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https://mathoverflow.net/questions/126555 | 3 | Two disjoint unit discs $D\_1$ and $D\_2$. Inside them there are random poisson points with intensity $\lambda$. For a given real $r>0$, what is the probability that there exist a poisson point $x\_1\in D\_1$ and a poisson point $x\_2\in D\_2$ such that $\|x\_1-x\_2\|\_2 \le r$?
Couldn't find any existing results on t... | https://mathoverflow.net/users/32828 | Distance between poisson points in two disjoint unit discs | If $M$ is the $4$-dimensional Lebesgue measure of $\{(x,y) \in D\_1 \times D\_2: \|x - y\|\_2 \le r\}$, then the number of such pairs of points is a Poisson random variable with parameter $M \lambda^2$, so the probability that there is at least one is $1 - \exp(-M \lambda^2)$.
EDIT: Oops, this is wrong. It is not Poi... | 1 | https://mathoverflow.net/users/13650 | 126567 | 70,651 |
https://mathoverflow.net/questions/126547 | 5 | It is known that no finite power of the closed unit interval I is homogeneous, while the countable power, i.e., the Hilbert cube, is. It seems that a power is not homogeneous for every successor cardinal. But, is it homogeneous for every limit cardinal, or is this dependent on some of the more sensitive cardinal proper... | https://mathoverflow.net/users/50650 | Which powers of the closed unit interval are homogeneous? | For $\kappa \geq \omega$, $I^\kappa$ is homeomorphic to $(I^\omega)^\kappa$, and product of homogeneous spaces is homogeneous, so $I^\kappa$ is homogeneous for every infinite $\kappa$.
| 11 | https://mathoverflow.net/users/11647 | 126568 | 70,652 |
https://mathoverflow.net/questions/117574 | 11 | My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity.
The wikipedia article on [constructive proof](http://en.wikipedia.org/wiki/Constructive_proof) begins, "a constructive proof is a method of proof that demonstrates the existence of a mat... | https://mathoverflow.net/users/25028 | Non-constructive proofs vs. efficient algorithms | As others have noted there are several different meanings for constructive.
I. **Constructive proof in the sense of constructive mathematics**
This meaning views an object as existing if we have a description of how to *construct* the objects (though we don't really need to carry it out), and there are several dis... | 13 | https://mathoverflow.net/users/7507 | 126582 | 70,660 |
https://mathoverflow.net/questions/123577 | 5 | In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse then the $k+1$-strandard parameter is $k+1$-solid/universal over $\mathcal M$.
The proof is not too complicated but on page 76 they state ... | https://mathoverflow.net/users/3859 | A question about Mitchell/Steel Fine Structure and Iteration Trees | The assumption is that $\rho=\rho\_{k+1}^M>\text{lh}(E)$ for \emph{all} $E$, not just some $E$. But looking at your message of March 6/7, it looks like you are indeed reading it as ``all''. Regarding the question in that message, $H$ agrees with $M$ below $\rho$ because $\rho\leq\alpha\_s$, and all ordinals below $\alp... | 1 | https://mathoverflow.net/users/15883 | 126593 | 70,666 |
https://mathoverflow.net/questions/126577 | 4 | Let $A = \big[{1\ 1\atop 1\ 0}\big]$, and let $G\_n$ be the graph whose adjacency matrix is
$A^{\otimes n}$. Also let $\kappa(G)$ denote the number of spanning trees of $G$. From a significant amount of computational evidence, it seems highly likely that that $\kappa(G\_n) \mid \kappa(G\_{n+1})$ always holds $n \geq 0... | https://mathoverflow.net/users/32835 | Divisibility Relation for Spanning Trees of a Graph | The answer is actually as nice as could be. The number of spanning trees of $G\_n$ is
$$
\frac{1}{3^n}2^{n 2^{n-1}}\prod\_{k=0}^{n-1} \big(1-(-2)^{k-n}\big)^{\binom{n}{k}}
$$
This follows directly from the theorems in the comment above. The divisibility (what a beautiful word!) properties then follow from the formu... | 8 | https://mathoverflow.net/users/3032 | 126600 | 70,668 |
https://mathoverflow.net/questions/126594 | 2 | The questions below are motivated by pure curiosity. I heard of the first question from my former advisor. I have no idea how difficult they are, since I have no experience with magic squares.
By a normal magic square of order $n$ I mean a $n\times n$ magic square whose terms are all of the numbers $0,1,\ldots,n^2-1$... | https://mathoverflow.net/users/2578 | Increasing sequence of normal magic squares | Here you are:
$$
\begin{array}{|c|c|c|c|c|}
\hline
8& 15& 21& 2& 14\cr
\hline
20& 7& 13& 19& 1\cr
\hline
12& 24& 0& 6& 18\cr
\hline
4& 11& 17& 23& 5\cr
\hline
16& 3& 9& 10& 22 \cr
\hline
\end{array}
$$
In fact, you may start with any weakly magic square (that is --- with no conditions on the diagonals);... | 5 | https://mathoverflow.net/users/17581 | 126604 | 70,670 |
https://mathoverflow.net/questions/126610 | 1 | Suppose we have a finitely presented group $G$ with free presentation
$$
R\hookrightarrow F \twoheadrightarrow G.
$$
To this presentation we may associate an extension with abelian kernel
$$
R/R' \hookrightarrow F/R' \twoheadrightarrow G.
$$
Here $R'$ denotes the commutator subgroup of $R$, and the $G$-module $R/R'$ is... | https://mathoverflow.net/users/8103 | Calculations with relation modules | Let $F$ be a finitely generated non-abelian free group with a non-trivial normal subgroup $R \lhd F$. Suppose that $G:=F/R$ is finitely presented. Then the group $F/R'$ is finitely presented if and only if $G$ is finite. In particular, if $G=C\_2\*C\_3$ then $F/R'$ is not finitely presented.
This is mentioned in the ... | 3 | https://mathoverflow.net/users/7644 | 126617 | 70,675 |
https://mathoverflow.net/questions/126609 | 12 | I am writing a paper about a flaw that I found in a published paper. There, the statement is called “Theorem 2”. In my paper, I am reproducing the other paper’s definitions, and steps leading towards that statement, and now I’d like to reproduce the statement, immediately followed by the counter example that I found.
... | https://mathoverflow.net/users/28027 | How to refer to a theorem that you have shown to be wrong | You can have a look to the paper "A counterexample to a 1961 “theorem” in homological algebra" by Neeman and use his style. By the way, I think that the paper is very very good.
| 15 | https://mathoverflow.net/users/7845 | 126628 | 70,681 |
https://mathoverflow.net/questions/126625 | 5 | Let $I\subset k[x\_1,\dots,x\_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one?
Gröbner bases come to mind, but for example, $$I = \langle x-y-z, y^4z^2+2y^3z^3+y^2z^4 \rangle = \langle x-y-z,x^2y^2z^2\rangle$$ has $$\left... | https://mathoverflow.net/users/5495 | Is there an algorithm to decide if an ideal contains monomials? | Computing colon ideals is pretty quick. You could colon out the
variables in order. If the ideal changes, record the variable that
worked, and go back to the beginning of the list.
Either you get to the unit ideal, in which case you found
the lex-first monomial that's in the ideal, or you make it to the
end of the lis... | 7 | https://mathoverflow.net/users/391 | 126637 | 70,685 |
https://mathoverflow.net/questions/126638 | 2 | We can find many examples of smooth Riemannian manifolds with boundaries whose boundaries are convex. But it seems to me I know no any example of smooth Riemannian manifold with concave boundary. So my question is What kinds of smooth Riemnnian manifolds admit concave boundaries? Do they exist? Thank you in advance!
| https://mathoverflow.net/users/29480 | What kinds of manifolds admit concave boundary? | I think the answer is yes and in a strong way.
Precisely, let $M$ be a compact manifold, and put *any* Riemann metric on $\partial M$ Then I claim there is a Riemann metric on $M$, extending the Riemann metric on $\partial M$ with $\partial M$ convex in the sense above -- that the geodesic curvature of boundary curv... | 4 | https://mathoverflow.net/users/1465 | 126658 | 70,693 |
https://mathoverflow.net/questions/126516 | 15 | Recall that a connected semisimple algebraic group $G$ over an algebraically closed field $K$ of arbitrary characteristic was defined by Chevalley to be *simply connected* if the character group $X(T)$ of a maximal torus $T$ in $G$ is "as large as possible": equal to the abstract weight lattice associated to the root l... | https://mathoverflow.net/users/4231 | Simply connected algebraic groups and reductive subgroups of maximal rank | Surely it's easier to check whether $H'$ is simply-connected by inspecting the co-root lattice...? For the example of $G\_2$ containing $SO(4)$ that Allen mentions, we have a pseudo-Levi subalgebra of type $A\_1\times \tilde{A}\_1$ where $\{(3\alpha+2\beta),\alpha\}$ is a basis of simple roots. Now the cocharacter $(3\... | 5 | https://mathoverflow.net/users/26635 | 126661 | 70,696 |
https://mathoverflow.net/questions/105128 | 5 | This is basically a reference request. Does anyone know if the structure of the homotopy category of spectra (or maybe just the model, i.e. w/o the homotopy, category), localized at infinite wedges of Morava K-theories, or BP, is either well described somewhere or somehow stupid and uninteresting? This question is moti... | https://mathoverflow.net/users/11546 | Localization at Infinite Wedges of K-theories or BP | I mentioned that I proved this in the comment above but am "answering" just for closure. A link to the proof is here:
<http://chromotopy.org/?p=1110>
| 1 | https://mathoverflow.net/users/11546 | 126671 | 70,699 |
https://mathoverflow.net/questions/126595 | 7 | In studying the papers of Abbes and Saito on ramification theory in the imperfect residue field case, I come to the following questions. Let $\overline{\mathbb{Q}}\_p\supset\mathbb{Q}{}^{t}\_p\supset \mathbb{Q}{}^{nr}\_p$ be an algebraic closure of the field $\mathbb{Q}\_p$ of $p$-adics with its maximal tamely ramified... | https://mathoverflow.net/users/17988 | Algebraic $p$-adic integers mod $p$ | This is an extended comment.
Let $\Gamma=\mathrm{Gal}(\overline{\mathbf{Q}}\_p|\mathbf{Q}\_p)$. Your question pertains to the filtration $V\subset T\subset\Gamma$, where $T$ is the inertia subgroup and $V$ the ramification subgroup, so that $\overline{\mathbf{Q}}\_p^T$ is the maximal unramified extension of $\mathbf{... | 1 | https://mathoverflow.net/users/2821 | 126692 | 70,710 |
https://mathoverflow.net/questions/126673 | 3 | Is the geodesic exponential map for a Lie group with the (-)-connection a diffeomorphism? This connection is one of two flat connections introduced by Cartan and Schouten on a Lie group and has torsion $T(X,Y)=-[X,Y]$.
| https://mathoverflow.net/users/14454 | The (-)-Connection on a Lie Group | The $(+)$/0/$(-)$-connections on a Lie group $G$ are the left-invariant connections (I will identify connections with covariant derivatives) defined on left-invariant vector fields by $\nabla\_XY=c[X,Y]$ with respectively $c=1,\frac{1}{2},0$. Their name is related to the fact that their torsion is $T(X,Y)=d[X,Y]$ with ... | 12 | https://mathoverflow.net/users/32864 | 126695 | 70,712 |
https://mathoverflow.net/questions/126708 | 4 | Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$\_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes.
Consider now a degree-$n$ irreducible representation of SL$\_d(\mathbb C)$ with $d\ge n$, and restrict it to the subgroup o... | https://mathoverflow.net/users/10481 | Representations of Sym(n) and SL_d | This is given in Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the restrictions in terms of plethysm and inner product.
| 6 | https://mathoverflow.net/users/3992 | 126711 | 70,716 |
https://mathoverflow.net/questions/126627 | 9 | Here by «algorithm» I mean a (halting) Turing machine with finite alphabet and memory.
Is it possible to obtain by purely existential (*i.e.* non-constructive) means the existence of an algorithm performing some required task, while beeing unable to explicit it (*i.e.* write it down or describe it constructively) ?
... | https://mathoverflow.net/users/24309 | Existence of unknowable algorithms ? | Yes.
We take a fixed diophantine equation in variables $y,x\_1, x\_2, \ldots, x\_k$.
Task: for an input $n \in \mathbb{Z}$, output either
1. Five values of $y$ for which solutions exist to the equation.
2. A solution to the equation for which $y=n$.
3. "No", in which case there must be no solution with $y=n$.
It ... | 11 | https://mathoverflow.net/users/2294 | 126716 | 70,719 |
https://mathoverflow.net/questions/126706 | 3 | One defines the **finite dual** of a Hopf algebra $H$ as
$$
H^o := \{f \in H^\* ~|~ f(I) = 0, \text{ for some ideal $I$ of $H$ with } \dim\_C(H/I) < \infty \}.
$$
As is well-known, $H^o$ has a well-defined Hopf algebra structure obtained by dualizing the Hopf structure of $H$.
On the other hand, for any finite-dimen... | https://mathoverflow.net/users/12653 | Hopf Duals and Matrix Coefficients | The equality always holds, for any algebra (not just Hopf algebra) $H$. See Kapitel IV, Bemerkung 3.4 **5)** in [Schneider's Hopf algebra class](https://sites.google.com/site/darijgrinberg/hopfalgebren?pli=1) if you can read German (which I assume you can, given your location); it is currently on page 483 (as of versio... | 4 | https://mathoverflow.net/users/2530 | 126719 | 70,721 |
https://mathoverflow.net/questions/126704 | 4 | Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx A\*\rho$. We form a graph $G$ by connecting any pair of points with an edge if they are within unit length distance of one-an... | https://mathoverflow.net/users/32871 | The probability distribution for vertex degree in a unit disc graph generated from random points on a plane | The random placement of points you described is called [Poisson point process](http://en.wikipedia.org/wiki/Point_process#Poisson_point_process) (in the limit). The resulting graph is called [Random Geometric Graph](http://en.wikipedia.org/wiki/Random_geometric_graph). Searches will yield a good crop of papers about th... | 5 | https://mathoverflow.net/users/1061 | 126723 | 70,722 |
https://mathoverflow.net/questions/126656 | 3 | Let $X$ be a projective variety over an algebraically closed field $k$ and $\mathscr L$ a line bundle on $X$. Its section ring,
$$
R(X,\mathscr L) = \bigoplus \_{n=0}^\infty H^0(X,\mathscr L^{\otimes n}),
$$
is a finitely generated graded $H^0(X,\mathscr O\_X)\simeq k$-algebra.
I wonder if the following condition has... | https://mathoverflow.net/users/10076 | Linearly generated embedding? | *This was a comment originally.*
The generation in degree 1 is the same as projective normality at least for normal varieties.
For simplicity, assume that $X$ is normal.
By Hartshorne, Chapter II, Exercise 5.14, we know that if $X$ is projectively normal (with embedding associated to the complete linear system of ... | 4 | https://mathoverflow.net/users/3521 | 126725 | 70,723 |
https://mathoverflow.net/questions/126722 | 11 | Let $S$ be a graded ring with $A := S\_0$. Set $X := \textrm{Proj} (S)$. Then the projective coordinate ring of $X \times\_A X$ is the graded ring $ \bigoplus\_{n \geq 0} S\_n \otimes\_A S\_n $, cf Hartshorne, Exercise II.5.11. (This matches with geometric intuition because this ring corresponds [at least when $S$ is n... | https://mathoverflow.net/users/1528 | Why does the naive choice of homogeneous coordinate ring of a product of projective schemes not work? | A ($\mathbb{Z}$-)grading on a ring $S$ is equivalent to an action of the group $\mathbb{G}\_m$ on $U=\operatorname{Spec} S$. This is an exercise in algebra; the statement is that the symmetric monoidal category of comodules over the Hopf algebra $A[t,t^{-1}]$ is equivalent to the category of graded $A$-modules. The con... | 22 | https://mathoverflow.net/users/75 | 126728 | 70,724 |
https://mathoverflow.net/questions/126720 | 0 | My current **question** is concerned with a reference (paper or book) containing a proof of this result: The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros.
| https://mathoverflow.net/users/25947 | The $k^{th}$ derivative of a L-function has necessarily infinitely many zeros | This is a result of fairly standard complex analysis, probably due to Hadamard (entire functions of finite order strictly larger than one have necessarily infinitely many zeros). I don't have the book handy but this is certainly in Titchmarsh's "Theory of functions". It has very little to do with number theory.
| 3 | https://mathoverflow.net/users/32882 | 126732 | 70,725 |
https://mathoverflow.net/questions/126739 | 0 | **I changed the title and added revisions and left the original untouched**
For this post, $k$ is defined to be the square root of some $n\geq k^{2}$. Out of curiousity, I took the sum of one of the factorials in the denominator of the binomial theorem;
$$\sum \_{k=1}^{\infty } \frac{1}{k!} \equiv e-1$$
[OEIS A09113... | https://mathoverflow.net/users/16888 | $\sum _{k=0}^{\infty } \frac{1}{(k+m) k!} \equiv 1$ for $m=2$ | A way to get this, and also to understand the behavior for other values would be like so (though I do not know if this is not overly indirect):
Recall that
$$
e^x = \sum\_{k=0}^{\infty} \frac{x^k}{k!}
$$
so
$$
x^{m-1}e^x = \sum\_{k=0}^{\infty} \frac{x^{k+m-1}}{k!}
$$
Now 'integerate', then
$$
F(x) = \sum\_{k=0}^{\... | 10 | https://mathoverflow.net/users/nan | 126740 | 70,727 |
https://mathoverflow.net/questions/126745 | 3 | If C and D are irreducible, affine varieties over an algebraically closed field, and I form the product variety CxD, is the projection morphism from CxD to C necessarily an open map? That is, is the projection of each Zariski open subset of CxD necessarily Zariski open in C?
| https://mathoverflow.net/users/32891 | Are the projection morphisms from a product of varieties necessarily open? | Yes. More generally, flat morphisms locally of finite presentation are universally open (EGA IV2, Théorème 2.4.6).
| 3 | https://mathoverflow.net/users/2841 | 126749 | 70,730 |
https://mathoverflow.net/questions/126754 | 3 | I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, where $X$ is a compact Riemann surface and $\mathcal{O}$ is the sheaf of holomorphic functions. Forster proves this by r... | https://mathoverflow.net/users/27040 | Finiteness theorem for first-cohomology group of sheaf of holomorphic functions on compact Riemann surfaces | Wells, Differential Analysis on Complex Manifolds, gives the complete proof using the standard approach for complex manifolds, through Hermitian metrics. Forster's book is great, but odd. Forster avoids the use of Hermitian metrics entirely, trying to rely as much as possible on sheaf cohomology. In the algebraic categ... | 4 | https://mathoverflow.net/users/13268 | 126767 | 70,738 |
https://mathoverflow.net/questions/126489 | 8 | As the title. Geometrically, is there a projective complex manifold(or more generally an projective algebraic variety) accepting only infinite nontrivial cover(which may not be projective)?
Thanks.
| https://mathoverflow.net/users/15124 | Is there non-simple-connected projective variety(over C) with trivial etale fundamental group? | There are several classes of spaces for which this question can be asked, here are the answers:
1. Compact complex-projective manifolds (also frequently called *manifolds admitting flat complex-projective structures*): These are n-manifolds admitting an atlas where transition maps are elements of $PGL(n+1, C)$. For s... | 12 | https://mathoverflow.net/users/21684 | 126774 | 70,739 |
https://mathoverflow.net/questions/126513 | 21 | I have a couple of conjectures on recursive functions, that I feel must have been proved or refuted by someone else, but I don't know where to look. In short:
*1. The primitive recursive functions form a pseudoinitial small finite product category with natural number object.*
*2. The partial recursive functions for... | https://mathoverflow.net/users/3603 | Categories of recursive functions | The relevant piece of categorical folklore here is the notion of **arithmetic universe**.
This was studied by André Joyal in 1973 with the goal of proving Gödel's Incompleteness Theorems in a categorical fashion. However, André never published anything and many people have tried without success to obtain any notes fr... | 34 | https://mathoverflow.net/users/2733 | 126782 | 70,741 |
https://mathoverflow.net/questions/126784 | 2 | Let $X =$ {0, 1, 2, ...} and $T$ = { $\emptyset$, $X$, {0}, {1}, {0,2}, {0,1,3}, {0,1,2,4}, {0,1,2,3,5}, ... } $\cup$ {{0,1,2}, {0,1,2,3}, {0,1,2,3,4}, ... }. It is easily verified that $T$ forms a topology on $X$. Burdick has shown (Amer. Math. Monthly January 2006 p. 83) that the singleton {0} generates infinitely ma... | https://mathoverflow.net/users/5090 | Closure-complement-union: countable space, finite seed, infinite family, space unique? | If you replace each point in your space with 2 points (or more), and use the induced topology, then you still have the same generating property, but the resulting space has no isolated points, and hence is not homeomorphic to your space.
Perhaps you want to insist upon a weak separation axiom.
| 1 | https://mathoverflow.net/users/1946 | 126793 | 70,744 |
https://mathoverflow.net/questions/126742 | 1 | Hi, given a function $f:X \rightarrow Y$, not necessarily invertible, is there a conventional name for the function $$g\_f := f^{-1} \circ f:X \rightarrow \mathcal{P}(X),$$ where $\mathcal{P}(X)$ denotes the power set of $X$?
| https://mathoverflow.net/users/5887 | Name convention for the composition of the preimage of a function and the function itself | I do not know any name for it. However, note that the concept you are definining does not really depend on the function $f$, only on the equivalence relation induced by $f$ (which is sometimes called the kernel of $f$, unless you work with groups).
A set $S$ is called "saturated" with respect to an equivalence relat... | 6 | https://mathoverflow.net/users/14915 | 126800 | 70,748 |
https://mathoverflow.net/questions/126775 | 1 | Let $\mathcal{P}$ be a finite presentation of some group. When we apply some Nielsen transformations on $\mathcal{P}$, will the homotope type of the presentation complex $K\_{\mathcal{P}}$ of $\mathcal{P}$ always be preserved?
| https://mathoverflow.net/users/27253 | Do Nielsen transformations on a presentation preserve the homotopy type of the corresponding presentation complex? | Yes. In fact, slightly more is true: the **simple** homotopy type of the presentation 2-complex is preserved under Nielsen transformations. For a proof of this fact, see
Micheal N. Dyer and Allan J. Sieradski, *Trees of homotopy types of two-dimensional CW-complexes. I.*, Comment. Math. Helv. **48** (1973), 31–44. [MR0... | 5 | https://mathoverflow.net/users/17846 | 126806 | 70,752 |
https://mathoverflow.net/questions/126808 | 0 | This topic was created to discuss how many ways we know to create piecewise linear functions with smooth transitions between the phases. An alternative is presents by Bacon & Watts (1971):
the idea is build the model by using the signal operator and then replace it by a smooth approximation. Let
$y(x\_i) = \theta\_... | https://mathoverflow.net/users/32869 | How many ways we know to join two line segments with a smooth transitional function? |
>
> How many ways we know to join two line segments with a smooth transitional function?
>
>
>
We can do this in a lot of ways. If I understand your question, some of the more common ways are mentioned on Wikipedia.
<http://en.wikipedia.org/wiki/Heaviside_step_function#Analytic_approximations>
| 0 | https://mathoverflow.net/users/32907 | 126815 | 70,755 |
https://mathoverflow.net/questions/126817 | 1 | How would you say that a small additive category $C$ embedds (contravariantly) into the category of exact functors from a 'large' abelian $C'$ into abelian groups (this is something like Yoneda's embedding, but $C$ does not map canonically into $C'$)? My problem is that I do not want to consider all functors from $C'$ ... | https://mathoverflow.net/users/2191 | How would you say that a small category is embedded into functors from a large $C'$ to abelian groups? | In your example, there likely exists a cardinal $\kappa$ such that $C'$ is [$\kappa$-accessible](http://ncatlab.org/nlab/show/accessible+category), as are the functors $C'\to Ab$ associated to each object of $C$ (i.e., they preserve $\kappa$-filtered colimits). In this case, you can consider $C$ embedded into the categ... | 1 | https://mathoverflow.net/users/75 | 126820 | 70,757 |
https://mathoverflow.net/questions/126818 | 0 | Suppose I have the limit
$\lim\_{m\rightarrow \infty}\frac{\sum\_{k=0}^ma\_{k,m}}{\sum\_{k=0}^mb\_{k,m}}$.
When can I write this as
$\lim\_{n\rightarrow \infty}\lim\_{m\rightarrow \infty} \frac{\sum\_{k=0}^ma\_{k,n}}{\sum\_{k=0}^mb\_{k,n}}$?
To be specific, both sums converge to exponentials, which tend to zero... | https://mathoverflow.net/users/nan | splitting one limit into two? | The first (single) limit is totally blind to terms $a\_{k\ n}\ \ b\_{k\ n}$ for all $k > n$, while the second (double) limit depends on them. Thus the two limits are hardly related at all.
In other words, the single limit considers finite segments, and the double limit the infinite segments. To compare these ... | 1 | https://mathoverflow.net/users/8385 | 126827 | 70,761 |
https://mathoverflow.net/questions/124998 | 22 | This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that such a method is "a most inconsistent way of arguing, and such as would not be allowed in Divinity". This passage is quo... | https://mathoverflow.net/users/28128 | Was the early calculus inconsistent? | Coming back to the B.Berkeley critics, there is a common denominator of all known getarounds, both the two mainstream ones (Wstrass and NSA) and exotic ones like the SDG interpretation.
That is, one considers an extension - call it $R^+$ - of the true reals R and a map
$R^+ \to R\cup \{\infty\}$ - call it the valua... | 10 | https://mathoverflow.net/users/32916 | 126846 | 70,767 |
https://mathoverflow.net/questions/126843 | 4 | Hi all; I just ended to write a file which collects some cases of Mihailescu theorem that are solvable directly with elementary tools, and that can be useful to a student following math contests; in particular, given the equation in integer $x^p-y^q=1$, the following cases are studied: $2\mid p$, $2\mid q$ (both are hi... | https://mathoverflow.net/users/32898 | Elementary cases of Mihailescu theorem | Of course, you can just consider the case when $p$ and $q$ are primes. A good reference for your question is, I think, Schoof's monograph on Catalan's equation. The case $q = 2$ is solved in Ch. II, and it involves some arithmetic in the ring of Gaussian integers (the argument goes back to V.A. Lebesgue). The case $p =... | 2 | https://mathoverflow.net/users/16537 | 126860 | 70,773 |
https://mathoverflow.net/questions/126865 | 2 | Let the first-order language ${\mathcal{L}}$ have a single binary predicate $P$. Consider the structure whose underlying set is ${\mathbb{Z}}$, the integers, and an ordered pair $(m,n)$ is in $P$ if and only if $m=n+1$ for some nonzero $n$.
Is the subset of positive integers defineable in $({\mathbb{Z}},P)$, that is... | https://mathoverflow.net/users/nan | Definability in a language with a single binary predicate | No, the structure is definitionally equivalent with $(\mathbb Z,0,S)$ (that is, you make the successor function a function rather than a predicate), which is well-known to have elimination of quantifiers: every formula is equivalent to a Boolean combination of formulas of the form $y=S^n(x)$, where $x,y$ are either var... | 5 | https://mathoverflow.net/users/12705 | 126867 | 70,775 |
https://mathoverflow.net/questions/126862 | 4 | I have a few questions concerning Kostant's work on principal three-dimensional subalgebras (TDS). Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and $\mathfrak{a}\subseteq\mathfrak{g}$ a principal TDS. Is it true that the centralizer $Z\_{\mathfrak{g}}(\mathfrak{a})=\{\xi\in\mathfrak{g}:\te... | https://mathoverflow.net/users/25358 | Kostant's theorem on principal 3-dimensional subalgebras | It's helpful to point out the original source, in one of Kostant's influential early papers: "The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group", *Amer. J. Math.* 81 (1959), available online [via JSTOR](https://www.jstor.org/stable/2372999). (Note also the Bourbaki report by K... | 6 | https://mathoverflow.net/users/4231 | 126879 | 70,782 |
https://mathoverflow.net/questions/126856 | 0 | In the book I am reading they write that for Hurwitz zeta function, $\zeta(x,s)=\sum\_{n=0}^{\infty} \frac{1}{(x+n)^s}$, the next sum in the RHS converges for $\Re(s)>-1$, and I don't see how exactly?!
$$\zeta(x,s)-(\zeta(s)-sx\zeta(s+1)= x^{-s} + \sum\_{n=1}^{\infty} n^{-s}[(1+x/n)^{-s} - (1-x/n)]$$
I think the su... | https://mathoverflow.net/users/13904 | This might be a trivial question on Hurwitz's zeta function. | Looks like there's a typo in your post. Andrews has $\left(1-\frac{sx}{n}\right)$ where you have $\left(1-\frac{x}{n}\right)$ in the rightmost term.
If you expand $\left(1+\frac{x}{n}\right)^{-s}$ as a power series in $\frac{x}{n}$, the series begins $1-s\frac{x}{n}+O(x^2/n^2)$. The first two terms are cancelled by t... | 3 | https://mathoverflow.net/users/5263 | 126882 | 70,784 |
https://mathoverflow.net/questions/126881 | 30 | In Katz's article *p-adic properties of modular schemes and modular forms* in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
>
> By an elliptic curve over a scheme $S$, we mean a proper smooth morphism $p: E \to S$, whose geometric fibres are connected curves... | https://mathoverflow.net/users/6779 | Elliptic curve over a scheme is a group scheme? | The argument that allows you to show that an elliptic curves defined as you say is a group scheme, and even a commutative one is the construction of a functorial and natural
isomorphism $E(T) \rightarrow Pic\_{E/S}^0(T)$ for every $S$-scheme $T$. This allows
to see the functor $T \mapsto E(T)$ as a funtor in group, an... | 25 | https://mathoverflow.net/users/9317 | 126884 | 70,786 |
https://mathoverflow.net/questions/126712 | 5 | Suppose $G$ is an elementary abelian $p$-group of rank n (for simplicity we can assume n=1). Denote by $\beta$ the well-known Bockstein boundary map from $H^1(G,\mathbb F\_p)$ to $H^2(G,\mathbb F\_p)$. I am looking for an explicit formula for $\beta(f)$ on $[g|h]$ if we know the value of $f$ on $G$.
| https://mathoverflow.net/users/25701 | Explicit formula for Bockstein hom in group cohomology of elementary abelian p-groups | Write $G=\langle \sigma\rangle\cong \mathbb{Z}/p$. $f\in H^1(G,\mathbb{F}\_p)$ can be taken as group homomorphism $f: G \to \mathbb{F}\_p$. If $B$ denotes the bar resolution then $\beta(f)$ is represented by the cocycle
$$B\_2 \to \mathbb{F}\_p,\;\; [\sigma^i,\sigma^j] \mapsto
\begin{cases}f(\sigma) & , & i+j\ge p \n... | 6 | https://mathoverflow.net/users/27895 | 126896 | 70,791 |
https://mathoverflow.net/questions/126890 | 5 | Let
$$R(x)=\sum\_{n\leq x}\phi(n)-\frac{3x^2}{\pi^2}.$$
Montgomery has shown that $R(x)=\Omega\_{\pm}(x\sqrt{\log\log x})$, which is the best known lower bound. It seems interesting therefore that
$$\int\_0^\infty \frac{R(x)\,dx}{x^2}=0,$$
because it tells us that the oscillations (which continue indefinitely) are part... | https://mathoverflow.net/users/10980 | On the oscillation of the summatory totient about its average | I am not sure if this result is explicitly mentioned in the literature, but it certainly is classical.
Let
$$R(x) = \sum\_{n \leq x}{\varphi(n)} - \frac{3x^2}{\pi^2}, \qquad H(x) = \sum\_{n \leq x}{\frac{\varphi(n)}{n}} - \frac{6x}{\pi^2}.$$
Then by partial summation,
$$\int^{x}\_{0}{\frac{R(t)}{t^2} \: dt} = H(x) ... | 5 | https://mathoverflow.net/users/3803 | 126902 | 70,793 |
https://mathoverflow.net/questions/126851 | 11 | It is well-known that finite subgroups of $PGL\_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are classical.
If $K$ is an arbitrary field then theres are Beauville's notes about $PGL\_2(K)$:
<http://arxiv.org/abs/0909.3942>... | https://mathoverflow.net/users/32920 | Finite subgroups of $PGL(3,K)$ | *Edited in view of Derek Holt's comment on Schur indices*: These things are well studied in the literature. You probably want to restrict to irreducible subgroups, and it's probably just as well to work with ${\rm GL}(3,K).$ In such a low dimensions, the Schur index usually will not play much of a role. The Schur index... | 7 | https://mathoverflow.net/users/14450 | 126909 | 70,794 |
https://mathoverflow.net/questions/126829 | 30 | **Clarification:** My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the *compact-open* $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ for definiteness and simplicity. I am not particularly interested in any specific value of $k$.$\newcommand{\Diff}{\... | https://mathoverflow.net/users/21095 | Is the space of diffeomorphisms homotopy equivalent to a CW-complex? | Here is an example where ${\rm Diff}(M)$ with the compact-open topology is not homotopy equivalent to a CW complex. Take $M$ to be a surface of infinite genus, say the simplest one with just one noncompact end. I will describe an infinite sequence of diffeomorphisms $f\_n:M\to M$ converging to the identity in the compa... | 28 | https://mathoverflow.net/users/23571 | 126912 | 70,796 |
https://mathoverflow.net/questions/126780 | 20 | The following is a problem from our department algebra competition for
students:
>
> **Non-question.**
> An experimental-math geek was trying to raise all matrices $17\times17$
> over the field with 17 elements to the power of 100, sum the returns,
> and observe the result, when his computer broke. Help him.
>
... | https://mathoverflow.net/users/24165 | The sum of same powers of all matrices modulo p | [corrections applied, per Ilya and Anton]
Consider the formal series
$$
f(x) = \sum\_{k=0}^\infty (\sum\_A A^k) x^k
$$
where $A$ runs through $p \times p$ matrices. It is equal to $\sum\_A (I - Ax)^{-1}$, a rational function with values in scalar matrices. Thus, for some $d$, if the first $d$ coefficients vanish all ... | 14 | https://mathoverflow.net/users/29980 | 126918 | 70,799 |
https://mathoverflow.net/questions/126911 | 21 | Consider a number $a\_1a\_2a\_3a\_4 \dots a\_n$ in some base $b$, such that for each $k, 1\leq k \leq n$, the subnumber $a\_1a\_2\dots a\_k$ is a multiple of $k$.
For instance $1836$ is such a number in base $10$, because $1$ is a multiple of $1$, $18$ is a multiple of $2$, $183$ is a multiple of $3$, and $1836$ is a... | https://mathoverflow.net/users/18060 | How long can this string of digits be extended? | Following links at the OEIS entry mentioned above takes one in a step or two to [this page](http://www.mapleprimes.com/posts/43769-Ponder-This) where there are posts (from 2005) with Maple code and results out to base $23$. The values $N(b)$ for $2 \le b \le 23$ are reported to be
$2, 6, 7, 10, 11, 18, 17, 22, 25, 2... | 11 | https://mathoverflow.net/users/8008 | 126919 | 70,800 |
https://mathoverflow.net/questions/126900 | 4 | While inverting a Laplace transform using Post's inversion formula I found the following expression:
$$
\sum\_{k=1}^n S^n\_k \ x^k(\alpha)\_k
$$
where $S^n\_k$ is a Stirling number of second kind and $(\alpha)\_k$ is a Pochhammer symbol. This formula seems a mixture of the definition of these Stirling numbers and that ... | https://mathoverflow.net/users/23871 | Relations involving Stirling numbers of second kind | Let $f\_n$ denote your expression. Using the well-known formula $\sum\_n S\_k^n\frac{t^n}{n!}
=\frac{1}{k!}(e^t-1)^k$, we get the generating function
$$ \sum\_{n\geq 0}f\_n\frac{t^n}{n!} = (1-x(e^t-1))^{-\alpha}. $$
This generating function suggests that there will be no simpler expression for $f\_n$ than its definiti... | 4 | https://mathoverflow.net/users/2807 | 126924 | 70,801 |
https://mathoverflow.net/questions/126916 | 3 | How does one prove existence decomposition of $R^2$ for countable many subsets $A(n)$
s
.t.$\forall$ $x,y$ $\epsilon$ $A(n)$ $|x-y|$ is nonrational?
I tried thinking of $R^2$ as infinite tree with $2^\omega$ levels of cardinality $2^\omega$ but I don't see how to make this nonrationality condition hold.
| https://mathoverflow.net/users/32849 | Countable coloring of a plane | This is a result of Erdos: Problems and results in chromatic graph theory, Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) , pp. 27--35, Academic Press, New York, 1969, see p. 32. Available at the Renyi Institute's collection of Erdos papers:
<http://www.renyi.hu/~p... | 7 | https://mathoverflow.net/users/6647 | 126937 | 70,806 |
https://mathoverflow.net/questions/126938 | 5 | Let $A$ be a commutative ring with $1$ and $M,N$ be $A$-modules. Can you give a quick proof that $\textrm{Tor}\_i(M,N) \cong \textrm{Tor}\_i(N, M)$ using derived categories?
In his Homological algebra book, Weibel proves this with an argument via a double complexes: the so-called "acyclic assembly lemma", and from wh... | https://mathoverflow.net/users/25854 | Commutativity of Tor | Yes, the derived category of a commutative ring is symmetric monoidal $M\otimes\_A^{\mathbb L}N=N\otimes\_A^{\mathbb L}M$ and Tor is the homology of this tensor product.
| 12 | https://mathoverflow.net/users/12166 | 126939 | 70,807 |
https://mathoverflow.net/questions/126946 | 1 | I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of the principal $GL(n,\mathbb{R})$-bundle $F$ (the frame bundle) on $M$. Set now $S=F/G$ which carries a smooth manifold str... | https://mathoverflow.net/users/32947 | Torsion-free $G$-Structures | The bundle $P$ is made out of frames, being a subbundle of the frame bundle $F$. So each point in $P$ is a basis of a tangent space of $M$. We can take any metric on $M$, and use it to parallel transport the vectors of that basis. In general, for an arbitrary metric, such a parallel transport will take these vectors in... | 6 | https://mathoverflow.net/users/13268 | 126948 | 70,809 |
https://mathoverflow.net/questions/126940 | 1 | The problem of determining a set of generators of the ring of invariants of the group $\textrm{SL}\_2$ acting on the complex $n+1$-dimensional vector space of binary $n$-ic forms is known to be very hard and is open for $n > 10$. However we *do* know thanks to Hilbert that this is a finitely generated ring over $\mathb... | https://mathoverflow.net/users/15704 | Can one pick generators for the ring of invariants of binary n-ic forms which have rational coefficients? | Sure. If you have an algebraic group $G$ defined over $\mathbb Q$ acting on a $\mathbb Q$-algebra $A$, the $\mathbb Q$-algebra of invariants $A^G$ is the equalizer of two usual homomorphisms of algebras $A \to A \otimes\_{\mathbb Q}\mathbb Q[G]$. Tensoring with $\mathbb C$ is exact, so $(A \otimes \mathbb C)^{G\_{\math... | 5 | https://mathoverflow.net/users/4790 | 126955 | 70,812 |
https://mathoverflow.net/questions/126935 | 5 | I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap in one of the proofs (starting at page 331). The goal is to prove :
>
> For any non simply connected compact surface $(M,g)$, $Area(M,g... | https://mathoverflow.net/users/8887 | Fixing a proof of the systolic inequality for higher genus surfaces | You don't really need to reattach chopped off fingers to get the estimate. The point is that for almost all (in the sense of Sard) $r$, the points $\gamma(-r)$ and $\gamma(r)$ are in the same connected component of the $r$-level curve of the distance function. Applying the coarea formula and the triangle inequality the... | 3 | https://mathoverflow.net/users/28128 | 126959 | 70,814 |
https://mathoverflow.net/questions/126411 | 6 | Background
----------
[It is known](https://mathoverflow.net/questions/107980/convolution-of-sequences) that given real sequences $a = (a\_n)\_{n \in \mathbb Z} \in \ell\_p$ and $b = (b\_n)\_{n \in \mathbb Z} \in \ell\_q$, their convolution defined as
$$ a \* b (n) = \sum\_{k \in \mathbb Z} a\_{n-k} b\_k $$
is in $\e... | https://mathoverflow.net/users/16988 | Convolution in $\ell_p$ when $0<p<1$ | If $0< p<1$, then $(u+v)^p\leqslant u^p+v^p$ for any non-negative numbers $u$ and $v$. Consequently, working first on finite sums and then after taking the limit, we can see that
$$|a\*b(n)|^r\leqslant \sum\_{k\in\mathbb Z}|a(k)|^r\cdot |b(n−k)|^r$$
for any integer $n$ and any $r\geqslant \max\{p,q\}$. Then, by an appl... | 2 | https://mathoverflow.net/users/17118 | 126964 | 70,817 |
https://mathoverflow.net/questions/126947 | 3 | In their famous book, Faltings and Chai constructed, among other things, minimal and toroidal compactification of the Siegel moduli space. The reason for the use of the word toroidal is clear, since toroidal geometry is used to define such compactification, it is not clear to me the use of the word minimal. So the ques... | https://mathoverflow.net/users/32948 | Minimal compactification | Given the symmetric space $D$ of non-compact type for the real semi-simple Lie group $G$, there are $2^r-1$ Satake compactifications $\bar{D}$ for $D$ up to homeomorphism, where $r$ is the real rank of $G$.
These compactifications correspond to the non-empty subsets of a set $S$ of simple roots, and as such they for... | 1 | https://mathoverflow.net/users/7460 | 126968 | 70,819 |
https://mathoverflow.net/questions/126971 | 2 | Let $E^n$ be a Hamming cube of dimension $n$, and $\phi$ be a mapping from $E^n$ to $E^n$ that preserves Hamming distance, i.e. $d(x,y)=d(\phi (x),\phi (y))$. The question is the following: show that $\phi$ can only be constructed as a permutation of coordinates plus a constant vector, i.e. $\phi (x)=\pi (x)+v$, where ... | https://mathoverflow.net/users/32953 | Isometry on a Hamming cube | This might not be research-level, but here's an answer anyway. I'll take the cube to be $\{0,1\}^n$. Let an isometry $\phi$ be given. Adding a constant vector, if necessary, we can assume that $\phi(\vec0)=\vec 0$. Then the unit vectors (the vectors with exactly one component equal to 1), being the vectors at distance ... | 2 | https://mathoverflow.net/users/6794 | 126975 | 70,822 |
https://mathoverflow.net/questions/126923 | 24 | The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C\_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is dense in $C\_0$. Everything I have read/heard says that the range is "difficult to describe".
But, since $\mathcal{F}$... | https://mathoverflow.net/users/6342 | Image of L^1 under the Fourier Transform | Here is an attempt, somewhat rough around the edges but I think it works:
Claim: The range of the Fourier transform $\mathcal{F}:L^1(\mathbb
R)\to C\_0(\mathbb R)$ is a Borel set in $C\_0(\mathbb R)$ of the form
\begin{equation}
\bigcap\_{k=1}^\infty \bigcup\_{N=1}^\infty \bigcap\_{m,n\geq N} E\_{m,n,k}
\end{equati... | 11 | https://mathoverflow.net/users/13360 | 126977 | 70,824 |
https://mathoverflow.net/questions/126966 | 6 | Let $p$ be prime, and $z\_0, z\_1, ..., z\_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$
Is it true or false that a combination of two (or more, in general) of the roots can give us another root of the same order?
In mathematical terms, does there exist indices $i\_1,i\_2,...,i... | https://mathoverflow.net/users/32922 | Can the sum of two roots of unity be a root of unity? | If $p$ is intended to be prime, you should say so explicitly. I'll assume $p$ is prime. Then the subset sums are distinct except that the sum of all $p$th roots of unity is $0$, the sum over the empty set. Any coincidence of subset sums $\sum\_{i \in I} \zeta\_p^i = \sum\_{j\in J} \zeta\_p^j $produces a polynomial of d... | 18 | https://mathoverflow.net/users/2954 | 126979 | 70,826 |
https://mathoverflow.net/questions/126932 | 4 | It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}\_q)$. (Thanks in advance!!!)
1. The **number** of conjugacy classes in $U(n,\mathbb{F}\_q)$.
(I saw some papers of Vera-Lopez, which give the number for $n\leq 5$; but it... | https://mathoverflow.net/users/6761 | Finite Unipotent Groups: References | I guess U(n,F\_q) - upper triangular matrices over F\_q.
I am not an expert in the field, but was recently interested in similar question, so I'll put these remarks.
There is certain amount of quite recent works related to representations of U(n,F\_q) and some "big names" involved. There are certain conjectures which... | 4 | https://mathoverflow.net/users/10446 | 126982 | 70,828 |
https://mathoverflow.net/questions/126965 | 4 | I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k .
I am wondering if anyone can give me more information about a practical solution applicable to a large set, particularly what we can say about the greedy algorithm?
Thanks in advance,... | https://mathoverflow.net/users/27060 | Maximizing supermodular functions | Maximizing a supermodular function is like minimizing a submodular function. This is a polynomial time activity, for which several algorithms are known (search google for **min norm algorithm** and you'll find many hits, including to a paper by Fujishige).
There are several other approaches available for solving subm... | 0 | https://mathoverflow.net/users/8430 | 126988 | 70,830 |
https://mathoverflow.net/questions/123529 | 2 | Given a symmetric, real $n \times n$-matrix $M$, is there a way to find all $m \times m$-submatrices ($1 < m < n$) that are nilpotent?
By the [Cauchy interlacing theorem](http://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem), I know that $M$ must have both negative and positive eigenvalues, which w... | https://mathoverflow.net/users/13767 | How to find the nilpotent submatrices of a symmetric, real matrix? | Every real symmetric matrix is similar to a diagonal matrix by the spectral theorem. The only nilpotent matrix which
is similar to a diagonal matrix is the zero matrix. Hence if you mean principal submatrices then the answer is
trivial, as Chris Godsil has pointed out.
If you consider arbitrary submatrices, the ques... | 1 | https://mathoverflow.net/users/32332 | 126992 | 70,832 |
https://mathoverflow.net/questions/126989 | 1 | I have the following question: Let $R \subset \Lambda^{\*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega\_{0}$ and $\omega\_0$ where $\Omega\_{0}=dz^{1}\wedge ... \wedge dz^{n}$ and $\omega\_{0}=\frac{i}{2}\sum\_{i=1}^{n}dz^{i}\wedge ... | https://mathoverflow.net/users/32947 | $SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$ | Actually, as you have written it, the ring $R$ is generated by $\omega\_0$ and the real and imaginary parts of $\Omega\_0$, since $R$ consists of real-valued $1$-forms. You should probably consider, instead the ring $R^\mathbb{C} = \mathbb{C}\otimes R$ of complex-valued exterior forms on $\mathbb{R}^{2n}=\mathbb{C}^n$.... | 3 | https://mathoverflow.net/users/13972 | 127001 | 70,835 |
https://mathoverflow.net/questions/127030 | 6 | Do there exist real numbers whose integer powers are bounded away from integers? More precisely, for an arbitrary constant 0 < $\epsilon$ < 1/2, does there exist real x such that for all positive integer n,
$ \epsilon < (x^n)$ mod 1 < $1-\epsilon $?
Pisot numbers more or less provide the opposite of this behavior,... | https://mathoverflow.net/users/24681 | Reals with integer powers bounded away from integers? | We should be able to construct such a real number $x$. Let $\epsilon\in(0,1/2)$ be fixed. If $S\subset \mathbb{R}\_{>0}$ and $r>0$, we write $S^r$ for the set of positive $r$-th powers of elements of $S$. Recursively define a sequence of intervals $I\_n$ of the form $[N\_n+\epsilon,N\_n+1-\epsilon]$ where the $N\_n$ ar... | 6 | https://mathoverflow.net/users/5263 | 127031 | 70,846 |
https://mathoverflow.net/questions/121371 | 25 | Question
--------
>
> Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable?
>
>
>
By efficient here I am willing to consider anything with smaller expected complexity than the exponential mess one gets by naively testing all possible orderin... | https://mathoverflow.net/users/18263 | Testing simplicial complexes for shellability | Since there were no answers for a few months, I asked this question to my colleague and triangulation expert [Frank Lutz](http://page.math.tu-berlin.de/~lutz/). Since his response was wonderful and exhaustive, I am reproducing it here for the benefit of others who find such matters interesting.
*Spoiler alert*: it is... | 13 | https://mathoverflow.net/users/18263 | 127039 | 70,849 |
https://mathoverflow.net/questions/127038 | 4 | Hi!
I'm trying to make headway on a question for my undergraduate honors thesis, specifically the question of which rings of integer-valued polynomials if any satisfy the QR-property; that is, the property that all overrings are rings of quotients with respect to some multiplicative subsets.
But, this occurred to, ... | https://mathoverflow.net/users/27445 | Example of an overring of an integral domain which is not a ring of quotients? | In $\S 22.2.2$ of [my commutative algebra notes](http://alpha.math.uga.edu/%7Epete/integral.pdf) I discuss overrings of Dedekind domains. In particular I discuss a beautiful theorem of Oscar Goldman: in a Dedekind domain $R$, every fractional ideal has some positive integer power which is principal -- in other words $\... | 8 | https://mathoverflow.net/users/1149 | 127042 | 70,850 |
https://mathoverflow.net/questions/127050 | 5 | Consider the Segre embedding
$$
\mathbb P^n\times \mathbb P^m \hookrightarrow \mathbb P^N.
$$
It seems to me that from the definition it is clear that
$$
H^0(\mathbb P^N, \mathscr O\_{\mathbb P^N}(1)) = H^0(\mathbb P^n, \mathscr
O\_{\mathbb P^n}(1))\otimes H^0(\mathbb P^m, \mathscr O\_{\mathbb P^m}(1)),
$$
but ... | https://mathoverflow.net/users/31847 | Is the Segre embedding projectively normal? | Yes. The Segre embedding $i:P:=\mathbb P^n \times \mathbb P^m\to \mathbb P^N$ is defined by the sections of the line bundle $O\_P(1,1):=pr\_1^\*O\_{\mathbb P^n}(1)\otimes pr\_2 O\_{\mathbb P^m}(1)$ on $\mathbb P^n \times \mathbb P^m$ and by definition of $i$ we have $i^\*O\_{\mathbb P^N}(1)=O(1,1)$. In other words, the... | 6 | https://mathoverflow.net/users/3996 | 127053 | 70,857 |
https://mathoverflow.net/questions/127010 | 4 | Is there any classification for coadjoint orbits of lower or upper triangular matrices in general case $n\times n$. Is there any reference?
| https://mathoverflow.net/users/nan | classification for coadjoint orbits of lower or upper triangular matrices | Let me first say, that I am not an expert, but was interested in the same question recently, so I would also be happy if someone provides more info on the question. Let me collect some facts, which I know.
1. As far as I understand the general classification of orbits is in certain sense "wild" problem.
2. Classifica... | 2 | https://mathoverflow.net/users/10446 | 127064 | 70,863 |
https://mathoverflow.net/questions/127068 | 1 | From "Models and Games" by Jouko Vaananen (Cambridge studies in advanced mathematics), I quote
>
> The Hardy-Ramanujan asymptotic formula says that the number of equivalence relations on a fixed set of $n$ elements is asymptotically
> $$\frac{1}{4\sqrt{3}n}e^{\pi\sqrt{2n/3}}$$
> So this is also an asymptotic uppe... | https://mathoverflow.net/users/20343 | Does the Hardy-Ramanujan Asymptotic Formula Partition Sets or Integers? | If $R$ and $S$ are two equivalence relations on a set $X$ of card. $n$, we say that they are equivalent if there exists a bijection $\varphi:X\rightarrow X$ such that:
$\varphi(x)R\varphi(y)\quad\iff\quad xSy$
Then the partition function $n$ is counting the number of equivalence relations on $X$ modulo this identif... | 3 | https://mathoverflow.net/users/1049 | 127069 | 70,864 |
https://mathoverflow.net/questions/127055 | 3 | EDIT: I'm bumping this, because while Joel ruled out some naive options, my question in bold below is not yet answered.
Suppose I have a directed partially ordered set $(\Gamma,\leq)$ with a bottom element (every pair of objects has *some* upper bound), such that $\Gamma$ is [well-quasi-ordered](http://en.wikipedia.... | https://mathoverflow.net/users/4177 | How much of ZFC do I need to construct this cofinal, order-preserving class function? | **Update: A counterexample.** I've realized that the claim you are trying to generalize from sets to classes is not actually true for sets. Specifically, there are directed well-quasi-ordered sets that do not admit any linearly ordered cofinal subset. And a similar counterexample exists for classes.
**Theorem.** Ther... | 6 | https://mathoverflow.net/users/1946 | 127073 | 70,866 |
https://mathoverflow.net/questions/127078 | 5 | Is the group $\mathcal{U} (n)$ of all $n\times n$ unitary matrices over $\mathbb{C}$ a (local) path-connected space?
If so, what are the connected components of the unitary matrix group $\mathcal{U}(n)$? Is the number of components finite? What is the representative for each component? Is each components closed in n... | https://mathoverflow.net/users/32709 | Is the unitary matrix group path-connected? | Every matrix Lie group is a smooth manifold, hence it is path-connected if and only if it is connected.
And $U(n)$ is compact and connected as a topological space (any unitary matrix can be diagonalized by a unitary matrix, this gives a path from it to the identity). It is not simply-connected, though.
We have $\pi\_1(... | 9 | https://mathoverflow.net/users/32332 | 127079 | 70,867 |
https://mathoverflow.net/questions/127080 | 13 | Simpson's book shows $\mathsf{ACA}\_0$ is conservative over $\mathsf{PA}$ in the natural way by model theory using definable subsets. Of course, $\mathsf{ACA}\_0$ being conservative over PA is interesting even apart from consistency strength. So one might not be explicit about the metatheory. And I am sure it is not a ... | https://mathoverflow.net/users/38783 | What metatheory proves $\mathsf{ACA}_0$ conservative over PA? | The conservativity of $\mathrm{ACA}\_0$ over $\mathrm{PA}$ is provable in $I\Delta\_0+\mathit{SUPEXP}$ by a cut elimination argument. It is not provable in $I\Delta\_0+\mathit{EXP}$, since $\mathrm{ACA}\_0$ has superexponential speedup over $\mathrm{PA}$ (a result attributed to Solovay, Pudlák, and Friedman).
EDIT: L... | 18 | https://mathoverflow.net/users/12705 | 127084 | 70,871 |
https://mathoverflow.net/questions/127076 | 16 | What is the length of the shortest word $w\in F\_2$ such that $w(x,y)$ is trivial for every $x,y\in S\_n$?
There is a simple argument showing that we must have $\ell(w)\geq n$. See [here](https://mathoverflow.net/questions/20471/why-are-free-groups-residually-finite/20485#20485) for instance. It seems likely however ... | https://mathoverflow.net/users/20598 | What is the length of the shortest law of $S_n$? | As far as I know, the state of the art gives some improvements to those bounds, but they are not huge. For a better lower bound: given a nontrivial element $w \in F\_2$ of word length $\ell$, using a result of Buskin (*Economical separability in free groups*, Sib. Math. J., 50 (2009), 603-608) there exists a subgroup, ... | 15 | https://mathoverflow.net/users/3970 | 127091 | 70,876 |
https://mathoverflow.net/questions/127086 | 7 | I am struggling with an integral pretty similar to one already resolved in MO (link: [Integration of the product of pdf & cdf of normal distribution](https://mathoverflow.net/questions/101469/integration-of-the-product-of-pdf-cdf-of-normal-distribution) ). I will reproduce the calculus bellow for the sake of clarity, b... | https://mathoverflow.net/users/32977 | Integral of the product of Normal density and cdf | [The horror, the horror](http://www.youtube.com/watch?v=aNUr__-VZeQ)... :-)
Recall that $\Phi(x)=P[X\leqslant x]$ for every $x$, where the random variable $X$ is standard normal, and that, for every suitable function $u$,
$$
\int\_{-\infty}^{+\infty}u(x)\phi(x)\mathrm dx=E[u(Y)],
$$
where the random variable $Y$ is s... | 10 | https://mathoverflow.net/users/4661 | 127094 | 70,879 |
https://mathoverflow.net/questions/127035 | 10 | Hi,
I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the uniform distribution on the unit disc. I was wondering if there is an intuitive way of understanding why this should be the case.
| https://mathoverflow.net/users/20381 | Intuition behind the spectral density of random matrices | I don't know of a fully intuitive derivation, but there are some informal arguments that give the circular law with a relatively small amount of calculation.
Let $M$ be a matrix where the entries are iid with mean zero and variance one. One can begin with the determinant formula
$$ \log |\det( M - z )| = \sum\_{j=1... | 14 | https://mathoverflow.net/users/766 | 127101 | 70,883 |
https://mathoverflow.net/questions/127100 | 0 | The number of non-isomorphic equivalence relations on a set of $n$ elements is the partition function
$$p(n) =\frac{1}{\pi\sqrt{2}} \sum\_{k=1}^{\infty} \sum\_{h=1}^{k} \delta\_{\gcd(h,k),1} \text{exp}\left(\pi i \sum\_{j=1}^{k-1} \frac{j}{k}\left(\frac{hj}{k} - \left\lfloor \frac{hj}{k} \right\rfloor - \frac{1}{2}\rig... | https://mathoverflow.net/users/20343 | Asymptotics of the Number of Non-Isomorphic Equivalence Relations and the Number of Non-Isomorphic Relations | You don’t need either of the two fancy formulas. Since every equivalence relation is the kernel of a function from the $n$-element set into itself, their number is at most $n^n$ (and taking them up to isomorphism can only make it smaller). On the other hand, there are $2^{n^2}$ binary relations in total, and each isomo... | 3 | https://mathoverflow.net/users/12705 | 127105 | 70,885 |
https://mathoverflow.net/questions/69074 | 26 | In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^\*M$ and the "raising indices" operator by $\sharp:T^\*M \to TM$. These isomorphisms are sometimes referred to as *musical isomorphisms*, as stated on [Wikipedia](http://en.wikipedia.org/wiki/Musical_isomorphism) and in [several](http... | https://mathoverflow.net/users/8452 | The Origin of the Musical Isomorphisms | Marcel Berger, on p. 696 of his Springer-book: [A panoramic view of Riemannian Geometry](http://rads.stackoverflow.com/amzn/click/3540653171), writes "Next we define the canonical musical duality", referring to a footnote: "These dualities are called *musical* because they are often written with symbols like ♭: V $\rig... | 19 | https://mathoverflow.net/users/nan | 127112 | 70,889 |
https://mathoverflow.net/questions/127108 | 19 | If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+x+x,y+y+y$)$-$min($x,y$),$\:x+y$)$\ =\ $min($x+x,y+y$)?
| https://mathoverflow.net/users/3621 | Do all subtraction-free identities tropicalize? | It suffices to show that whenever $F$ is a function $\mathbb R\_{\geq 0}^k\to\mathbb R\_{\geq 0}$ defined using $\times,/,+,1$, and $f,g$ is the corresponding tropicalization $\mathbb R^k\to\mathbb R$, for all real $x\_1,\dots,x\_k$ we have $$F(\exp(-\beta x\_1),\dots,\exp(-\beta x\_k))^{1/\beta}\to \exp(-f(x\_1,\dots,... | 18 | https://mathoverflow.net/users/408 | 127118 | 70,891 |
https://mathoverflow.net/questions/126969 | 2 | Let $N\geq 1$ be an integer and let $S\_2(\Gamma\_0(N))$ be the cusp forms of weight 2 for the usual congruence subgroup $\Gamma\_0(N)\subset SL\_2(\mathbb Z)$.
Let $a\_n(f)$ denote the n-th Fourier coefficient of $f\in S\_2(\Gamma\_0(N))$, $n\geq 1$.
Let $X\_0(N)$ be a smooth projective model over $\mathbb Q$ of the... | https://mathoverflow.net/users/32952 | Lower bounds for Petersson inner products of cuspforms with integral Fourier coefficients | It's not hard to see that the answer to (a) is yes. There is a basis of $S\_2^{\textrm{new}}(\Gamma\_0(N))$ consisting of newforms. These newforms come into Galois orbits $\{f^\sigma\}\_{\sigma}$. Here $\sigma$ runs through the embeddings of $K\_f$ into $\mathbf{C}$, where $K\_f$ is the field generated by the Fourier c... | 2 | https://mathoverflow.net/users/6506 | 127123 | 70,894 |
https://mathoverflow.net/questions/127114 | 10 | *Given two $0$-symmetric convex bodies $K \subset L \subset \mathbb{R}^n$, is it true that the Loewner ellipsoid of $K$ is contained in the Loewner ellipsoid of $L$?*
I have just finished proving a lemma stating that the Loewner ellipsoid depends continuously on parameters and the proof is a bit more elaborate than I... | https://mathoverflow.net/users/21123 | Monotonicity of Loewner ellipsoid? | No, the Loewner ellipsoid is not monotone w.r.t. inclusion. Let $K$ be a square, whose Loewner ellipsoid is its circumcircle. Let $L$ be any other ellipse through the four vertices of $K$. The Loewner ellipsoid of $L$ is $L$ itself but it does not contain the circle. (I assume that the Loewner ellipsoid is the minimal ... | 16 | https://mathoverflow.net/users/4354 | 127127 | 70,896 |
https://mathoverflow.net/questions/127013 | 6 | Suppose that $k$ is a field of characteristic $p$ such that $k$ is not a finite $k^p$-module. For example, $k = \mathbb{F}\_p(x\_1, x\_2, x\_3, ...)$.
Is it true that $k[[x]]$ is a free $(k[[x]])^p$-module? We know that it is flat by a theorem of Kunz. However, it is certainly not finite, so flat is not the same as ... | https://mathoverflow.net/users/3521 | $k[[x]]$ as a $(k[[x]])^p$ module for ugly fields | The answer to the question is no.
Let $R = k[[x]]$, $S = k[[y]]$, and $f:R \to S$ is the absolute Frobenius. We must show that $S$ is not a free $R$-module. By assumption on $k$, we know that $S$ must have infinite rank if it were free. On the other hand, $S$ is $x$-adically complete (since $x^p = y \in S$). Hence, ... | 5 | https://mathoverflow.net/users/25792 | 127131 | 70,898 |
https://mathoverflow.net/questions/127115 | 1 | Let $f:ℝ→ℝ$ be a real analytic function with infinitely many isolated zeros. Let us define the function:
$$h(s₁,s₂,...,s\_{r+1})=\prod\_{k=1}^{r+1}f^{(k+1)}(\left(1-2\prod\_{j=1}^{k}s\_{j}\right)$$
Also, all the $k$-th derivatives of $f$ have infinitely many isolated zeros. Does $h$ have infinitely many **isolated** ze... | https://mathoverflow.net/users/25947 | Does $h$ have infinitely many isolated zeros? | I assume $f$ was a real-valued analytic function on $(0,1)$, otherwise I do not understand the notation for $h$. But then no zero of $h$ can be isolated. Indeed, the zero set of $h$ is the union of the zero sets of its $r+1$ factors, and all of them vanish on some $r$ dimensional submanifold of $(0,1)^{r+1}$.
| 2 | https://mathoverflow.net/users/6101 | 127132 | 70,899 |
https://mathoverflow.net/questions/127128 | 21 | Why do [currents](http://en.wikipedia.org/wiki/Current_%28mathematics%29), functionals on compactly supported differentiable *n*-forms, bear the name they do?
I've assumed that it has something to do with an electrical current being formalized as a vector field along a curve or other submanifold, but this is nothing ... | https://mathoverflow.net/users/32961 | Why are currents named currents? | George de Rham motivates the name "*current*" for his functionals by the fact that in three dimensions electric currents are represented by "*1-dimensional currents*".
The history of this concept is described by Jesper Lützen in [De Rham’s Currents](http://dx.doi.org/10.1007/978-1-4613-9472-3_6).
| 16 | https://mathoverflow.net/users/11260 | 127140 | 70,903 |
https://mathoverflow.net/questions/127139 | 6 | I have three related questions.
I understand homotopy pushouts via the standard model structure on the diagrams - and taking the derived functor of the pushout.
I'm not sure, but I believe that in classical algebraic topology, we use implicitly at least three model structures in Top: Quillen structure, Hurewicz st... | https://mathoverflow.net/users/18017 | Homotopy excision and homotopy pushout | 1) I assume your three model structures have the same weak equivalences, correct me if I'm wrong. Let $\mathcal C$ be a model category and $I$ a small category, e.g. $I=\bullet\leftarrow \bullet\rightarrow\bullet$ if you're interested in push-outs. The homotopy colimit functor $\operatorname{hocolim}\_I\colon\operatorn... | 5 | https://mathoverflow.net/users/12166 | 127147 | 70,907 |
https://mathoverflow.net/questions/127150 | 1 | I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma\_{i,j}= \exp(-h\_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\partial \log(\det(\Sigma))}{\partial \rho} \text{ and } \frac{\partial \Sigma^{-1}}{\partial \rho}.$$
Someone can help m... | https://mathoverflow.net/users/32990 | Derivative of log determinant and inverse | [*modified according to the clarification given in comments*].
In general, for an invertible square matrix $\Sigma=\Sigma(\rho)$, differentiably depending on the real variable $\rho$, we have: $(\Sigma^{-1})'=-\Sigma^{-1} \Sigma' \Sigma^{-1}$, and $\big(\det(\Sigma)\big)'=\operatorname{tr} (\Sigma^{-1} \Sigma')\det(\... | 2 | https://mathoverflow.net/users/6101 | 127152 | 70,909 |
https://mathoverflow.net/questions/127120 | 1 | I would like to understand the syzygies of the determinantal ideal $I\_r$, generated by the $r\times r$ minors of a matrix $(X\_{ij})$ of indeterminantes in the polynomial ring over an algebraically closed field of characteristic zero. The original resource for this object of study is the paper *"Syzygies des variétés ... | https://mathoverflow.net/users/9947 | Syzygies of determinantal varieties: Looking for English text | Weyman's book is a good reference. If you want other references, you can see the paper by myself joint with Snowden and Weyman: <http://arxiv.org/abs/1209.3509>
One can view the coordinate ring of the determinantal variety as a ring of invariants for a natural group action and in that paper we calculate the syzygies ... | 5 | https://mathoverflow.net/users/321 | 127153 | 70,910 |
https://mathoverflow.net/questions/127154 | 1 | Let $Sset$ denote the category of simplicial sets with its Quillen model structure,
when is a functor $F: (ho Sset)^{op} \to Ab$ representable? With $Ab$ category of Abelian groups. There is probably some classical references but my googlefu wasn't strong enough. I am hoping it would just be the direct translation of ... | https://mathoverflow.net/users/20196 | Brown representability for the standard model category of simplicial sets | There is a second classical paper by Brown himself which abstracts his original paper:
Brown, Edgar H., Jr.
Abstract homotopy theory.
Trans. Amer. Math. Soc. 119 1965 79–85.
I think you will find that it applies directly. Of course, that
was well before model categories.
| 3 | https://mathoverflow.net/users/14447 | 127164 | 70,913 |
https://mathoverflow.net/questions/127116 | 13 | Two greedy chocolate eaters play the following game involving $n$ pieces of chocolate
and an additional parameter $\alpha$ with initial value $1$: Each player eats either $\alpha$
pieces of chocolate or he increments $\alpha$ by $2$ (replacing thus $\alpha$ by $\alpha+2$)
and eats then $\alpha$ pieces of chocolate (he ... | https://mathoverflow.net/users/4556 | An unfair game involving an odd number of pieces of chocolate | Let $n$ be odd. Inductively assume that $n-2$ is a first player win. Suppose eating $3$ is not a winning play for the first player. Then we should eat $1$ first, of course. If the second player responds by eating $1$, we use the winning strategy for $n-2$. If the second player responds by eating $3$, then we use the se... | 15 | https://mathoverflow.net/users/2954 | 127165 | 70,914 |
https://mathoverflow.net/questions/127157 | 26 | $\DeclareMathOperator\GL{GL}$Apologies if this question has already been dealt with on MO. I am wondering about the status of the global Langlands conjectures for $\GL\_2$ over the rational numbers. How close is humanity to the proof of these conjectures?
I guess Langlands picture is related to (or, should I say, inc... | https://mathoverflow.net/users/12168 | Status of (global) Langlands conjecture for $\mathrm{GL}_2$ over $\mathbb{Q}$ | This question has already been discussed here, though perhaps not exactly on these terms (but
see [What makes Langlands for n=2 easier than Langlands for n>2?](https://mathoverflow.net/questions/74472/what-makes-langlands-for-n2-easier-than-langlands-for-n2)).
So first there is an ambiguity of what id global Langland... | 24 | https://mathoverflow.net/users/9317 | 127166 | 70,915 |
https://mathoverflow.net/questions/127159 | 1 | Hello all.
I have the following (perhaps basic) question: Let $X$ be a separable metric space. Does there necessarily exist a countable set $\mathcal{C}$ of Borel sets in $X$ such that any two probability measures which agree on $\mathcal{C}$ must agree on the whole of $\mathcal{B}(X)$?
(And slightly more generally... | https://mathoverflow.net/users/15570 | "Uniqueness of extension" results for measures on separable spaces | Yes. This can be proved using [Dynkin's $\pi$-$\lambda$ theorem](http://en.wikipedia.org/wiki/Dynkin_system). The collection $\mathcal{L} := \{ B \in \Sigma : \mu(B) = \nu(B)\}$ is a $\lambda$-system. By Dynkin's theorem, if $\mathcal{L}$ contains a $\pi$-system which generates $\Sigma$ then $\mathcal{L} = \Sigma$, i.e... | 1 | https://mathoverflow.net/users/4832 | 127171 | 70,917 |
https://mathoverflow.net/questions/127182 | 1 | Suppose you are facing an infinitely-long wall. Somewhere in the wall is a door, but you can only see the door if you are right next to it. You want to go through the door.
You don't know whether the door is to your right or to your left.
How far in each direction should you walk to minimize the total (expected) di... | https://mathoverflow.net/users/11960 | What is the optimal distance to walk back and forth if you don't know how far away or which side the target is on? | To be well posed you need to specify the probability distribution which specifies where the door is placed (this is the old problem of not being able to put a uniform distribution on a line).
Once you specify this, this is known as the linear search problem. There is a fair amount of literature on this problem. As a... | 9 | https://mathoverflow.net/users/630 | 127183 | 70,922 |
https://mathoverflow.net/questions/97690 | 11 | Consider the global projective model category
of simplicial presheaves on some category
(the category of smooth manifolds is particularly interesting to me).
In Section 9.1 of Dugger's paper “[Universal homotopy theories](https://arxiv.org/abs/math/0007070)”
one can find a sufficient condition for a simplicial preshe... | https://mathoverflow.net/users/402 | Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves | Let $\mathscr C$ be a small category. Necessary and sufficient conditions for a presheaf $F$ to be cofibrant in the global projective model structure on $[\mathscr C^\mathrm{op}, [\Delta^\mathrm{op}, \mathbf{Set}]]$ are that:
(1) Each $F(-)(n) \colon \mathscr C^\mathrm{op} \to \mathbf{Set}$ is projective (i.e., a cop... | 19 | https://mathoverflow.net/users/33001 | 127187 | 70,923 |
https://mathoverflow.net/questions/127185 | 3 | Hi!
I am looking for notions of general position that are stronger than linear general position.
To illustrate, 3 points in linear general position don't lie on a line. I want a notion that would claim that 6 points in general position don't lie on a quadric.
To be more specific: Assume I have an ideal $I$ in $\m... | https://mathoverflow.net/users/32999 | Strong notions of general position | The defining equation of a degree $d$ hypersurface in $\mathbb P^n$ has $n+d\choose d$ coefficients and hence these hypersurfaces may be parametrized by a projective space of dimension ${n+d\choose d} -1$. Picking a point to be contained by the hypersurface is a linear equation on the coefficients of these defining equ... | 4 | https://mathoverflow.net/users/10076 | 127188 | 70,924 |
https://mathoverflow.net/questions/127174 | 5 | Let $p$ be an odd prime. Let $K\_{\infty}$ be the field extension of $\mathbb{Q}$ generated by all $p^n$-th roots of unity. Let $M/K\_{\infty}$ be the maximal abelian $p$-extension of $K\_{\infty}$ unramified outside $p$. Denote its Galois group by $\mathcal{X}$. Then the Iwasawa main conjecture concerns the structure ... | https://mathoverflow.net/users/32994 | A question on Iwasawa theory | By Theorem 13.31 in L. Washington *Introduction to Cyclotomic Fields*, Second Edition, GTM 83 we know that $\mathcal{X}\sim \Lambda^{r\_2}\oplus(\Lambda-$torsion), where $r\_2$ is the number of complex embeddings of $\mathbb{Q}(\zeta\_p)$, so $r\_2=(p-1)/2$, and $\sim$ means "pseudo-isomorphism". In particular, since w... | 7 | https://mathoverflow.net/users/18238 | 127195 | 70,925 |
https://mathoverflow.net/questions/127178 | 2 | Suppose a number $a=\sum\_{r\in R,s\in S}r^{-s}$ where R is a subset of natural numbers with positive density and S is a subset of natural numbers of density 0. Is $a$ transcendental?
| https://mathoverflow.net/users/7360 | Transcendency of numbers of a special form. | If you allow $S$ to be finite, then the answer is no: *any* real number $x\in(0,\frac{\pi^2}6-1)$ can be written as $\sum\_{r\in R} r^{-2}$ for some set $R$ of positive integers with positive density.
To see this, first choose $m\ge2$ such that $m^2x > \frac{\pi^2}6$, and let $R\_1 = m\mathbb N$, so that $\sum\_{r\in... | 7 | https://mathoverflow.net/users/5091 | 127201 | 70,927 |
https://mathoverflow.net/questions/127194 | 0 | Let $p$ be a Merssene prime, i.e. $p=2^a-1$, where $a$ is a prime.
Let $R$ be a 2-group of order $2(p+1)=2^{a+1}$. Also we know that $|Z(R)|=2$ and $R/Z(R)$ is abelian.
Can we conclude that $R$ has no automorhism of order $p$?
I know that there is a theorem that says that if $p$ is a prime and $G$ is a $p$ -group w... | https://mathoverflow.net/users/31045 | The automorphisms of a 2-group of nilpotency class 2 | For $a=2$, you can take $R=Q\_8$, which does indeed have an automorphism of order 3.
For $a>2$ there are no groups $R$ satisfying your hypothesis. In such a group, the commutator map would be a non-generate alternating (in fact symmetric in this case) bilinear map $G/Z(G) \times G/Z(G) \to Z(G)$, which forces $G/Z(G)... | 4 | https://mathoverflow.net/users/35840 | 127209 | 70,930 |
https://mathoverflow.net/questions/124091 | 1 | For a standard Brownian motion, the generator of the diffusion is
$$
L = \frac12 \frac{d^2}{dx^2}.
$$
Is there a nonstandard definition of this generator?
| https://mathoverflow.net/users/nan | Nonstandard definition for the generator of a standard Ito diffusion | try F. Herzberg at <http://link.springer.com/chapter/10.1007/978-3-642-33149-7_7>
| 0 | https://mathoverflow.net/users/28128 | 127225 | 70,936 |
https://mathoverflow.net/questions/127238 | 0 | Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductive group. Roughly speaking $[X//G]$ is the fine moduli space for $M$; hence it carries a universal family and there's a correspondence betwee... | https://mathoverflow.net/users/4096 | universal families and maps to quotient stacks | First a couple corrections. If $M$ is represented by quotient stack, it better be a contravariant functor, and moreover, it should probably take values in groupoids, not set. Anyway, here's what going on:
$X$ does not carry a universal family, but it carries a "locally" universal family $v \in M\left(X\right)$, in th... | 3 | https://mathoverflow.net/users/4528 | 127242 | 70,942 |
https://mathoverflow.net/questions/127244 | 2 | Hi!
I'm sure it is well known but i don't know enough algebraic topology... Let $k\geq 2$,
$$S^{2k-1}\hookrightarrow \mathbb{C}^{k}$$
be the unit sphere and $G$ be a finite subgroup of $U(k)$ acting linearly (the action is induced by the one on $\mathbb{C}^{k}$) and freely on $S^{2k-1}$ so the quotient $S^{2k-1}/G$ ... | https://mathoverflow.net/users/4971 | Calculation of $H^{2}(S^{2k-1}/G,\mathbb{Z})$ | Yes there is, and it goes by the name of the *Cartan-Leray spectral sequence*. See Ken Brown's "Cohomology of Groups", VII.7.9 for a textbook reference.
This spectral sequence in your setup has
$$
E\_2^{p,q} = H^p(G;H^q(S^{2k-1};\mathbb{Z}))
$$
and converges to a graded group associated to $H^\ast(S^{2k-1}/G;\mathbb{... | 8 | https://mathoverflow.net/users/8103 | 127247 | 70,945 |
https://mathoverflow.net/questions/127248 | 13 | I know that in 1952 [Jitsuro Nagura](https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-28/issue-4/On-the-interval-containing-at-least-one-prime-number/10.3792/pja/1195570997.full) was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k ... | https://mathoverflow.net/users/15915 | At what point would an elementary generalization of Bertrand's Postulate be interesting? | Current results are able to yield such results. Depending on how generous one is regarding what $X$ is. If it is just the optimal value can be calculated *exactly* this will work for many more $k$ and if one is happy with an explicit bound for all $k$.
For example Dusart showed that
$$
\frac{x}{\log x - 1} \le \pi(... | 20 | https://mathoverflow.net/users/nan | 127262 | 70,952 |
https://mathoverflow.net/questions/127250 | 5 | I have some general questions on Sturm-Liouville theory. We are planning to introduce a graduate course on Sturm-Liouville theory and every one has been asked to propose topics which might be suitable for the course.
I would like to know the following.
1. Is it worth to have a course exclusively on just Sturm-Lio... | https://mathoverflow.net/users/8974 | A graduate course on Sturm Liouville theory? | A more advanced/comprehensive course can be based on Atkinson's book Discrete and continuous boundary
problems. No functional analysis is required, neither for Hilbert-Courant nor for Atkinson.
(When Courant wrote the first volume of HC, functional analysis did not exist yet:-)
Another book which studies some of thes... | 5 | https://mathoverflow.net/users/25510 | 127276 | 70,958 |
https://mathoverflow.net/questions/127237 | 0 | Is there any result like the mean value theorem for harmonic functions on ellipsoids (instead of sphere)?
| https://mathoverflow.net/users/15197 | Mean value theorem for harmonic functions on ellipsoid | Let me expand Aaron's answer: there is a mean value theorem with any centrally symmetric surface.
You integrate your harmonic function on the surface
against the harmonic measure at the center, and you recover
the value of your function at the center. You can also generalize this to non centrally symmetric
surfaces, b... | 2 | https://mathoverflow.net/users/25510 | 127277 | 70,959 |
https://mathoverflow.net/questions/127281 | 3 | Suppose I have a topological category $\mathcal{C}$ of the following form: The object space consists of just two points $p\_1, p\_2$. The endomorphism space of $p\_1$ contains just the identity. The endomorphism space $End(p\_2)$ of $p\_2$ is a group $G$ and we have a restriction map $End(p\_2) \to hom(p\_1, p\_2)$ whi... | https://mathoverflow.net/users/3995 | classifying space of a strange category | I think it is equivalent to $BG$. Picking an $x\_0 \in X := hom(p\_1, p\_2)$ we get a map
$$X \to G$$
sending x\_0 \* g to g. Using this on morphism sets we get a map $F: \mathcal{C} \to G$ which you easily check to be a functor. Furthermore, the element $x\_0$ can be used to get a natural transformation $id\_\math... | 6 | https://mathoverflow.net/users/318 | 127286 | 70,963 |
https://mathoverflow.net/questions/127253 | 1 | Dear mathematicians,
I want to know how much advance there has been in complex dynamics of several variables. I am at present reading Carleson's book on Complex Dynamics on one variables.Curious to know about several variables case.t
Specifically,what are the best papers in this area.
| https://mathoverflow.net/users/30081 | complex dynamics in several variables | There has been indeed much development in the dynamics of several complex variables in the last 20 years or so. The directions which the subject took focus on different aspects. E.g., is one interested in maps in the affine space, projective space or more general complex manifolds? Are the maps holomorphic diffeomorphi... | 5 | https://mathoverflow.net/users/14493 | 127287 | 70,964 |
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