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https://mathoverflow.net/questions/141596 | 7 | Let $D(s)$ be a Dirichlet series with abscissa of convergence $\sigma\_c=\sigma\_a$. Does it follow that the Dirichlet series defined by $P(s)=D(s)\bar D(\bar s)$ has the same abscissa of convergence?
In light of Noam D. Elkies answer below, I would like to know about conditions on the coefficients that also lead to ... | https://mathoverflow.net/users/10980 | Abscissa of convergence of Dirichlet series | That's not true even for power series with real coefficients.
Let
$$
D(s) = \sqrt{1-2^{-s}}
= 1 - \frac12 2^{-s} - \frac18 4^{-s} - \frac1{16} 8^{-s} - \frac5{128} 16^{-s}
- \cdots .
$$
Then $P(s) = 1 - 2^{-s}$, so
$\sigma\_c(D) = \sigma\_a(D) = 0$ but
$\sigma\_c(P) = \sigma\_a(P) = -\infty$.
| 11 | https://mathoverflow.net/users/14830 | 141628 | 77,234 |
https://mathoverflow.net/questions/141626 | 2 | I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of course, the question is interesting only for $n\leq 2m,$ since every map can be approximated by an immersion for higher $n$.... | https://mathoverflow.net/users/23935 | Understanding maps from M to R^n, for n>dim M | For example if $m=2$ and $n=3$ then generic maps will have finite number of Whitney umbrella points. In this case generic maps are stable, i.e. stable maps form an open dense subset in
$C^\infty(M,R^n).$
Similar statements can be formulated for higher dimensions until we are in the so called nice dimensions of Mather... | 1 | https://mathoverflow.net/users/36950 | 141630 | 77,236 |
https://mathoverflow.net/questions/141620 | 12 | This is my first post here :)
I have the following two related questions. While looking in Conway's ATLAS (the 1985 one) for $SL\_3(Z/pZ)$, where he does the cases $p=2,3,5,7$, I saw that the dimension of the biggest irreducible representation is ~$p^3$. More precise, as the order of the group is $p^3(p^3-1)(p^2-1)$,... | https://mathoverflow.net/users/39717 | Dimensions and number of complex irreducible representations for SL3(Z/pZ) | To add another viewpoint to what Paul has already said, it's important to realize that a lot of general theory exists by now for these particular groups and for others of Lie type (especially the work inspired by Lusztig). The Atlas authors relied on only some of this work, but started with older literature on special ... | 10 | https://mathoverflow.net/users/4231 | 141635 | 77,238 |
https://mathoverflow.net/questions/141639 | 0 | In a cartesian closed category, the exponential object [A,B] basically internalizes the collection of morphisms from A to B.
Is there some similar notion that internalizes the isomorphisms between A and B? What about the unique isomorphisms?
It seems like you could get somewhere with subobject classifiers, but mayb... | https://mathoverflow.net/users/39732 | What does an object of isomorphisms look like? | There is a map $[A, B] \times [B, A] \to [A, A] \times [B, B]$ given by composition in each of the possible directions. There is also a map $1 \to [A, A] \times [B, B]$ which picks out $(\text{id}\_A, \text{id}\_B)$. These fit together into a diagram $[A, B] \times [B, A] \to [A, A] \times [B, B] \leftarrow 1$ and the ... | 11 | https://mathoverflow.net/users/290 | 141641 | 77,240 |
https://mathoverflow.net/questions/141500 | 4 | Let $G$ be a connected, simply-connected complex semisimple Lie group, and let $P\subseteq G$ be a parabolic subgroup. Suppose that $V$ is a $1$-dimensional complex $P$-representation and consider the associated complex line bundle $G\times\_P V\rightarrow G/P$. Let $X\subseteq G\times\_P V$ denote the complement of th... | https://mathoverflow.net/users/25358 | The Gysin Sequence for an Associated Bundle over a Partial Flag Variety | Here is my comment as an answer.
An excellent reference / example for these ideas is Juteau's paper "Cohomology of the minimal nilpotent orbit" (<http://arxiv.org/abs/0704.3417>). The minimal nilpotent orbit can be described as the the complement of the zero section in a line bundle on $G/P$, where $P$ is the standar... | 3 | https://mathoverflow.net/users/919 | 141643 | 77,242 |
https://mathoverflow.net/questions/141651 | 9 | What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
| https://mathoverflow.net/users/34859 | Largest convex hull of a unit length path | The answer seems to be $\frac{1}{2\pi}$, using a semi circle. See
[Moran, P. A. P. "On a problem of S. Ulam." Journal of the London Mathematical Society 1.3 (1946): 175-179.](http://jlms.oxfordjournals.org/content/s1-21/3/175.full.pdf)
| 16 | https://mathoverflow.net/users/39495 | 141653 | 77,244 |
https://mathoverflow.net/questions/141650 | 9 | Let $\alpha:\mathbb R\to U(H)$ be a strongly continuous action of the reals on some Hilbert space, and let $A=-i\frac d{dt}\alpha(t)|\_{t=0}$ be its infinitesimal generator, so that $\alpha(t)=e^{itA}$. Finally, let $D\subset H$ be the domain of $A$.
A subset $D\_0\subset D$ is called a core of $A$ if the closure of ... | https://mathoverflow.net/users/5690 | why is this a sufficient condition for a domain to be a core of an unbounded operator? | This is stated and proved as Proposition 2.20 here:
<https://isem-mathematik.uibk.ac.at/isemwiki/images/9/94/ISEM15_Lecture2.pdf>
| 8 | https://mathoverflow.net/users/12120 | 141655 | 77,245 |
https://mathoverflow.net/questions/141662 | 4 | Deligne has a theorem in "Theorie de Hodge II" as follows:
>
> Let $S$ be a smooth separated scheme, and $f:X\to S$ be a smooth proper morphism.
> Let $\bar{X}$ be a non singular compactification of $X$. Then the canonical morphism :
> $$
> H^n(\bar{X},\mathbf{Q)}\to H^0(S,\mathbf{R}^nf\_\* \mathbf{Q})
> $$
> i... | https://mathoverflow.net/users/4504 | global section of local system from direct image | Yes, this is still true, if we assume that $E$ is the underlying local system of a polarized variation of Hodge structure on $\overline X$, which takes care of most local system of "algebro-geometric origin".
Deligne's result comes from a combination of the following three results:
1. The Leray spectral sequence fo... | 9 | https://mathoverflow.net/users/1310 | 141665 | 77,253 |
https://mathoverflow.net/questions/141503 | 7 | For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself.
For each finite measurable partition $\mathcal{A}=\{A\_i\}\_{i=1}^{m}$ of $X$, we can define
$h(\mathcal{A},T,m)$ as $\lim\_{n\rightarrow \infty}\frac{1}{n}H(\bigvee\_{i=0}^{n-1}T^{-i}\mathcal{A})$.
and measurable h(T,m) is ... | https://mathoverflow.net/users/11966 | Intuition of Kolmogorov-Sinai entropy | A good way to understand measurable entropy is via the Shannon-McMillan-Breiman Theorem. Roughly speaking it says that there is a constant $c$ so that most atoms $A$ in $\bigvee\_{i=0}^{n-1} T^{-i}\mathcal{A}$ have measure $m(A)\approx e^{-cn}$, and the value of $c$ is the measure entropy $h(\mathcal{A},T,m)$. More pre... | 9 | https://mathoverflow.net/users/8112 | 141669 | 77,254 |
https://mathoverflow.net/questions/141673 | 1 | Consider an $n \times n \times n \times\dots\times n$ torus board of total size $n^k$ with $n > 4$ either even or odd.
Consider the basic cube of size $1 \times 1 \times \dots \times 1$ at a lattice point on the board and label the vertices from $\Bbb F\_2$ with the vertices of the cube taking value $1$ and rest of t... | https://mathoverflow.net/users/10035 | Linear combinations of basic cubes on a torus board | Note that the set of available configurations is in fact a $\mathbb{F}\_2$-vector space.
In one dimension (that is, for $k=1$), it's easy to see that that vector space has dimension $n-1$: you can get all but one of the lattice points to be whatever you want, and then the last one is forced to be the sum of the ones ... | 3 | https://mathoverflow.net/users/14901 | 141678 | 77,259 |
https://mathoverflow.net/questions/141647 | 1 | John Milnor proofed the existence of a Denjoy counterexample in <http://www.math.sunysb.edu/~jack/DYNOTES/dn15.pdf> page 15-4.
I could follow his arguments until "It is now reasonably straightforward to check that the map f [...] is $C^1$-smooth with derivative equal to +1 [...]."
I tried calculating the Difference q... | https://mathoverflow.net/users/39737 | Construction of a Denjoy Counterexample | I think the idea is that you choose a point $x$ and a really small interval $[x,x+h]$ or $[x-h,x]$ on one side of it. This interval will be partitioned into some $I\_{n\_i}$'s (these are dense in the circle) and some other stuff. There may also be a piece of some $I\_k$. The length of the other stuff won't change becau... | 1 | https://mathoverflow.net/users/11054 | 141681 | 77,260 |
https://mathoverflow.net/questions/141683 | 0 | Considering pure QR algorithm (without shifts and preliminary tridiagonal reduction) are there sufficient conditions for algorithm to converge to quasi-diagonal form?
For the the following matrix
$$
A = \left(\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right).
$$
with eigenvalues $\lambda\_1 = 1$ and $\lambda\_2 =... | https://mathoverflow.net/users/39752 | QR alogrithm for eigenvalue problem | This example is explained in section 11.5 [here](http://www.math.iit.edu/~fass/477577_Chapter_11.pdf).
| 2 | https://mathoverflow.net/users/39754 | 141684 | 77,261 |
https://mathoverflow.net/questions/141613 | 8 | Let $G$ be a reductive algebraic group defined over an algebraically closed field $k$ of characteristic p, let assume p is good prime for simplicity. Fix $B$ a Borel subgroup of $G$. Then for every $B$-variety $X$, we can define an associated bundle $G\times^B X$. Suppose $X$ has the dualizing sheaf $\omega\_X$. My que... | https://mathoverflow.net/users/39715 | Dualizing sheaf of an associated bundle | Yes, there is a formula for this due to Brion (it's Lemma 2 in his paper Multiplicity-Free Subvarieties of Flag Varieties). First, for any $B$-equivariant coherent sheaf $\mathcal F$ on $X$, there is a natural $G$-equivariant coherent sheaf $ G \times^B \mathcal F $ on $G \times^B X$. This assignment in fact is an equi... | 6 | https://mathoverflow.net/users/1528 | 141687 | 77,263 |
https://mathoverflow.net/questions/141660 | 8 | Three question concerninng metrics on the real line:
Is there a metric $d$ on $\Bbb{R}$ such that a function $f : (\Bbb{R},d) \longrightarrow (\Bbb{R},d)$ ( or $f : \Bbb{R} \longrightarrow (\Bbb{R},d)$ or $f : (\Bbb{R},d) \longrightarrow \Bbb{R}$) is continuous if and only if $f : \Bbb{R} \longrightarrow \Bbb{R}$ is ... | https://mathoverflow.net/users/nan | Continuous functions as uniformly continuous function | EDIT: The answer now applies to arbitrary topologies, using an idea by Pietro Majer from the comments.
**Proposition:** There are no topologies $\tau\_0,\tau\_1$ on $\mathbb R$ such that $f\colon\mathbb R\to\mathbb R$ is uniformly continuous in the Euclidean metric iff $f\colon(\mathbb R,\tau\_0)\to(\mathbb R,\tau\_1... | 9 | https://mathoverflow.net/users/12705 | 141691 | 77,264 |
https://mathoverflow.net/questions/129986 | 5 | Let $R = \mathbb{Z}[x\_{1}, \dots, x\_{r}]$.
Let $X$ be $n \times n$ matrix with entries in $R$.
Let $Y$ be $m \times m$ matrix with entries in $R$ formed from $\mathbb{Z}$-linear or $\mathbb{R}$-linear combinations of entries in $X$.
Let $m \ge n$ and $r \ge n^{2}$.
Do there always exist $A$ and $B$ such that $AXB ... | https://mathoverflow.net/users/10035 | Solve for $A$ and $B$ in $AXB=Y$ | This is not possible. Let each entry of $X$ be a distinct monomial. Then we can write each entry of $Y$ as a $\mathbb Z$-linear combination of these $n^2$ monomials, then the set of possible $Y$ can be seen as $(\mathbb Z^{n^2})^{m^2}= \mathbb Z^{n^2m^2}$. The possible values of $A$ and $B$ are both $\mathbb Z^{nm}$, s... | 0 | https://mathoverflow.net/users/18060 | 141697 | 77,267 |
https://mathoverflow.net/questions/141689 | 6 | Let $\cal C$ be a model category which is also additive. Suppose that the homotopy category $\operatorname{Ho}\mathcal C$ is additive, for example this is true when the weak equivalences in $\cal C$ is closed under biproducts (see [this question](https://mathoverflow.net/questions/44047/localizing-an-arbitrary-additive... | https://mathoverflow.net/users/38418 | A model category which is an additive category | If by "additive" you mean an $\mathbf{Ab}$-enriched category with a zero object and biproducts, then yes. Let $\mathcal{M}$ be model category that is additive in this sense, let $\mathcal{M}\_c$ be the full subcategory of cofibrant objects, let $\mathcal{M}\_f$ be the full subcategory of fibrant objects, and let $\math... | 4 | https://mathoverflow.net/users/11640 | 141699 | 77,269 |
https://mathoverflow.net/questions/141679 | 2 | Let
$$I(x) = \frac{\sigma(x)}{x}$$
be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example,
$$\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.$$
My question is this: What proportion of the positive integers satisfy the inequality
$$I(n) < \frac{2... | https://mathoverflow.net/users/10365 | What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2) < 2$? | The function $I(n^2)$ has a continuous limiting distribution (look up the Erd\H{o}s--Wintner theorem). Since your lower limit $2n/(n+1)$ converges to your upper limit of $2$ as $n\to\infty$, the continuity of the distribution function shows that the limiting proportion of $n$ satisfying your inequality is $0$.
| 5 | https://mathoverflow.net/users/16510 | 141709 | 77,272 |
https://mathoverflow.net/questions/141368 | 13 | Consider a sequence of independent uniform $[0,1]$ random variables, and for nonnegative real $t$, let $m(t)$ be the expected number of terms in the first partial sum that exceeds $t$. For instance it's folklore that $m(1)=e$, meaning we need on average $e$ terms to get the sum above 1. It follows from renewal theory t... | https://mathoverflow.net/users/14302 | Error term for renewal function | Given $x$, let $P\_n=P\_n(x)$ denote the probability that $X\_1+\ldots+X\_n \le x$
where $X\_i$ are independent and uniform in $[0,1]$. You are asking for $P\_0+P\_1+\ldots$. Now for any $c>0$ the integral
$$
\frac{1}{2\pi i} \int\_{c-i\infty}^{c+i\infty} \frac{y^s}{s} ds
$$
equals $1$ if $y>1$ and $0$ if $0\le y<... | 12 | https://mathoverflow.net/users/38624 | 141710 | 77,273 |
https://mathoverflow.net/questions/141702 | 10 | Let $p$ be a prime number. Is there a non-commutative reduced ring of order $p^2$? (Note that any ring of order $p^2$ with identity is commutative).
| https://mathoverflow.net/users/nan | Non-commutative reduced rings of order $p^2$ | Let $A$ be a reduced ring of order $p^n$. Then $(pa)^n = p^n a^n = 0$ for all $a \in A$ so $pA = 0$ and $A$ is an algebra over $\mathbb{F}\_p$.
Adjoining an identity element to $A$ if necessary, we may view $A$ as an ideal (of codimension at most $1$) in a finite dimensional unital $\mathbb{F}\_p$-algebra $B$. Let $... | 13 | https://mathoverflow.net/users/6827 | 141711 | 77,274 |
https://mathoverflow.net/questions/141693 | 12 | Let $A,B\subseteq\omega$. We write $A\subseteq^\*B$ if $A\setminus B$ is finite, if additionally $B\setminus A$ is infinite then we write $A\subsetneq^\*B$, otherwise we write $A=^\*B$.
We say that a $\cal A\subseteq P(\omega)$ is almost disjoint if for every two distinct $A,B\in\cal A$ we have $A\cap B=^\*\varnothin... | https://mathoverflow.net/users/7206 | Are there insane families in $L$? | Your requirements are inconsistent; there is no insane family.
Suppose towards contradiction that we have an insane family
$\mathcal{B}=\{B\_\alpha\mid\alpha\lt\kappa\}$, witnessed by tower
$\langle A\_\alpha\mid\alpha\lt\kappa\rangle$. For finite $k$, let $b\_k$ be any
element in $[(A\_\omega-A\_k)\cap B\_k]-\bigcup... | 12 | https://mathoverflow.net/users/1946 | 141712 | 77,275 |
https://mathoverflow.net/questions/141696 | 4 | This is a partial duplicate of [this Stack Exchange question](https://math.stackexchange.com/questions/469128/terminology-for-blow-ups-in-algebraic-geometry) which unfortunately got no answer.
All schemes are Noetherian and of finite type, although they need not be normal.
With $Z \subset X$ a closed subscheme, con... | https://mathoverflow.net/users/18403 | Terminology for blowups in algebraic geometry | Here are some thoughts
1. I think this should be called the "true center", because it is where something interesting happens. Plus, if you blew up,say, the singular point of a nodal curve, then according to your definition the "true center" would be empty which doesn't sound right. As Karl suggests calling it the "tr... | 4 | https://mathoverflow.net/users/10076 | 141715 | 77,277 |
https://mathoverflow.net/questions/141705 | 3 | Consider a renewal process whose holding times are given by a continuous random variable $X$ supported on $[0,1]$. It is known (e.g. [Stone '65](http://www.jstor.org/stable/2238132)) that the renewal function $m(t)$ converges to $t/\mathbb{E} X + \mathbb{E}[X^2]/2\mathbb{E}^2[X]$ exponentially fast. I am looking for a ... | https://mathoverflow.net/users/7732 | Uniform bound on the rate of convergence of the renewal measure | If the density is bounded by a constant $C$ then I think the rate of convergence can be bounded as $\exp(-\delta t /C^2)$ for some fixed constant $\delta >0$. If the density is unbounded then I don't think there needs to be a uniform version of exponential convergence.
To elaborate on this, I will use my answer to t... | 2 | https://mathoverflow.net/users/38624 | 141733 | 77,284 |
https://mathoverflow.net/questions/141734 | 5 | Let $R$ be a commutative ring. A vector $(c\_1,\ldots,c\_n) \in R^n$ is **unimodular** if $Rc\_1 + \cdots + Rc\_n = R$. Say that a vector $\vec{v} \in R^n$ is a **basis element** if there exists a free basis for $R^n$ containing $\vec{v}$. It is clear that all basis elements of $R^n$ are unimodular. Moreover, if $\vec{... | https://mathoverflow.net/users/39771 | Ring $R$ such that $R^n$ contains unimodular elements that are not part of a free basis for all $n \geq 2$ | Start with the integers. Adjoin variables $X\_{in}$ and $Y\_{in}$ for all $1\le i \le n$. Mod out by all relations of the form
$$\sum\_{i=1}^nX\_{in}Y\_{in}=1$$
Call the resulting ring $R$.
Then, by construction, any $(X\_{1n},X\_{2n},\ldots,X\_{nn})$ is a unimodular row over $R$. I claim it's not a basis element. ... | 4 | https://mathoverflow.net/users/10503 | 141736 | 77,285 |
https://mathoverflow.net/questions/141743 | 12 | $X\in \mathbb{R}$. Which distribution $P(X)$ has the highest possible entropy given its expected value, variance, skewness, and kurtosis? Is it an exponential family distribution of the form $P(X) \propto \exp(a\cdot x +b\cdot x^2 + c\cdot x^3 +d\cdot x^4)$ in analogy to the normal distribution being the maximum-entrop... | https://mathoverflow.net/users/39776 | What is the maximum-entropy distribution given mean, variance, skewness, and kurtosis? | Yes, your $P(X) \propto \exp(a\cdot x +b\cdot x^2 + c\cdot x^3 +d\cdot x^4)$ maximises the entropy $-\int P(X){\rm log} P(X)dX$ for prescribed first four moments, if the skewness and curtosis lie in a certain range:
*M. Rockinger and E. Jondau, [Entropy densities with an application to autoregressive conditional skew... | 10 | https://mathoverflow.net/users/11260 | 141752 | 77,289 |
https://mathoverflow.net/questions/141753 | 2 | In <https://www.google.com/#q=tensor+product+of+complexes%2Benochs> a new tensor product of complexes is defined which characterizes flatness in the category of complexes of $R$-modules. That is, a complex $F$ is flat if and only if $F\otimes -$ is exact when $\otimes$ denotes the new tensor product of complexes. Also ... | https://mathoverflow.net/users/38585 | pure sub-complexes of exact subcomplexes | Yes:
Let $0\rightarrow A\rightarrow B \rightarrow C\rightarrow0$ be a pure exact sequence of complexes with $H\_\*(B)=0$. Consider the complex
$$S^n = \cdots \rightarrow 0 \rightarrow R \rightarrow 0 \rightarrow\cdots,$$
concentrated in degree $n$, and the complex
$$D^{n+1} = \cdots \rightarrow 0 \rightarrow R ... | 1 | https://mathoverflow.net/users/12166 | 141754 | 77,290 |
https://mathoverflow.net/questions/140750 | 0 | Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}\_{R} S < \infty$ (e.g. $S=R\*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a finitely generated $S$-module.
What are sufficient criteria we can impose on $R \hookrightarrow S$ that $\mathrm{rk}\_... | https://mathoverflow.net/users/24231 | $\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be? | Two for my applications important cases turned out to have quite easy solutions. For the sake of completeness I'll post them, although I'm still wondering if there is a more general theory for ranks under base change.
In one case $x \mapsto x^{p^k}$ provided a map $S \rightarrow R$, hence $\mathrm{tor}\_R M = \mathrm... | 0 | https://mathoverflow.net/users/24231 | 141758 | 77,292 |
https://mathoverflow.net/questions/141751 | 2 | In their book "Non-Archimedean analysis", when BGR refer to an epimorphism in the context of $k$-Banach algebras do they actually require (or does it follow that) such maps are surjections? For example, after their definition of $k$-affinoid algebras, they seem to use Banach's Open mapping theorem to deduce such algebr... | https://mathoverflow.net/users/25854 | Terminology: Epimorphism in non-archimedean analysis | The authors mean a surjective homomorphism, "epimorphism" is too weak. An affinoid $k$-algebra is defined to be a Banach $k$-algebra of the form $k\langle x\_1,\dotsc,x\_n \rangle/\mathfrak{a}$ for some closed ideal $\mathfrak{a}$. This is analogous to affine $k$-algebras which have the form $k[x\_1,\dotsc,x\_n]/\mathf... | 3 | https://mathoverflow.net/users/2841 | 141765 | 77,294 |
https://mathoverflow.net/questions/141756 | 1 | The coefficient of Selberg Class L-function satisfy:
$a\_n <M\_{\epsilon} n^{\epsilon}$ (for any $\epsilon >0$) and the $a\_n$ are multiplicative.
So I would like to know if it can be shown that we also have the following partial sum bounded by a constant (maybe using also other properties of $a\_n$ coefficients):
... | https://mathoverflow.net/users/38290 | On properties of coefficients of Selberg Class L-function | I think this fails for the Dirichlet coefficients of $L(s):=\zeta(s+i\gamma)\zeta(s-i\gamma)$ whenever $\gamma>0$ satisfies $2^{i\gamma}+2^{-i\gamma}\neq 1$. Indeed, for such $\gamma$, the inequality would imply that $S(x):=\sum\_{n\leq x}a\_n$ is bounded by a constant either from below or from above. However, $L(s)$ h... | 3 | https://mathoverflow.net/users/11919 | 141775 | 77,297 |
https://mathoverflow.net/questions/141766 | 3 | A point set $P$ is said to be embedded in $\mathbf{Z}^2$ in *general position*, if no three points lie on a common line. Assume that $|P|=n$, I am interested in the smallest $k \times k$ integer grid in which $P$ can be embedded in general position? What can be said about $k$ as a function in $n$?
One idea is to plac... | https://mathoverflow.net/users/39300 | Points in general position on a small grid | This is the "no three in a line" problem, and you will find many discussions of it if you type the quoted phrase into the internet and stand back. A starting place is the [Wikipedia](http://en.wikipedia.org/wiki/No-three-in-line_problem) page on the problem.
| 5 | https://mathoverflow.net/users/3684 | 141783 | 77,298 |
https://mathoverflow.net/questions/141791 | 2 | I was wondering if it was possible to prove existence of a unitary operator $A$ such that:
$\langle Au,u\rangle=0$ for all $u$.
In 2-dimensions it clearly is (just a 90 degrees rotation) and similarly in other finite dimensions. However is it possible to prove this for infinite dimensions?
For all $u\in L^{2}$, say?
| https://mathoverflow.net/users/39803 | Existence of an "Orthogonalizing" Operator | To find such an operator in separable real Hilbert space just take a countable direct sum of rotations of $\mathbb{R}^2$. More formally:
Let $(e\_n)\_{n \in \mathbb{N}}$ be a countable basis for Hilbert space and define an operator $A$ by $Ae\_{2n}=e\_{2n+1}$ and $Ae\_{2n+1}=-e\_{2n}$. If $u=\sum\_{k=1}^\infty a\_ke\... | 3 | https://mathoverflow.net/users/1840 | 141798 | 77,302 |
https://mathoverflow.net/questions/141784 | 4 | If $X$ is a curve over a field of characteristic zero, then $X$ has a rational function, i.e., a finite morphism to the projective line.
Question. Suppose that $X$ is a Deligne-Mumford (or just algebraic) stack of dimension one over a field of characterisic zero. Does there exist a finite morphism to some $\mathbf P^... | https://mathoverflow.net/users/39801 | Do Deligne-Mumford curves also have rational functions | I suspect the following works. Let $\mathcal{X}$ be a proper, smooth, finite type Deligne-Mumford stack over $k$ that is one-dimensional and that has a dense open substack $U$ that is a scheme. Let $u:\mathcal{X}\to X$ be the coarse moduli space. Let $\{p\_1,\dots,p\_r\}\subset X$ be the complement of $U$. For each poi... | 2 | https://mathoverflow.net/users/13265 | 141803 | 77,304 |
https://mathoverflow.net/questions/141778 | 5 | we guess there is no maximal space which is also a P-space. Am I right? Do u know a counter example?
clarifications:
Maximal space is that space with topology $\tau$ which is maximal crowded topology on X.
crowded: a topology with no isolated point = dense in itself.
P-space:every $G\_\delta$ set is open = every prime ... | https://mathoverflow.net/users/38926 | Is There a maximal space that is a P-space? | Given a space $(X,\tau)$, the collection of all $G\_\delta$-subsets of $X$ form a base for a stronger topology $\tau\_\omega$ on $X$. It is easy to see that $(X,\tau\_\omega)$ is a $P$-space (sometimes called the $G\_\delta$ modification of $X$). If the original space $(X, \tau)$ is maximal, then there are two possibil... | 6 | https://mathoverflow.net/users/17836 | 141806 | 77,305 |
https://mathoverflow.net/questions/141801 | 23 | Famously, Solovay showed that, if $\textrm{ZFC}$ plus $\textrm{IC}$ (the existence of an inaccessible cardinal) is consistent, then so is $\textrm{ZF}$ plus $\textrm{DC}$ (dependent choice) plus $\textrm{LM}$ (all subsets of $\mathbb{R}$ are Lebesgue measurable). And Shelah showed that, conversely, the consistency of $... | https://mathoverflow.net/users/13506 | How strong is "all sets are Lebesgue Measurable" in weaker contexts than ZF? | The consistency of ZFC + IC is perhaps a little bit too much to ask, but I believe the next best thing is true:
**Conjecture.** Every boolean topos1 with dependent choice in which every set of reals is Lebesgue measurable contains a well-founded model of ZFC. In fact, every real is contained in a well-founded model o... | 19 | https://mathoverflow.net/users/2000 | 141811 | 77,309 |
https://mathoverflow.net/questions/141742 | 4 | Sufficient background:
Let $\mathcal{M}=(M,...)$ be an $\mathcal{L}$-structure and $X\subset M$.
**Definition**. $X$ is *large* if there exists a function $f:\mathcal{M}^n \overset {\leq k} \rightarrow \mathcal{M}$ definable in $\mathcal{M}$ such that $f(X^n)=M$ for some $n$, $k$. Otherwise, $X$ is *small*.
Here... | https://mathoverflow.net/users/31979 | "Small" subfields of algebraically closed fields | Let $X$ be a large subfield of an ACF $M$.
By quantifier elimination, every such definable function is piecewise algebraic, that is, there is a $d$ such that every element of $M$ is algebraic of degree at most $d$ over $X$. In particular, $M$ is the algebraic closure of $X$.
In characteristic $0$, every finite exte... | 4 | https://mathoverflow.net/users/12705 | 141817 | 77,310 |
https://mathoverflow.net/questions/141814 | 1 | Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times X$ be the diagonal embedding and $p\_1: X\times X\rightarrow X, ~p\_2: X\times X\rightarrow X$ be the projections to the first and second components. Let $E$ be a finite dimensional vector bundle on $X$. We define
$$
E\_{\Delta}:=\Delta... | https://mathoverflow.net/users/24965 | Do we have the following isomorphism for $\mathcal{Ext}$? | Note that $E \cong \Delta^\*p\_1^\*(E)$ since $p\_1\circ\Delta = 1\_X$. Therefore by the projection formula
$\Delta\_\*E = \Delta\_\*\Delta^\*p\_1^\*(E) = p\_1^\*(E)\otimes\Delta\_\*O\_X$, hence
\begin{align\*}
\mathcal{Ext}(\Delta\_\*E,\Delta\_\*E) &
= \mathcal{Ext}(p\_1^\*(E)\otimes\Delta\_\*O\_X,p\_1^\*(E)\otimes\D... | 5 | https://mathoverflow.net/users/4428 | 141819 | 77,311 |
https://mathoverflow.net/questions/141833 | 6 | The question in the title arises from a problem in Stewart's "Galois Theory, Third Edition" (and possibly elsewhere) which has been bugging me for a few days since reading it:
Problem 19.5 (p. 224) asks:
```
Use the equations
$641 = 5^4+2^4 = 5\cdot 2^7+1$
to show that 641 divides $F_5$.
```
Now the latter exp... | https://mathoverflow.net/users/12301 | Using the decomposition $641 = 5^4 + 2^4$ to factor $F_5$ | There is no deep mathematics involved here. The proof goes as follows (see, for example, W.A. Coppel, Number Theory: An Introduction to Mathematics, Springer, 2009, p. 160). Since $641=5\cdot2^7+1$ $=5^4+2^4$, we have $5\cdot 2^7\equiv -1 \;(\mathrm{mod}\; 641)$ and $2^4\equiv -5^4 \;(\mathrm{mod}\; 641)$. Thus
$$2^{32... | 12 | https://mathoverflow.net/users/32389 | 141834 | 77,316 |
https://mathoverflow.net/questions/141578 | 5 | Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top dimensional simplices of $A(S)$ the map $\lambda$ is induced by a homeomorphism $\Phi$ of $S$.
**Question: How to show that... | https://mathoverflow.net/users/9485 | Injective simplicial maps between Arc complexes | I'm not as familiar with this area as I should be, but it seems as though the desired result follows from the main theorem of the following [paper](http://journals.tubitak.gov.tr/math/issues/mat-10-34-3/mat-34-3-5-0812-16.pdf):
Irmak, McCarthy. *Injective simplicial maps of the arc complex*, Turk J Math, 34 (2010) , ... | 3 | https://mathoverflow.net/users/18263 | 141836 | 77,318 |
https://mathoverflow.net/questions/141854 | 7 | First note to the following well known theorems:
**Theorem (1):** The notion of "$x$ is a strongly inaccessible cardinal" is first order expressible and $\Pi\_{1}$.
**Theorem (2):** The notion of "$x$ is a measurable cardinal" is first order expressible but not $\Pi\_{1}$... | https://mathoverflow.net/users/nan | Are larger large cardinals less expressible? | For question (1), there are many exceptions. For example, being [superstrong](http://cantorsattic.info/Superstrong#Superstrong_cardinal) is $\Sigma\_2$ expressible, since it is witnessed inside a sufficiently large $V\_\theta$, but this is stronger than strong in consistency strength, and being strong is $\Pi\_3$. Simi... | 9 | https://mathoverflow.net/users/1946 | 141858 | 77,326 |
https://mathoverflow.net/questions/141857 | 5 | Note to the following well known theorem:
**Theorem (1):** If $\kappa$ be a "measurable" cardinal and $\mathcal{F}$ be a "non-principal $\kappa$-complete normal" ultrafilter on it then: $\langle V\_{\kappa +1},\in \rangle \cong \prod\_{\mathcal{F}}\lbrace ... | https://mathoverflow.net/users/nan | Is there a truth approximation on a cumulative hierarchy? | For question (1), the answer is the truth approximation property at $\delta$ implies the existence of a measurable cardinal. This is simply because the filter $\mathcal{F}$ witnessing your isomorphism must be countably complete, or else the ultraproduct on the right hand side will have an ill-founded $\omega$, preventi... | 4 | https://mathoverflow.net/users/1946 | 141859 | 77,327 |
https://mathoverflow.net/questions/141838 | 12 | It is known that if $\alpha,\beta,\gamma$ are three partitions then the Littlewood-Richardson coefficient $c\_{\alpha \beta}^{\gamma}$ is positive when the triple ($\alpha,\beta,\gamma$
) occurs as eigenvalues of Hermitian $n \times n$
matrices $A, B, C$ with $C = A + B$ which can be seen from the following paper.
<h... | https://mathoverflow.net/users/39829 | calculating Littlewood-Richardson coefficients | It's hard to prove that there isn't a way to do something, but I think the answer is no.
The [saturation conjecture](http://arxiv.org/abs/math/9810180), now a theorem of Knutson and Tao, says that $c\_{(N \alpha) (N \beta)}^{N \gamma} >0$ implies $c\_{\alpha \beta}^{\gamma} >0$ for any positive integer $N$ and any pa... | 10 | https://mathoverflow.net/users/297 | 141865 | 77,330 |
https://mathoverflow.net/questions/141876 | 0 | Let $G$, be a Lie Group and $\mathfrak{g}$ be its Lie algebra ,i.e, $Lie(G)=\mathfrak{g}$. Let $\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^\*}$. Here $X\in \mathfrak{g} $ and $F\in \mathfrak{g^\*}$ . So we can construct orbit of $\zeta$. by
$\mathfrak{G}=\{ ( Ad(g)X,Ad^\*(g)F\ ): g\in G \}$. By the action of $... | https://mathoverflow.net/users/nan | fiber bundle on an orbit of $\mathfrak{g}\oplus\mathfrak{g^*}$ | Yes. If a Lie group $G$ acts smoothly on a manifold $X$, and $H$ is the $G$-stabilizer of a point $x\_0 \in X$, then $H$ is a closed subgroup of $G$, clearly, and the orbit is identified with $G/H \cong Gx\_0$ by the map $gH \mapsto gx\_0$. Every Lie group $G$ is an $H$-bundle over any of its homogeneous spaces $G/H$: ... | 2 | https://mathoverflow.net/users/13268 | 141877 | 77,334 |
https://mathoverflow.net/questions/141875 | 1 | It is basic fact that for a holomorphic (or mermorphic) map $f$, the family of iterates $\{f^n\}\_{n=1}^{\infty}$, is normal if and only if $\{f^{mn}\}\_{n=1}^{\infty}$, is normal $\forall m\geq 1$.
For a sequence of integers $\{n\_i\}\_{i=1}^{\infty}$,
There is natural question to ask under which condition:
$\{f^{... | https://mathoverflow.net/users/11966 | Normal family and arithmetic progression | This is true for every subsequence. Indeed, if $z$ is on the set of normality
(where $f^n$ is normal), then evidently every subset of $f^n$ is normal. If $z$ is on
the Julia set, then there is a repelling periodic point in every neighborhood of $z$.
At a repelling periodic point, any subsequence of iterates is evidentl... | 4 | https://mathoverflow.net/users/25510 | 141880 | 77,336 |
https://mathoverflow.net/questions/141870 | 9 | Let $X$ be a smooth complex projective variety such that the restriction of $TX$ on any curve $C$ in $X$ is ample. Is true in this case that $X$ is isomorphic to $\mathbb CP^n$?
I guess the above condition implies that that $TX$ is nef (i.e. $O(1)$ is nef on $\mathbb P(TX)$), but it is not clear for me that this cond... | https://mathoverflow.net/users/13441 | Variety $X$ such that $TX$ is ample on any curve in $X$ | In this very famous paper:
Mori, Shigefumi,
Projective manifolds with ample tangent bundles.
Ann. of Math. (2) 110 (1979), no. 3, 593–606.
it is proven that over an algebraically closed field of characteristic 0 $\mathbb P^n(K)$ is the only manifold $X$ with ample tangent bundle.
In the introduction the author... | 17 | https://mathoverflow.net/users/10610 | 141881 | 77,337 |
https://mathoverflow.net/questions/141886 | 6 | Let $\mathcal{O}\_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}\_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which will ensure that the image of the group of units $(\mathcal{O}\_k)^{\ast}$ in $\mathcal{O}\_k/\mathfrak{p}$ is all of $(\m... | https://mathoverflow.net/users/39856 | Reduction mod $p$ of units in a ring of integers | I don't know of any result that specifically looks at units but there are a lot of results on looking at the image of a fixed finitely generated subgroup $G$ of $k^\*$ in the units of the residue fields. The granddad of these questions is the Artin conjecture for primitive roots. There are results of Gupta and Murty (w... | 6 | https://mathoverflow.net/users/2290 | 141889 | 77,339 |
https://mathoverflow.net/questions/141845 | 9 | A well-known theorem of Mills asserts that there is a model of Peano Arithmetic $M$ in an uncountable language such that $M$ has no elementary end extension (e.e.e.). I ask whether every complete extension of $PA$ in an uncountable language can have models of every cardinality with arbitrary large e.e.e.'s. To put it i... | https://mathoverflow.net/users/27034 | Elementary end extensions of models of Peano Arithmetic in uncountable languages | Apologies for completely rewriting this answer. It took a long time to organize and verify all the details.
The following outlines a proof of the following:
**Theorem.** *Every model $N$ of $T$ which has an expansion to a model $(N,\mathcal{X})$ of $T^+$ for which there is an admissible $N$-saturated ultrafilter on... | 5 | https://mathoverflow.net/users/2000 | 141892 | 77,340 |
https://mathoverflow.net/questions/141782 | 14 | Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time? The convex hull itself might have an exponential number of facets so we cannot afford explicitly to compute it.
My main interest is not in computer prec... | https://mathoverflow.net/users/45564 | Efficiently determine if convex hull contains the unit ball | The problem is NP hard. Here is a proof sketch.
The problem is to determine if there is a point $y$ with $\|y\|=1$ outside of the convex hull of given points $x\_1,\dots, x\_n$. Note that such point exists if and only if there is hyperplane at distance less than $1$ from the origin such that all points $x\_1, \dots,... | 14 | https://mathoverflow.net/users/26349 | 141894 | 77,342 |
https://mathoverflow.net/questions/141717 | 6 | Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is a standard cross-cap. I believe this question to be a standard one, and I have two questions about it.
First, I think ... | https://mathoverflow.net/users/36108 | Knotted projective planes and fake complex projective space | I haven't been to Melvin's talks, but I suspect he's using the cyclic 2-sheeted branch cover construction. Specifically, the cyclic 2-sheeted branched cover of $S^4$ branched over the unknotted embedded $\mathbb RP^2$ is either $\mathbb CP^2$ or its mirror reflection depending on the normal Euler class of the $\mathbb ... | 5 | https://mathoverflow.net/users/1465 | 141897 | 77,344 |
https://mathoverflow.net/questions/139886 | 2 | Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small non zero real number. Now define the
following sequence of functions:
$$ v\_{0}(x,y) : \equiv 0 $$
$$ v\_{n+1}(x,y):= \... | https://mathoverflow.net/users/4463 | Does a particular iteration produce a weak solution to a non linear pde? | This is
$$
(\partial\_x\nu+\frac12)^2=\frac14+e^{2ty}-1=e^{2ty}-\frac34.
$$
The parameter $ty$ should be chosen so that $2ty\ge \ln 3-\ln 4$, and then a solution is given by
$$
\partial\_x\nu+\frac12=\alpha,\quad \alpha^2=e^{2ty}-\frac34,
$$
e.g.
$
\nu=(\alpha -\frac12)x+\phi(y).
$
| 0 | https://mathoverflow.net/users/21907 | 141902 | 77,345 |
https://mathoverflow.net/questions/141764 | 17 | As the title said, I would like to know if constructive measure theory has been developed somewhere ?
I am more precisely interested in the (constructive) theory of completely continuous valuation on locale, or eventually in countably continuous valuation on locale.
I know how to do constructively the integration o... | https://mathoverflow.net/users/22131 | reference request : constructive measure theory | Steve Vickers’ [*A monad of valuation locales*](http://www.cs.bham.ac.uk/~sjv/papersfull.php#Riesz) presents a strong monad on the category of locales, a localic analogue of the Giry monad. It is commutative, i.e. product valuations exist and a Fubini Theorem holds. Concrete representations are given for the tensor pro... | 10 | https://mathoverflow.net/users/25122 | 141904 | 77,346 |
https://mathoverflow.net/questions/141909 | 1 | I need some help on a problem on combinatorics.
Let $n$ be a natural number greater than $1$ and $k,m$ be two fixed natural numbers not exceeding $n$ with $m\leq\frac{k(k+1)}{2}$.
Let $N=\{{1,2,...,n}\}$ and $S\_i=\{a\_{i1},a\_{i2},...,a\_{ik}\}$ with $S\_i\subseteq{N}$ .
Denote the sum of elements of $S\_i$ by $... | https://mathoverflow.net/users/38851 | Number of subsets with fixed cardinality k, and sum of elements a multiple of m | It's not clear whether Gaitanas wants his sum to divide $m$ or be a multiple of $m$. If he wants the sum to be a multiple of $m$, then
some related questions are studied in the paper "[Enumeration of Power Sums Modulo a Prime](http://math.mit.edu/~rstan/pubs/pubfiles/35.pdf)" by Andrew M. Odlyzko and Richard P. Stanley... | 3 | https://mathoverflow.net/users/10744 | 141916 | 77,351 |
https://mathoverflow.net/questions/141922 | 4 | Is there any direct formula or algorithm better than the brute force (O(n) algorithm by iterating from 1 to n) way to calculate the sum
\begin{equation}
S = \sum\limits\_{i=1}^n [{\frac{n}{i}}]
\end{equation}
where [x] denotes the integral part of x?
I tried out calculating the sum by considering i's that are fa... | https://mathoverflow.net/users/39871 | sum of integral part of n/k | For questions like this searching for the first few terms
in OEIS might help [3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41](https://oeis.org/search?q=3%2C+5%2C+8%2C+10%2C+14%2C+16%2C+20%2C+23%2C+27%2C+29%2C+35%2C+37%2C+41&sort=&language=&go=Search)
This is A006218.
There are a lot of references and bounds for th... | 6 | https://mathoverflow.net/users/12481 | 141926 | 77,353 |
https://mathoverflow.net/questions/141920 | 7 | Are there any consistency results in set theory (or in mathematics) that can be proved using nonstandard models of ZFC but not using transitive models of ZFC?
| https://mathoverflow.net/users/11115 | Consistency results using nonstandard models | Yes. Harvey Friedman has identified several (natural) combinatorial statements that are equivalent to the $1$-consistency of $\mathsf{ZFC}$ or strengthenings of it via large cardinals (here, $1$-$\mathrm{Con}(T)$ is the assertion that all $\Sigma^0\_1$ consequences of $T$ are true). In particular, this means that the s... | 8 | https://mathoverflow.net/users/6085 | 141928 | 77,354 |
https://mathoverflow.net/questions/141933 | 3 | Taking a modular form such that we have Fricke involution:
$\sum\_{n=1} a\_n e^{-\pi nx^2} = \frac{A}{x^k} \sum\_{n=1} a\_n e^{-\pi \frac{n}{x^2}}$ [1]
I would like to know if there exists results on possible formula with $a\_n$ coeficient which will be like the classical Poisson summation formula (it works for Dir... | https://mathoverflow.net/users/38290 | Possible to have Poisson Summation formula with coefficient of modular forms? (for some functions) | The functional equation of the Riemann Zeta function is equivalent to the Poisson summation formula. This should be adoptable to the setting of automorphic $L$-functions. I am not sure where the abelian Fourier Analysis should happen here, probably in higher rank though(?)
Here is a reference, which is even more gene... | 1 | https://mathoverflow.net/users/10400 | 141954 | 77,363 |
https://mathoverflow.net/questions/141882 | 2 | The following question stems from a question I already asked on MO:
[Nakai-Moishezon theorem for abelian varieties](https://mathoverflow.net/questions/140654/nakai-moishezon-theorem-for-abelian-varieties)
I would like to prove that if $L\_0$ is an ample line bundle on an abelian variety $A$ of dimension $n$ (define... | https://mathoverflow.net/users/14143 | Nef divisors on abelian varieties | I think it is probably easier to prove that $L$ is ample when all the inequalities are strict:
The assumption for $i=n$ implies that $K(L)$ is finite, i.e., $L$ is non-degenerate, by the second statement in the Riemann-Roch theorem in Mumford's "Abelian Varieties" and then the index theorem together with Riemman-Roc... | 5 | https://mathoverflow.net/users/519 | 141960 | 77,365 |
https://mathoverflow.net/questions/141942 | 3 | This is the exact question:
>
> You are given black-box which returns a random number between 0 and 1(uniform distribution).You keep generating random numbers X1,X2,X3 and so on and store the sum of all those random numbers. You stop as soon as the sum exceeds 1.What is the expected number of random variables used ... | https://mathoverflow.net/users/36640 | What is the expected number of random numbers (generated uniformly) such that their sum of numbers exceeds one? | For any random variable $X$ taking values in $\mathbb{N}$, $E[X]= \sum\_{n=0}^\infty P(X\gt n)$. In this case, the probability that the sum of the first $n$ numbers is less than $1$ is $1/n!$, the volume of the simplex with vertices at the origin and the standard basis vectors in $n$ dimensions. So, the expected number... | 8 | https://mathoverflow.net/users/2954 | 141962 | 77,367 |
https://mathoverflow.net/questions/141937 | 10 | Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it true that $R \cong S$?
| https://mathoverflow.net/users/nan | Are these rings of functions isomorphic? | They are not ring isomorphic, because e.g. $R$ has the following property of a ring $X$, and $S$ does not:
>
> There is a non zero element $u \in X$ such that for any
> invertible $f\in X$ either $uf$ or $-uf$ is a square, and for some $g\in X$
> neither $ug$ nor $-ug$ is a square.
>
>
>
| 17 | https://mathoverflow.net/users/6101 | 141964 | 77,369 |
https://mathoverflow.net/questions/141958 | 2 | As I understand it, the Nakai-Moishezon criterion gives conditions for the existence of an ample divisor class on an arbitrary proper scheme, and Kleiman's criterion does the same for arbitrary projective schemes (in fact, more generally for 'quasi-divisorial' schemes, as defined in Kleiman's paper).
Are there simila... | https://mathoverflow.net/users/22975 | Do versions of the Nakai-Moishezon and Kleiman criteria hold for Moishezon manifolds, or other 'nice' spaces? | This is adding to ulrich's comment. One reference for this result is the following article of Kollár.
MR1064874 (92e:14008) Reviewed
Kollár, János(1-UT)
Projectivity of complete moduli.
J. Differential Geom. 32 (1990), no. 1, 235–268.
14D22 (14H10 14J10)
<http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=D... | 3 | https://mathoverflow.net/users/13265 | 141971 | 77,371 |
https://mathoverflow.net/questions/141950 | 8 | I've been wondering about the following "finiteness statement" concerning etale covers for a while.
Let $K$ be a field of characteristic zero, not necessarily algebraically closed. A variety over $K$ is a smooth quasi-projective geometrically connected scheme over $K$.
Let $X$ be a variety, and let $d$ be an intege... | https://mathoverflow.net/users/39889 | Do all varieties have only finitely many etale covers of fixed degree | Just to add one word to ulrich's answer: for a geometrically irreducible $K$-variety $X$, there is a short exact sequence of etale fundamental groups.
$$
0 \to \pi\_1^{\text{et}}(X\otimes\_K \overline{K}) \to \pi\_1^{\text{et}}(X) \to \text{Gal}(\overline{K}/K) \to 0.
$$
An étale degree $d$ cover is equivalent to a hom... | 7 | https://mathoverflow.net/users/13265 | 141978 | 77,373 |
https://mathoverflow.net/questions/141980 | 3 | Assume that $f\in C^{\infty}$ and that $M\_n$ is a sequence such that $$\sum\_{n=0}^{\infty}\frac{M\_n}{(n+1)M\_{n+1}}=\infty$$
and for certain compact neighborhood of the origin $U$ of $\mathbb{R}$, there is a constant $A$ such that for every $x\in U$, and $n\in\mathbb{N}$, \begin{equation}|g^{(n)}(x)|\leq n!A^nM\_n... | https://mathoverflow.net/users/5506 | If $f(x)+f(2x)$ is quasianalytic, is $f(x)$ necessarily quasianalytic? | Assuming $V:=U=(-\alpha,+\alpha)$, we want the sequence of real numbers
$$B\_n:=\bigg( \frac{\| f^{(n)} \|\_{\infty,U}} {n!M\_n} \bigg) ^{1/n}$$
to be bounded (in which case $B:=\sup\_{n\in\mathbb{N}}B\_n$ is the best constant for the quasianalytic bounds to hold for $f$ on $U$ ).
Indeed, since $f(x)=g(x/2)-f(x/2)$,... | 6 | https://mathoverflow.net/users/6101 | 141987 | 77,375 |
https://mathoverflow.net/questions/141934 | 6 | It is a basic fact in the weak-\* topology, the set of invariant measures for a dynamical system is closed, compact, and convex in the weak-\* topology. Furthermore, the set of ergodic measures is equal to the set of extremal points of the the set of invariant measure.
In the symbolic space, the set of ergodic measur... | https://mathoverflow.net/users/11966 | inverse problem for ergodic measures | Let $X$ be a compact metric space and $T\colon X\to X$ be a continuous map. The set of $T$-invariant Borel probability measures $\mathcal{M}\_T(X)$ is well known to be non-empty, convex, compact, and metrizable. Moreover, its extreme points $\mathcal{M}^e\_T(X)$ coincide with ergodic invariant measures and every invari... | 12 | https://mathoverflow.net/users/24676 | 141992 | 77,376 |
https://mathoverflow.net/questions/142005 | 4 | Let $X$ and $Y$ be two irreducible, affine $\newcommand{\C}{\mathbb C}\C$-varieties. Let $f:X\to Y$ be a morphism. Denote by $u:\tilde X\to X$ and $v:\tilde Y\to Y$ their normalizations. Now, if $f$ is dominant, I get an induced morphism $\tilde f: \tilde X\to \tilde Y$ such that $v\circ\tilde f= f\circ u$. This follow... | https://mathoverflow.net/users/9947 | When is normalization functorial? | Closed immersions obviously don't work (think about the node, if I include the singular point into the node, where does it go in the normalization).
Open immersions are fine for obvious reasons (if it works on schemes, it works for open immersions of schemes).
**EDIT:** I'm going to make what I wrote here more pre... | 5 | https://mathoverflow.net/users/3521 | 142012 | 77,382 |
https://mathoverflow.net/questions/142019 | 7 | Here is a cute observation: Let $F,G : \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functors between symmetric monoidal categories, and let $\eta : F \to G$ be a monoidal transformation. Then for every dualizable object $V \in \mathcal{C}$ the morphism $\eta\_V : F(V) \to G(V)$ is actually an isomorphism! The i... | https://mathoverflow.net/users/2841 | Monoidal transformations are isomorphisms at dualizable objects | This goes back at least to Saavedra-Rivano "Categories Tannakiennes." In fact, this has a generalisation to Frobenius functors, in Day-Pastro "Note on Frobenius monoidal functors," and to more general settings, as monoidal bicategories: Lopez Franco, Street, Wood; "Duals Invert," (App. Categor. Str. Vol 19).
| 9 | https://mathoverflow.net/users/8482 | 142021 | 77,387 |
https://mathoverflow.net/questions/142020 | 3 | Suppose $G$ is a sufficiently nice (maybe locally compact and abelian) group which acts on the separable Hilbert space $\mathcal{H}$ by unitary transformations. Is there a generalization of the spectral theorem to this context? Specifically, what ought the spectrum of such a group action be, and, do generalizations of ... | https://mathoverflow.net/users/38023 | A version of the spectral theorem for group actions | This is true for locally compact abelian $G$; you ought to be able to find it in any text on abstract harmonic analysis (a reference I have at hand is Theorem 4.44 in Folland's *A Course in Abstract Harmonic Analysis*).
Here's a bit of general perspective that may be helpful. Unitary representations of $G$ are the sa... | 5 | https://mathoverflow.net/users/75 | 142028 | 77,391 |
https://mathoverflow.net/questions/142036 | 3 | This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically connected curve over $K$ and let $P$ a $K$-rational point of $C$. Let $X$ be a smooth projective geometrically connected surfa... | https://mathoverflow.net/users/39889 | Are Isom-schemes geometrically connected | If $X\to C$ is trivial, i.e. $X=C\times\_K F$, then $\underline{\mathrm{Isom}}\_C(X,C\times\_K F)$ is just $C\times\_K \underline{\mathrm{Aut}}\_K(F)$ which is geometrically connected if and only if $\underline{\mathrm{Aut}}\_K(F)$ is trivial.
| 5 | https://mathoverflow.net/users/7666 | 142047 | 77,399 |
https://mathoverflow.net/questions/142058 | 4 | Let $f\colon \mathbb{R}\_{\geq0} \to \mathbb{R}\_{\geq0}$ be a function. We say that $f$ has the property of inducing metric spaces, whenever for all metric space $(X,d)$, $(X, f \circ d)$ is also a metric space. In other words, for all distances $d$, $f \circ d$ is also a distance. In this question, I am trying to est... | https://mathoverflow.net/users/23434 | Inducing metric spaces | These functions are called metric-preserving functions and are well-studied, as just one minute of Googling would have told you. For instance:
<http://pcorazza.lisco.com/papers/metric-preserving.pdf>
| 4 | https://mathoverflow.net/users/39948 | 142061 | 77,402 |
https://mathoverflow.net/questions/142060 | 2 | A program P takes a string as an input and returns a string of same length as output.
>
> **Q** Given two strings A and B how fast can a program tell weather string B cannot be obtained by a recursive application of P over the initial string A ?
>
>
> **Q** Given P , length L of the strings and the initial string... | https://mathoverflow.net/users/34859 | String transformer : Polynomial time approximation schemes? | **Theorem.** There is $P$ for which the reachability problem in your first question is NP hard.
Proof. Suppose we have any NP decision problem $A$, where for any string $a$, we have $a\in A$ if and only if there is a string $b$ of the same length such that a fixed polynomial time program $p$ accepts $(a,b)$ (this fo... | 6 | https://mathoverflow.net/users/1946 | 142064 | 77,403 |
https://mathoverflow.net/questions/141999 | 11 | Let $n,p \in \mathbb{N}\_+$ with $p \leq n.$ Let $\mathcal{P}$ denote the set of partitions of $\{1, \ldots, n\}$ into $p$ nonempty sets. How can I efficiently sample uniformly from $\mathcal{P}$?
| https://mathoverflow.net/users/2586 | How to efficiently sample uniformly from the set of p-partitions of an n-set? | The sets you're interested in are counted by [Stirling Numbers of the Second Kind](http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind), which satisfy the recursion
$$\left\{{n \atop k}\right \}=\left\{{n-1 \atop k-1}\right \}+k \left\{{n-1 \atop k}\right \}$$
Here the first term represents those partition... | 9 | https://mathoverflow.net/users/405 | 142080 | 77,409 |
https://mathoverflow.net/questions/142078 | 2 | Let $X\_1, X\_2$ be two smooth complex manifold and $C\_1 \subset X\_1, C\_2 \subset X\_2$ be two smooth projective curves. Assume that $C\_1 \simeq C\_2$ as complex curves and their normal bundles are isomorphic.
Q. Is it possible to take two analytic open neighbourhoods $U\_1 \subset X\_1$ and $U\_2 \subset X\_2$ ... | https://mathoverflow.net/users/12390 | Determine complex analytic germ along a smooth compact curve via normal bundle? | This is equivalent to asking whether for any smooth curve $C$ on a complex manifold $X$, there is an analytic neighborhood of $C$ in $X$ that is equivalent to a neighborhood of $C$ in the normal bundle $N$. However, this implies that the exact sequence
$$
0 \longrightarrow T\_C \longrightarrow T\_X\mid\_{C} \longrighta... | 5 | https://mathoverflow.net/users/4790 | 142084 | 77,410 |
https://mathoverflow.net/questions/139708 | 6 | **Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through?**
*i.e. which are the necessary and sufficient conditions for a planar compact set C to pass through a closed interval in a plane?*
The matter has been studied in the 1982 paper by Gilbert Strang, "[The ... | https://mathoverflow.net/users/34859 | Passing C through a slot | Concerning nonconvex $C$, I posed it as an open-problem exercise in *[Computational Geometry in C](http://cs.smith.edu/%7Ejorourke/books/compgeom.html)* to determine the worst polygon (or, equivalently, the worst polygonal
curve $C$), worst in the number of "moves," to get $C$ through the doorway interval
(p.321, Ex.4)... | 7 | https://mathoverflow.net/users/6094 | 142097 | 77,414 |
https://mathoverflow.net/questions/126454 | 2 | Let $R$ be a ring of characteristic $p$. Let $G$ be the kernel of the natural map $\pi\_1^{et} (\operatorname{Spec} R[[x]] [1/x]) \to \pi\_1^{et}( \operatorname{Spec} R)$. $G$ has a natural map to $\prod\_{l\neq p} \mathbb Z\_l(1)$, coming from the etale coverings adjoining the $n$th roots of $x$ for $n$ prime to $p$. ... | https://mathoverflow.net/users/18060 | Etale fundamental group of punctured formal neighborhood | The answer is no.
Take $R=\mathbb F\_2[y]$ and consider the cover defined by the equation $t^3+yt+x=0$. By computing the discriminant, this cover is etale. The inertia group of this cover is a subgroup of $S\_3$. By reducing mod $(y)$, it contains a $3$-cycle. By reducing mod $y-1+x^2$, the equation factors into $(t^... | 0 | https://mathoverflow.net/users/18060 | 142099 | 77,415 |
https://mathoverflow.net/questions/142083 | 10 | A *bramble* in a graph $G$ is a set of connected subgraphs $H\_1, \dots, H\_m$ such that for every $i, j$, either $H\_i$ intersects $H\_j$ in a vertex, or there exists an edge of $G$ with one end in $V(H\_i)$ and one end in $V(H\_j)$. The *order* of the bramble is the minimum $|X|$ such that $X \subseteq V(G)$ and $X \... | https://mathoverflow.net/users/20940 | What is the relationship between the bramble number and the strict bramble number of a graph? | **Yes**, $sBr(G) \geq Br(G)/2$ for all graphs $G$.
To see this, let $\mathcal{Y}:=Y\_1, \dots, Y\_m$ be a bramble of $G$ with a minimum hitting set $X$ such that $|X|=br(G)$. It will be slightly more convenient to let $Y\_i$ be sets of vertices which induce connected subgraphs. For each $x \in X$, let $\mathcal{Y}\_... | 8 | https://mathoverflow.net/users/2233 | 142100 | 77,416 |
https://mathoverflow.net/questions/141981 | 3 | My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
canonical G-invariant symplectic structure.
Kirillov defined the coadjoint orbit by the natural way as follows,
Let $G... | https://mathoverflow.net/users/nan | Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $ | There is no hope for such a construction. The orbits of $G$ on $\frak{g}\oplus\frak{g}^\ast$ are not always of even dimension, so they will not support symplectic structures.
For example, take $G=\mathrm{SO}(3)$. Because $G$ is simple, $\frak{g}^\ast$ is isomorphic to $\frak{g}$ as a $G$-module, so the action of $G$... | 5 | https://mathoverflow.net/users/13972 | 142109 | 77,419 |
https://mathoverflow.net/questions/142025 | 1 | Suppose a compact Kähler manifold $(M,\omega)$ admits a smooth circle action $g\_t,~t\in S^1$. So the pull back of the Kähler form $g\_t^{\ast}(\omega)$ is a nondegenerate two-form. Since the circle action is only smooth and may not preserve the complex structure, $g\_t^{\ast}(\omega)$ may not be $(1,1)$-from. My quest... | https://mathoverflow.net/users/36974 | pull back of a Kähler form by a smooth circle group action on a compact Kähler manifold | Now that I fully understand your question, which (apparently) is whether *every* circle action on a compact Kähler manifold $(M,J,\omega)$ must preserve the $J$-type of $\omega$, I can answer it. The answer is 'no'.
Consider the $4$-dimensional case. Observe that a circle action that preserves *all* of the $(1,1)$-fo... | 2 | https://mathoverflow.net/users/13972 | 142113 | 77,420 |
https://mathoverflow.net/questions/142115 | 7 | I was told that the signature of $S\_1\times F\_3$ is zero, where $F\_3$ is a compact oriented 3-manifold. Let $M\_4$ be a fibre bundle with $S\_1$ as a the base manifold and $F\_3$ as the fibre. Assume $M\_4$ is oriented. Can one show that the signature of $M\_4$ is zero?
| https://mathoverflow.net/users/17787 | Signature of compact oriented 4-manifold | Yes, in several ways. It can be proven by cohomological methods (Meyer, ''Die Signatur von Faserbündeln'', PhD thesis in Bonn, early 1970s), L-theory (Lück-Ranicki ''Surgery obstructions if fibre bundles'') or index theory (a footnote in Atiyah ''The signature of fibre-bundles'', the details worked out by myself (arXiv... | 14 | https://mathoverflow.net/users/9928 | 142119 | 77,422 |
https://mathoverflow.net/questions/142004 | 2 | Browder proved the following fixed point theorem in his 1968 Mathematische Annelen paper (Theorem 1):
**Theorem.** Let $K$ be a non-empty compact convex subset of a topological vector space $E$ (where we assume that $E$ is separated but not necessarily locally convex). Let $T$ be a mapping of $K$ into $2^K$, where fo... | https://mathoverflow.net/users/36696 | Browder's fixed point theorem in non-Hausdorff topological vector spaces | It is true indeed without assuming the Hausdorff separation axiom on the topological vector space $E$, for the reason that we can quotient over the closure of the origin, $\overline{\{0\}}$.
Let's denote $\pi:E\to \tilde E:=E/\overline{\{0\}}$ the quotient projection, and consider $\tilde K:=\pi (K)$, a compact conve... | 1 | https://mathoverflow.net/users/6101 | 142129 | 77,424 |
https://mathoverflow.net/questions/142128 | 4 | Let $X$ be the real line with the usual topology. Then clearly $|C(X)| = c = |X|$ and on the other hand $|X| = 2^{\aleph\_0}$.
Now my question is as in the title: **Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$** (where $C(X)$ is the set of all real valued continuous... | https://mathoverflow.net/users/nan | Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$ | The answer is yes.
Let $\kappa$ be any singular [strong limit cardinal](http://cantorsattic.info/Strong_limit#strong_limit_cardinal) of uncountable cofinality, such as the cardinal $\beth\_{\omega\_1}$ for a specific example, and let $X=\kappa+1$, the ordinals up to and including $\kappa$ itself. Under the order top... | 12 | https://mathoverflow.net/users/1946 | 142134 | 77,426 |
https://mathoverflow.net/questions/141979 | 11 | It is well known that if $(P, \leq)$ is a partial order then $\leq$ can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by Banach, Kuratowski and Tarski.
Now suppose that $f$ is an automorphism of $(P, \leq)$ and we want to extend $\leq$ to a l... | https://mathoverflow.net/users/17836 | Extending a partial order while preserving an automorphism | I couldn't find anything in the literature either, but the answer to the first question is positive. Let $G$ be a group acting on a space $X$. Say that $G$'s action on $X$ has the invariant order-extension property provided that every $G$-invariant partial order on $X$ (i.e., partial order $\le$ such that $x\le y$ iff ... | 5 | https://mathoverflow.net/users/26809 | 142139 | 77,428 |
https://mathoverflow.net/questions/142157 | 2 | Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?
For each free homotopy classes $\gamma$ of mappings of the circle into $M$ with "trivial monodromy" I can construct the embedding of torus $\gamma \times S^1$ into E. So, I think that is ... | https://mathoverflow.net/users/37807 | isotopy classes of embeddings of the torus | As stated, your question is rather hopeless. Consider the 3-sphere, which also happens to admit the Hopf fibration over the 2-sphere. Then classifying 2-tori in the 3-sphere is essentially equivalent to classifying all knots. Clearly, the vertical tori as in your questions are insufficient.
However, if you assume th... | 6 | https://mathoverflow.net/users/21684 | 142163 | 77,438 |
https://mathoverflow.net/questions/142170 | 6 | Suppose that $V$ is a model of $\sf ZFC$, and for concreteness I should point that at this point I am interested in $V=L$ as a ground model.
Suppose that $V[c]$ is a Cohen extension of $V$ where $c$ is a real number, and $\{A\_n\mid n\in\omega\}$ is an almost disjoint family in $V[c]$ of subsets of $\omega$. Can we a... | https://mathoverflow.net/users/7206 | Separation of almost disjoint families by ground model almost disjoint families | Perhaps I am misunderstanding your question, but I think you intend that $A\_n'\in V$ and also that the family $\{A\_n'\mid n\in\omega\}$ is in $V$. Is that right?
In this case, the answer is no. Let $A\_0$ consist of every other element of $c$, and $A\_1$ consist of every other element of the remaining elements of $... | 5 | https://mathoverflow.net/users/1946 | 142174 | 77,442 |
https://mathoverflow.net/questions/141931 | 2 | I want to know the fundamental representation of classical Lie algebra of type $D\_{n}$ over complex numbers with the following informations. For example, $L(\omega\_i)$ be a fundamental rep of fundamental weight $\omega\_i$, supposed $i\neq 1$(easy) and $i\neq n$(no idea what the rep looks like for now, and I am readi... | https://mathoverflow.net/users/38283 | Representation theory, classical Lie algebra, D_{n} | First, note that your edited question still has $i=2n$ when you mean $i=n$. As was pointed out by others, $n$ is the rank here and $2n$ the dimension of the first fundamental representation (the natural one for the Lie algebra in the even orthogonal case).
One useful source (if you can locate it), based on lectures ... | 5 | https://mathoverflow.net/users/4231 | 142175 | 77,443 |
https://mathoverflow.net/questions/142192 | 5 | In a 1995 [paper](http://www.new1.dli.ernet.in/data1/upload/insa/INSA_2/20005a1b_1057.pdf), Choudhry gave a table of solutions to the quartic Diophantine equation,
$a^4+nb^4 = c^4+nd^4\tag{1}$
for $n\leq101$. Seiji Tomita recently [extended](http://www3.alpha-net.ne.jp/users/fermat/dioph121e.html) this to $n<1000$... | https://mathoverflow.net/users/12905 | On $a^4+nb^4 = c^4+nd^4$ and Chebyshev polynomials | Question 2:
The reason must be that Chebyshev polynomials solve Fermat-Pell equations.
The difference between the two sides of equation (2),
$$
(n+nx+y)^4 + n(1−nx−xy)^4 = (n−nx+y)^4 + n(1+nx+xy)^4,
$$
factors as $8nx(n+y) \phantom. ((x^2-1)y^2 + 2n(x^2-1)y + 1 - n^2)$.
The last factor is a quadratic in $n$ with leadin... | 6 | https://mathoverflow.net/users/14830 | 142193 | 77,449 |
https://mathoverflow.net/questions/142132 | 0 | Let $M\_{n+1}$ be a fibre bundle with $S\_1$ as the base and $n$-dimensional CW complex $F\_n$ as the fibre.
Assume $M\_{n+1}$ is oriented.
(1) Can one show that **$M\_{n+1}$ is always a boundary of a CW complex $M\_{n+2}$,
where $M\_{n+2}$ is a fibre bundle with $S\_1$ as the base and $(n+1)$-dimensional CW comple... | https://mathoverflow.net/users/17787 | fibre bundle as a boundary of a fibre bundle | I'll answer your questions with "CW complex" replaced by "manifold".
The answer to question (1) is no. If I understand you correctly, you want the boundary of $F\_{n+1}$ to be $F\_n$. But some manifolds cannot be realized as boundaries of other manifolds. (The simplest examples are 4-manifolds with non-zero signature... | 6 | https://mathoverflow.net/users/284 | 142199 | 77,451 |
https://mathoverflow.net/questions/142208 | 9 | I don't know much of algebraic topology so the following question could be very silly. Let $G$ a finite subgroup of $U(n)$ that acts linearly (the action induced by the action of $U(n)$ on $\mathbb{C}^{n}$) and freely on the unit sphere $S^{2n-1}\subset \mathbb{C}^{n}$. Let $X$ be the quotient manifold $$X:=S^{2n-1}/G$... | https://mathoverflow.net/users/4971 | Second Stiefel Whitney class of quotients of odd spheres | The second Stiefel-Whitney class of $TX$ will vanish if and only if $X$ admits a spin structure and this will be the case if and only if the action of $G$ on the oriented orthonormal frame bundle of $X$ lifts to an action on the (unique) spin bundle of the round sphere. The total space of the oriented orthonormal frame... | 12 | https://mathoverflow.net/users/394 | 142211 | 77,456 |
https://mathoverflow.net/questions/142200 | 3 | **Background:**
The complete infinite binary tree (CIBT) has path cardinality of continuum size, where path cardinality refers to the size of the set of all paths from the root.
If we consider the random infinite binary tree with constant degree $1+\epsilon$, for an infinitesimal constant $\epsilon$, we get a parti... | https://mathoverflow.net/users/1320 | What is the path cardinality of an infinite binary tree with fractal boundary of constant dimension below 1? | Your title question seems to be answered by the following theorem:
**Theorem.** The set of infinite paths through any finitely branching tree is either countable or size continuum.
Proof. This is a consequence of the Cantor-Bendixson theorem. To summarize, suppose that $T$ is a finitely branching tree. Since we are... | 3 | https://mathoverflow.net/users/1946 | 142212 | 77,457 |
https://mathoverflow.net/questions/142205 | 13 | I was reading about the monster group, and how hard it was to do calculations in it, and I wondered: Is there a known presentation of the monster group? I know that it is a hurwitz group, but other than that I don't know. If we have two generators a and b such that $a^2=b^3=(ab)^7=1$, what are the possible orders of $[... | https://mathoverflow.net/users/38744 | Presentation of the Monster Group | There's a 12-generator 80-relator presentation for the Monster group. Specifically, we have 78 relators for the Coxeter group Y443:
* $12$ relators of the form $x^2 = 1$, one for each node in the Coxeter-Dynkin diagram;
* $11$ relators of the form $(xy)^3 = 1$, one for each pair of adjacent nodes;
* $55$ relators of ... | 23 | https://mathoverflow.net/users/39521 | 142216 | 77,460 |
https://mathoverflow.net/questions/142220 | 30 | Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original proof? If not, where can I find it?
Thank you for viewing.
Note: I am not expecting to find Fermat's handwritin... | https://mathoverflow.net/users/38851 | Fermat's proof for $x^3-y^2=2$ | Fermat never gave a proof, only announced he had one (sounds familiar?). Euler did give a proof, which was flawed, see Franz Lemmermeyer's [lecture notes,](http://www.fen.bilkent.edu.tr/~franz/ant/ant01.pdf) or see page 4 of David Cox's [introduction.](http://math.stanford.edu/~lekheng/flt/cox.pdf)
For a discussion w... | 32 | https://mathoverflow.net/users/11260 | 142225 | 77,465 |
https://mathoverflow.net/questions/142206 | 2 | More specifically, consider the following particular situation: Let $I=[0,1]$ with the standard Borel $\sigma$-algebra. Consider functions $y:I\times I\to I$. Say that $y$ is *scalarly* measurable iff $y(\cdot,s)$ is measurable for every fixed $s\in I$. Say that $y$ is *simple* iff there is a partition of $I$ into fini... | https://mathoverflow.net/users/12643 | Is scalarly measurable simply measurable? | Since the total cardinality of the set of sequences of finite Borel partitions is continuum, you might just as well ask if there is a universal pointwise approximation scheme for all Borel functions on $[0,1]$ by simple functions. The answer is "No". Suppose that $P\_k$ is any fixed family of Borel partitions. WLOG $P\... | 6 | https://mathoverflow.net/users/1131 | 142226 | 77,466 |
https://mathoverflow.net/questions/141872 | 3 | Consider an $N\times K$ random matrix $X$ (defined on a probability space $(Ω,F,μ)$) with i.i.d. entries having zero mean and variance $1/K$.
There are a lot of results regarding the asymptotic behavior of the empirical distribution of eigenvalues of $XX^T$, or more precisely, the asymptotic behavior of the Stieltjes... | https://mathoverflow.net/users/39846 | Determining the asymptotic behavior of random matrices with vanishing ratio dimensions | There are many ways to see that, maybe the simplest is the following: let $W=XX^T/K$.
Compute $E Tr W^k$ as $K,N\to\infty$. The combinatorics is easy - essentially, the only terms that survive passage to the limit are the terms that involve only diagonal terms
of $W$, which converge to $1$. This shows that that the exp... | 3 | https://mathoverflow.net/users/35520 | 142229 | 77,468 |
https://mathoverflow.net/questions/142234 | 9 |
>
> **Question:** Let $S$ be a graded ring and $ f \in S\_+$. Does the ring $ S\_{(f)}$ which consists of degree $ 0$ elements of $ S\_f$ represent a nice functor?
>
>
>
**Motivation:** Let $ X = {\rm Spec} A$. Assume that $D(f) \subseteq D(g)$. It is messy to think about the restriction map $ A\_g \to A\_f ... | https://mathoverflow.net/users/4002 | Is there a universal property for graded localization? | As for your motivation: First of all, $S\_f$ satisfies the same universal property in the category of graded commutative rings (in fact, there is a theory of localization for objects or algebras in arbitrary cocomplete tensor categories). From this one deduces a natural isomorphism of graded rings $S\_{fg} \cong (S\_f)... | 10 | https://mathoverflow.net/users/2841 | 142238 | 77,470 |
https://mathoverflow.net/questions/142244 | 5 | Let $V$ be an $n$-dimensional vector space and $L$ be a lattice in $V$, i.e. a free $\mathbb{Z}$-module of full rank. Define the following two numbers. Let $\lambda$ be the minimal positive real number such that the ball centered at the origin of radius $\lambda$ contains a basis of the lattice. Let $\mu$ be the minima... | https://mathoverflow.net/users/38767 | a question on lattice invariants | Yes, as long as $n \geq 5$. The standard example is the lattice generated by
${\bf Z}^n$ and $(\frac12,\frac12,\ldots,\frac12)$. Then $\mu=1$ and
$\lambda = n/4$.
| 7 | https://mathoverflow.net/users/14830 | 142248 | 77,476 |
https://mathoverflow.net/questions/142243 | 24 | Terms like "impractical" and "unfeasible" are used to say the Robertson, Sanders, Seymour, and Thomas proof of the four color theorem needs computer assistance. Obviously no precise measure is possible, for many reasons.
But is there an informed rough estimate what a graph theorist would need to verify the 633 reduci... | https://mathoverflow.net/users/38783 | Can one measure the infeasibility of four color proofs? | To answer the question it is important to disentangle the proof as follows.
**Theorem 1.** Every minimum counterexample to the 4CT is an internally 6-connected triangulation.
**Theorem 2.** If $T$ is a minimum counterexample to the 4CT, then no *good configuration* appears in $T$.
**Theorem 3.** For every intern... | 36 | https://mathoverflow.net/users/2233 | 142250 | 77,478 |
https://mathoverflow.net/questions/124868 | 5 | Say that a preordering $\le$ on a set of subsets of some space preserves strict inclusion provided that $A\lt B$ whenever $A\subset B$ (where $A\lt B$ iff $A\le B$ and $B \not\le A$).
Let the space be the unit circle $T$. Then it's easy to see that there is no preorder on all countable subsets of $T$ that preserves ... | https://mathoverflow.net/users/26809 | Rotation-invariant strict-inclusion-preserving preorderings on subsets of the circle | The answer is positive, given Choice. It turns out that whenever $G$ is an abelian group acting on a set $X$, then there is a $G$-invariant preorder $\le$ on $2^X$ such that if $A$ is a proper subset of $B$, then $A<B$.
The proof uses the fact that $G$-invariant partial orders on $\Omega$ extend to $G$-invariant line... | 1 | https://mathoverflow.net/users/26809 | 142257 | 77,479 |
https://mathoverflow.net/questions/142235 | 1 | From Pollack's table on his homepage, I have the values of mu invariant of elliptic curves 38B1 & 38B2 (labeled as in Cremona table). But I need to know the values of mu invariants of 38A1, 38A2, 38A3. Also please inform me, how to calculate the mu invariants of elliptic curves in general.
| https://mathoverflow.net/users/30999 | Determining $\mu$-invariant of elliptic curves over $\mathbb{Q}$ | At what prime ? Well, let $p$ be an odd prime at which $E$ has good ordinary reduction. There are two ways:
* Use sage or magma to compute the analytic $p$-adic L-function via modular symbols. In sage this is E.padic\_lseries(p).series(n) where $n$ is the precision to which the series is computed. The $\mu$-invariant... | 6 | https://mathoverflow.net/users/5015 | 142271 | 77,483 |
https://mathoverflow.net/questions/142284 | 7 | Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with vertices at the singular points of the euclidean structure of $S$).
**Question 1**: can $S$ be realized (as a polyhedra... | https://mathoverflow.net/users/36575 | Embedding of flat surfaces | It may be that this theorem of Burago & Zalgaller (partially) answers your question?
>
> **Theorem** (Burago-Zalgaller 1.7).
> Every polyhedron $M$ admits an isometric piecewise-linear $C^0$
> immersion into $\mathbb{R}^3$.
> If $M$ is orientable or has a nonempty
> boundary, then $M$ admits an isometric piece... | 9 | https://mathoverflow.net/users/6094 | 142286 | 77,489 |
https://mathoverflow.net/questions/142269 | 2 | **Definition (1):** An $\mathcal{L}$ - structure $\mathcal{M}$ called "rigid" iff there is no non-trivial automorphism on $\mathcal{M}$.
**Definition (2):** An $\mathcal{L}$ - structure $\mathcal{M}$ called "non-rigid" iff it is not a rigid model.
**Def... | https://mathoverflow.net/users/nan | Can we flex the rigid models by enough power? | $\textbf{Question 1}$ For question $1$, the answer is affirmative.
Suppose that $\mathcal{M}$ is an infinite structure over some language $\mathcal{L}$. Then take a sufficiently large ultrapower $\mathcal{M}^{\mathcal{U}}$ (if necessary) such that there exists distinct points $a,b\in\mathcal{M}^{\mathcal{U}}$ such th... | 2 | https://mathoverflow.net/users/22277 | 142298 | 77,492 |
https://mathoverflow.net/questions/142290 | -2 | We need to plot the real and imaginary parts of a complex function $k(\omega)$, and cannot find a good way to do this without using "ad hoc tricks."
**Definitions**
* $k$ is a complex-valued function where both the real and the
imaginary parts are functions of $\omega$ so that $k(\omega) =
\beta(\omega) - i\alpha(... | https://mathoverflow.net/users/30924 | Howto plot a specific complex function | You can transform (1) into the equation of the flow of a 2-dimensional real vector field with time $\omega$ (just differentiate it with respect to $\omega$). Then you plug the resulting system into any solver of differential equation. If $\omega$ is to take complex values then matters can get harder, but as I understan... | 2 | https://mathoverflow.net/users/24309 | 142306 | 77,495 |
https://mathoverflow.net/questions/142297 | 10 | Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction?
The answer does not follow from mere existence of Tarski's decision routine since the natural definition of constructible quantifies ov... | https://mathoverflow.net/users/38783 | Can Tarski decide constructibility in elementary geometry? | Here is a partial affirmative answer, where we consider existence-and-uniqueness assertions rather than just existence assertions.
Consider the collection of algebraic reals, which form a real-closed field, and give each such algebraic real a finite name, of the form, "the $k^{th}$ solution of $p(x)=0$", for a specif... | 6 | https://mathoverflow.net/users/1946 | 142307 | 77,496 |
https://mathoverflow.net/questions/142301 | 11 | Let $n$ be a positive integer, we consider partitions of the following form :
$$n = d^{2}\_{1} + d^{2}\_{2} + ... + d^{2}\_{r}$$ such that :
* $d\_{i}\vert n$
* $1=d\_{1}<d\_{2} \le d\_{3} \le ... \le d\_{r}$
* $gcd(d\_{2},d\_{3},...,d\_{r}) = 1$
We call $r$ the **rank** of the partition.
**Examples** :
* ... | https://mathoverflow.net/users/34538 | A problem on a specific integer partition | **Update:** Sebastien and I just found out that the equation $X\_1^2+\dots+X\_r^2=mX\_1\dots X\_r$ which evolved in the comments below is a classical topic, named the *Hurwitz equation*. See [this encyclopedia entry](http://www.encyclopediaofmath.org/index.php/Hurwitz_equation). It was actually discussed on MO before, ... | 20 | https://mathoverflow.net/users/18739 | 142309 | 77,498 |
https://mathoverflow.net/questions/142305 | 0 | I would like to calculate the limit value of a linear functional
\begin{equation}
\lim\_{n\rightarrow\infty}\mathcal{I}\_n=\lim\_{n\rightarrow\infty}\frac{1}{n}\sum\_{i=1}^n f(\lambda\_i)=\lim\_{n\rightarrow\infty}\int f(x)\mathrm{d}G\_n(x),
\end{equation}
where $\lambda\_i$ are eigenvalues of a $n\times n$ matrix and ... | https://mathoverflow.net/users/40063 | Can I obtain the limit value of a linear spectral statistics using Stieltjes transform? | Let me write the limit value $I$ which you are seeking as
$$I=\int f(x)\rho(x)dx$$
where $\rho=dG/dx$ is the eigenvalue density, with Stieltjes transform $S\_\rho$. The [Stieltjes-Perron inversion formula](http://en.wikipedia.org/wiki/Stieltjes_transformation) reads
$$\rho(x)=\lim\_{\epsilon\rightarrow 0^+}\frac{... | 0 | https://mathoverflow.net/users/11260 | 142327 | 77,505 |
https://mathoverflow.net/questions/142319 | 3 | Let $M$ and $N$ be topological spaces. Are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?
For $M$ compact and $N$ a metric space, the space is obviously metrizable using the uniform convergence topology, $d(f,g)=\sup\_{x\in M}d(f(x),g(x))$.
An... | https://mathoverflow.net/users/40076 | Metrization of spaces of functions | As to the compact-open topology of $C(X,Y)$, it is metrizable if and only if $Y$ is metrizable, and $X$ is [hemicompact](https://en.wikipedia.org/wiki/Hemicompact_space).
| 11 | https://mathoverflow.net/users/6101 | 142328 | 77,506 |
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