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https://mathoverflow.net/questions/145634 | 1 | Let $X$ be a fixed algebraic manifold over $\mathbb{C}$ , $\{E\_{a}\}$ be vector bundles over $X$. We can construct moduli space of $\{E\_{a}\}$ by classical theory. My question is that if we consider the category $\mathcal{C}$ of two-term complexes$\{E\_1 \xrightarrow{\phi} E\_2\}$ where $E\_1,E\_2$ are vector bundles... | https://mathoverflow.net/users/40042 | moduli space of two-term complexes of vector bundles over a fixed variety | paper:Moduli spaces of holomorphic triples over compact Riemann surfaces
| 1 | https://mathoverflow.net/users/40042 | 145814 | 78,907 |
https://mathoverflow.net/questions/145811 | 4 | Let $G$ be an abelian locally compact group and $H$ be its closed subgroup. It is known from Pontryagin duality theory that every unitary character of $H$ can be extended to $G$. I think this is true for any character. Am I right? What is the reference?
| https://mathoverflow.net/users/41837 | Extension of characters of abelian locally compact groups | If you mean that every continuous homomorphism $H\to\mathbf{R}$ can be extended to a continuous homomorphism $G\to\mathbf{R}$, this is contained in Theorème 5 of [1] (the "unitary" version is when $\mathbf{R}$ is replaced twice by $\mathbf{R}/\mathbf{Z}$).
[1] J. Dixmier. *Quelques propriétés des groupes abéliens loc... | 4 | https://mathoverflow.net/users/14094 | 145819 | 78,909 |
https://mathoverflow.net/questions/145475 | 14 | Namely, the following one
>
> "All problems appeared once in the [American Mathematical] *Monthly*."
>
>
>
I remember reading it several years ago... When I first posed the question, I believed that I had read it somewhere in Krantz' *Mathematical Apocrypha* but according to [Carlo Beenakker](https://mathoverf... | https://mathoverflow.net/users/1593 | What is the source of this E̶r̶d̶ő̶s̶ quote? | In [Index to Mathematical Problems, 1980-1984 - Page xi](http://books.google.de/books?id=n-F52zK2UAgC&pg=PR11&dq=%22all%20%20problems%22%20%20%22american%20mathematical%20monthly%22&hl=en&sa=X&ei=n0ZqUrn0JYLWtAa3roDwDw&redir_esc=y#v=onepage&q=%22all%20%20problems%22%20%20%22american%20mathematical%20monthly%22&f=false)... | 22 | https://mathoverflow.net/users/39495 | 145824 | 78,912 |
https://mathoverflow.net/questions/145822 | 6 | If you sample $n$ integers from the range $1$ to $n$ inclusive it seems intuitive that you are likely to get a lot of numbers exactly once. Call $X\_n$ the number of integers you get that occur exactly once in your sample. Is there a nice simple way of showing the following for all $n\geq 2$.
$$\exists a,b >0\text{ s... | https://mathoverflow.net/users/45564 | Probability of having many unique elements | $X\_n$ is a random variable on the product space $\Omega = \{1,\ldots,n\}^n$ such that changing the value of any one coordinate of $\omega \in \Omega$ changes $X\_n(\omega)$ by at most 2. This is a classic case in which [martingale concentration inequalities](https://en.wikipedia.org/wiki/Azuma%27s_inequality) can be a... | 8 | https://mathoverflow.net/users/25485 | 145825 | 78,913 |
https://mathoverflow.net/questions/145799 | 8 | I'm essentially trying to figure out exactly what the title asks for. I've been scouring old Seminaires Henri Cartan and books by Stong to try to see exactly how to do this, but the combination of French and older terminology has made it rough going.
In particular, I've been able to work out (I think!) that, modulo ... | https://mathoverflow.net/users/11546 | The Image of the Mod 2 Homology of BSp in the Homology of BSO | We have $H^\*(BO, \mathbb{F}\_2) = \mathbb{F}\_2[w\_1, w\_2, \ldots]$, where the $w\_i$ are the Stiefel--Whitney classes. If $f: \mathbb{RP}^\infty \to BO$ classifies the reduced universal real line bundle, and $e\_i \in H^i(\mathbb{RP}^\infty;\mathbb{F}\_2)$ is the nontrivial class, we let $a\_i = f\_\*(e\_i)$ and the... | 7 | https://mathoverflow.net/users/318 | 145834 | 78,916 |
https://mathoverflow.net/questions/145830 | 1 | ***Notations and terminology***: Let $k$ be a field and $X$ be a $k$-scheme. Denote by $X\_L$ the scheme $X\times\_k\rm Spec(L)$. For a field extension $L/k$, *a $L/k$ form* is a $k$-scheme $Y$ such that there is a $L$ isomorphism $X\_L\cong Y\_L$. For a galois extension $L/k$ with the galois group $\rm Gal(L/k)$, we d... | https://mathoverflow.net/users/31747 | $L/k$ forms for affine schemes of finite type | See [Serre, Galois cohomology], Chapter III, §1 Forms, and the references there ([Serre, Local Fields], X.§2, and [Giraud, Cohomologie Non Abélienne]).
| 2 | https://mathoverflow.net/users/nan | 145836 | 78,917 |
https://mathoverflow.net/questions/145862 | 24 | Let $(A\_i)\_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A\_kX^k+\cdots+A\_1X+A\_0=0\_n$ where $X\in\mathcal{M}\_n(K)$ is unknown. Let $g = \det(\lambda^kA\_k+\cdots+\lambda A\_1+A\_0)\in K[\lambda]$.
Question: is it true that, if $B$ is a solutio... | https://mathoverflow.net/users/9091 | Cayley-Hamilton revisited | Yes, this follows from known facts on matrix polynomials. There is a full characterization of spectral divisors of matrix polynomials in Gohberg, Lancaster, Rodman, *Matrix Polynomials*. They treat monic polynomials (i.e., $A\_k=1$), but this is not a restriction, since one can make a Möbius transform to enforce it unl... | 16 | https://mathoverflow.net/users/1898 | 145871 | 78,924 |
https://mathoverflow.net/questions/145877 | 7 | In Unsolved Problems in Number Theory, 3rd edition, section D1, 2004, R. K. Guy says, "Andrew Bremner has computed the rational rank of the elliptic curve x^3 + y^3 = Taxicab(n) as equal to 2, 4, 5, 4 for n = 2, 3, 4, 5, respectively." Since Taxicab(1) = 2, my question can be restated:
What is the rational rank of th... | https://mathoverflow.net/users/11930 | What is the rational rank of the elliptic curve x^3 + y^3 = 2? | It's isomorphic to $y^2 = x^3 - 27$, which has rank $0$.
| 13 | https://mathoverflow.net/users/2698 | 145879 | 78,926 |
https://mathoverflow.net/questions/145880 | 4 | Questions a bit similar to this one have already appeared I think on the forum but I couldn't find the answer to my question using those answers. I must say from the beginning that my knowledge of group theory is very very basic so please adapt your answers.
Let $G$ be a reductive group over an algebraically closed f... | https://mathoverflow.net/users/1328 | Centralizer of a subtorus in a reductive group is Levi? | Yes, it is true. You can look at the book of Digne-Michel Proposition 1.22 for a proof.
| 7 | https://mathoverflow.net/users/41870 | 145892 | 78,934 |
https://mathoverflow.net/questions/145898 | 12 | We know the following facts:
**(1)** For all $1\leq n\leq 2$ the equation $x\_{1}^{n}+x\_{2}^{n}=x\_{3}^{n}$ has a solution in $\mathbb{N}$.
**(2)** For all $3\leq n$ the equation $x\_{1}^{n}+x\_{2}^{n}=x\_{3}^{n}$ has no solution in $\mathbb{N}$.
**Question:** Is the following generalization true?
For all $2\l... | https://mathoverflow.net/users/nan | On Generalizations of Fermat's Conjecture | No this is not true. For $n=4$ one has
$$2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$$
found by Elkies (as part of an infinite family of solutions).
Also earlier it was known, for example, for $n=5$,
$$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$
by Lander and Parkin.
But part 2 was a [conjecture of Euler](http://en... | 25 | https://mathoverflow.net/users/nan | 145899 | 78,937 |
https://mathoverflow.net/questions/145868 | 2 | Consider a matrix function $A(x)$, *analytically* depending on single parameter $x$.
Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$.
The question is whether we can construct the analytic functions $\lambda(x)$ and $w(x)$ that are the eigenvalue/eigenvector of $A(x)$.
(I assume th... | https://mathoverflow.net/users/41863 | Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix | This is completely answered in [Kato's Perturbation theory for linear operators,](http://rads.stackoverflow.com/amzn/click/354058661X) chapter 1 (he treats the non-hermitian case). Eigenvalues are generally analytic, eigenvectors not so much (in the neighborhood of values of $A(x)$ with multiple eigenvalues.
| 4 | https://mathoverflow.net/users/11142 | 145901 | 78,939 |
https://mathoverflow.net/questions/145904 | 6 | I wonder if the following simple generalization of Johnson and Kneser
graphs has a name? Let the vertex set of the graph $G(n,k,t)$ be the
set of $k$-element subsets of an $n$-set, with two $k$-sets adjacent if their
intersection has cardinality $t$.
So for $t=0$ we have the Kneser graphs, and for $t=k-1$ we have the... | https://mathoverflow.net/users/39684 | Name for Kneser/Johnson-like graphs? | Chen and Lih call such a $G(n,k,t)$ (with the same notation) a *uniform subset graph*; see ["Hamiltonian uniform subset graphs" (1987)](http://www.sciencedirect.com/science/article/pii/009589568790044X). As you point out, the Johnson graph $J(n,k)$ is then $G(n,k,k-1)$. And the Kneser graph $KG(n,k)$ corresponds to $G(... | 2 | https://mathoverflow.net/users/4137 | 145905 | 78,940 |
https://mathoverflow.net/questions/145495 | 1 | Suppose $G$ is a simple graph and $V(G)=V(C)\bigcup \{u\_1,...,u\_n\}$,where $C$ is a $2n$-cycle in $G$ and $V(C)=\{a\_1,...,a\_n,b\_1,...,b\_n\}$ such that
$(1)V(C)\bigcap \{u\_1,...,u\_n\}=\varnothing$;
$(2)E(G)=\{a\_1u\_1,u\_1b\_1,a\_2u\_2,u\_2b\_2,...,a\_nu\_n,u\_nb\_n\}\bigcup E(C)$.
I think there must exis... | https://mathoverflow.net/users/40096 | a conjecture about Hamiltonian graph | The conjecture is right.It is equivalent to the claim below:
Let G be a simple graph which is a $2n$-cycle equipped with $n$ chords such that $G$ is $3$-regular,in other words,the set of the $n$ chords is a perfect matching of $G$(that is,every vertex of $G$ is matched).Then there must exist at least two different $2... | 2 | https://mathoverflow.net/users/40096 | 145908 | 78,943 |
https://mathoverflow.net/questions/145500 | 1 | We use $N^+$ to denote the set of positive integers. For any finite array $A:(a\_1,b\_1),...,(a\_k,b\_k)$, where every $(a\_i,b\_i)\in N^+\times N^+$, we call $A$ is good if for every $i\in \{a\_1,b\_1,...,a\_k,b\_k\}$, $i$ appears exactly two times in $a\_1,b\_1,...,a\_k,b\_k$. For example, $A\_1:(1,1)$,$A\_2:(1,2),(1... | https://mathoverflow.net/users/40096 | The "compact good array" in a "good $N^+$-cycle" | The conjecture is right. It is equivalent to the claim below:
Let G be a simple graph which is a $2n$-cycle equipped with $n$ chords such that $G$ is $3$-regular, in other words, the set of the $n$ chords is a perfect matching of $G$ (that is, every vertex of $G$ is matched). Then there must exist at least two differ... | 0 | https://mathoverflow.net/users/40096 | 145909 | 78,944 |
https://mathoverflow.net/questions/145913 | 5 | To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is *Condition C* of Palais and Smale (note that this is automatically satisfied for $M$ compact). It ultimately implies (for instance... | https://mathoverflow.net/users/12310 | Failure of Palais-Smale Condition C and the Mini-Max Principle | There are some works on this, but maybe THE expert on this topic is Abbas Bahri. Take a look at his works or google "critical point at infinity".
See, for example: [this paper](http://link.springer.com/chapter/10.1007/BFb0100779) or his book ""Critical Points at Infinity in Some Variational Problem".
| 2 | https://mathoverflow.net/users/21123 | 145929 | 78,947 |
https://mathoverflow.net/questions/145969 | 0 | Let $aSS$ / $abSS$ be the category of augmented bi/simplicial sets (one can also consider $SS$/$bSS$ be the usual bi/simplicial sets, the results should related in some reasonable way.)
There is an obvious functor $n$-th row $R\_n$ sending an augmented bisimplicial set $X\_{\bullet,\bullet}$ to $X\_{n,\bullet}$. What... | https://mathoverflow.net/users/7341 | What is the left adjoint for taking rows of a bisimplicial set? | I'll answer the second question, as it subsumes the first. Indeed, if $[n]: 1 \to \Delta$ names the simplex of dimension $n$, then the $n^{th}$-row functor of the question is $f^\ast$ where $f = \Delta \simeq 1 \times \Delta \stackrel{[n] \times 1\_\Delta}{\to} \Delta \times \Delta$.
If $A$ and $B$ are small categor... | 3 | https://mathoverflow.net/users/2926 | 145975 | 78,962 |
https://mathoverflow.net/questions/47944 | 9 | Could you give me an example of a compact Kähler manifold which analytically deforms to a non Kähler one?
For example, there is no hope to find a complex structure on a Hopf manifold in order to make it Kähler because of topological obstructions (the second Betti number is zero).
For instance, I think that the Iwas... | https://mathoverflow.net/users/9871 | Deform a compact Kähler manifold to a non Kähler one | (Just so this question has an answer. All manifolds are compact.)
In **dimension one** every deformation of Kähler manifolds is Kähler because every Riemann surface is Kähler.
In **dimension two** the same is true but for less trivial reasons. A two-dimensional complex manifold is Kähler if and only if its first B... | 7 | https://mathoverflow.net/users/21564 | 145976 | 78,963 |
https://mathoverflow.net/questions/145962 | 5 | Let prime $p$ and given $\zeta\_p = e^{2\pi i/p}$. It is well-known that the minimal polynomial of $x = \zeta\_p + \zeta\_p^{p-1}$ has a constant term either $\pm 1$ and, for certain $p$, the sum of four terms $x = \zeta\_p +\zeta\_p^{a}+\zeta\_p^{p-a}+\zeta\_p^{p-1}$ have as well. However, we also have,
$$F(x) = x^5... | https://mathoverflow.net/users/12905 | Minimal polynomial of sums of roots of unity with constant term $\pm1$ | This is not a complete answer, but a reformulation of your question in a way that removes the algebraic number theory, which translates the question into the realm of additive combinatorics. I assume that you also require the order of $a$ in $(\mathbf{Z}/p\mathbf{Z})^\*$ to be $h$, as in your examples and Noam's commen... | 3 | https://mathoverflow.net/users/30412 | 145984 | 78,966 |
https://mathoverflow.net/questions/145775 | 9 | Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they connect their assigned faces by a path (potentially using diagonal moves). Once a vertex is chosen, it cannot be chosen b... | https://mathoverflow.net/users/31437 | Infinite-dimensional hex | It seems the claim is false, though I don't have a good sense for what the actual counterexample looks like. With luck, I've made an error somewhere.
Consider the normed space $\ell^{\infty}$. [One can show](http://www.ams.org/journals/proc/1983-088-03/S0002-9939-1983-0699410-7/S0002-9939-1983-0699410-7.pdf) that th... | 3 | https://mathoverflow.net/users/31437 | 145985 | 78,967 |
https://mathoverflow.net/questions/145983 | 1 | Let $G$ be a connected reductive group acting on a projective variety $X$. Let
$L$ be a $G$-linearized (very) ample line bundle on $X$, and let $R$ be the ring
of sections (or homogeneous coordinate ring of $(X, L)$). Let $R\_k = H^0(X,
L^{\otimes k})$ denote the $k$-th graded piece of $R$, and $R\_{k, \lambda}$ its $\... | https://mathoverflow.net/users/21491 | Continuity of volume of GIT quotients | Let's start with the case of a torus. Since you don't require $X$ to be smooth, we can reduce to this case by replacing $X$ by $X//N$. (Though actually inferring results about the nonabelian case from the abelian looks a bit hard.)
Since the DH function is piecewise continuous (even polynomial) on polyhedral chambers... | 4 | https://mathoverflow.net/users/391 | 145986 | 78,968 |
https://mathoverflow.net/questions/145941 | 2 | If you look through papers on the Ellipsoid Method, there is a large agreement, that the Ellipsoid Method, although theoretically polynomial, is in practice way slower than the Simplex Method. Apparently, I was not able to find a single article that really backs up this claim by a survey, a meta-survey, some instances ... | https://mathoverflow.net/users/41187 | Survey on Compared Running Time: Ellipsoid Method vs. Simplex Method | It’s not a survey. Nevertheless, [this paper](http://www.sciencedirect.com/science/article/pii/016754198490019X) can be interesting as we can read in the abstract:
“The computer experiment compared the two algorithms performance in solving SX exponential time problems. This represented a best possible case (for KA) a... | 1 | https://mathoverflow.net/users/34050 | 145987 | 78,969 |
https://mathoverflow.net/questions/145989 | 8 | (To clarify, my interest is mainly *lightface*, that is, $\Pi^1\_2$ instead of **$\bf \Pi^1\_2$**, although it doesn't particularly matter.)
This is just an idle curiosity. In logic, I find myself frequently working with the countable ordinals; this is a $\Pi^1\_1$ set of structures, in the sense that the set of real... | https://mathoverflow.net/users/8133 | Natural $\Pi^1_2$ (or worse) classes of structures? | An example that comes to mind is the set of recursive dilators; this is a $\Pi^1\_2$-complete set (Theorem 4.1 in Girard 1985, [Introduction to $\Pi^1\_2$-logic](http://dx.doi.org/10.1007/BF00486046)). Dilators have some use outside of logic (e.g.: Some uses of dilators in combinatorial problems [I](http://dx.doi.org/1... | 7 | https://mathoverflow.net/users/2004 | 145993 | 78,972 |
https://mathoverflow.net/questions/145991 | 3 | This is probably fairly elementary, but does someone know how to prove the following or know a reference.
Let $X$ be a Kaehler manifold. Let $\theta$ be a closed $(1,1)$-form and $T$ be a closed positive $(1,1)$-current cohomologous to $\theta$. Then there is a quasi-plurisubharmonic function $\psi$ so that
$$ T=\... | https://mathoverflow.net/users/10749 | ddbar lemma for positive closed (1,1)-currents | If I remember correctly, all classic Kähler identities between operators $d, d^c \dots $ of differential forms are also satisfied by the corresponding operators of currents. A reference for this could be L. SCHWARTZ, Lectures on Complex Analytic Manifolds, Tata Inst. Fund. Res. Lectures on Math. and Phys. 4, Springer, ... | 1 | https://mathoverflow.net/users/1246 | 145995 | 78,974 |
https://mathoverflow.net/questions/145961 | 1 | Let $C = \{1,...,n\}$ be a set of $n$ colors. Let $S\_1,...,S\_k$ be non-empty subsets of $C$, that is, $S\_i \subseteq C$ for all $i \in \{1,...,k\}$. It is helpful to think of the $S\_i$ as urns with colored balls.
We now draw one ball from each $S\_i$.
Given $b\_1,...,b\_n$ with $b\_1 +...+ b\_n = k$. How many w... | https://mathoverflow.net/users/17188 | Draws from multiple non-disjoint urns | The answer is the coefficient of $x\_1^{b\_1}\cdots x\_n^{b\_n}$ in
$$\prod\_{i=1}^k \biggl(\sum\_{j\in C\_i} x\_j\biggr).$$
It's unlikely that anything more useful can be said in the general case.
| 3 | https://mathoverflow.net/users/10744 | 145996 | 78,975 |
https://mathoverflow.net/questions/146000 | 6 | Let $F$ be a p-adic field (finite extensions of $\mathbb{Q}\_p$ for some prime $p$), and $E/F$ be a quadratic extension. Use $\sigma$ to denote the nontrivial element in the Galois group $Gal(E/F)$. For any $x\in E$, $N(x)=x\sigma(x)$ denotes the norm of $x$.
Now consider $U(1)=\{x\in E; N(x)=1 \}$, the group of elem... | https://mathoverflow.net/users/1832 | structure of norm one group for quadratic extension of p-adic fields | The answer will depend upon whether your quadratic extension is unramified, tamely ramified, or wildly ramified. A good place to start would be the chapter on the norm map in Fesenko-Vostokov (Ch. III in [Local fields and their extensions](https://www.maths.nottingham.ac.uk/personal/ibf/book/book.html)) or the correspo... | 9 | https://mathoverflow.net/users/2821 | 146003 | 78,977 |
https://mathoverflow.net/questions/145951 | 3 | Does fixing the reparameterization invariance of the string action, for example by choosing the light-cone gauge
$$
X^{+} = \beta\alpha' p^{+}\tau
$$
$$
p^{+} = \frac{2\pi}{\beta} P^{\tau +}
$$
correspond to some kind of orbifolding?
This [answer](https://physics.stackexchange.com/a/82141) explains that gauge ... | https://mathoverflow.net/users/30967 | Does fixing the reparameterization invariance of the string action correspond to some kind of orbifolding? | Maybe you are confusing the worldsheet and the spacetime. What is usually called an "orbifold" in the string theory context is a spacetime orbifold. The corresponding perturbative string theory is constructed from a sigma model whose target space is an orbifold, obtained as the quotient of the action of a finite group ... | 3 | https://mathoverflow.net/users/2183 | 146014 | 78,981 |
https://mathoverflow.net/questions/104533 | 6 | Let me start saying that a similar question can be stated for general locally Noetherian Grothendieck categories but I state it for categories of modules as it is simpler. So we fix a right Noetherian ring (non-commutative) and we let $Mod(R)$ be the category of right $R$-modules.
Recall the following theorem due to ... | https://mathoverflow.net/users/24891 | (Co)localization of the derived category | I think the answer is no in the following very simple example.
Let $R$ be the path algebra of the quiver $\bullet\rightarrow\bullet$ over a field $k$. Then since $R$ is hereditary and of finite representation type, the objects of ${\bf D}(R)$ are just coproducts of shifts of copies of the three indecomposable represe... | 2 | https://mathoverflow.net/users/22989 | 146024 | 78,986 |
https://mathoverflow.net/questions/146025 | 5 | Theorems $16.1$ and $16.3$ in Billingsley Convergence of measures.
$16.1$ reads : Random variables $u\_1,\ldots$ on $(\Omega,\mathcal{B},\mathbb P)$
and are i.i.d. with $0$ mean and finite variance $\sigma^{2}$, define $X\_n(t,\omega) = \frac{1}{\sigma\sqrt{n}}S\_{[nt]}(\omega)$ (here $t\in [0,1]$). Then $X\_n \to W$... | https://mathoverflow.net/users/nan | Donsker Theorem Billingsley | I think the statement of Theorem 16.3 Billingsley meant is
>
> Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $(\xi\_i,i\geqslant 1)$ be i.i.d. zero mean random variables (for $\mathbb P$), and $X\_n(t,\omega):=\frac 1{\sigma\sqrt n}\sum\_{i=1}^{\lfloor nt\rfloor}\xi\_i(\omega)$. Assume that $\mathbb ... | 5 | https://mathoverflow.net/users/17118 | 146029 | 78,988 |
https://mathoverflow.net/questions/146011 | 2 | Consider *injective* homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$.
Thus for $|z|\leq 1$ we have $ f(z)=\sum\_{k=0}^{\infty} a\_k z^k $ with $a\_0=0$ and $a\_1=1$.
Ludwig Bieberbach conjectured in 1916 and Louis de Branges proved ... | https://mathoverflow.net/users/30081 | hayman's result for $ A^2(D) $ | I am not sure what "analogous" means but the estimate $|a\_n|\leq n$ is best possible
when you restrict to $H^2$, and even when you restrict to polynomials. For the simple reason that the extremal function can be approximated by polynomials injective in the unit disc.
| 3 | https://mathoverflow.net/users/25510 | 146045 | 78,997 |
https://mathoverflow.net/questions/145750 | 0 | Let us call the sum of the first half and the latter half of the cyclic numbers of an irreducible fraction '**a division sum**' when the period of a repeating decimal is **even**. Also, let $\lambda(l)$ be the length of the repeating digits of $\frac 1l$ in decimal expansion.
**Example 1** : The division sum of $\fra... | https://mathoverflow.net/users/34490 | About the sum of the first half and the latter half of the cyclic numbers of a repeating decimal | I'm posting an answer just to inform that the question has received an answer by mercio on MSE.
<https://math.stackexchange.com/questions/529871/about-the-sum-of-the-first-half-and-the-latter-half-of-the-cyclic-numbers-of-a-r>
| 0 | https://mathoverflow.net/users/34490 | 146049 | 78,998 |
https://mathoverflow.net/questions/145938 | 2 | It seems to me that the ["indization"](http://ncatlab.org/nlab/show/ind-object) process of a category can be formulated in the language of sketches (by sketch I mean what is defined in [LPAC].2.F, Def. 2.55); in particular, see [this](https://mathoverflow.net/a/80017/7952) answer by T. Johnson-Freyd.
Expressing the i... | https://mathoverflow.net/users/7952 | Is $\text{Ind-}\bf C$ the category of models for a sketch? | As suggested I repost my comment as an answer.
The paper Adámek, Borceux, Lack, Rosický, *A classification of accessible categories addresses exactly these types of questions*. In particular, Section 3 is about the Ind construction and Section 4 is about sketches. Everything is relative to some nice class of small ca... | 3 | https://mathoverflow.net/users/12547 | 146058 | 79,004 |
https://mathoverflow.net/questions/146056 | 2 | It is known that if $\sum\_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 1/2$ then RH holds. My question is:
>
> 1. Under RH why is it not $\sum\_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 0$?
> 2. Is it known that $\sum\_{k=1}^{\infty} \frac{\mu(k)}{k^s}$ diverges... | https://mathoverflow.net/users/2865 | On the convergence of Dirichlet series over the Mobius Mu function | We can not hope to the estimate of $M(n):=\sum\_{k\leq n} \mu(k)$ better than $M(n)=O(n^{1/2+o(1)})$. Take $s\in (0,1/2)$. Rewrite partial sum of our series as
$$\sum\_{k=n}^m (M(k)-M(k-1))/k^s=-M(n-1)/n^s+M(m)/m^s+\sum\_{k=n+1}^{m-1} M(k)(k^{-s}-(k+1)^{-s}).$$
Assume that both $n$, $m$ are large, $M(m)$ is much great... | 3 | https://mathoverflow.net/users/4312 | 146060 | 79,005 |
https://mathoverflow.net/questions/145957 | 17 | In short: **If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?**
Now more carefully, with some notation: Suppose $(X, d\_X)$ and $(Y, d\_Y)$ are metric spaces, with Borel sets $\mathcal B\_X$ and $\mathcal B\_Y$, respectively. Let $f:(X, \ma... | https://mathoverflow.net/users/8106 | Continuity on a measure one set versus measure one set of points of continuity | As Jason and Gerald predicted, the answer is *yes* for Polish $X, Y$. (Indeed, it is sufficient for $X$ to be merely separable and metrizable and for $Y$ to be merely complete and metrizable.)
As Nate observed, we may assume that $D$ is $G\_\delta$. Here is the key fact:
**Lemma.** *If $D \subseteq X$ is a nonempty... | 10 | https://mathoverflow.net/users/2000 | 146063 | 79,006 |
https://mathoverflow.net/questions/146062 | 3 | Let $X$ be a smooth compact Kahler manifold and let $Y\subset X$ be a smooth complex submanifold of complex codimension at least $2$. Let
$$j:Y\hookrightarrow X$$
the natural (holomorphic) embedding map. Let $E\rightarrow Y$ a holomorphic vector bundle over $Y$. Are there conditions (on $E$, $j$,$Y$,$X$) under which t... | https://mathoverflow.net/users/41933 | Push forward of a Vector bundle is a coherent sheaf? | The answer is that $j\_\*E$ is *always* coherent under your hypotheses, provided that $Y$ is closed in $X$.
This is a consequence of the the so-called *Extension Principle*, that can be stated as follows:
>
> **Proposition.** An analytic sheaf $\mathscr{S}$ on a closed complex subspace $Y$ of a complex space $X$... | 5 | https://mathoverflow.net/users/7460 | 146064 | 79,007 |
https://mathoverflow.net/questions/146015 | 1 | Let $V$ be an affine algebraic variety defined over $\mathbb R$.
We assume that $V\subset \mathbb A\_n$(affine $n$ space).
Suppose that for any algebraic curve $C$ in $V$ defined over $\mathbb R$, the real points $C(\mathbb R)$ is contained in a finite union of proper affine $\mathbb R$-subspaces of $\mathbb A\_n$. Is... | https://mathoverflow.net/users/11056 | curves in varieties | Yes. We can assume without loss of generality that the real points of $V$ are Zariski dense. Choose some linear projection $\rho: V \to \mathbb R^{\dim V}$ whose image includes some open ball, e.g. a generic linear projection. Then if $L$ is a linear subspace of codimension $1$ in $\mathbb R^n$, $\rho(V \cap L )$ is a ... | 3 | https://mathoverflow.net/users/18060 | 146078 | 79,014 |
https://mathoverflow.net/questions/146085 | 5 | I want to maximize a quadratic form $\mathbf x^T\mathbf Q\mathbf x$ and also want to find out which vector $\mathbf x$ maximizes the quadratic form when
1. $\mathbf Q$ is an $n\times n$ [positive definite](http://en.wikipedia.org/wiki/Positive-definite_matrix) matrix, and
2. $\mathbf x$ is a vertex of an $n$-dimensio... | https://mathoverflow.net/users/11361 | Maximizing quadratic form on the hypercube | This maxQP problem is hard, it includes MaxCUT as a special case---see this paper [by M. Charikar on MAXQP](http://www.cs.princeton.edu/~moses/papers/maxqp.ps). Having $Q$ be positive definite does not really help (take the MAXQP problem in Charikar's paper, and since $x\_i \in \{\pm1\}$, you can add a suitable multipl... | 7 | https://mathoverflow.net/users/8430 | 146087 | 79,017 |
https://mathoverflow.net/questions/146047 | 2 | A metric is said to have harmonic curvature if the exterior derivative of the Ricci tensor vanishes. It is known that there exists manifolds with dRic=0 that are not Einstein. My question is whether or not there are examples of manifolds that admit a metric with harmonic curvature but don't admit an Einstein metric.
| https://mathoverflow.net/users/15856 | harmonic curvature versus Einstein metric | There are many $3$-manifolds that admit a metric with harmonic curvature that don't admit an Einstein metric because in dimension $3$, 'Einstein' is equivalent to constant sectional curvature, while in dimension $3$, harmonic curvature is the same as conformally flat with constant scalar curvature.
Since one can tak... | 7 | https://mathoverflow.net/users/13972 | 146098 | 79,021 |
https://mathoverflow.net/questions/146101 | 7 | Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be said? In particular, is the following true?
>
> Every abelian Hausdorff topological group has a nontrivial continuo... | https://mathoverflow.net/users/41946 | A Hausdorff abelian group with no character? | The infinite-dimensional sphere $S^\infty$ (the evident colimit of finite-dimensional spheres) admits a topological group structure whose underlying abelian group is a torsion group (in fact a structure of $\mathbb{F}\_2$-vector space). See this answer here: <https://mathoverflow.net/a/43047/2926>.
If $\phi: S^\inft... | 14 | https://mathoverflow.net/users/2926 | 146104 | 79,025 |
https://mathoverflow.net/questions/146079 | 2 | I have two questions regarding Non-holomorphic Eisenstein series:
$E\_s(\tau)=\sum\_{m,n}\frac{Im(\tau)^s}{|m+n \tau|^{2s}}$
where the sum runs over all the integers and we exclude $(0,0)$. The questions are:
1.- Does anyone know the expression for $E\_s(i)$? I feel this should be known.
2.- Are the $E\_s$ orth... | https://mathoverflow.net/users/41940 | Non-holomorphic Eisenstein series | Hecke and Maass already knew that (your normalization of) $E\_s(i)$ is (4 times) the Dedekind zeta function of the Gaussian integers $\mathbb Z[i]$ evaluated at $s$.
You are correct that these Eisenstein series are not in $L^2$ for the $SL\_2(\mathbb R)$-invariant measure $dx\,dy/y^2$ descended to the (finite-volume)... | 4 | https://mathoverflow.net/users/15629 | 146106 | 79,027 |
https://mathoverflow.net/questions/146090 | 2 | Let $\overline{\mathcal{M}}\_{1,1}$ be the DM compactification of the moduli stack of elliptic curves. Its Picard group is $\mathbb{Z}$. Let us now consider stack of $r$-prym curves $\overline{\mathcal{M}}\_{1,1}^r$, parametrizing elliptic curves plus an $r$-root of $\mathcal{O}$. It is compactified naturally as an Hur... | https://mathoverflow.net/users/4096 | picard group of moduli of elliptic r-prym curves | The stack $\overline{\mathcal M}\_{1,1}^r$ is usually denoted by $X\_1(r)$. It is a modular curve. In fact one can write down by hand an isomorphism between the moduli functors for the Hurwitz stack interpretation of $\overline{\mathcal M}\_{1,1}^r$ that you mention, and the modular interpretation of $X\_1(r)$ describe... | 1 | https://mathoverflow.net/users/1310 | 146122 | 79,031 |
https://mathoverflow.net/questions/145032 | 4 | Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||\_{\infty} < \epsilon\right\}\right)$?
In finite dimensions this is straightforwardly equivalent to Brouwer's fixed point theorem. In the infinite dimensional case t... | https://mathoverflow.net/users/31437 | An approximate infinite-dimensional fixed point theorem | In fact the property fails on any non-compact closed convex set $K$ of a Banach space : there exists a Lipschitz map $f:K\to K$ with no approximate fixed point, that is $\inf\_{x\in K}\|f(x)-x\|=\delta > 0$ (See [*Geometric Nonlinear Functional Analysis- Part 1*](http://books.google.it/books?id=lXZ95EKwjYUC&printsec=fr... | 1 | https://mathoverflow.net/users/6101 | 146125 | 79,033 |
https://mathoverflow.net/questions/146119 | 6 | In Control of Distributed Singular Systems p 236, JL Lions makes the conjecture :
Let $\Omega$ be a domain in $\mathbb{R}^n$, $Q = \Omega \times ]0,T[$ and consider
$\phi'' - \triangle \phi = F$
$\phi(x,0) = \phi^0(x), \phi'(x,0) = \phi^{1}(x), \phi^0 \in H^1(\Omega), \phi^1 \in L^2(\Omega)$
$\frac{\partial \p... | https://mathoverflow.net/users/41952 | On a conjecture of Lions for the wave equation | You will find your answer in this article:
<http://www.sciencedirect.com/science/article/pii/0022247X89902059>
| 4 | https://mathoverflow.net/users/12120 | 146134 | 79,037 |
https://mathoverflow.net/questions/146048 | 0 | For $N\in\mathbb N$, let
$$P\_l(N)=\# \{(n,m)|0\le n\le N, 0\le m\le n,\binom{n}{m}\not\equiv 0 \mod l\}.$$
Suppose that $\binom{n}{0}=1$ for $n\ge 0$ and that $\# S$ represents the number of the elements of a set $S$.
Then, here are my questions.
**Question 1** : Find $\lim\_{N\to\infty}\log\_N P\_6(N)$.
**Ques... | https://mathoverflow.net/users/34490 | About the number of the elements of a set related with binomial coefficients | ${a \choose b} \not \equiv 0 \mod 6$ iff either ${a \choose b} \not \equiv 0 \mod 2$ or ${a \choose b } \not \equiv 0 \mod 3$. So, $$P\_3(N) \le P\_6(N) \le P\_2(N) + P\_3(N).$$
Since $P\_2(N) = \Theta(N^{\log\_2 3}) = \Theta(N^{1.585})$ is asymptotically negligible compared with $P\_3(N) = \Theta(N^{\log\_3 6}) = \... | 4 | https://mathoverflow.net/users/2954 | 146143 | 79,040 |
https://mathoverflow.net/questions/146115 | 4 | For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question to do so is to consider the composition of the following functors (yet is it "optimal"?).
1. Using the methods of secti... | https://mathoverflow.net/users/2191 | Yoneda embeddings of stable model categories; composition with Bousfield localizations | Mikhail, I think that what you're looking for is what is known as a *presentation* of a model category. There's a beautiful paper by Dugger proving that any *combinatorial* model category has a presentation:
Dugger, Daniel(1-PURD)
Combinatorial model categories have presentations. (English summary)
Adv. Math. 164 (2... | 3 | https://mathoverflow.net/users/12166 | 146144 | 79,041 |
https://mathoverflow.net/questions/146146 | 3 | For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety).
What can be said about singular varieties? I am mostly interested in the case where the ambient variety is singular but the subvariety to ... | https://mathoverflow.net/users/36922 | Is there a blow-up formula for the derived category of a singular ambient variety? | In the simplest example of a 2-dimensional quadratic cone, the derived category of the blowup is indecomposable over the base. On the other hand, in this paper <http://link.springer.com/article/10.1007/s00029-008-0052-1> there are examples of interesting semiorthogonal decompositions of such blowups.
| 4 | https://mathoverflow.net/users/4428 | 146154 | 79,042 |
https://mathoverflow.net/questions/146157 | 4 | Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x\_1,x\_2,x\_3,...,x\_n,...]$. Is the following question true?
**Question:** *Is there any maximal ideal in $S$ such that it contains a **chain** with uncountable number of prime ideals*?
PS: We recall that a ... | https://mathoverflow.net/users/41971 | chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,...,x_n,...]$ | Pick your favorite bijection $\phi$ from the natural numbers to the rational numbers. For each real number $\alpha$, let $I\_\alpha$ be the ideal generated by all $x\_n$ with $\phi(n)<\alpha$.
Then $I\_\alpha\subset I\_\beta$ if and only if $\alpha\le \beta$, so the $I\_\alpha$ form an uncountable chain, clearly cont... | 10 | https://mathoverflow.net/users/10503 | 146163 | 79,047 |
https://mathoverflow.net/questions/146135 | 1 | Let $(A, +, \preceq)$ be an ordered group, namely $(A, +)$ is a group and $\preceq$ is a total order on $A$ such that $x + z \prec y + z$ and $z + x \prec z + y$ for all $x,y,z \in A$ with $x \prec y$.
>
>
> >
> > **Q1.** If $x,y,z \in A$ and $x \preceq z$, is it true that $x+y \ne y + z$ unless $x = z$?
> >
> ... | https://mathoverflow.net/users/16537 | Is $x + y \ne y+nx$ for $x \ne 0$ and $n \ge 2$ (in an ordered group)? | As I pointed out in the comments, Q1 has a negative answer in general and actually has a positive answer only for abelian groups. Your second question (in the version of October 28 2013, 14:54 GMT) also has a negative answer. Indeed, the Baumslag-Solitar group $G=\langle x,t\mid txt^{-1}=x^n\rangle=\mathbf{Z}[1/n]\rtim... | 4 | https://mathoverflow.net/users/14094 | 146176 | 79,052 |
https://mathoverflow.net/questions/146128 | 4 | By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that,
$$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum\_{n\in \mathbb Z} |\hat{f}(n)| < \infty \}).$$
My question is:
Let $f \in L^{2}(\mathbb T) \ast L^{2}(\mathbb T)$ that is, $f=g \ast h$ for some $g, h \i... | https://mathoverflow.net/users/33018 | If $f$ is non-prime, can we say $|f|$ is also a non-prime; in convolution algebra? | The answer is no. Let $A$ be the space of real values functions with absolutely convergent
Fourier series, and $F$ is a real function on some interval $I$.
Theorem (Katznelson). If $F\circ f\in A$ for all $f\in A$ with range in $I$, then $F$
is analytic on $I$.
Taking $F(x)=|x|$ we obtain that there is $f\in A$ suc... | 3 | https://mathoverflow.net/users/25510 | 146182 | 79,056 |
https://mathoverflow.net/questions/146187 | 6 | **Motivation**: In his utterly famous paper, Rezk ([here, (pag. 7)](http://www.math.uiuc.edu/~rezk/rezk-ho-models-final-changes.pdf)) defines a structure called "Quillen ring". I'm wearing my algebraist's hat today, so I was wondering if this definition is chosen to suggest that somewhere a "true" ring structure is hid... | https://mathoverflow.net/users/7952 | The pushout product as an operation | Whenever you start with a *biclosed* monoidal category $(\mathcal C, \otimes, I)$, the push-out product is a (biclosed) monoidal structure on the category of arrows $\operatorname{Mor}(\mathcal C)$. The trick is to think of $\operatorname{Mor}(\mathcal C)$ as the category of functors $\mathbf 2\rightarrow\mathcal C$, w... | 12 | https://mathoverflow.net/users/12166 | 146202 | 79,063 |
https://mathoverflow.net/questions/146150 | 4 | Let $\mathcal A = (X, Q, \delta, q\_0, F)$ be a deterministic finite automata with the following acceptance condition on infinite words:
The automata accepts $\xi \in X^{\omega}$ with respect to $F$ iff
$$
\forall i : \delta(q\_0, \xi[0...i]) \in F.
$$
Meaning that every prefix $\xi[0...i]$ of $\xi$ goes to an acce... | https://mathoverflow.net/users/37580 | Subsets of $\omega$-regular lanuages accepted by automata with special acceptance condition | Your first conjecture — the claim that does not insist on eventually periodic input — is not true.
For a counterexample, consider the language consisting of all infinite binary strings $\xi$, such that every infix maximal finite block of $0$s has even length. This language can be accepted by a machine according to y... | 3 | https://mathoverflow.net/users/1946 | 146203 | 79,064 |
https://mathoverflow.net/questions/105655 | 10 | It is easily shown that, for any uncountable infinite cardinal $\kappa$, $\square\_\kappa$ implies that for any stationary $S\subseteq \kappa^+$, there exists a stationary $T\subseteq S$ such that $T$ does not reflect at (i.e. is not stationary in) any $\alpha<\kappa$ of uncountable cofinality. The standard proof does ... | https://mathoverflow.net/users/26002 | Square and stationary reflection | The answer is yes (at least for $\kappa = \omega\_1$): Shelah and Harrington showed that you can force that every stationary subset of $S\_{\omega}^{\omega\_2}$ reflects starting with a Mahlo cardinal. See Theorem A in [Some exact equiconsistency results is set theory](http://projecteuclid.org/euclid.ndjfl/1093870823).... | 7 | https://mathoverflow.net/users/41953 | 146207 | 79,067 |
https://mathoverflow.net/questions/146183 | 2 | How to do analytic continuation for following function?
$$f(z) = \prod\_{n=0}^{+\infty} {(1+z^{4^n})}$$
Evidently it satisfies $f(z)f(z^2)=\dfrac{1}{1-z}$...
| https://mathoverflow.net/users/22954 | Analytic continuation for PI(1+z^(4^n)) | All the answers above are absolutely fine. Just to give a completely obvious argument:
Take a the line $z=t\exp(i\phi)$ with $\phi=2\pi/{4^{n\_0}}$ for some $n\_0\in \mathbb{N}$.
Then
$$
f(z)=\Pi\_{n=0}^{n\_0-1}\left(1+t^{4^n}\exp\left(i\phi^{4^n}\right)\right)\Pi\_{n=n\_0}^{\infty}(1+t^{4^n})
$$
Since $4^{n\_0}/2... | 1 | https://mathoverflow.net/users/40120 | 146210 | 79,068 |
https://mathoverflow.net/questions/145999 | 6 | A maximum [independent set](http://en.wikipedia.org/wiki/Independent_set_%28graph_theory%29) is a largest independent set for a given graph $G$ and its size is denoted $\alpha(G)$. And the [Lovász number](http://en.wikipedia.org/wiki/Lov%C3%A1sz_number) of $G$ is denoted $\vartheta(G)$. $\vartheta(G)\geq \alpha(G)$ by ... | https://mathoverflow.net/users/38722 | What is the maximum of the ratio $\vartheta(G)/\alpha(G)$? | It is infinite, in fact much stronger versions are also true, see e.g., Theorem 1 here:
<http://arxiv.org/abs/cs/0608021>
(Shannon capacity is between $\alpha$ and $\vartheta$.)
| 6 | https://mathoverflow.net/users/955 | 146212 | 79,069 |
https://mathoverflow.net/questions/146208 | 5 | A number $n \in \mathbb{N}$ is squarefree if for every divisor $d | n, d > 1$, we have $d^2 \nmid n$. It is known that the squarefree numbers have a density of $6/{\pi^2}$ over $\mathbb{N}$. It is a question of interest to know whether polynomials take on infinitely many squarefree values.
I am asking for an explicit... | https://mathoverflow.net/users/10898 | Are there any known non-trivial functions that takes on squarefree values with the right density? | Indeed the Piatetski-Shapiro numbers are such an example: see Theorem 4 of this paper <http://arxiv.org/pdf/1203.5884.pdf> by Baker, Banks, Brudern, Shparlinski and Weingartner.
Note: I understood the question to ask for the density to be exactly $\pi^2/6$; which the PS example gives. One can also compute the densit... | 7 | https://mathoverflow.net/users/38624 | 146213 | 79,070 |
https://mathoverflow.net/questions/145937 | 3 | Let $f$ be a continuous function on complex Grassmannian $G(k, 2n+1)$. Is it true to say that there is a $k$-plane $Y$ such that $Y$ has nontrivial intersection with $f(Y)$?
A motivation for this question is the following alternative proof for fixed point
property of $CP^{2n}$:
Assume that $f$ is a map on $CP^{2n}$... | https://mathoverflow.net/users/36688 | A weak fixed point property for Grassmannian | As proved by Bob Stong [Robert E. Stong, Splitting the universal bundles over Grassmannians, Algebraic and Differential Topology - Global Differential Geometry, Occas. 90th Anniv. M. Morse’s Birth, Teubner-Texte Math. 70, 275-287 (1984)], over the complex Grassmann manifolds, the canonical bundles never contain proper ... | 4 | https://mathoverflow.net/users/41997 | 146215 | 79,072 |
https://mathoverflow.net/questions/146219 | 3 | Define the surface $X$ to be the total space of $\mathcal{O}\_{\mathbb{P}^1}(-5)$.
By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the equation (or the set of equations) that describes this singularity (as a surface in some $\mathbb{C}^n$, possibly just $\mat... | https://mathoverflow.net/users/5259 | Which isolated surface singularity comes from a -5 curve? | The weighted projective plane $\mathbb{P}(1,1,n)$ can be viewed as $\mathbb{P}(1,1,n)=\mathbb{C}^3\setminus \{0\}/(x,y,z)\sim (\lambda x,\lambda y,\lambda^n z)$. For $n=1$ we obtain the standard projective plane. For $n>1$, the point $(0,0,1)$ is the unique singular point, and the blow-up of this point is a Hirzebruch ... | 10 | https://mathoverflow.net/users/23758 | 146224 | 79,073 |
https://mathoverflow.net/questions/146199 | 5 | Let ${\bf Fin}$ denote the category of finite sets. If $X$ is a topological space, then for any natural number $k\in{\mathbb N}$, the slice category ${\bf Fin}/X$ contains the configuration space $C\_k(X)$ of $k$ distinct points in $X$. It also contains topological information about how these configuration spaces fit t... | https://mathoverflow.net/users/2811 | Configuration topos? | If $X$ is finite, then as a mere category $\mathbf{Fin}\_{/ X}$ is equivalent to the cartesian power $\mathbf{Fin}^X$. If $X$ is infinite, then $\mathbf{Fin}\_{/ X}$ should be thought of as the full subcategory of $\mathbf{Set}^X$ spanned by the $\aleph\_0$-compact objects. This is not a particularly nice category – fo... | 3 | https://mathoverflow.net/users/11640 | 146229 | 79,075 |
https://mathoverflow.net/questions/146156 | 14 | Numerical evidence suggests the following.
For $c \in \mathbb{N}, c > 2$ define the sequence $a\_n$ by
$a\_0=0,a\_1=1, \; a\_n=c a\_{n-1} - a\_{n-2}$
For $ 5 < n < 500, \; 2 < c < 100$ there are no primes in $a\_n$ though
semiprimes exist.
>
> Is it true that $a\_n$ is always composite for $n > 5$
>
>
> If ye... | https://mathoverflow.net/users/12481 | Is the sequence $a_n=c a_{n-1} - a_{n-2}$ always composite for $n > 5$? | Here is another approach to show that $a\_n$ is not prime when $c \gt 2$ and $n \gt 2$
We have (proof at end) $$a\_{n+m}=a\_na\_{m+1}-a\_{n-1}a\_{m} \tag{\*}$$
So, by induction on $j \ge 1$, $$a\_{n+jn}=a\_{n}a\_{jn+1}-a\_{n-1}a\_{jn}$$ is always divisible by $a\_n.$
Hence the only question is for $p \gt 2$ prime. ... | 6 | https://mathoverflow.net/users/8008 | 146235 | 79,079 |
https://mathoverflow.net/questions/145849 | 11 | Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by $(zf)(g):=f(z^{-1}g)$. Then let $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ be the algebra (multiplication by composition) ... | https://mathoverflow.net/users/27923 | Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra? | The following is an abstract (Banach) algebraic take on Werner's construction. Let $A=\ell^1(\mathbb Z)$ with convolution (but in general $A$ is any Banach algebra). We turn the dual space $A^\*$ into an $A$-bimodule (though in our example, $A$ is commutative) by dualising the actions:
$$ (a\cdot f)(b) = f(ba), \quad (... | 5 | https://mathoverflow.net/users/406 | 146253 | 79,081 |
https://mathoverflow.net/questions/146251 | 2 | This question is inspired by a [recent one](https://mathoverflow.net/questions/146156) : Let $c$ be a variable and define a sequence by $a\_0=0$ $a\_1=1$ and $a\_{n+1}=a\_{n}c-a\_{n-1}$ . So
$$\begin{align\*}
a\_2 &= c
\\ a\_3 &={c}^{2}-1= \left( c-1 \right) \left( c+1 \right)
\\ a\_4 &= {c}^{3}-2\,c=c \left( {c}^{2... | https://mathoverflow.net/users/8008 | Irreducible Polynomials from a Reccurence | The polynomials $s\_p$ and $d\_p$ are indeed irreducible: We have $a\_p(c)=s\_p(c)d\_p(c)$ with $s\_p$ and $d\_p$ monic of degree $m=(p-1)/2$. Let $z$ be another variable. Then $z^{2m}a\_p(z+1/z)=(z^ms\_p(z+1/z))(z^md\_p(z+1/z))$. The two factors on the right hand side are monic polynomials, and if we show that they ar... | 2 | https://mathoverflow.net/users/18739 | 146259 | 79,084 |
https://mathoverflow.net/questions/146258 | 9 | Let $p\_1, p\_2, \dotsc, p\_n$ be distinct primes, and let $\epsilon\_1, \epsilon\_2, \dotsc, \epsilon\_n$ be an arbitrary sequence of $1$ and $-1$.
There is an integer $a$ such that $\left( \frac{a}{p\_1} \right) = \epsilon\_1, \left( \frac{a}{p\_2} \right) = \epsilon\_2, \dotsc, \left( \frac{a}{p\_n} \right) = \eps... | https://mathoverflow.net/users/9147 | Quadratic residues and nonresidues of arbitrary patterns | Noam Elkies argument [here](https://mathoverflow.net/a/123636/297) shows that some such $A$ must occur among any consecutive $\prod (p\_i+3)/2 +1$ integers. In Elkies notation, take $a\_i = p\_i$; take $A\_i$ to be the $(p\_i-1)/2$ residue classes modulo $p\_i$ which are "good" and take $Z\_i$ to be the $(p\_i+1)/2$ re... | 4 | https://mathoverflow.net/users/297 | 146270 | 79,087 |
https://mathoverflow.net/questions/129184 | 3 | Has anyone an idea where one can read more about Deepam Patel's talk "Motivic structure on higher homotopy of non-nilpotent spaces" <http://www.ihes.fr/~abbes/SGA/patel.html> ?
Edit/Answer: The video is here: <http://www.ihes.fr/~abbes/SGA/suron-kika-passe.html>
| https://mathoverflow.net/users/451 | "Motivic structure on higher homotopy of non-nilpotent spaces" ? | The preprint is available on his site at the IHES.
<http://www.ihes.fr/~patel/motivichom.pdf>
| 1 | https://mathoverflow.net/users/12156 | 146290 | 79,100 |
https://mathoverflow.net/questions/146296 | 0 | Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x\_1,x\_2,x\_3,...,x\_n,...]$. My Question is about the **depth** of this ring.
**Question**: *Could we find any maximal ideal in $S$ such that it contains an **anti-chain** with uncountable number of prime ide... | https://mathoverflow.net/users/41971 | Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$ | In the same spirit as the answer to your [earlier question](https://mathoverflow.net/questions/146157/chain-of-prime-ideals-in-polynomial-ring-s-bbbrx-1-x-2-x-n), pick a bijection $\phi$ from the natural numbers to the rationals. For each real number $\alpha$, let $I\_\alpha$ be the ideal generated by all $x\_n$ such t... | 4 | https://mathoverflow.net/users/10503 | 146300 | 79,103 |
https://mathoverflow.net/questions/146294 | 2 | The 6 and 15 dimensional representations of $SU(3)$ are irreducible. The 90 dimensional tensor product representation $6\times 15$ decomposes into a sum of irreducible representations. What factors occur and with what multiplicity?
Note: by 6 I mean the 2 index symmetric representation and not its complex conjugate (... | https://mathoverflow.net/users/41312 | Decomposition of $SU(3)$ representation $6\times 15$ into irreducibles? | This question is borderline between what is on topic and what isn't; if you want to do a number of computations like this you should pick up a book on representation theory. My standard recommendations for $SL\_n$ rep theory are Chapter 8 of Fulton's *Young Tableaux* or Appendix II (by Fomin) in Stanley's *Enumerative ... | 5 | https://mathoverflow.net/users/297 | 146301 | 79,104 |
https://mathoverflow.net/questions/145193 | 3 | **Question** : Is the following true for any $n,N\in\mathbb N$?
$$\sum\_{k\_1+k\_2+\cdots+k\_N=n,\ k\_i\ge0\in\mathbb Z}\frac1{\prod\_{j=1}^{N}\{(N-1)k\_j+1\}}\le 1$$
**Motivation** : I've known the $N=3$ case :
$$\sum\_{k\_1+k\_2+k\_3=n,\ k\_i\ge0\in\mathbb Z}\frac1{(2k\_1+1)(2k\_2+1)(2k\_3+1)}\le 1$$
I proved... | https://mathoverflow.net/users/34490 | An inequality about the sum of some unit fractions with a property | I'm posting an answer just to inform that the question has received an answer by Ivan Loh on MSE.
<https://math.stackexchange.com/questions/520220/sum-k-1k-2-cdotsk-n-n-k-i-ge0-in-mathbb-z-frac1-prod-j-1n-n-1k>
| 1 | https://mathoverflow.net/users/34490 | 146310 | 79,109 |
https://mathoverflow.net/questions/146307 | 2 | I see adjointness between the two concepts of "being a definable *set*" and "being a set-builder *formula*":
---
>
> A set $X$ is **definable** when there is a formula $\phi(x)$ such that $X = \lbrace x : \phi(x)\rbrace$.
>
>
>
---
>
> A formula $\phi(x)$ is a **set-builder** when there is a set $X$ ... | https://mathoverflow.net/users/2672 | Another adjoint pair: Definable sets and set-builder formulas | Given any pair of sets $X, Y$ (let me not be too specific about what "sets" means because what follows is robust with respect to changes in the definition), any relation $R : X \times Y \to 2$ whatsoever induces a contravariant adjunction between the poset of subsets of $X$ and the poset of subsets of $Y$ given by
$$... | 7 | https://mathoverflow.net/users/290 | 146313 | 79,110 |
https://mathoverflow.net/questions/146317 | 6 | Let $G$ be a group containing a monoid $M$ that spans $G$ as a group. Is it possible to have a proper quotient $\varphi \colon G \to Q$ of $G$ such that the restriction of $\varphi$ to $M$ is injective?
More specifically I'm interested in the following: if $M$ is an Ore monoid (cancellative and admitting least common... | https://mathoverflow.net/users/5339 | Monoids and groups of fractions | For an Ore monoid the universal group is the group of right fractions. This is proved just as the universal property for localization of commutative rings is proved. It is irrelevant whether $H=\iota(M)\iota(M)^{-1}$, you can simply send a fraction $(m,n)$ to $\iota(m)\iota(n)^{-1}$ and check that this gives a well def... | 7 | https://mathoverflow.net/users/15934 | 146318 | 79,111 |
https://mathoverflow.net/questions/146315 | 5 | In Müger's article "[Conformal Field Theory and Doplicher-Roberts Reconstruction](http://arxiv.org/abs/math-ph/0008027)", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore every premodular category, which is a special case, see Bruguières articles) is the... | https://mathoverflow.net/users/13767 | Is every premodular category the *full* subcategory of a modular category? | I believe the answer to your first question is no: If you take a symmetric tensor category, Müger's construction should give the trivial tensor category.
The answer to your second question is yes. There is always a full inclusion of a braided tensor category $\mathcal{C}$ into its center $\mathcal{Z}(\mathcal{C})$, w... | 3 | https://mathoverflow.net/users/396 | 146321 | 79,113 |
https://mathoverflow.net/questions/146272 | 6 | I was wondering if there are any known (upper and lower) bounds for the complexity of computing the class-number of a finite extension of the rationals. (A general bound should be in function of the discriminant, I guess.)
This would also be of interest in special cases like fields with a given degree or signature. T... | https://mathoverflow.net/users/32210 | How long does it take to compute a class number? | In the case of quadratic fields there is a very nice paper of Andy Booker (see <http://www.ams.org/journals/mcom/2006-75-255/S0025-5718-06-01850-3/home.html> ) which uses the Burgess bounds on character sums (among other ideas) to compute the class number. The running time is always $O(D^{1/2+\epsilon})$ and if the GRH... | 6 | https://mathoverflow.net/users/38624 | 146327 | 79,115 |
https://mathoverflow.net/questions/146233 | 17 | The Paving Conjecture, which is equivalent to the famous Kadison--Singer Problem, was spectacularly settled in the affirmative by Marcus--Spielman--Srivastava ([arxiv:1306.3969](http://arxiv.org/abs/1306.3969)). Let $E$ denote the canonical projection from ${\mathcal B}(\ell\_2{\mathbb N})$ onto the subalgebra $\mathca... | https://mathoverflow.net/users/7591 | Operator Valued Kadison--Singer Problem | The case $M = L^\infty[0,1]$ seems like a simple measurability question. If every $x \in B(l^2)$ can be $K$-paved to $\|\sum P\_ixP\_i\| < \epsilon \|x\|$, then every $x \in B(l^2)\otimes L^\infty[0,1] \cong L^\infty([0,1],B(l^2))$ has a pointwise a.e. $K$-paving that does the same thing. So we just need a measurable s... | 8 | https://mathoverflow.net/users/23141 | 146331 | 79,117 |
https://mathoverflow.net/questions/146277 | 5 | In a topos ${\mathcal{A}}$, given a group object $G$ and a subgroup $H$, the object $H\backslash G$ of right cosets is the coequalizer of two maps $G\times H\rightrightarrows G$, namely the group multiplication and the projection onto $G$. Denote the coequalizing map by $H\backslash -$:
$$G\times H\rightrightarrows G\x... | https://mathoverflow.net/users/nan | Does the internal axiom of choice imply Lagrange's theorem? | Consider the topos $\mathcal E$ of $\mathbb Z/2$-sets. It satisfies the internal axiom of choice but not the external one. The Klein four-group $(\mathbb Z/2)\times(\mathbb Z/2)$ admits an action of $\mathbb Z/2$, namely interchanging the two factors, which makes it an object of $\mathcal E$. Let $H$ be the diagonal su... | 14 | https://mathoverflow.net/users/6794 | 146334 | 79,118 |
https://mathoverflow.net/questions/145615 | 5 | Let $(I,\leq)$ be a poset. Recall that the Krull dimension of $I$ is defined as follows:
-- $K.dim(I)=-1$ if and only if $I=\{0\}$;
-- if $\alpha$ is an ordinal and we already defined what it means to be a poset with Krull dimension $\beta$ for any ordinal $\beta<\alpha$, we say that $K.dim(I)=\alpha$ if and only i... | https://mathoverflow.net/users/24891 | Do constructible sets have Krull dimension? | For an arbitrary noetherian topological space $X$, I think the lattice $\mathcal{U}(X)$ of open subsets and the lattice $\mathcal{C}(X)$ of constructible subsets (the boolean algebra generated by open subsets) have the same Krull dimension (I use the standard ordering, to avoid confusion (see Ramiro's comment), so the ... | 9 | https://mathoverflow.net/users/14094 | 146350 | 79,126 |
https://mathoverflow.net/questions/146343 | 14 | I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would be great if somebody of you can help me.
For a first-order theory $T$, let $Mod(T)$ denote the category of all models... | https://mathoverflow.net/users/nan | The interplay between certain aspects of interpretability, model theory and category theory | Here is a negative answer to question 2 and the converse of question 3.
Let $T\_1$ be the theory of the integers under successor $\langle\mathbb{Z},S\rangle$. This theory asserts that $S$ is bijective and has no cycles of any finite length. That theory is complete, since all models of size $\aleph\_1$ are isomorphic,... | 10 | https://mathoverflow.net/users/1946 | 146354 | 79,128 |
https://mathoverflow.net/questions/109026 | 3 | Let $\boldsymbol p=(p\_1,p\_2,\ldots)$ be a distribution over $\mathbb{N}$
and suppose that $S=(X\_1,X\_2,\ldots,X\_n)$ are sampled iid according to $\boldsymbol p$. Define the
indicator variable $\xi\_j$ to be $0$ if $j$ occurs in the sample $S$ and $1$ otherwise:
$$
\xi\_j=\boldsymbol{1}\_{j\notin S},
\qquad j\in\mat... | https://mathoverflow.net/users/12518 | Large deviations for missing mass | Background: this question is motivated by the OP's paper <http://arxiv.org/pdf/1111.2328.pdf>. Amir Dembo and I had answered this question privately to Aryeh a long time ago. I paste the answer here, adapted from email correspondence, in case anybody still needs it. I used the question as a take home exam in a large de... | 4 | https://mathoverflow.net/users/35520 | 146360 | 79,130 |
https://mathoverflow.net/questions/146357 | 4 | Let $G$ be a connected, simply-connected complex semisimple linear algebraic group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$ and let $\Delta\subseteq Hom(T,\mathbb{C}^\*)$ be the resulting collection of roots. Choose positive and negative roots, $\Delta\_{+}$ and $\Delta\_{-}$, respectively. Let $... | https://mathoverflow.net/users/25358 | Moving Between Weight Spaces in Highest-Weight Representations | They're the ones perpendicular to $\lambda$.
To see this, note, first off, that $\mathfrak g\_\alpha v\_\lambda \in V\_{\lambda+\alpha}$ so if $g\_\alpha v\_\lambda \neq 0$ then $\lambda+\alpha$ must be a weight of $V$. But then so too would be $s\_\alpha(\lambda+\alpha)=s\_\alpha(\lambda)-\alpha$. Now if $\langle \l... | 3 | https://mathoverflow.net/users/430 | 146364 | 79,133 |
https://mathoverflow.net/questions/146239 | 4 | Some not very clever questions on closed model categories.
1. For a (left or right) Quillen functor $F:C\to D$ what arguments does one usually use for proving that $Ho F$ is fully faithful when restricted to a full subcategory $C'$ of $Ho C$? I suspect that there should exist some very 'classical' reasonings of this ... | https://mathoverflow.net/users/2191 | On closed model categories: standard arguments and fibrantly cogenerated categories | 1. It may be easier to prove that the map induced on the mapping spaces between fibrant and cofibrant objects is a weak equivalence, provided that $C$ and $D$ are enriched over some closed symmetric monoidal model category and $F$ is continuous. Of course this is not a necessary condition, but it may be considered a "h... | 2 | https://mathoverflow.net/users/30641 | 146370 | 79,134 |
https://mathoverflow.net/questions/146353 | 25 | This is a question about the history of commutative algebra. I'm curious why the Koszul complex from commutative algebra is called the Koszul complex? All of Koszul's early papers are about Lie algebras and Lie groups, in particular about the Chevalley-Eilenberg complex. He never published papers on commutative algebra... | https://mathoverflow.net/users/21029 | History of Koszul complex | Although the germ of the idea might've appeared in Koszul's earlier work on the cohomology of Lie algebras and homogeneous spaces, it seems that the first full-fledged appearance of the Koszul complex/resolution is in Koszul, *Sur un type d'algèbres différentielles en rapport avec la transgression*, Colloque de topolog... | 24 | https://mathoverflow.net/users/430 | 146371 | 79,135 |
https://mathoverflow.net/questions/146295 | 6 | Let $X$ be an algebraic variety over $\bar{\mathbb{Q}}$ of dimension $d$, then there is the l-adic cycle map $\mathrm{cl}\_{et}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X,\mathbb{Q}\_\ell(i))$ from the Chow group into the l-adic etale cohomology group, where $\mathbb{Q}\_\ell(i)$ refers to the Tate twist. Its definit... | https://mathoverflow.net/users/42042 | Comparison of cycle maps | Everything you could wish for is true :-). Passing to the inverse limit in Milne's theorem 21.1 one gets an isomorphism
$$ H^i\_{et}(X, \mathbf{Z}\_\ell) = H^i\_{sing}(X(\mathbf{C}), \mathbf{Z}) \otimes \mathbf{Z}\_{\ell}$$
(you don't have to extend scalars to $\mathbf{C}$ or $\mathbf{C}\_\ell$ or anything nasty li... | 7 | https://mathoverflow.net/users/2481 | 146373 | 79,137 |
https://mathoverflow.net/questions/146147 | 2 | suppose $\overline u(r)=\frac{1}{\omega\_{n-1}}\int\_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is whether (edited)
$$ u(r,w) \leq C\overline u(r),$$
where $C$ is independent of $u$. If this inequality is true, How can I pro... | https://mathoverflow.net/users/38739 | A general inequality about spherical mean of a function | You cannot use directly the inequality for subharmonic functions (as it is superharmonic), but if you know also that a Harnack inequality holds for this problem, that is,
$$
\max\_B u \leq C \min\_B u
$$
(possibly locally etc) then you are in business because then of course
$$
u \leq \max\_B u \leq C \bar{u}.
$$
Lookin... | 1 | https://mathoverflow.net/users/40120 | 146377 | 79,139 |
https://mathoverflow.net/questions/145383 | 3 | Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order.
What would be an example of an arithmetic Fuchsian lattice that is not of finite index in an Eichler order?
| https://mathoverflow.net/users/41401 | Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders? | I hate to see an easy question without a proper answer, so I'll elaborate a bit on few\_reps comment which really contains all there is to say about this question; his reference to Reid and MacLachlan's book also stands for this answer.
If $\Gamma$ is any Fuchsian group derived from a quaternion algebra then there a... | 3 | https://mathoverflow.net/users/32210 | 146378 | 79,140 |
https://mathoverflow.net/questions/146338 | 2 | I am confused by two examples of spinor bundles over 4-manifolds, which I saw in various places:
(1) The spinor bundle $S = S\_+ \oplus S\_-$ associated to a spin or spinc structure of Riemannian four-manifold $M$
(2) If the manifold $M$ has almost complex structure, then the bundle $\tilde S \equiv \Lambda^{0,\*}T... | https://mathoverflow.net/users/15884 | How to understand two examples of spin bundle | First let me point out that the bundle $\Lambda^2\_+ T^\*M$ makes sense only when $\dim M=4$. Maybe you should add this assumption.
As for your question, I think that you can find the answer in Example 1.3.3 page 30 of [these notes.](http://www3.nd.edu/~lnicolae/swnotes.pdf)
| 3 | https://mathoverflow.net/users/20302 | 146380 | 79,141 |
https://mathoverflow.net/questions/146368 | 1 | Let $X\_n$ be random elements of $D$ (space of cad lag functions on $[0,1]$ as domain). $X\_n$ has asymptotically independents if $0\leq s\_1 \leq t\_1 \leq s\_2 \leq \ldots < s\_r \leq t\_r \leq 1$, then for all linear Borel sets $H\_1,\ldots,H\_r$ we have
$P\{X\_n(t\_i)-X\_n(s\_i)\in H\_i, i = 1,\ldots,r \}-\prod\_... | https://mathoverflow.net/users/nan | Asymptotically independent increments random elements: Billingsley Ch:$4$ | Assume that we have proved that if $0\leqslant s\_1\leqslant t\_1\lt s\_2\leqslant t\_2\lt\dots\lt s\_r\leqslant t\_r\leqslant 1$, we have that $(X\_{t\_i}-X\_{s\_i})\_{i=1}^r$ is independent. Then using $\mathbb P\{X\in C\}=1$, we will deduce the independence of $(X\_{t\_i}-X\_{t\_{i-1}})\_{i=1}^r$ where $0\leqslant t... | 0 | https://mathoverflow.net/users/17118 | 146381 | 79,142 |
https://mathoverflow.net/questions/146223 | 6 | I would like to know the intuition behind the holomorphic bisectional curvature of Hermitian manifolds. I already know that the classical sectional curvature of a Riemannian (not necessarily complex) manifold roughly tells us how geodesics spread apart. What is the analogue for holomorphic bisectional curvature? Do you... | https://mathoverflow.net/users/39122 | Intuition for holomorphic bisectional curvature | Ngaiming Mok, [The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature,](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214441778) J. Differential Geom. Volume 27, Number 2 (1988), 179-214.
Mok improves Mori's and Siu-Yau's proof... | 2 | https://mathoverflow.net/users/3377 | 146389 | 79,147 |
https://mathoverflow.net/questions/146393 | 0 | Let $(V,< \cdot, \cdot >)$ be an inner product space over a field ${K}$. As usual, we can extend $< \cdot, \cdot >$ to a mapping on the exterior algebra of $V$ using the usual matrix determinant, and we then get a corresponding Hodge map $\ast$. Now is it true in general that $\ast$ is unitary, that is, does it hold th... | https://mathoverflow.net/users/37003 | Is the Hodge Map Unitary? | Yes. A cheap way is to choose an orthonormal basis $(e\_1,\ldots ,e\_n)$ of $V$ (replacing $K$ by its algebraic closure). Then the $e\_I$ for $I\subset [1,n]$ and $\#I=p$ form an orthonormal basis of $\wedge^pV$, and $\ast$ maps $e\_I$ to $\pm e\_{I^c}$ (as usual, I put $e\_I=e\_{i\_1}\wedge\ldots \wedge e\_{i\_p}$ for... | 2 | https://mathoverflow.net/users/40297 | 146395 | 79,151 |
https://mathoverflow.net/questions/146397 | 16 | I am interested in concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables.
Let $X\_1,..., X\_n$ be i.i.d random variables, $S\_n$ their centered sum and $M\_n$ the maximum of the partial sums:
$$ S\_n = \sum\_{k=1}^{n}{(X\_k - \mu)} \ , \quad M\_n = \max\_{1 \leq k \leq n}{... | https://mathoverflow.net/users/42087 | Concentration inequalities for the maximum of the rescaled/normalized sum of iid random variables | One can use Birnbaum and Marshall inequality:
**Theorem(Theorem 2.1. in [1](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177704964)).** If $\left(S\_k,k\geqslant 1\right)$ is a non-negative sub-martingale and $(c\_k,k\geqslant 1)$ a non-decreasing sequence of positive numbers... | 9 | https://mathoverflow.net/users/17118 | 146400 | 79,153 |
https://mathoverflow.net/questions/146329 | 0 | I have been wondering about the following (and allready posted a similar question, see [Dimension of ring completion wrt to a decreasing chain of ideals](https://mathoverflow.net/questions/143048/dimension-of-ring-completion-wrt-to-a-decreasing-chain-of-ideals)):
Let $R$ be the ring of formal power series in $n$ ind... | https://mathoverflow.net/users/36804 | Chain of Ideals of same height | No, it is not true that $\mathfrak{p}$ must have height $s$. For one counterexample, let $R$ be $\mathbb{C}[[s,t]]$, and let $I\_k$ be $\langle t(t-s)(t-2s)\cdots (t-(k-1)s) \rangle$.
| 3 | https://mathoverflow.net/users/13265 | 146407 | 79,156 |
https://mathoverflow.net/questions/146404 | 7 | Is the following consistent?
There are definable class forcing notions $\lbrace \mathbb{P}\_{n}\rbrace\_{n\in \omega}$ such that:
1. The product of any finitly many of them preserves $\text{ZFC}$ and all cardinals.
2. The $\prod\_{n\in \omega}\mathbb{P}\_{n}$ collapses all uncontable cardinals to $\omega$.
| https://mathoverflow.net/users/nan | Countable Product of Class Forcing Notions | Yes, this can happen.
Start with $V=L$ (but $V=HOD+GCH$ suffices), and assume there are no inaccessible cardinals. For each regular cardinal $\delta$, partition the ordinals of cofinality $\delta$ below $\delta^+$ into $\omega$ many disjoint stationary sets $\text{Cof}\_\delta\cap\delta^+=\bigsqcup\_n S^\delta\_n$. ... | 10 | https://mathoverflow.net/users/1946 | 146412 | 79,160 |
https://mathoverflow.net/questions/146396 | 16 | Suppose that $p,q>1$ are two relatively prime integers. Are there infinitely many positive integers $N$ such that
1. $N$ is relatively prime to $p$ and $q$;
2. there exists positive integers $k,l$ such that $p^k\equiv q\mod N$ and $q^l\equiv p \mod N$?
| https://mathoverflow.net/users/7360 | Infinitely many $N$ such that $\langle p\rangle=\langle q\rangle$ mod $N$ | The answer is yes for all $p$ and $q$. We shall assume, without loss of generality, that $p<q$. It suffices to show that for any positive integer $M$ which is coprime with $p$, there exists a prime $N\nmid M$ with the required two properties (since the first property implies $N\nmid p$).
Let $k>Mq$ be a prime such th... | 13 | https://mathoverflow.net/users/11919 | 146423 | 79,165 |
https://mathoverflow.net/questions/134841 | 1 | We are looking for a proof or counter-examples for the following hypothesis.
Two combinators $M$ and $N$ are solvable and equivalent in [the HP-complete sensible $\lambda$-theory](http://mathgate.info/cebrown/notes/barendregt.php#4) iff either
$$
\exists n \in \mathbb N: \langle\varnothing\ |\ \Gamma(M, x) \cup \Gamm... | https://mathoverflow.net/users/22795 | Interaction-based approximation for HP-complete λ-theory? | One counterexample for the forward implication ($\not\Rightarrow$) is solvable $M \equiv N \equiv \lambda x.x\ \Omega$. Indeed,
$$
\langle\varnothing\ |\ \Gamma(M, x) \cup \Gamma^\*(N, x)\rangle \not\rightarrow^\* \langle\varnothing\ |\ x\_1 = x\_1, x\_2 = x\_2, \Delta\rangle.
$$
Another counterexample ($\not\Leftarr... | 0 | https://mathoverflow.net/users/22795 | 146439 | 79,170 |
https://mathoverflow.net/questions/146413 | 6 | It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if we ask for exponential decay from only one side. More precisely :
**The context :** In *Fourier Analysis, Self-Adjointn... | https://mathoverflow.net/users/6187 | What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$ | Let me summarize what has been already said. If $f$ is a locally integrable function, you can break it into two parts: $f=f\_1+f\_2$, where $f\_1$ is supported on the positive ray, and
$f\_2$ on the negative ray. Then you consider two "halves" of the Fourier transform:
$$F^-(z)=\int\_0^\infty e^{-izt}f\_1(t)dt$$
and
$... | 7 | https://mathoverflow.net/users/25510 | 146452 | 79,175 |
https://mathoverflow.net/questions/146445 | 14 | Throughout this question, I shall let $A^{\mathcal{U}}$ denote the ultrapower of a structure $A$ by an ultrafilter $\mathcal{U}$. Suppose that $T$ is an Aronszajn tree and $\mathcal{U}$ is an ultrafilter on a countable set $I$. Let $T\_{\alpha}$ denote the $\alpha$-th level of a tree $T$. Let $T^{(\mathcal{U})}$ denote... | https://mathoverflow.net/users/22277 | Does an ultrapower of an Aronszajn tree have an $\omega_{1}$-branch? | Joel David Hamkins asked for the saturation version of Martin Goldstern's argument; here it is. Like Martin, I assume that every node of $T$ has successors at all higher levels. I'll build, by induction on ordinals $\xi<\omega\_1$, a branch $(x\_\xi)$ in $T^{(U)}$, where $x\_\xi$ is at level $\xi$. Of course, I start w... | 11 | https://mathoverflow.net/users/6794 | 146468 | 79,186 |
https://mathoverflow.net/questions/35927 | 26 | Let $\pi\_k(x)=|\{n\le x:n=p\_1p\_2\cdots p\_k\}|$ be the counting function for the *k*-almost primes, generalizing $\pi(x)=\pi\_1(x)$. A result of Landau is
$$\pi\_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\log x}\qquad\qquad(1)$$
but this approximation is very poor for $k>1$.
For $\pi(x)$ much more is known. A (dive... | https://mathoverflow.net/users/6043 | Asymptotic density of k-almost primes | I'll address Joel's edited question of getting good asymptotics on RH. The argument below is essentially due to Selberg, but this is not quite what he does and I haven't seen it presented this way in the literature. The natural problem is to consider the coefficients of $(\log \zeta (s))^k/k!$ rather than numbers with ... | 14 | https://mathoverflow.net/users/38624 | 146469 | 79,187 |
https://mathoverflow.net/questions/146459 | 1 | Googled the name, but almost all result pointed to Berger's preprint.
Is there any reference for this?
| https://mathoverflow.net/users/1190 | What is "Berger's isembolic inequality"? | See [Chris Croke's 2007 paper](http://www.ams.org/journals/proc/2008-136-02/S0002-9939-07-09079-X/S0002-9939-07-09079-X.pdf) for all you ever wanted to know.
| 5 | https://mathoverflow.net/users/11142 | 146473 | 79,188 |
https://mathoverflow.net/questions/126962 | 1 | We are looking for a proof or counter-examples to the following
**Hypothesis.** In [interaction calculus](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.3594) $\langle \varnothing\ |\ \Gamma(M, x) \cup \Gamma(N, x)\rangle \downarrow \langle \varnothing\ |\ x\_1 = x\_1, \dots, x\_n = x\_n \rangle$, where t... | https://mathoverflow.net/users/22795 | Hypothesis: interaction-based model for λKβη | One counterexample for the forward implication ($\not\Rightarrow$) is $M \equiv \lambda x.x\ I\ I$ and $M \equiv \lambda x.x\ I\ F$. Indeed,
$$
\langle \varnothing\ |\ \Gamma(M, x) \cup \Gamma(N, x)\rangle \rightarrow^\* \langle \varnothing\ |\ x = x\rangle,
$$
but $M$ and $N$ are different $\beta\eta$-normal forms... | 0 | https://mathoverflow.net/users/22795 | 146487 | 79,192 |
https://mathoverflow.net/questions/146485 | 0 | Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous.
Under what kind of extra condition for $f'$, (not $C$) holds the following relation?
$$
\Big | \frac{1}{|I\_{1}|}\int\_{I\_{1}}f'(x)dx- \frac{1}{|I\_{2}|}\int\_{I\_{2}}f'(x)dx\Big|\overset{|J|\rightarrow 0}{\longrightarrow} 0,
$$
for any $I\_{1}\cap I\_{2}=... | https://mathoverflow.net/users/40719 | Let f:J→R be an absolutely continuous and f'\in...? | Modification after the comments from below.
Let I\_1=(a,b) and I\_2=(c,d), where a< b<=c< d. Then
\frac{1}{|I\_1|}\int\_{I\_1}f'(t)dt=\frac{f(b)-f(a)}{b-a}:=J\_1,
similarly,
\frac{1}{|I\_2|}\int\_{I\_2}f'(t)dt=\frac{f(d)-f(c)}{d-c}:=J\_2.
Let |J| \to 0,
J\_1 \to f(b-), while J\_2 \to f(b+).
So one natural assumpti... | 0 | https://mathoverflow.net/users/26608 | 146489 | 79,193 |
https://mathoverflow.net/questions/118527 | 19 | Some background on (compact) Belyi surfaces
-------------------------------------------
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a **Belyi surface** if there exists a branched covering map $f:X\to \Ch$ such that $f$ is branched over at most three points of $\Ch$. Here $\Ch$ denotes... | https://mathoverflow.net/users/3651 | Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles | As you may possibly already be aware, there is a parallel phenomenon in circle packing riemann surfaces.
Those compact riemann surfaces admitting full circle packings are a countable dense subset of the moduli space, where by full circle packing a riemann surface, I mean finding a circle packing $C$ such that the ca... | 6 | https://mathoverflow.net/users/20516 | 146503 | 79,197 |
https://mathoverflow.net/questions/146500 | 5 | Consider a complex $K3$ surface $X$ and take its group of automorphisms $Aut(X)$. It is a known fact that the action of $Aut(X)$ on the set of rational $-2$ curves of $X$ has only finite number of orbits.
**Questions.** What kind of ideas one has to use to prove this fact? Is there some nice exposition? Does this fac... | https://mathoverflow.net/users/13441 | Action of automorphisms of a $K3$ surface on its $(-2)$-curves | The group of symplectomorphisms $Aut(X)$ of a K3 is the group $O(\Lambda)$ of
automorphisms of its period lattice $\Lambda=H^{1,1}(M,{\Bbb Z})$. For each
(-2)-cohomology class $\eta\in H^{1,1}(M,{\Bbb Z})$, either $\eta$ or $-\eta$ is
represented by a curve (this follows from the Riemann-Roch formula). This curve
is ... | 13 | https://mathoverflow.net/users/3377 | 146514 | 79,201 |
https://mathoverflow.net/questions/146512 | 2 | Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
| https://mathoverflow.net/users/40038 | Dual of a Complex 2-Torus | For non-algebraic tori, $T$ and $T^\*$ are (usually) not isomorphic; for algebraic ones, they are isogeneous, and for the principally polarized abelian varieties, $T$ and $T^\*$ are isomorphic.
This is obvious if you realize that a complex torus is a totally imaginary 2-dimensional complex subspace in ${\Bbb C}^4$ u... | 6 | https://mathoverflow.net/users/3377 | 146515 | 79,202 |
https://mathoverflow.net/questions/146495 | 12 | We say that a group $(A, \cdot)$ is bi-orderable if there exists a *total* order $\preceq$ on $A$ such that $xz \prec yz$ and $zx \prec zy$ for all $x,y,z \in A$ with $x \prec y$.
Let $m,n$ be non-zero integers, and let ${\rm BS}(m,n)$ denote the Baumslag-Solitar group $\langle a, b \mid a^{-1} b^m a = b^n\rangle$. I... | https://mathoverflow.net/users/16537 | Bi-orderability of Baumslag-Solitar group $\langle a,b \mid a^{-1} b^m a = b^n\rangle$ and of $\langle a,b \mid a^{-1} b a^m = b^n\rangle$ | (After the discussion in the comments.) $BS(m,n)$ is bi-orderable iff $mn>0$ and $\min(|m|,|n|)=1$.
Since it was already mentioned that $mn<0$ implies $BS(m,n)$ not bi-orderable and that $BS(1,n)$ is bi-orderable for all $n\ge 2$, all remains is to check that $2\le m\le n$ implies $BS(m,n)=\langle t,x\mid tx^mt^{-1}=... | 10 | https://mathoverflow.net/users/14094 | 146518 | 79,204 |
https://mathoverflow.net/questions/146525 | 9 |
>
> Take a compact connected simple centreless Lie group $G$.
> Can the *commutator map* $G\times G\to G$ sending $(x,y)$ to $[x,y]$ be homotopic to a constant map?
>
>
>
I am interested mostly in the case, where $G={\rm PSU}(n)$.
---
As far as I understand, the commutator map is homologically trivial (ri... | https://mathoverflow.net/users/24165 | Are compact simple groups homotopically non-abelian? | The following is the main theorem in
Araki, S.; James, I. M.; Thomas, E., *"Homotopy-abelian Lie groups"*, Bull. Amer. Math. Soc. 1960.
>
> **Theorem:** A compact connected Lie group is homotopy-abelian only if it is abelian.
>
>
>
| 7 | https://mathoverflow.net/users/17836 | 146526 | 79,206 |
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