is an infinite signature | Yes. If they are not isomorphic, then for each bijection of $A$ with $B$, there is an atomic formula that the bijection does not respect, that is, a reason that it is not an isomorphism. Since there are only finitely many bijections, we therefore reduce to the case of a finite sublanguage, in which the structures are n... | 6 | https://mathoverflow.net/users/1946 | 238579 | 110,196 |
https://mathoverflow.net/questions/238500 | 5 | (Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal characteristics of the continnum. It is a refined version of an [earlier question](https://mathoverflow.net/questions/238424/what-i... | https://mathoverflow.net/users/2415 | When is there an unbounded tower in $[\mathbb{N}]^\infty$? | A model for Question 1: Let $V \models$ MA + $2^{\aleph\_0} = \kappa \geq \aleph\_2$. Using MA, construct a $\supseteq^{\star}$-chain $\overline{A} = \langle A\_i : i < \kappa \rangle$ such that $\mathcal{A} = \{p\_{A\_i} : i < \kappa\}$ (where $p\_X(k)$ is the $k$th member of $X$) is a dominating family in $\omega^{\o... | 4 | https://mathoverflow.net/users/2689 | 238586 | 110,197 |
https://mathoverflow.net/questions/238519 | 6 | In Victor Reiner's *[Quotients of Coxeter Complexes and $P$-Partitions](http://www.math.umn.edu/~reiner/Papers/MyThesis.pdf)*, we have the below definition for the quotient complex of a Coxeter complex by a finite subgroup of the Coxeter group. I think there is something wrong with the definition, and would like your h... | https://mathoverflow.net/users/12419 | Quotient of Coxeter complex in terms of double cosets? | I think this is the same issue that I noticed later myself, and addressed in Remark 2.13 on page 228 of the paper that I wrote with Eric Babson: "Coxeter-like complexes" (Discrete Mathematics and Theoretical Computer Science 6, 2004, 223–252). Let me know, if that is not the issue!
| 7 | https://mathoverflow.net/users/91525 | 238592 | 110,198 |
https://mathoverflow.net/questions/199738 | 6 | Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma\_1,\sigma\_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$).
Let $f\in H^{1/2}(\partial\Omega)=H^1(\Omega)/H^1\_0(\Omega)$.
Let $u\_i\in H^1(\Omega)$, $i=1,2$, be the solution of
$$
\begin{cases}
\operatorname{div}(\sigma\_i\nabla u... | https://mathoverflow.net/users/55893 | How do solutions of a PDE depend on parameters? | If $p=2$ and $\sigma\_1, \sigma\_2 \in L^\infty(\Omega)$ one has the estimate
$$\|\nabla u\_1 - \nabla u\_2\|\_{L^2(\Omega)} \leq C \|f\|\_{H^{1/2}(\partial\Omega)} \|\sigma\_1 -\sigma\_2\|\_{L^\infty(\Omega)}$$ where $C = C(\Omega,c)$.
To see this, write $0 = (u\_1-u\_2) \operatorname{div}(\sigma\_1 \nabla u\_1 - \s... | 6 | https://mathoverflow.net/users/91324 | 238595 | 110,201 |
https://mathoverflow.net/questions/238582 | 4 | Let $n$ be an even positive integer and $W\_n$ be the class of all $n\times n$ matrices with entries from the set $\{-1,0,1\}$ such that all row sums and column sums are equal to $0$.
For each $M\in W\_n$, let $k(M)$ be the number of ones in $M$. I think the number of $M\in W\_n$ with $k(M)$ even is much bigger than ... | https://mathoverflow.net/users/58096 | The class of $(-1,0,1)$-matrices with all row sums and column sums equal to $0$ | Yes, this should be true. Here is an explanation why. Equivalently, we want to assign weights $-1,0,1$ to the edges of $K\_{n,n}$ such that the sum of the weights at each vertex is $0$. One way to do this is to choose edge-disjoint cycles $C\_1, \dots, C\_\ell$ in $K\_{n,n}$ with $\bigcup\_{i=1}^\ell E(C\_i)=E(K\_{n,n}... | 6 | https://mathoverflow.net/users/2233 | 238598 | 110,203 |
https://mathoverflow.net/questions/238581 | 5 | Let $F(\mathbf{x}) \in \mathbb{Z}[x\_1, ..., x\_n]$ be a degree $d$ homogeneous form. Let
$$
I(\alpha) = \int\_{[0,1]^n} e^{2 \pi i F(\mathbf{x}) \alpha} dx\_1...dx\_n.
$$
Then the singular integral is defined as
$$
\sigma\_{\infty} = \int\_{\mathbb{R}} I(\alpha) d\alpha.
$$
It says in an article I am reading that $... | https://mathoverflow.net/users/84272 | Showing the positivity of a singular integral that came up in circle method | Here's a sketch. First of all, proceeding formally, if we start with the $\alpha$ integral, then we obtain
$$
\sigma\_{\infty} = \int \delta(F(x))\, dx ,
$$
and near an $a$ with $F(a)=0$, $\nabla F(a)\not= 0$, this looks like $\delta(v\cdot t)$, so we obtain a positive contribution.
To make a proof out of this, fix a... | 2 | https://mathoverflow.net/users/48839 | 238601 | 110,205 |
https://mathoverflow.net/questions/238526 | 1 | For a principal bundle $\pi: P \to M$ we have the following Atiyah sequence that can be used to define a connection on it
\begin{equation}
0 \to V{P} \to T{P} \to \pi^\*{T{M}} \to 0
\end{equation}
A connection is defined as a $G$-equivariant splitting of the above short exact sequence.
My question is, is there a sim... | https://mathoverflow.net/users/82645 | Atiyah-sequence-like definition of connection on vector bundles? | Yes, there is a general notion of a connection, which makes sense for any fiber bundle, namely, Ehresmann connection, see <https://en.wikipedia.org/wiki/Ehresmann_connection>
In the case of vector bundle $E\to M$, you have to require additionally that for $x\in M$, $e\in E\_x$ the splitting of
$$
0 \to Vert\_{x,e}\,E... | 5 | https://mathoverflow.net/users/40950 | 238610 | 110,208 |
https://mathoverflow.net/questions/238616 | 0 | In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting:
$$ \int\limits\_{\mathbb{Q}\cap [0,1]} f(x) dx = \lim\_{Q \to \infty} \frac{2}{(Q+1)Q} \sum\limits\_{q =1}^{Q} \sum\limits\_{... | https://mathoverflow.net/users/38290 | Formula for an integration on $\mathbb{Q} \cap [0,1]$ | All you need is that $f$ is continuous on $[0,1]$.
Letting $$S(q) = \dfrac{1}{q} \sum\_{a=0}^{q-1} f(a/q)$$
(which is a Riemann sum for $J = \int\_0^1 f(x)\; dx$) and
$$ R(Q) = \sum\_{q=1}^Q \dfrac{2q}{Q(Q+1)} S(q)$$ you are defining
$$ \int\_{\mathbb Q \cap [0,1]} f(x)\; dx = \lim\_{Q \to \infty} R(Q)$$
For any $\... | 3 | https://mathoverflow.net/users/13650 | 238619 | 110,212 |
https://mathoverflow.net/questions/238604 | -2 | If $l \equiv 1$ mod 3 then $\mathbb{Q}(\zeta\_l)$ has a unique cubic subextension. I've been getting this field with the following magma code
>
> F:=CyclotomicField(l);
> S:=Subfields(F);
> for i:=1 to #S do
> if Degree(S[i][1]) eq 3 then
> return S[i][1];
> end if;
> end for;
>
>
>
but this is very slo... | https://mathoverflow.net/users/56362 | fast way to get subextensions in magma? | You have a typo there and you want $\ell \equiv 1 \pmod 3$. With that, this has little to do with Magma. Gauss proved that the cubic extension is generated by $g=\sum\_{j=1}^{\ell-1} e^{2\pi i j^3/\ell}$. In pari-gp, if you compute $g$ with sufficient precision and do algdep(g,3) you get the cubic satisfied by $g$ and ... | 0 | https://mathoverflow.net/users/2290 | 238620 | 110,213 |
https://mathoverflow.net/questions/238618 | 4 | For $i=1,2,\dots,l$, let $\mathbf{v}\_i =(v\_{i1},v\_{i2},\dots,v\_{in}) \in \mathbb{F}\_2^n$ be a sparse vector in GF(2) such that all $v\_{ij}$'s are independent for all $1 \le i \le l, 1 \le j \le n$ and $$\mathrm{Pr}[v\_{ij}=1] = \frac{\log(n)}{n}, \qquad 1 \le i \le l, 1 \le j \le n \\\mathrm{Pr}[v\_{ij}=0] = 1-\f... | https://mathoverflow.net/users/82990 | Probability of existence of a base in the span of sparse vectors in GF(2) | Let $V$ be the $n \times l$ matrix whose columns are your vectors.
For a given vector $x \in \mathbb F\_2^l$ with $k$ $1$'s, and $1 \le i \le n$, the number of
$j$ for which $x\_j = 1$ and $v\_{ij} = 1$
is a binomial random variable $W\_i(x)$ with parameters $(k,p)$, with $W\_i(x)$ for different $i$ independent. Thu... | 4 | https://mathoverflow.net/users/13650 | 238622 | 110,214 |
https://mathoverflow.net/questions/238541 | 6 | I'm searching for a counterexample for $C^\*$-algebras $A$ and $B$ and essential ideals (I assume an ideal to be closed and only two-sided ideals) $I\subseteq A$, $J\subseteq B$ , such that the ideal $I\otimes\_{max} J$ is not essential in the (maximal tensor product) -$C^\*$-algebra $A\otimes\_{max} B$. I'm not sure i... | https://mathoverflow.net/users/75338 | is the maximal tensor product of compact operators an essential ideal? | My suggestion (I hope that nothing is wrong):
Let $$F:B(l^2(\mathbb{N}))\otimes\_{max} B(l^2(\mathbb{N}))\to B(l^2(\mathbb{N}))\otimes\_{min} B(l^2(\mathbb{N}))$$ be the canonical map and define $M:=\ker(F)$. Since $F$ is not injective, $M$ is nontrivial.
Claim: $K(l^2(\mathbb{N}))\otimes\_{max} K(l^2(\mathbb{N}))... | 2 | https://mathoverflow.net/users/75338 | 238634 | 110,219 |
https://mathoverflow.net/questions/238535 | 1 | For a finite group $G$ and complex representation V of degree $n$, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants qualify to be called primary ? Or does the set need to satisfy additional conditions ?
I would like to know specifica... | https://mathoverflow.net/users/91492 | Primary invariants of a finite group | You should trust Max Horn. Definition 2.4.6 in our book has the finiteness condition explicitly. (This is the numbering in the first edition of the book.) For finite groups, this is equivalent to saying that the variety given by the primaries consists only of the origin (see Propos. 3.3.1).
A counter example arises i... | 2 | https://mathoverflow.net/users/82616 | 238637 | 110,221 |
https://mathoverflow.net/questions/238631 | 1 | Is there any way to solve the following system of non-linear differential equations exactly?
$x'(t) = x\times(y - \frac{1}{3(t + C)})$
$y'(t) = -\frac{1}{3}x^2 - \frac{y}{t + C}$
Here $x$ and $y$ are functions of $t$, and $C$ is some constant.
| https://mathoverflow.net/users/91548 | Is this non-linear system of differential equations tractable by other means than numeric approximation and dynamic analysis? | By changing the origin of time, we may assume $C=0$. One special family of
solutions is $x = 0, y = c/t$. But the general solution is
$$\eqalign{x(t) &= 12\,{\frac {a b^2{t}^{b-1}}{{a}^{2}{t}^{2\,b}+
12\,{b}^{2}}}
\cr
y(t) &={\frac {144\,{b}^{5}- a^4 \left( b+2/3 \right) {t}^{4\,b}-
16\,a^2 b^2 {t}^{2\,b}-96\,{b}^{4... | 1 | https://mathoverflow.net/users/13650 | 238641 | 110,222 |
https://mathoverflow.net/questions/238628 | 3 | This following problem is from my Conjecture many years ago,
Question :
Let $a,b>0,n\in N^{+},n\ge 3$,such
$$a^n+b^n+(2n+2)(ab)^n\le 2n$$
Conjecture: then $a+b\le 2$
or
$a+b>2.a>0.b>0,n\ge 3$,then we prove
$$a^n+b^n+(2n+2)a^nb^n-2n>0$$
or
it suffuce to prove
$$a^n+b^n+(2n+2)a^nb^n-2n>0\rm{when}~ a+b=2$$
Th... | https://mathoverflow.net/users/38620 | Prove this conjecture inequality | Take $a=1+x,b=1-x$, $|x|\leqslant 1$. We have to prove $(1+x)^n+(1-x)^n+(2n+2)(1-x^2)^n\geqslant 2n$. By Bernoulli inequality we have $(1-x^2)^n\geqslant 1-nx^2$ and by binomial expansion $(1+x)^n+(1-x)^n=2+n(n-1)x^2+2\binom{n}4 x^4+\dots$
So, it suffices to prove that $$2+n(n-1)x^2+2\binom{n}4 x^4+2n+2-n(2n+2)x^2\geqs... | 3 | https://mathoverflow.net/users/4312 | 238642 | 110,223 |
https://mathoverflow.net/questions/238646 | 12 | Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod\_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43} - q^{61} - q^{67} - q^{73} - q^{79} + q^{91} - q^{97} - q^{103} \dots = \sum\_{n=1}^\infty a\_n q^n$$
Then $a\_p + ... | https://mathoverflow.net/users/91552 | On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$ | Here's an answer to 2. You can tell me if it's the "best method".
Euler's Pentagonal number theorem gives that
$$ \eta(24z) = \sum\_{n \in \mathbb{Z}} (-1)^{n} q^{(6n+1)^{2}}. $$
This yields the formula
$$ \eta(6z) \eta(18z) = \sum\_{m, n \in \mathbb{Z}} (-1)^{m+n} q^{\frac{(6n+1)^{2} + 3(6m+1)^{2}}{4}}. $$
If $m=x$ ... | 15 | https://mathoverflow.net/users/48142 | 238656 | 110,226 |
https://mathoverflow.net/questions/238652 | 2 | Let $T\_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT\_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT\_{X}.$
1. How do we construct the isomorphism between the relative tangent bundle $T\_{\pi}$ and $R=L^{\*}\otimes \pi^{\*}T\... | https://mathoverflow.net/users/54760 | Relative tangent bundle and trivilization, tautological foliation | There is nothing special about the tangent bundle: for any vector bundle $E \rightarrow X$ with projectivisation $\pi : \mathbb{P}E \rightarrow X$, and tautological bundle $\mathcal{L} \rightarrow \mathbb{P}E$, we have $T\mathbb{P}E \cong (\pi^\*E / \mathcal{L}) \otimes \mathcal{L}^\*$. In fact the base $X$ is not real... | 2 | https://mathoverflow.net/users/nan | 238657 | 110,227 |
https://mathoverflow.net/questions/238664 | 21 | Every maximal subgroup of infinite index of a free non-cyclic group $F\_k$ is free of countable rank. Thus even though the set of maximal subgroups of $F\_k$ is uncountable, there are only countably many isomorphism classes of such subgroups (including subgroups of finite index). The same is true for surface groups of ... | https://mathoverflow.net/users/nan | The number of maximal subgroups up to isomorphism | Yes. Let $F$ be a nonabelian f.g. free group, then $F\times F$ has this property. Indeed, for each normal subgroup $N$ of $F$, denote $H\_N$ the subgroup $\{(x,y), xN=yN\}$ of $F\times F$ (this is a fibre product $F\times\_{F/N} F$). Then $H\_N$ contains the normal subgroup $N\times N$, and the quotient $H\_N/(N\times ... | 26 | https://mathoverflow.net/users/14094 | 238670 | 110,230 |
https://mathoverflow.net/questions/238667 | 2 | Throughout the proof of injectivity of the section conjecture, for example in Appendix B of <https://arxiv.org/abs/0809.0017>, one uses Mordell--Weil Theorem and for that embeds hyperbolic curve into an abelian variety. So, it may natural to take the embedding into the Jacobian variety. However, is it still possible to... | https://mathoverflow.net/users/91570 | Embedding of punctured projective line to abelian variety | The Jacobian is G\_m^2 and the embedding sends t to (t,t-1).
| 4 | https://mathoverflow.net/users/431 | 238675 | 110,232 |
https://mathoverflow.net/questions/238674 | 4 | Consider a vector field $F:\mathbb{R}^2\rightarrow \mathbb{R}^2$ of the following form $F(y\_1,y\_2)=(y\_2,\mu(y\_1))$, where $\mu\in\mathscr{C}^1(\mathbb{R})$ has appropriate growth so that the solutions of $Y'=F(Y)$ exists globally for any initial condition. Assume for some $a\in\mathbb{R}$ that $\mu(a)=0>\mu'(a)$.
... | https://mathoverflow.net/users/27767 | Periodic orbits in the plane | The period of the compact, non-singular orbit $\gamma$ is given by $$T(\gamma)=\oint\_\gamma \tau$$ where $\tau$ is any differential $1$-form such that $\tau(F)=1$, *e.g.* $\tau:=\frac{\mathrm{d}y\_2}{\mu(y\_1)}$. From this expression you may be able to study the boundedness of $T$ as $\gamma$ closes on another station... | 2 | https://mathoverflow.net/users/24309 | 238680 | 110,236 |
https://mathoverflow.net/questions/238668 | 3 | Let $F\_1(\mathbf{x}), F\_2(\mathbf{x}) \in \mathbb{Z}[x\_1, ..., x\_n]$ be a degree $d$ homogeneous polynomial.
For the system of equations $$F\_1(\mathbf{x})= F\_2(\mathbf{x}) =0,$$
we have the following singular integral
$$
\sigma\_{\infty} = \int\_{\mathbb{R}^2} I(\alpha\_1, \alpha\_2) d\alpha\_1d\alpha\_2,
$$
whe... | https://mathoverflow.net/users/84272 | Is the singular integral that come up in circle method independnet of the representatin of the equations? | Formally (ignoring issues of integrability), one has (writing $e(\theta) := e^{2\pi i\theta}$)
$$ \sigma'\_\infty = \int\_{{\bf R}^2} \int\_{[0,1]^n} e( F\_1({\bf x}) \alpha\_1 + g({\bf x}) F\_2({\bf x}) \alpha\_1 + F\_2({\bf x}) \alpha\_2)\ d{\bf x} d\alpha\_1 d\alpha\_2$$
$$ = \int\_{{\bf R}} \int\_{[0,1]^n} e( F\_... | 6 | https://mathoverflow.net/users/766 | 238687 | 110,237 |
https://mathoverflow.net/questions/238686 | 11 | Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = \mu(A)$ for every $t \in \Bbb{R}$ and for all borelian set A. In particular, the time-one map $\varphi^1$ also preserves... | https://mathoverflow.net/users/85681 | Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geodesic flow is recurrent? | Take a compact, connected Riemannian manifold $M$ with negative sectional curvature. Then choose any cover $M'$ of $M$ which is connected, Galois, and whose group of deck transformations is $\mathbb{Z}$ or $\mathbb{Z}^2$.
In dimension $2$ and for a $\mathbb{Z}$ cover, this can be done by taking $\mathbb{Z}$ copies o... | 12 | https://mathoverflow.net/users/75670 | 238699 | 110,242 |
https://mathoverflow.net/questions/238693 | 1 | I've been reading [this article](http://arxiv.org/pdf/math/0501401.pdf) on the overhand shuffle. In it the author uses a simplied mathematical model of the shuffle:
>
> Pemantle’s model for the overhand shuffle is
> parameterized by a probability $p \in (0,1)$. The transition rule is the following:
> Each of the ... | https://mathoverflow.net/users/41669 | Difficulty with a formula for a probability related to card shuffling | Perhaps its easiest to first ask for the probability $P\_0$ that the card does not move at all ($k=0$). One way to achieve this is to cut immediately below and above that card, with probability $1/2\times 1/2=1/4$. But any symmetric pair of cuts will do, for example you could cut at two cards below and two cards above,... | 1 | https://mathoverflow.net/users/11260 | 238700 | 110,243 |
https://mathoverflow.net/questions/238697 | 1 | Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-1}\left(\frac{1}{1+|.|^2} \right)\*f$
This is all well-defined in the sense of $S'(\mathbb{R}^n).$
Obviously, for $... | https://mathoverflow.net/users/91579 | Solution to inhomogenous PDE | This kernel is called the [Bessel potential](https://en.wikipedia.org/wiki/Bessel_potential#Representation_in_Fourier_space). It is smooth away from $0$, and in your case, this kernel is locally integrable.
| 2 | https://mathoverflow.net/users/37103 | 238704 | 110,244 |
https://mathoverflow.net/questions/238692 | 5 | I know that $F\_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F\_1:L^1 \rightarrow C\_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that there is also $F\_p: L^p \rightarrow L^q$ for $p \in (1,2).$ Here, the dual map trick does not work, so this transfo... | https://mathoverflow.net/users/91579 | Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$? | If $1\leq p<2$ then $\mathscr{F}: L^p \to L^{p'}$ is not surjective. I had this as a homework problem a week back.
The reason is the bounded inverse theorem: $\mathscr{F}: L^p \to L^{p'}$ is injective, (by fourier inversion on the dense subspace of schwarz functions). If the map were surjective then there would be an... | 6 | https://mathoverflow.net/users/68932 | 238707 | 110,246 |
https://mathoverflow.net/questions/160201 | 2 | We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c\_0+c\_1x+c\_2y+c\_3xy+c\_4x^2+c\_5y^2+c\_6xy^2 \\
G(x,y) &=& c\_k\cdot\exp\left(\frac{-(x^2+y^2)}{2\sigma^2}\right) \\
\square\_a(x,y) &=& \mathbf{1}\_{[-a,a]\times[-a,a]}(x,y)
\end{eqnarray}$
all fu... | https://mathoverflow.net/users/10668 | How is the deconvolution of a fat gaussian from a polynomial derived? | In a lax enough environment, the Fourier transform of $x^n$ is the distributional derivative of the $\delta$ distribution. The relevant behavior of the resulting distribution is: $$\delta^{(n)}[\phi]=(-1)^{n}\cdot \phi^{(n)}(0).$$
From this the result quickly follows.
| 0 | https://mathoverflow.net/users/10668 | 238710 | 110,248 |
https://mathoverflow.net/questions/238705 | 1 | Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the infinite point. We assume that $P$ is not torsion.
The point $P$ can be written as $(\frac{x}{d^2}, \frac{y}{d^3})$ for s... | https://mathoverflow.net/users/76332 | Heights of multiples of rational points on elliptic curves | You need to use Siegel's theorem (which is quite deep and relies on Roth's theorem or some such). This is in Chapter IX of *Arithmetic of Elliptic Curves*, specifically Theoream IX.3.1. You'll need to unsort the definitions a bit, since it's stated for number fields, but in your notation, one has
$$
\lim\_{n\to\infty} ... | 5 | https://mathoverflow.net/users/11926 | 238711 | 110,249 |
https://mathoverflow.net/questions/238703 | 11 | If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$.
Let $M$ and $M'$ be a simply connected manifolds of dimensions $m>0$. It is true that if $M$ is homotopic to $M'$, then for $k\geq m$, the spaces $Imm\left(M,\mathbb{R}^{m+k}\right)$ and $Imm\... | https://mathoverflow.net/users/84363 | Homotopy of space of immersions, Smale-Hirsch theorem | No. For example, if $M$ is a Moebius band then, at least for even $k$, $Imm(M,\mathbb R^{2+k})$ is not homotopy equivalent to $Imm(S^1\times \mathbb R,\mathbb R^{2+k})$.
The latter is equivalent to the space of all maps from $S^1$ to $O(2+k)/O(k)$. Its first non-trivial homotopy group is $\pi\_{k-1}=\pi\_k(O(2+k)/O(... | 17 | https://mathoverflow.net/users/6666 | 238713 | 110,250 |
https://mathoverflow.net/questions/238648 | 9 | I'm a newcomer to operads so apologies if this is a naive question.
The standard picture of an operad is of a collection of $n$-ary operations, thought of as objects with $n$ upward-pointing legs (inputs) and one downward-pointing leg (output), which can be composed in a sensible way. "Algebras over operads" then giv... | https://mathoverflow.net/users/nan | In what sense are operads "better" than PROPs? | I was friends with Frank Adams and Saunders Mac Lane, who invented PROPs in one of the world's most extensive unpublished collaborations. Saunders once showed me a box full of their correspondence. One reason they never published is that they lacked a way of showing the PROPs they were interested in acted on the things... | 19 | https://mathoverflow.net/users/14447 | 238717 | 110,251 |
https://mathoverflow.net/questions/238725 | 5 | It seems to be true that OEIS sequences A001222 and A257091 are closely related. First one is the number of prime divisors of n counted with multiplicity. The second one is the logarithms to the base 5 of the denominators of the Dirichlet series of zeta(s)^(1/5). Actually first terms of these sequences satisfy A257091(... | https://mathoverflow.net/users/91596 | The number of prime factors of a natural number | Robert Israel's guess is almost correct, actually for $n=\prod p\_j^{e\_j}$ we have
$$
A257091(n)−A001222(n)=\sum\_j \nu\_5(e\_j!)=\sum\_j \sum\_{m=1}^{\infty} \lfloor e\_j/5^m\rfloor
$$
(where $\nu\_5(N)$ denotes the largest $k$ for which $5^k$ divides $N$).
Indeed, expanding as binomial each Euler multiple in $\zet... | 11 | https://mathoverflow.net/users/4312 | 238728 | 110,255 |
https://mathoverflow.net/questions/238732 | 1 | Recently I came across a functional equation which always has a polynomial with integer coefficients solution.
Let
$$
L\_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod\_{i=1}^n(x+i).
$$
**Problem:**
For any positive integer $n$, there exists a polynomial $f(x) $with integer coefficients of order $n$ satisfying... | https://mathoverflow.net/users/6104 | Integer Polynomial solutions to functional equation | Note that $L(f):=(2x+1)^2f(x+1)-4x(x+n+1)f(x)$ is a linear operator which maps the $\mathbb{Q}$-linear space $\pi\_n$ of polynomials $h(x)\in \mathbb{Q}[x]$ of degree at most $n$ to itself. Assume that $L(f)=0$. Substituting $x=0$ we get $f(1)=0$, then substituting $x=1$ we get $f(2)=0$ and so on, thus $f\equiv 0$. So,... | 12 | https://mathoverflow.net/users/4312 | 238738 | 110,257 |
https://mathoverflow.net/questions/238740 | 12 | I would like to know whether there is some computer algebra software that can be used to verify if a group, given by a finite presentation, is hyperbolic (in the sense that it terminates with "yes" if the group is hyperbolic and otherwise might not terminate). I read that the kbmag software package for GAP could be use... | https://mathoverflow.net/users/15302 | Verify that a group is hyperbolic via computer algebra | The KBMAG package can be used to verify hyperbolicity of a group defined by a finite presentation. It does it by verifying that geodesic bigons in the Cayley graph are uniformly thin. Then a result of Paposoglu implies that the group is hyperbolic. So it proves hyperbolicity, but it does not provide any useful informat... | 12 | https://mathoverflow.net/users/35840 | 238746 | 110,258 |
https://mathoverflow.net/questions/238749 | 2 | In [this post](https://mathoverflow.net/questions/231202/pulled-back-foliation-is-completely-integrable) I already ask a similar question.
Assume $M$ is a symplectic manifold of dimension $2n$. Assume $G$ is a Liegroup, $\mathfrak{g}$ be the Liealgebra and $\mathfrak{g^\*}$ the corresponding dual. Endow the $\operato... | https://mathoverflow.net/users/75382 | multiplicity free actions - Guillemin&Sternbergy collective integrability | Maybe, I misunderstand your problem. After doing the math I get even an equality
$$
\dim M=2\dim G.x-\dim G.\Phi(x)
$$
for any coisotropic action. More precisely, the multiplicity freeness of $M$ is equivalent to
$$
M/G\to W/G
$$
being bijective (i.e., all symplectic reductions are points). This translates into
$$
\dim... | 2 | https://mathoverflow.net/users/89948 | 238756 | 110,262 |
https://mathoverflow.net/questions/238741 | 1 | Given a compact Riemann surface $M$ and two double coverings $\hat\pi\colon \hat M\to M$ and $\tilde\pi\colon \tilde M\to M$ which are branched over the same points $p\_1,..,p\_n\in M.$ As is well-known, the two coverings are related by an element in $H\_1(M,\mathbb Z\_2).$ Is there any (explicitely) known relationship... | https://mathoverflow.net/users/4572 | Jacobians of twisted coverings | They are not isogeneous in general. For an explicit example, take the case $n=0$ and $C$ hyperelliptic, so that we have a double covering $C\rightarrow \mathbb{P}^1$ branched along a subset $B$ of $\mathbb{P}^1$. Étale double coverings $\tilde{C}\rightarrow C $ correspond to partitions $B=B'\cup B''$, with $\#B'$ and $... | 3 | https://mathoverflow.net/users/40297 | 238763 | 110,264 |
https://mathoverflow.net/questions/238384 | 0 | As we all know, in RS code, when provide erasure (the position of error symbol), the decoding capacity of RS code is stronger. Specifically, $2e+v \leq (n-k)$, where $e$ is the number of errors, $v$ the number of erasures and $(n-k)$ the number of RS code characters (called nsym in the code) [1].
It has been seen in... | https://mathoverflow.net/users/88968 | How to analysis the relationship between the accuracy of erasure and the probability of decoding in RS code | Based on the comments below the original post, I am going to take a stab at rephrasing and answering the original question.
We wish to transmit symbols $$X\_1,...,X\_k,$$ where $X\_i$ lies in some finite field $\mathbb{F}$. We use a Reed-Solomon code to encode them as $$Y\_1,...,Y\_n,$$ with $n\geq k$. We receive obs... | 1 | https://mathoverflow.net/users/8938 | 238766 | 110,266 |
https://mathoverflow.net/questions/238733 | 7 | Does exists a proof of the Poincaré Duality version for non-compact manifolds without using the Zorn's Lemma? I know that there is a proof using the Whitney embedding theorem, but I don't know this theorem's proof, so I don't know if it uses the Zorn's Lemma too.
| https://mathoverflow.net/users/70864 | Poincaré Duality for non-compact manifolds without Zorn's Lemma | Often the definition of a manifold includes a condition to exclude strange manifolds that are "very large" such as the Prüfer surface. One condition commonly assumed is that the topology on the manifold is second countable. This is necessary for embedding the manifold in Euclidean space for example. Second countability... | 18 | https://mathoverflow.net/users/23571 | 238776 | 110,268 |
https://mathoverflow.net/questions/238629 | 8 | Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}\_{flat}(\Sigma)$ and $\overline {\mathcal A}\_{flat}(Y)$ denote the gauge equivalence classes of flat connections on the trivial bundles $\Sigma \times G$ and $Y \times... | https://mathoverflow.net/users/15197 | Flat connections on 3-manifold with boundary | Another good reference is Chris Herald's paper. Legendrian cobordism and Chern-Simons theory on 3-manifolds with boundary. Comm. Anal. Geom. 2 (1994), no. 3, 337–413.
It is an easy exercise with Poincare duality to see that the image of the map in cohomology (so at the level of "Zaraski" tangent spaces)
$$
H^1(Y;ad\... | 8 | https://mathoverflow.net/users/12605 | 238778 | 110,269 |
https://mathoverflow.net/questions/238764 | 0 | We have the unit sphere $S^2$ in $\mathbb{R}^3$ and two points, $X$ and $Y$ on the surface of the sphere. Then, a function is defined for any point $P$ inside of the unit ball as:
$$f(P) = R\,d(P, XY)$$
where $d(P,XY)$ is the Euclidean distance from the line containing $XY$, and $R$ is the radius of the small circl... | https://mathoverflow.net/users/91624 | Showing convexity of a function in the unit ball | Let $A$ be a midpoint of $XY$. We may restrict ourselves to the plane $\alpha$, which is perpendicular bisector to $XY$. Choose Cartesian and polar coordiantes in $\alpha$ centered in $A$, so that $O=(a,0)$ (Cartesian) and $P=(x,y)$ (Cartesian)$=(r,\varphi)$ (polar). Then $d(P,XY)=d(P,A)=r$, $R=\sqrt{1-d(O,PA)^2}=\sqrt... | 1 | https://mathoverflow.net/users/4312 | 238779 | 110,270 |
https://mathoverflow.net/questions/238669 | 7 | Let $p \ge 1$ be an integer. Does there exist a constant $C\_p$ such that for every random variable $X \ge 0$,
$$
\mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C\_p \mathbb{E} \left[ \left(X^p - \mathbb{E} \left[ X^p \right] \right)^{2} \right] \ \ ?
$$
| https://mathoverflow.net/users/56892 | Is this simple-looking moment inequality true? | The answer is yes. Let $Y$ be an independent random variable distributed as $X$. We have
$$
\|X - \mathbb{E}[X]\|\_{L^{2p}} = \|\mathbb{E}\left[X - Y \, | \, X\right]\|\_{L^{2p}} \le \|X - Y\|\_{L^{2p}}.
$$
Moreover, there exists a constant $C\_p$ such that for every $x,y \ge 0$,
$$
|x-y|^p \le C\_p |x^p - y^p|.
$$
Ind... | 6 | https://mathoverflow.net/users/56892 | 238780 | 110,271 |
https://mathoverflow.net/questions/238788 | 5 | Consider the diagonal action of the orthogonal group $O(n)$ on $\mathbb{R}^n\times\mathbb{R}^n$ defined as: $U\cdot (x,y) = (Ux,Uy)$ for $U\in O(n)$ and $x,y\in\mathbb{R}^n$. I am looking for a description of the algebra of polynomials in $\mathbb{R}[x,y]$ that are invariant under this action. That is, the subalgebra o... | https://mathoverflow.net/users/85652 | Invariant polynomials under diagonal action of the orthogonal group | The invariants are generated by the quadratic polynomials $(u,u)$, $(u,v)$, and $(v,v)$ where $(.,.)$ is the scalar product defining $O(n)$. This pattern generalizes to arbitrary many copies of $\mathbb R^n$. This is called the first fundamental theorem for the orthogonal group.
| 12 | https://mathoverflow.net/users/89948 | 238792 | 110,277 |
https://mathoverflow.net/questions/238786 | 1 | Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ?
$$F(x)= \sum\limits\_{q =1}^{\infty} \; \sum\limits\_{n =1}^{\infty} \; \sum\limits\_{\underset{(p,q)=1} {p=1}}^{q-1} f(n) \; e^{2 i \pi n (x+ \frac{p}{q}) }$$
First of all, the existence of the sum do... | https://mathoverflow.net/users/38290 | Is this function $\mathbb{Q}$ periodic? | Your sum diverges for most $x\in\mathbb{R}$. The innermost sum is
$$\sum\limits\_{\underset{(p,q)=1} {p=1}}^{q-1} f(n) \; e^{2 i \pi n (x+ \frac{p}{q}) } = f(n)e^{2 i \pi nx}\sum\limits\_{\underset{(p,q)=1} {p=1}}^{q-1}e^{2 i \pi n\frac{p}{q} } = f(n)e^{2 i \pi nx}\sum\_{d\mid (n,q)}d\mu\left(\frac{q}{d}\right),$$
by t... | 5 | https://mathoverflow.net/users/11919 | 238798 | 110,279 |
https://mathoverflow.net/questions/238790 | 3 | The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\text{sgn}(x)$ for $x \neq 0$. How to "correctly" define $f(0)$?
Here is the proposed solution in finite dimension. L... | https://mathoverflow.net/users/89064 | Extension of a function from almost everywhere to everywhere | You seem to be assuming that the RHS of Lebesgue differentiation theorem converges for every $x$.
Let $f:\mathbb{R}\to \mathbb{R}$ be the following function.
First divide $\mathbb{R}$ into dyadic intervals. More precisely, Let $D\_n = [-2^{n+1}, -2^n)\cup (2^{n}, 2^{n+1}]$. We have that $\mathbb{R} = \cup\_{n\in ... | 4 | https://mathoverflow.net/users/3948 | 238800 | 110,280 |
https://mathoverflow.net/questions/238807 | 5 | A tree $T$ on $\omega \times \lambda$ is weakly homogeneous if there is a countable set $\sigma$ of countably complete measures on ${}^{<\omega}\lambda$ so that $x \in p[T]$ if and only if there is a countably complete tower of measures $\bar{\mu} = \langle \mu\_i : i \in \omega \rangle$ so that each $T\_{x\upharpoonri... | https://mathoverflow.net/users/43354 | Martin-Solovay Tree of Weakly Homogeneous Tree under $\mathsf{AD}_\mathbb{R}$ | It is a theorem of $\text{ZF+DC}$ that if $T$ is a weakly homogeneous tree on $\omega\times \kappa$ some $\kappa$ with homogeneity system of measures $\vec{\mu}$ and $ms(T,\vec{\mu})$ is the Martin-Solovay tree associated to $T$ then $p[T]$ and $p[ms(T,\vec{\mu})]$ are complements. See Cabal reprints volume 1 Jackson's... | 3 | https://mathoverflow.net/users/3859 | 238814 | 110,287 |
https://mathoverflow.net/questions/238815 | 3 | Let me first fix some notations.
If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $$d(A,B)=\inf\{\|a-b\|:a\in A,b\in B\},$$$$\widehat{d}(A,B)=\sup\{d(a,B):a\in A\}.$$
Let $A$ be a bounded subset of a Banach space $X$. The Hausdorff measure of non-compactness of $A$ is defined by $$\chi(A)=\inf\{\... | https://mathoverflow.net/users/41619 | A question on the quantification of compact operators | A negative answer to your question follows from a well-known result of Schlumprecht (Israel J. Math 1991): there exist arbitrarily distortable Banach spaces. Schlumprecht's result implies that for every $\lambda>1$ there exist a Banach space $W$, a Banach space $Y$ isomorphic to $W$, and an isomorphism $T:W\to Y$ such ... | 3 | https://mathoverflow.net/users/37822 | 238827 | 110,289 |
https://mathoverflow.net/questions/238825 | 7 | Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of $\frac{i}{\pi}\partial \overline{\partial} \varphi$ coincides with the first Chern class $c\_1(L)$ of the line bundle.
... | https://mathoverflow.net/users/47692 | Chern classes and singular hermitian metrics on vector bundles | Given a complex hermitian vector bundle $E$ of complex rank n over a smooth manifold $M$, a representative of each Chern class $ck(E)$ of $E$ are given as the coefficients of the characteristic polynomial of the curvature form $Ω$ of $E$.
$$\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum\_k c\_k(E) t^k $$
Each C... | 7 | https://mathoverflow.net/users/nan | 238829 | 110,290 |
https://mathoverflow.net/questions/238824 | 2 | (**Preamble:** I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that *brilliant idea* can help with answering my questions at the end. If this is not appropriate for MO, please let me know and I will be more than happy to migrate t... | https://mathoverflow.net/users/10365 | On attempting a proof for $r > 1$, if $M = {2^r}{b^2}$ is an even almost perfect number which is not a power of two | Perhaps you might consider using the fact, pointed out in an answer to [one of your other questions](https://mathoverflow.net/questions/201002/does-there-exist-an-integer-that-is-both-solitary-and-almost-perfect?) that $\sigma(M)$ and $M$ are relatively prime, to impose further restrictions on $b$. Although this may no... | 2 | https://mathoverflow.net/users/85586 | 238830 | 110,291 |
https://mathoverflow.net/questions/238818 | 5 | **Introduction:** Let's assume we have a 2-form $\alpha=(1/2)\sum\_{j,k=1}^n a\_{jk}\ e\_j\wedge e\_k$, where $n=2m$, and $a\_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\alpha\ (m\ {\rm times}) = n!\ ({\rm Pf}\ A)\cdot e\_1\wedge\dots\wedge e\_n$, where ${\rm Pf}\ A$ is the Pfaffian... | https://mathoverflow.net/users/91644 | Pfaffian of several skew-linear transformations / matrices | For two 2-forms I can offer you a blueprint of an algorithm. Details would have to be filled in.
Consider two skewsymmetric $n\times n$-matrices $A$ and $B$. The task is to compute $\operatorname{Pf}(sA+tB)$ since this gives all $\alpha^p\wedge\beta^{m-p}$ simultaneously.
Let's assume first that $A$ is invertible. ... | 1 | https://mathoverflow.net/users/89948 | 238854 | 110,299 |
https://mathoverflow.net/questions/237768 | 2 | Let $1<p<\infty$. The James $p$-space $J\_{p}$ is the Banach space of all sequences of real numbers $(a\_{i})\_{i}\in c\_{0}$ such that
$$\|(a\_{i})\_{i}\|=\sup\{(\sum\_{j=1}^{n}|a\_{p\_{j-1}}-a\_{p\_{j}}|^{p})^{\frac{1}{p}}:p\_{0}<p\_{1}<\cdots<p\_{n}, n\in\mathbb{N}\}<\infty.$$
Let $(e\_{n})\_{n}$ be the unit vector... | https://mathoverflow.net/users/41619 | Two questions on the James $p$-space $J_{p}(1<p<\infty)$ | I think that this question should be considered as answered, following the comments of Bill Johnson as:
(1) One can follow the argument of Andrew and show that $z\_n^\*$ is equivalent to the unit vector basis of $\ell \_q$, where $q$ is conjugate to $p$.
(2) The well-known result of Rosenthal implies that each boun... | 0 | https://mathoverflow.net/users/85406 | 238864 | 110,303 |
https://mathoverflow.net/questions/238744 | 3 | I am still a newbie to $\Psi$DO-Operators. As far as i understood, one can easily compute the square root of the Laplace operator $\Delta$ by
$$(-\Delta)^{1/2} \ u=\mathcal{F}^{-1}(\|\xi\| \widehat{u}).$$
However if i want to compute the spare root of $(-c(x) \ \Delta)$ things get complicated (Lets assume $c(x)$ is... | https://mathoverflow.net/users/91608 | The square root of Laplacian with nonconstant coefficent | Note that your operator is not positive for the $L^2$ product. A better starting point might be $-Au=\nabla\cdot(c^2(x)\nabla u)$ which satisfies at least $(Au,u)\ge0$. That said, it depends how explicit you want your square root. For a selfadjoint operator $A$ with spectral measure $dE\_\lambda$ you have simply $A^{1/... | 1 | https://mathoverflow.net/users/7294 | 238872 | 110,307 |
https://mathoverflow.net/questions/238879 | 0 | My question pertains to the paper ["A Simplified Proof of the Divergence Theorem"](http://docdro.id/E0I6BtA) by Djairo Guedes de Figueiredo.
It's not a big question, actually, but it's confusing me a lot: In the statement of Lemma 2 in the paper, the author writes "$h(x') < x\_1 < 1$ and $-1 < x' < 1$", where $x'=(x\... | https://mathoverflow.net/users/89456 | Comparing vectors with numbers? | The statement $-1 < x' < 1$ means that every coordinate of $x'$ is between $-1$ and $1$. (The set of all such $x'$ is a cube.)
| 2 | https://mathoverflow.net/users/68305 | 238880 | 110,308 |
https://mathoverflow.net/questions/238901 | 3 | In some recent computation I came across certain cubic forms and was wondering about analogue of following result for quadratic forms.
If $k^\*/(k^\*)^2$ is finite then there are only finitely many quadratic forms up to equivalence. This can be easily seen because of diagonalisation.
I have been wondering if we can... | https://mathoverflow.net/users/69977 | cubic forms and finiteness of $k^*/(k^*)^3$ | To expand a bit on the comments. For simplicity, let's assume $k$ is characteristic $0$ (or at least, not characteristic 2 or 3). Let $f(X,Y)=aX^3+bX^2Y+cXY^2+dY^3$ be a cubic form. The case where $f$ is reducible comes down to classifying quadratic forms and linear forms, so let's assume that $f$ is irreducible. More ... | 3 | https://mathoverflow.net/users/11926 | 238916 | 110,318 |
https://mathoverflow.net/questions/238915 | 4 | I asked [this question at MSE](https://math.stackexchange.com/questions/1772552/some-examples-of-clean-topological-spaces) but I did not received any answer, so I repeat it here at MO:
What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$ of all real (or complex) valued c... | https://mathoverflow.net/users/36688 | Some examples of clean topological spaces | Well, the two point set will do. Also an infinite discrete set will do.
To give a less obvious example, let me argue that every compact space with a basis of clopen sets will do too.
Let $X$ be such and consider a real valued function $f$ on $X$.
Consider the compact subset $\{f \leq 1/3\}$ and cover it with clopen s... | 5 | https://mathoverflow.net/users/89334 | 238918 | 110,319 |
https://mathoverflow.net/questions/238924 | -5 | There is a notion of equivalence of categories which is the functor $F:\mathcal{C} \to \mathcal{D}$ such that there is a functor $G:\mathcal{D} \to \mathcal{C}$ such that $FG \cong id\_{\mathcal{D}}$ and $GF \cong id\_{\mathcal{C}}$ where $\cong$ means natural isomorphism (natural transformation with each component bei... | https://mathoverflow.net/users/24078 | Equivalence of categeories-variants of definition | Let $D$ be the category whose only object is $\mathbb{N}$ and the morphisms are all functions $\mathbb{N} \to \mathbb{N}$. Let $C$ be the subcategory whose morphisms are bijections $\mathbb{N} \to \mathbb{N}$.
The inclusion $F : C \to D$ has the properties you ask for.
| 7 | https://mathoverflow.net/users/1176 | 238925 | 110,322 |
https://mathoverflow.net/questions/238922 | 1 | In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):
Let $g$ be a positive multiplicative function and let $A$ and $B$ be two constants such that for all $y\geq 1,$
$$\sum\_{p\leq y} g(p) \log{p} \leq Ay \quad \textrm{ and ... | https://mathoverflow.net/users/76102 | Implicit constant in Tenenbaum's result | In the French edition, it is said that the considered constant does not exceed $4(1+9\lambda\_{1}+\lambda\_{1}\lambda\_{2}/(2-\lambda\_{2})^2)$ where $\lambda\_{1}>0$, $0\leq \lambda\_{2}<2$ are such that $g(p^\nu)\leq\lambda\_{1}\lambda\_{2}^{\nu-1}$.
| 5 | https://mathoverflow.net/users/13625 | 238930 | 110,325 |
https://mathoverflow.net/questions/237717 | 1 | A number $n \in \mathbb{N}$ is said to be [*superperfect*](http://mathworld.wolfram.com/SuperperfectNumber.html) if
$$\sigma(\sigma(n)) = 2n.$$
A number $m \in \mathbb{N}$ is said to be [*almost perfect*](http://mathworld.wolfram.com/AlmostPerfectNumber.html) if $$\sigma(m) = 2m - 1.$$
Here is my question:
>
> ... | https://mathoverflow.net/users/10365 | Is it possible to have an even superperfect number and an odd superperfect number whose product is an almost perfect number? | Guy's Unsolved Problems in Number Theory states that it is a result of Suryanarayana and Kanold that the even superperfect numbers are precisely the numbers $2^{n-1}$ where $2^n-1$ is a Mersenne prime. So let $r=n-1$. Since even almost perfect numbers which have an odd factor greater than one have the form $2^rb^2$ (wi... | 2 | https://mathoverflow.net/users/85586 | 238934 | 110,328 |
https://mathoverflow.net/questions/238920 | 17 | Following combinatorics problem is claimed to be an open problem in "[The Princeton Companion to Mathematics](https://books.google.com/books/princeton?hl=en&q=%22Will%20there%20always%20be%20a%20positive%20integer%22&vid=ISBN9781400830398&btnG.x=0&btnG.y=0&btnG=Search%20This%20Book#v=snippet&q=%22Will%20there%20always%... | https://mathoverflow.net/users/90056 | Combinatorics problem about sum of natural numbers | If the sequence $a\_1, a\_2,\dots$ has the property that each integer can be written in at most $g$ ways (counting order and allowing repetition), then we call the set $\{a\_1,a\_2,\ldots\}$ a $B^\ast[g]$ set. I wrote an extensive bibliography about 10 years ago, and published it in the [Electronic Journal of Combinato... | 11 | https://mathoverflow.net/users/935 | 238949 | 110,334 |
https://mathoverflow.net/questions/238948 | 1 | Let $(f\_1, f\_2, \ldots, f\_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative integers to non-negative integers such that:
$$[f\_1(a\_1) + f\_2(a\_2) + \cdots + f\_n(a\_n)] \bmod{m}= f[(a\_1 ... | https://mathoverflow.net/users/75293 | Linear functions | You can view $f$ and all the $f\_i$ as functions $\mathbb{N}\cup\{0\}\to\mathbb{Z}/m$.
Do so.
For $n=1$ every $f\_1$ will do. I am assuming now $n\geq 2$.
Note that replacing each $f\_i$ by $f\_i-f\_i(0)$ and $f$ by $f-\sum f\_i(0)$ will not change the desired property, so assume $f\_i(0)=0$.
Choosing $a\_i=x$, $a... | 3 | https://mathoverflow.net/users/89334 | 238960 | 110,337 |
https://mathoverflow.net/questions/238940 | 5 | If an operator $A$ on a Hilbert space $H$ generates a strongly continuous semigroup, does then the operator $B$ on $H \oplus H$ given by the matrix
$$ B := \begin{pmatrix} 0 & \mathrm{id} \\ A & 0\end{pmatrix}$$
generate a semigroup as well? This would then yield a solution to the wave equation $u^{\prime\prime} = A u$... | https://mathoverflow.net/users/16702 | Abstract Wave Equation and Semigroups | You need two additional assumptions: the operator $A$ has to be a so-called cosine function generator, and your product space has to be $V\times H$ with a space $V\subset H$.
Cosine function generator is more than sectorial, it is more or less when the numerical range of $A$ is in a parabola (selfadjoint negative de... | 4 | https://mathoverflow.net/users/12898 | 238963 | 110,339 |
https://mathoverflow.net/questions/238961 | 1 | Does anyone know of any PROOF WITHOUT WORDS for logarithmic functions?
The only one I've seen in calculus based and I need one for high school math kids in MATH 1,2,3.
Any suggestions would be appreciated!
| https://mathoverflow.net/users/91718 | proof without words for logarithms | Here are some:
<http://www.mat.uniroma2.it/~liverani/Inform04/Logarithm.pdf> (Proof without words: ln(a · b) = ln a + ln b)
<http://www.maa.org/sites/default/files/Ferlini-MaMag-2001.pdf> (Proof Without Words: Logarithm of a Number and Its Reciprocal).
<https://www.jstor.org/stable/2690586> (Proof without Words: ... | 2 | https://mathoverflow.net/users/32389 | 238964 | 110,340 |
https://mathoverflow.net/questions/238974 | 4 | It is well known that the problem of finding a maximal acyclic subgraph of a digraph is NP-complete.
Is this the case also when the digraph is symmetric ,i.e. if $(a,b)$ is a link, then $(b,a)$ is also a link?
| https://mathoverflow.net/users/91728 | Maximal acyclic subgraph | Arbitrarily label the vertices $1,\ldots,n$. Choose all edges $a\to b$ such that $b\gt a$. This is acyclic and exactly half the directed edges in the graph, which is obviously the best possible.
If the edges are weighted then it is again NP-complete, since the weights of $a\to b$ and $b\to a$ can be chosen with suffi... | 7 | https://mathoverflow.net/users/9025 | 238977 | 110,342 |
https://mathoverflow.net/questions/238971 | 5 | According to the article *[Second-order and Higher-order Logic](http://plato.stanford.edu/entries/logic-higher-order/)* from the [Stanford Encyclopedia of Philosophy](http://plato.stanford.edu/),
>
> there is no need to stop at second-order logic; one can keep going. [...] we can allow quantification over super-pre... | https://mathoverflow.net/users/91516 | Expressive power of $\omega$-order logic | Instead of an example, I give an existence proof:
Take any finite or countable language, for example the language of equality. Since all formulas (even in $\omega$-logic) are finite, there are only countably many formulas, hence at most $\mathfrak c:= 2^{\aleph\_0}$ many theories. Find more than continuum many cardi... | 8 | https://mathoverflow.net/users/14915 | 238979 | 110,343 |
https://mathoverflow.net/questions/238975 | 4 | [Pemantle 1989](https://www.math.upenn.edu/~pemantle/papers/overhand2.pdf) proves, among other things, that the Markov chain on $S\_n$ induced by repeatedly and independently performing an [overhand shuffle](https://en.wikipedia.org/wiki/Shuffling#Overhand_shuffle) on a deck of $n$ cards is ergodic and has limiting dis... | https://mathoverflow.net/users/41669 | Why does the overhand shuffle converge to the uniform distribution on $S_n$? | Shuffles like the overhand shuffle or riffle shuffle are not just random walks, they are symmetric random walks because you apply a random permutation drawn from the same distribution no matter what the initial configuration is. You can apply a shuffle with the cards face down. (That's not the case for non-shuffle rand... | 3 | https://mathoverflow.net/users/2954 | 238980 | 110,344 |
https://mathoverflow.net/questions/238773 | 9 | I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me:
let $E$ be an elliptic curve over $\mathbb{Q}$, $\ell$ a prime number, and consider $\rho=\rho\_{E,\ell}$ the Galois representati... | https://mathoverflow.net/users/57609 | Neron models and ramification | This result (and much more) is in Exposé IX *Modèles de Néron et monodromie* by A.Grothendieck (in SGA7), and more precisely in section 11.1. In particular what you want is exactly Proposition 11.2 thereof.
The proof is not hard but it also far from being obvious (to me, anyway). Let me recall it briefly. A separate... | 9 | https://mathoverflow.net/users/2284 | 238982 | 110,346 |
https://mathoverflow.net/questions/238877 | 5 | Let $\pi\_n$ be the poset of all set partitions of $\{1,...,n\}$ ordered by refinement, $\sigma = \{B\_1,...,B\_k\}$ be a set partition with blocks $B\_i$, and $\max(B\_i)$ be the maximum value in the block $B\_i$. I'm trying to prove the following:
$$\Sigma\_{\sigma\in\pi\_{n-1}}\Pi\_{B\_i\in\sigma}(-1)^{|B\_i|-1}(|... | https://mathoverflow.net/users/91671 | Combinatorics: set partitions of a poset | Let us prove even more general formula that Douglas Zare's one from the comments. Let $a\_1>\dots>a\_n$ be real numbers. Then
$$
\sum\_{\sigma\in\pi\_{n}}
\prod\_{B\_i\in\sigma}(-1)^{|B\_i|-1}(|B\_i|-1)!a\_{\min B\_i}
=\prod\_{i=1}^n(a\_i-i+1).
$$
(The required formula follows by setting $a\_i=2(n+1)-i$ and shifting... | 6 | https://mathoverflow.net/users/17581 | 238984 | 110,347 |
https://mathoverflow.net/questions/238986 | 0 | So I've been doing some self study on Modal logic and I would like some external input on how to present my proofs for some of the axioms
1) say for example I am told to prove that □phi implies ♢psi is valid in serial frames. Would I be correct in creating an arbitrary world and stating that phi is necessarily true i... | https://mathoverflow.net/users/91732 | Proving satisfiability in modal logic | A relation $\newcommand\toward\rightharpoonup\toward$ is *serial*, if $\forall x\exists y\ x\toward y$.
If the accessibility relation of a Kripke model is serial, then it is clear that $\square\varphi\to\Diamond\varphi$, since if $\square\varphi$ holds at a world $x$, then by seriality there is a world $y$ that $x$ ... | 0 | https://mathoverflow.net/users/1946 | 238992 | 110,352 |
https://mathoverflow.net/questions/239000 | 4 | Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$.
On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes
algébriques lineaires sur um corps de nombres", it is claimed that there exists a collection of finite field extensions $k \subset k\_i$ ... | https://mathoverflow.net/users/5101 | Adjoint semi-simple algebraic groups over non-algebraically closed fields | See Proposition 6.4.4 and Remark 6.4.5 in Brian Conrad's article "Reductive groups schemes" in "Autour des schemas en groupes, Vol. I" (alternatively, <http://math.stanford.edu/~conrad/papers/luminysga3smf.pdf>) for a proof (in a more general setting).
As you mention, this is special for adjoint groups (but would al... | 6 | https://mathoverflow.net/users/5498 | 239003 | 110,356 |
https://mathoverflow.net/questions/238998 | 1 | There's something I am missing comparing Descartes' theorem for three isometric circles [here](https://en.wikipedia.org/wiki/Descartes%27_theorem) and this wiki post on circle packing of 3 circles [here](http://mathworld.wolfram.com/CirclePacking.html).
From my calculation:
$$
r\_{ext} = \frac{r\_{int}}{{(3\pm2\sqr... | https://mathoverflow.net/users/91736 | Descartes' theorem and Circle Packing | My bad. I misunderstood the second article:
$$
1+2\frac{\sqrt{(3)}}{3}
$$
is the coefficient in a linear expression. Both the solutions are the same, you can derive (2) from (1) using square difference formula.
| 1 | https://mathoverflow.net/users/91736 | 239011 | 110,359 |
https://mathoverflow.net/questions/239002 | 4 | I know I met the following construction somewhere, but I cannot remember where. Let $A$ be
a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define multiplication by
\begin{equation\*}
(a,m)(b,n) := (ab,\,an+mb)~.
\end{equation\*}
The set $A\times N$ equipped with thi... | https://mathoverflow.net/users/56721 | Attaching an ideal whose square is zero: does this operation have a name and a notation? | In the representation theory of finite dimensional algebras, at least, it's called a "trivial extension algebra" (although that sometimes refers to the special case where $N$ is the vector space dual of $A$). Googling "trivial extension algebra" will find you several references.
| 2 | https://mathoverflow.net/users/22989 | 239015 | 110,361 |
https://mathoverflow.net/questions/239013 | 12 | Let $p\_1, \ldots, p\_n$ be a finite sequence of nonconstant polynomials with integer coefficients. Does there exist a finite sequence of integers $x\_1, \ldots, x\_n$ such that the integers $p\_1(x\_1), \ldots, p\_n(x\_n)$ have a nontrivial common divisor?
| https://mathoverflow.net/users/66833 | Noncoprime polynomial values | Here's an argument that might be a bit too high-brow (it uses Chebotarev's density theorem).
**Reduction 1.** Your question is equivalent to: for $f\_1, \ldots, f\_n \in \mathbb Z[x]$, does there exist a prime $p$ such that all $f\_i$ have a root in $\mathbb F\_p$. Indeed, taking lifts $x\_i \in \mathbb Z$ of roots $... | 20 | https://mathoverflow.net/users/82179 | 239018 | 110,363 |
https://mathoverflow.net/questions/238910 | 2 | Let $H$ be a bialgebra and $A$ a $H$-module algebra. The cross product $A \rtimes H$ is defined as follows. As a vector space $A \rtimes H = A \otimes H$. The multiplication on $A \rtimes H$ is defined as follows:
\begin{align}
(a \otimes h)(b \otimes g) = \sum a(h\_{(1)}.b) \otimes h\_{(2)}g,
\end{align}
where $a, b \... | https://mathoverflow.net/users/11877 | Is the cross product $A \rtimes H$ a bialgebra? | In the case that $A$ is a braided Hopf algebra in $\_H^H YD$ you can form $A \# H$ with the multiplication you already wrote down. The coproduct uses the fact that this kind of $A$ also has a left $H$ coaction as well given by $\delta r^{(2)} = r^{(2)}\_{(-1)} \otimes r^{(2)}\_{(0)}$
$$
\Delta ( r \# h) = ( r^{(1)} \... | 2 | https://mathoverflow.net/users/69850 | 239019 | 110,364 |
https://mathoverflow.net/questions/238976 | 4 | Let us consider a smooth (complex) cubic surface $X \subset \mathbb{P}^3$ and a general point $p \notin X$. Then it is classically well-known that linear the projection $$\pi\_p \colon X \longrightarrow \mathbb{P}^2$$ is a triple cover whose branch locus $B$ is a sextic plane curve with six cusps lying on a conic. Conv... | https://mathoverflow.net/users/7460 | Triple covers of $\mathbb{P}^2$ with fixed branch locus | A sextic with 6 cusps on a conic has a unique (up to obvious equivalence) *torus structure*, i.e., a representation of the equation in the form $f\_2^3+f\_3^2=0$, where $\deg f\_i=i$. (Note $\{f\_2=0\}$ is necessarily the conic and, in fact, the six cusps are the intersection of the two curves $\{f\_i=0\}$.) Then, it i... | 4 | https://mathoverflow.net/users/44953 | 239021 | 110,366 |
https://mathoverflow.net/questions/239034 | 2 | Let $M$ be a connected orientable **open** 4-manifold (noncompact, without boundary).
1. Is it possible for $M$ to be non-parallelizable ?
If yes, what example of such $M$ is there ?
[**EDIT** : The answer to this is yes : see the answer of Danny Ruberman]
2. Suppose now that $M$ admits a lorentzian metric **an... | https://mathoverflow.net/users/89425 | Open non-parallelizable 4-manifolds | Yes to your first question; the Stiefel-Whitney classes obstruct parallelizability, even for open manifolds. So for instance a non-orientable manifold (eg a Mobius band cross R^2) is not parallelizable. An oriented example would be $CP^2$ minus a point, which has nonzero $w\_2$.
I think you may have interpreted that ... | 8 | https://mathoverflow.net/users/3460 | 239036 | 110,370 |
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