parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/111618 | 3 | Any Hyperkähler manifold has 3 complex structures $I\_{1}, I\_{2}, I\_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI\_{1} + bI\_{2} + cI\_{3}$, where $(a,b,c) \in S^{2} \subset \mathbb{R}^{3}$? Hope the question is not too trivial :).
| https://mathoverflow.net/users/22073 | Rotation in Hyperkähler manifolds | Assume $J$ is compatible with $g$ so that $J^\dagger = -J$. Then $J$ is a linear combination of $I\_i$ as above if and only if $J I\_i + I\_i J = -2 a\_i \mathbf{1}$, where $\mathbf{1}$ is the identity endomorphism and $a\_i$ are real numbers satisfying $a\_1^2 + a\_2^2 + a\_3^2 = 1$.
**Proof**: Necessity is obvious,... | 4 | https://mathoverflow.net/users/14628 | 111695 | 64,013 |
https://mathoverflow.net/questions/111699 | 1 | Assume that we have an $n\times n$ matrix ${\bf A}$ with elements drawn i.i.d. Gaussian with mean zero and variance 1.
Are there any results on the asymptotic behavior of its $i$-th largest singular value?
I am mostly interested when $i=\sqrt{n}$ or $i=n/\log(n)$, or in general when $i$ is some increasing function of $... | https://mathoverflow.net/users/2763 | Scaling laws for singular values of random matrices | Yes, see [Roman Vershynin's notes.](http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf)
| 1 | https://mathoverflow.net/users/11142 | 111700 | 64,016 |
https://mathoverflow.net/questions/111676 | 3 | **A little bit of background:**
In a small course a friend of mine is attending, everybody shall give a presentation about some topic. Now for each presentation, here can be two *assistants* which are other students that inform themselves about the topic and help moderating the discussion etc.
Of course, optimally,... | https://mathoverflow.net/users/4080 | How to count 2-2 regular directed graphs with n vertices? | Here is how it can be done. Probably the answer appears somewhere already, but I don't recall it and I'm too lazy to search.
Draw $2n$ dots $A=\lbrace a\_1,a'\_1,\ldots,a\_n,a'\_n\rbrace$ and another $2n$ dots $B=\lbrace b\_1,b'\_1,\ldots,b\_n,b'\_n\rbrace$. Take a matching from $A$ to $B$. Now make a graph with vert... | 5 | https://mathoverflow.net/users/9025 | 111705 | 64,019 |
https://mathoverflow.net/questions/111312 | 9 | Suppose $a,b$ are two natural numbers relatively prime to $n$ and to each other. Assume $n\geq ab+1$. Suppose further that $\frac{a}{b}\equiv k \pmod{n}$ for some $k\in \lbrace 1,2,\dots, n-1\rbrace$ and $\frac{a}{b}\equiv k'\pmod{n+ab}$ for some $k'\in \lbrace 1,2,\dots, n+ab-1\rbrace$.
>
> **Question:** Is there ... | https://mathoverflow.net/users/2384 | Length of Hirzebruch continued fractions | Lets call expansions
$$\langle
x\_1,\ldots,x\_m\rangle:=\cfrac{1}{x\_1-{\atop\ddots\,\displaystyle{-\cfrac{1}{x\_m}}}}$$
(as in Perron's book) reduced regular continued fractions (RRCF).
Probably they are older then Hirzebruch.
We'll prove more precise statement.
**Theorem.** *If $(n,ab)=1$ and $n>ab$ then RRCF for... | 10 | https://mathoverflow.net/users/5712 | 111713 | 64,023 |
https://mathoverflow.net/questions/111706 | 3 | I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know.
Suppose $u: \bar D \to \Bbb R$ is a smooth function on the closure of the unit disc in $\Bbb C$. We introduce the fo... | https://mathoverflow.net/users/17965 | Subharmonic envelope | $h$ is the solution of the *obstacle problem*: it minimizes the Dirichlet energy
$$\int |\nabla v|^2$$
among all functions $v$ less or equal than $u$, and having the same boundary values as $u$. One can alo characterize $h$ as the solution of a free boundary problem, namely $h$ is harmonic on $\Omega \setminus \Gamma$,... | 3 | https://mathoverflow.net/users/8794 | 111716 | 64,025 |
https://mathoverflow.net/questions/109027 | 35 | A theorem of Frobenius states that if $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Frobenius conjectured that if the number of solutions is exactly $n$, then the set of solutions form a characteristic subgroup of $G$. The conjecture was eventually p... | https://mathoverflow.net/users/10146 | Applications of Frobenius theorem and conjecture | Danny Gorenstein, in his book *Finite Simple Groups, An Introduction to their Classification* discusses a very significant application of Frobenius' theorem (see p. 95). He is interested in the following central problem in the Classification of Finite Simple Groups:
>
> Let $G$ be a simple group in which the struct... | 14 | https://mathoverflow.net/users/801 | 111717 | 64,026 |
https://mathoverflow.net/questions/111712 | 1 | I am trying to understand a certain sentence in a paper that I am reading. Let me start with some notation/background. (For a motivation of why this should be interesting, see below, under the questions.)
Notation/background
-------------------
First the (pretty standard) setup:
* $k$ a field with a discrete valu... | https://mathoverflow.net/users/21815 | Understanding the determinant of the action of Frobenius on the character group of the toric part of the reduction of the Jacobian of a curve. | You need to work with just regular homomorphisms between those two groups, i.e. algebraic characters. That does two things.
First, note that $\operatorname{End}(\kappa\_s^\*)=\mathbb Z$.
Second, $f \circ \sigma$ is not in general a regular homomorphism. You need to take $\sigma^{-1} \circ f \circ \sigma$.
The key... | 4 | https://mathoverflow.net/users/18060 | 111735 | 64,031 |
https://mathoverflow.net/questions/111740 | 3 | look at Problem14.12 of chapter3 of "Aliprantis-Burkinshaw-Principles of real analysis-3ed.1998" ; 12. Let A be the collection of all measurable subsets of X of finite measure. That is, A = {B in X: m(A) < oo}. a. Show that A is a semiring. b. Define a relation ~ on A by B ~ C if m(B \Delta C) = 0. Show that ~ is an eq... | https://mathoverflow.net/users/27896 | Is P(X) a connected set for a set X with a $\sigma$-algebra P(X) and a measure function m on it to [0,$\infty$] when P(X) is equiped with meter d, that for every A,B in P(X), $d(A,B)=m(A \Delta B)$? | Note that to get a true distance, you have to restrict to measurable sets of finite measure, and to quotient over null sets. This way you get a closed subspace $M$ of $L^1(X)$ (the subset of binary functions). If the measure is atomless, then $M$ is even path-connected. A well-known result on atomless measures (essenti... | 5 | https://mathoverflow.net/users/6101 | 111742 | 64,034 |
https://mathoverflow.net/questions/110531 | 2 | It is not difficult to get a formula for the number of $n\times n$-matrices
$g\in\mathrm{Mat}\_{n}(\mathbb{F}\_{q})$ with $\mathrm{rank}(g)=k$. Namely we have got
\begin{align\*}
\mid\{g\in\mathrm{Mat}\_{n}(\mathbb{F}\\_{q})\mid \mathrm{rank}(g)=k\}\mid = \binom{n}{k}\\_{q}\cdot(q^{n}-1)\cdots(q^{n}-q^{k-1})
\end{ali... | https://mathoverflow.net/users/27514 | What is the number of $2n\times 2n$-matrices $g=\begin{pmatrix} A & B \\\ C & -A^{t} \end{pmatrix}$, $B$ and $C$ symmetric, over the finite field $\mathbb{F}\_{q}$ with $\mathrm{rank}(g)=k$? | Just for the (virtual) glory of possible reputation points, here is the comment as an answer.
The columns can be switched to put B and C on the main diagonal, and multiplication
of the bottom rows by -1 should produce a symmetric matrix from one of the desired form.
This should be a bijective rank preserving operatio... | 3 | https://mathoverflow.net/users/3493 | 111744 | 64,036 |
https://mathoverflow.net/questions/111746 | 1 | Is there a closed form for the characteristic polynomial of the graph cycle (of $n$ edges and $n$ summits)?
I know it for the graph path (it is a Chebyschev polynomial), but I couldn't find a closed form when adding the missing edge.
| https://mathoverflow.net/users/17164 | Is there a closed form for the characteristic polynomial of the graph cycle (of n edges and n summits) ? | If I use $\phi$ for characteristic polynomial, then
$$
\phi(C\_n,t) = \phi(P\_n,t) - \phi(P\_{n-2},t) - 2.
$$
This follows from the formulas in Section 4.1 of "Algebraic Combinatorics" by yours truly,
see in particular Exercise 5 in Chapter 4. (Sorry about the self-referencing, but I have
no recollection of seeing the... | 4 | https://mathoverflow.net/users/1266 | 111766 | 64,045 |
https://mathoverflow.net/questions/111281 | 2 | Let $A$ and $B$ be two matrices with $\det(A)=\det(B)=1$. Does it follow that
$\sqrt{\mathrm{tr}(A^TB^TBA-I)}\le\sqrt{\mathrm{tr}(A^TA-I)}+\sqrt{\mathrm{tr}(B^TB-I)}$
I suspect that this can be shown using the singular value decomposition, but I've not been able to write a proof yet. Scaling arguments suggest that ... | https://mathoverflow.net/users/1969 | Trace inequality for matrices with determinant 1 | As per fedja's comment, the version of the inequality written in the question does not hold. A version that does hold is given below.
$\newcommand{\trace}{\mathrm{tr}}$
Define $\trace\_n(X) := \frac{1}{n}\trace(X)$ be the normalized trace. Let $U, V$ be unitary matrices. Then, the following inequality can be shown t... | 5 | https://mathoverflow.net/users/8430 | 111782 | 64,051 |
https://mathoverflow.net/questions/111787 | 6 | Equations such as $\sqrt{x+1}+\sqrt{x+2}=x+3$ are easily solvable by squaring both sides. But if we increase an extra square root, like if trying to solve $\sqrt{x+1}+\sqrt{x+2}+\sqrt{x+3}=x+4$ we encounter a problem. Squaring both sides doesn't work now, since the L.H.S. would still end up with three terms involving s... | https://mathoverflow.net/users/24715 | Is there an analytical method of solving general square root equations? | Let us set $z\_j=\sqrt{P\_j(x)}$, so that the equation becomes a *system* of algebraic equations:
$$z\_1+\cdots+z\_n=Q(x),\qquad P\_j(x)=z\_j^2\qquad\forall j=1,\ldots,n.$$
Using the resultant, you may eliminate the $z\_j$'s. At the end, you obtain a single polynomial equation $R(x)=0$. Now, the difficulty is that $R$ ... | 10 | https://mathoverflow.net/users/8799 | 111788 | 64,055 |
https://mathoverflow.net/questions/111791 | 5 | Which theorems from classical theory of continued fractions have 3-(or multi-) dimesional analogs?
Of cause classical one is a periodicity of Klein polyhedra. Probably there are some more...
| https://mathoverflow.net/users/27916 | 3-D continued fractions | (1) There is 3-D analog of Vahlen's theorem, see <http://link.springer.com/article/10.1007%2Fs11006-006-0018-6?LI=true>
(2) 3-D isolation theorems and extemal Davenport forms (see Cassels "An Introduction to the Geometry of Numbers" and Swinnerton-Dyer, "On the product of three homogeneous linear forms" Acta Arith., ... | 2 | https://mathoverflow.net/users/5712 | 111793 | 64,056 |
https://mathoverflow.net/questions/111745 | 4 | Let $K$ be a commutative unital ring *field*. Let $\pi:A \to K$ be a surjective homomorphism of commutative $K$-algebras with nilpotent kernel. (Recall that this means $\operatorname{Ker}(\pi)^n=0$ for some integer $n$.) Notice that this implies that, as a $K$-module, $A\cong K \oplus M,$ where $M=\operatorname{Ker}(\p... | https://mathoverflow.net/users/4528 | Detecting elements of nilpotent extensions via finitely generated ones | What do you think of the following example :
$$ A = \frac{K[u,x\_0, x\_1, ...]}{(u^2) + (x\_i^2-u)\_{i\in\Bbb N} + (x\_i x\_j)\_{i\neq j}}$$
If $M$ is the ideal $(u, x\_0, x\_1, \dotsc)$, then $A = K \oplus M$ and $M^4 = (0)$.
Let $I$ be a proper ideal of $A$ such that $A/I$ is finite dimensional over $K$.
It is cl... | 6 | https://mathoverflow.net/users/19205 | 111798 | 64,059 |
https://mathoverflow.net/questions/111804 | 9 | Hello everyone,
Let $\mathcal{C}$ a category which has the pullbacks and $f$:$X\longrightarrow Y$ a morphism in $\mathcal{C}$ then I know that the functor pullback $f^{\star}$:$\mathcal{C}/Y\longrightarrow\mathcal{C}/X$ has a left adjoint which is just the postcomposition by $f$.
If I add that $\mathcal{C}$ is loc... | https://mathoverflow.net/users/nan | Right adjoint to pullback functor | Let's focus on the case where $C = Set$, since this will give the intuition for other cases.
An object $p: E \to X$ in the category $Set/X$ can be thought of as an $X$-indexed set, where over every $x \in X$ there is a fiber $p^{-1}(x)$. Similarly, a morphism in $Set/X$ from $p: E \to X$ to $q: F \to X$ is a global ... | 18 | https://mathoverflow.net/users/2926 | 111806 | 64,064 |
https://mathoverflow.net/questions/111731 | 12 | I need to find some masters-level exercises about numerical methods for solving ODEs. Are there any good references?
| https://mathoverflow.net/users/18763 | Any good books on numerical methods for ordinary differential equations? | Since nobody else has mentioned them, I will recommend [**Solving Ordinary Differential Equations** (volumes I](http://www.springer.com/mathematics/analysis/book/978-3-540-56670-0) [and II)](http://www.springer.com/mathematics/computational+science+%2526+engineering/book/978-3-540-60452-5) by Hairer, Nørsett, and Wanne... | 9 | https://mathoverflow.net/users/274 | 111812 | 64,066 |
https://mathoverflow.net/questions/111810 | 3 | Is this true? For any hyperfinite $n$ that isn't finite, there is a hyperfinite set $A$ such that $\mathbb R \subset A$ and $|A|\le n$ (that's the crucial part, of course)? Intuitively it seems right, but I haven't found a reference and I am not very good at NSA.
| https://mathoverflow.net/users/26809 | Hyperfinite set containing the reals, with specified upper bound on internal cardinality? | This depends on how strong your axioms for non-standard analysis are. Certainly any sufficiently saturated model will have an $A$ of the sort you ask about. [Proof: The collection of formulas consisting of "$x$ is a set", "$|x|\leq n$", and "$r\in x$" for all standard reals $r$ is finitely satisfiable. So by saturation... | 4 | https://mathoverflow.net/users/6794 | 111813 | 64,067 |
https://mathoverflow.net/questions/111578 | 6 | I have another exterior differential system for one forms $U^i$, where the $\theta^i$ are a cotangent basis on $SO(3)$, i.e. they satisfy $d \theta^i = \epsilon\_{ijk} \theta^j \wedge \theta^k$ for the antisymmetric tensor $\epsilon$. This is related to a previous question [here](https://mathoverflow.net/questions/1114... | https://mathoverflow.net/users/17660 | ANOTHER Exterior differential system on $SO(3;\mathbb R) \times \mathbb R$ | Jeanne's calculations give the right answer, i.e., that the solutions depend on two arbitrary functions of 2 variables.
It turns out, though, that, with the right choice of variables, one can reduce the problem to an *underdetermined* system of $2$ linear equations for a pair of tensors of rank $2$ on the $2$-sphere... | 16 | https://mathoverflow.net/users/13972 | 111817 | 64,068 |
https://mathoverflow.net/questions/111816 | 6 | I am reading some theory on partial orders and I wonder something which perhaps has a simple answer : Given two partial orders $G\_1,G\_2$ (by their hasse diagrams), is it possible to know in polynomial-time if it exists a injective order-preserving map from $G\_1$ to $G\_2$ ? (that is to say a function $f : G\_1 \righ... | https://mathoverflow.net/users/nan | Is it possible to decide in polynomial time if a poset is a subposet of another which is given ? | Completing Emil's observation: Take any subgraph isomorphism problem (well known to be NP-complete). Add a new vertex in the middle of each edge and then orient the new edges outwards from the new vertex. That is, replace each undirected edge $x-y$ by $x\leftarrow z\rightarrow y$. I think you get two posets (with two l... | 7 | https://mathoverflow.net/users/9025 | 111819 | 64,070 |
https://mathoverflow.net/questions/111765 | 16 | **Motivation:** Incompleteness (and various independence statements) is about unprovable statements. One natural way to make an unprovable statement provable is to assume it as a new axiom. But this feels like cheating, so people often look for "natural" axioms to add that will imply their favorite unprovable statement... | https://mathoverflow.net/users/38434 | Unprovable statements S where the only way to prove S is to assume S | The following is essentially Joel's answer and also essentially the last part of Francois's answer, but its "look and feel" seems different enough to make it worth pointing out. The main point is that, if $S$ is minimally unprovable over $T$ then $T\cup\{\neg S\}$ is consistent and complete. Consistency is just your re... | 28 | https://mathoverflow.net/users/6794 | 111820 | 64,071 |
https://mathoverflow.net/questions/111823 | 2 | Sylvester-Schur says: "if $n \ge 2k$, then there is a number in the list
$n − k + 1, n − k + 2,$ ... $, n$
divisible by a prime $p > k$."
Shouldn't it also be true that if $n \ge k$, then there is a number in the list $n + 1, n + 2, $ ... $, n+k$ divisible by a prime $p > k$.
Does anyone know of a theorem that come... | https://mathoverflow.net/users/15915 | Are there any theorems about a prime $p > k$ in a sequence stronger than Sylvester-Schur? | Yes, this is true.
Indeed, by a result of Denis Hanson (Canad. Math. Bull., 1973) the product (I use notation to match the papers not the question)
$$\Delta(n,k)= n (n+1) \dots (n+k-1)$$
for $n \ge k$ is divisible by a prime of size greater $3k/2$
with only the exception of $3\cdot 4$ , $8 \cdot 9$ and $6\cdot 7 \... | 3 | https://mathoverflow.net/users/nan | 111831 | 64,076 |
https://mathoverflow.net/questions/111808 | 0 | Hi,
I want to whether there is a vector generating function (/matrix) such that it can generate a m-dimensional vector which will always be linearly independent of the set of m-dimensional vectors the function has already generated.
My problem can be written in pseudocode format as follow. I therefore expect that a... | https://mathoverflow.net/users/27924 | Function (/matrix) to generate linearly independent vectors. | I don't quite follow the first paragraph of your question. But reading the rest of your post, if you only need a set $S$ of $N$ $m$-dimensional vectors over the finite field of some small order in which any subset $S' \subset S$ of cardinality $m$ is a set of linearly independent vectors, that's a parity-check matrix o... | 2 | https://mathoverflow.net/users/27829 | 111832 | 64,077 |
https://mathoverflow.net/questions/111821 | 0 | Does one need to work over an algebraic closed field in ordre to construct GIT quotients à la Mumford?
If yes, why?
| https://mathoverflow.net/users/4096 | Algebraic closure and GIT | Hey MBeasy,
you can construct the quotients over an arbitrary "not necessarily algebraically closed field", as Mumford states at the beginning of chapter 1.
Greetings
Daniel
| 1 | https://mathoverflow.net/users/24980 | 111835 | 64,078 |
https://mathoverflow.net/questions/111842 | 12 | Let $\omega$ be a [nonprincipal ultrafilter](http://en.wikipedia.org/wiki/Ultrafilter) on $\mathbb N$.
A standard construction gives an $\omega$-limit, say $x\_\omega$, for any bounded sequence $(x\_n)$ of real numbers.
Namely, there is unique real value $x\_\omega$ such that
$$\{\,n\in\mathbb N\mid |x\_\omega-x\... | https://mathoverflow.net/users/1441 | Ultralimit versus partial limit | No, only if $\omega$ is a p-point.
If $(A\_k:k\in \mathbb N)$ is a partition of the natural numbers into $\omega$-small sets such that there is no $\omega$-large set meeting each $A\_k$ in a finite set, then we can choose a sequence $x\_n$ by declaring $x\_n:= 1/k$ whenever $n\in A\_k$. On each set $J\in \omega$ the... | 18 | https://mathoverflow.net/users/14915 | 111843 | 64,082 |
https://mathoverflow.net/questions/111801 | 11 | Hello,
Morita Equivalences occur in various categories, such as rings, operator algebras, homotopical categories, groupoids, etc. I'd like to know: What is the correct and precise definition of Morita Equivalence and what is the fundamental concept behind it. Here I want to restrict to (possibly étale, i.e. r-discret... | https://mathoverflow.net/users/27923 | What is the correct notion of Morita Equivalence between topological groupoids | I'll attempt to answer some of your questions. First off, 1) and 3) are equivalent. This is because the bicategory of fractions in 3) is the Morita bicategory of topological groupoids, which is equivalent to the bicategory of topological stacks. What maps are you inverting in 3)? Well, if you have an internal functor $... | 7 | https://mathoverflow.net/users/4528 | 111845 | 64,084 |
https://mathoverflow.net/questions/111851 | 11 | Jerry Shurman has a [lovely set of notes](http://people.reed.edu/%7Ejerry/361/lectures/heckechar.pdf) explaining the classical definition of Hecke characters, the idelic definition of Hecke characters, their relationship, and the classification of algebraic Hecke characters for $\mathbb{Q}$ as Dirichlet characters. He ... | https://mathoverflow.net/users/683 | Explicit examples of algebraic Hecke characters with infinite image? | The "most obvious" algebraic Hecke characters of a field $K$ are the characters of the ideal class group of $K$, which have trivial infinity-type and trivial conductor. There might be no non-trivial examples (as in the case of Q). But you can get more examples by:
* beefing up the conductor (which gets you Dirichlet ... | 23 | https://mathoverflow.net/users/2481 | 111854 | 64,088 |
https://mathoverflow.net/questions/91827 | 13 | **Question 1:** Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I have found only some vague mentions of control theory and maybe some approximations of functions, but don't know the d... | https://mathoverflow.net/users/18376 | Application of polynomials with non-negative coefficients | One answer to **question 2** is Polya's theorem on forms positive on an orthant: Let a form (i.e. homogeneous polynomial) in several variables be given which is (strictly) positive whenever evaluated on non zero tuples of nonnegative reals. Then you can multiply it with a high power of the sum of the variables such tha... | 5 | https://mathoverflow.net/users/15506 | 111872 | 64,097 |
https://mathoverflow.net/questions/111830 | 11 | My understanding is Atiyah-Patodi-Singer solved the index theorem for manifold with boundary only for certain types of Dirac operators, correct?
There is still no (or no hope to get) uniform theorem for the Dirac operator associated with any Dirac bundle (in the sense of Gromov-Lawson)?
| https://mathoverflow.net/users/1190 | Is there any general index theorem for manifold with boundary? | 1. The APS theorem works for any Dirac-type operator; see e.g. the excellent monograph by Booss-Wojchiecowski on this topic.
2. More than four decades ago, Boutet de Monvel has described a general set-up for dealing with boundary value problems that mimicks the K-theoretic approach to the index theorem on closed manifo... | 11 | https://mathoverflow.net/users/20302 | 111884 | 64,102 |
https://mathoverflow.net/questions/111890 | -1 | What is the definition of a geometrically connected curve?
| https://mathoverflow.net/users/27959 | Geometrically connected curve | For a variety over a non-algebraically closed field, "geometrically connected" means connected over the algebraic closure.
As an example where this fails, note that the curve $x^2+1=0$ in $\mathbb{A}^2$ is connected over $\mathbb{Q}$, but not over $\mathbb{Q}[i]$, where is becomes $(x+i)(x-i)=0$, which is a union of ... | 5 | https://mathoverflow.net/users/5101 | 111892 | 64,105 |
https://mathoverflow.net/questions/111882 | 0 | I was looking for an article mimicing result of Pelczynski for $\ell\_p$. I have found this one
Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c\_0(I)$ and $\ell\_p(I)$ for $1 < p < \infty$, Atti Sem.
Mat. Fis. Univ. Modena, 42, 399–402.
but I can't download it anywhere. Could someone provide me w... | https://mathoverflow.net/users/19593 | Complemented subspaces of $\ell_p(I)$ for uncountable $I$ | This is just an exercise in the reflexive range once you know the proof in the separable case. Suppose $X$ is a subspace of $\ell\_p(I)$ with density character $\aleph$. Since each vector in $X$ has countable support, WLOG $I=\aleph$. If $X$ is complemented, you just need to check that $X$ has a complemented subspace i... | 1 | https://mathoverflow.net/users/2554 | 111896 | 64,107 |
https://mathoverflow.net/questions/111897 | 0 | Is there any research work about automorphism group of non-commutating graph?
| https://mathoverflow.net/users/27962 | non-commutating graph | Given that you're based in Esfahan, and given that there are a number of Iranian mathematicians working on the non-commuting graphs of groups, I'll assume this is what you're asking about.
So the **non-commuting graph of a group** is the graph with vertices equal to elements of $G$, and two elements connected if they... | 4 | https://mathoverflow.net/users/801 | 111901 | 64,110 |
https://mathoverflow.net/questions/111902 | 4 | Call a real $x$ limit computable iff there is a Turing machine $T$ such that, for any $i\in\omega$, there is $t(i)\in\omega$ such that the $i$th entry on the tape is not changed after time $t(i)$ and is from that time on equal to $1$ if $i\in x$ and otherwise equal to $0$. Intuitively, this means that the output of $T$... | https://mathoverflow.net/users/15814 | Transfinitely iterated limit computability | Yes, this notion has been studied. It is considerably weaker than the model of [infinite time Turing machines](http://jdh.hamkins.org/infinitary-computability-publications/). To see this, observe that your machines can never do anything after time $\omega^2$, because whatever local configuration the machine is in at ti... | 6 | https://mathoverflow.net/users/1946 | 111903 | 64,111 |
https://mathoverflow.net/questions/111869 | 5 | I remember hearing this in other contexts, but I encountered it again when reading Elkies' paper "Shimura Curve Computations", where on page 10, he says that:
"We now return to the Shimura curves $\mathcal{X}(1),\mathcal{X}^\*(1)$ obtained from arithmetic groups $\Gamma = \Gamma(1),\Gamma^\*(1)$. These curves also ha... | https://mathoverflow.net/users/15242 | How does a moduli interpretation give an analytic object an algebraic structure? | Even on the level of sets, the idea that any compact Riemann surface gives rise to an algebraic curve over $\mathbf{Q}$ should feel resoundingly false. There are uncountably many compact Riemann surfaces and only countably many algebraic curves over $\mathbf{Q}$.
I think you may be confusing one or more of the follo... | 6 | https://mathoverflow.net/users/3384 | 111907 | 64,113 |
https://mathoverflow.net/questions/111917 | 2 | Consider an infinite set $S$, of positive integers.
If all the finite subsets of $S$ have GCD $>$ $1$, is it necessary that the GCD of $S$ is greater than $1$ as well?
| https://mathoverflow.net/users/19453 | Is it necessary that gcd > 1 of an infinite set? | Consider the set of finite linear combinations with integral coefficients from $S$. It is an ideal in $\mathbb{Z}$, hence it equals the set of multiples of some integer $d>0$. The integer $d$ divides every element of $S$, hence it cannot exceed the GCD of $S$. On the other hand, $d$ is a finite linear combination with ... | 5 | https://mathoverflow.net/users/11919 | 111919 | 64,118 |
https://mathoverflow.net/questions/111910 | 2 | The marcum Q-function is defined by
$$ Q\_m(a,b) = \int^\infty\_b x \left(\frac{x}{a}\right)^{m-1} \exp\left(-\frac{x^2+a^2}{2}\right)
I\_m\left(a x\right)
\:\mathrm{d} x,$$
where $m\in\mathbb{N}$ , $b\in\mathbb{R}^+$ , $a\in\mathbb{R}^+$ , and $I\_m(.)$ is the $m$-th order modified Bessel function of the first type... | https://mathoverflow.net/users/19493 | Dertivative of a Special Function with respect to Order | For large $x$, if $a>0$, $I\_m$ behaves asymptotically like $I\_m(ax)\approx e^{ax}/\sqrt{ax}$. Therefore for large $x$ the integrand will look like $x^{m-1/2}e^{-(x-a)^2/2}$. This dies off fast enough that the improper integral converges uniformly in $m$ and you can differentiate inside the integral.
| 2 | https://mathoverflow.net/users/20729 | 111920 | 64,119 |
https://mathoverflow.net/questions/111694 | 7 | Let $M$ be (for example) a Calabi-Yau threefold with Kaehler form $\omega$ and holomorphic 3-form $\Omega$. We say that a submanifold $L$ of $M$ is a special Lagrangian submanifold if $L$ is Lagrangian with respect to symplectic form $\omega$ and also $\mathrm{Im}\Omega|\_L=0$. I would like to know geometric intuition ... | https://mathoverflow.net/users/25812 | What is geometric intuition of special Lagrangian manifolds? | Let $(M,g,J,\Omega)$ be a Calabi-Yau $n$-fold. Then $\textrm{Re }\Omega$ is a calibration on $(M,g)$. Let $L\subset M$ be a real submanifold with $\dim\_{\mathbb{R}}L=n$. You have the following
**Proposition**
$L$ is a special Lagrangian if and only if it admits an orientation making it into a calibrated (for $\te... | 7 | https://mathoverflow.net/users/6278 | 111932 | 64,126 |
https://mathoverflow.net/questions/111934 | 0 | It is often stated that a single-valued analytic function f(z) is uniquely and
completely determined if (1) it is analytic at all points of a convergent sequence of
points in the complex plane and at their limit point and (2) one is given the points
of the sequence and the values of f(z) at each of these points.
Let ... | https://mathoverflow.net/users/4423 | A question that arises in trying to make mathematically precise a well known informal statement about analytic functions | If I am not misunderstanding your question, you read off the power series pretty much directly from the given data. You know f(0). You also know f'(0) by using the definition of the derivative. The higher derivatives can all be determined by using higher order difference equations <http://en.wikipedia.org/wiki/Finite_d... | 2 | https://mathoverflow.net/users/27968 | 111936 | 64,128 |
https://mathoverflow.net/questions/111943 | 11 | Gelʹfand, I. M. and MacPherson, R. D. "A combinatorial formula for the Pontrjagin classes" Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 304–309.
In the above paper the authors claimed a construction of combinatorial formula for the Pontrjagin classes. According to mathscinet review: MR1129313, it seems the constr... | https://mathoverflow.net/users/1190 | Does combinatorial formula for the Pontrjagin classes exist? | I believe there is a formula like the one you seek together with a good survey of the previous literature in the paper "Local formulae for combinatorial Pontrjagin classes" by Gaifullin, available [here](http://arxiv.org/abs/math/0407035).
| 13 | https://mathoverflow.net/users/317 | 111947 | 64,133 |
https://mathoverflow.net/questions/111905 | 3 | Let $\mu:Y\to X$ be a birational morhism between normal projective (complex) varieties.
Suppose, furthermore, that $Y$ is smooth. Let $D$ be a Weil divisor on $X$ and let $\mathcal{O}\_X(D)$ be its corresponding reflexive sheaf.
If we denote by $\widetilde{D}$ the strict transform of $D$, is it true that
the push-forwa... | https://mathoverflow.net/users/6430 | push-forward and strict transforms | No, the is definitely not true. Let me give you an example, suppose that $X = \mathbb{A}^2$ and that $\pi : Y \to X$ is the blowup of $X$ at the origin with exceptional divisor $E$. Suppose that $D$ is the Cartier divisor corresponding made up of $n$ lines through the origin, $n \geq 2$. Then $\widetilde{D}$ is just a ... | 8 | https://mathoverflow.net/users/3521 | 111958 | 64,137 |
https://mathoverflow.net/questions/111949 | 2 | Let $X$ be a compact metric Borel space. Suppose $\mu\_{n}(A)\rightarrow\mu(A)$
for all $\mu-$continuity sets $A$ (sets with zero boundary measure), where $\mu\_{n}$ is a sequence of probability measures.
(some people call it weak other weak\* convergence)
If $E$ is a measurable set such that
$\mu(E)>0$ and the Cesa... | https://mathoverflow.net/users/18384 | Weak convergence, and Cesaro convergence (of mu_n (E) ) imply convergence (of mu_n (E))? | In general, the answer is "no", because $\mu\_n(E)$ can be an arbitrary sequence of zeros and ones.
Take on the real line $E$={$0$}, and $\mu\_n=\delta\_{x\_n}$, where $x\_n$ tends to $0$.
We have $\mu\_n\to\delta\_0$ weakly.
Then $\mu\_n(E)=1$ or $0$ depending on whether $x\_n=0$ or not. So you can choose such sequenc... | 1 | https://mathoverflow.net/users/25510 | 111960 | 64,138 |
https://mathoverflow.net/questions/111935 | 3 | The standard weak formulation of the Neumann problem for the Poisson equation
is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$:
$$ \int\_{\Omega} \nabla u \nabla v d x = \int\_{\Omega} fv d x +
\int\_{\partial \Omega} gv d s $$
for given $f \in L^2 ( \Omega)$ and $g \in H^{- 1 / 2} ( \Ome... | https://mathoverflow.net/users/27967 | Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation | Actually the operator is in divergence form. You do not see it in those coordinates because you are missing the composition with $\phi^{-1}$ and because the computation is indeed a mess.
I think it is best to write the variational formulation and transform the functional. Then the divergence form will be apparent. Le... | 3 | https://mathoverflow.net/users/26672 | 111962 | 64,140 |
https://mathoverflow.net/questions/111893 | 0 | To avoid case distinction overload, I also call (say) $Z^3-Z^4$ cyclotomic. Just divide out the $Z=0$ solutions in the following if they offend.
In the following, all exponents are assumed to be positive integers.
Assume that $P=Z^a+Z^b+Z^c-Z^d-Z^e-Z^f$ is cyclotomic. The "standard" form (since $Z-1$ factors out ... | https://mathoverflow.net/users/11504 | Cyclotomic polynomial question | To start with your second question, cyclotomic polynomials have a central symmetry or antisymmetry, If $f(x)$ is a cyclotomic polynomial of degree $N$ then $f(Z)=\pm Z^nf(\frac1Z).$ So $Z^n\pm Z+1.$ would only be cyclotomic if $n=2$
In your construction $P'$ you could replace $Z-1$ by $Z^k-1$. Looking at $(Z-1)\Phi\... | 2 | https://mathoverflow.net/users/8008 | 111964 | 64,141 |
https://mathoverflow.net/questions/111966 | 7 | Consider some flat metric on $S^2$ with a fixed finite number of conical singularities $p\_1,\ldots,p\_n$.
What is the moduli space of such metrics up to isometry? In particular what is its dimension?
| https://mathoverflow.net/users/26250 | flat metrics on the 2-sphere with conical singularities | There is a necessary and sufficient condition that such a metric exist (at least when the cone angles are all less than $2\pi$), which is the simple linear condition on the cone angles imposed by Gauss-Bonnet.
This allows for $n-1$ degrees of freedom. There is also an extra $2n$-dimensional freedom from moving the p... | 11 | https://mathoverflow.net/users/17969 | 111967 | 64,142 |
https://mathoverflow.net/questions/108055 | 2 | Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice differentiable?
| https://mathoverflow.net/users/8380 | Function with all but mixed second partial derivatives twice differentiable? | Here is an explicit counterexample in $B\_1$,
$$f(x,y) = x\, y \, \log(-\log(x^2+y^2))$$
Both $\partial\_{xx} f$ and $\partial\_{yy} f$ are continuous but $f$ is not twice differentiable at the origin.
| 3 | https://mathoverflow.net/users/26672 | 111969 | 64,144 |
https://mathoverflow.net/questions/111916 | 6 | I am reading a proof on p.51 of Robert Steinberg in his book "Endomorphisms of Algebraic Groups" and I am having a bit of difficulty understanding one point in the proof.
The setting is as follows. Consider an algebraic group $G$ over an algebraically closed field of arbitrary characteristic. We say that an automorph... | https://mathoverflow.net/users/1437 | Confusing Point in Proof: Semisimple Automorphism Fixes Torus | Consider yourself fortunate to be having only "a bit of difficulty" in reading Steinberg's concise proofs. His style typically involves numerous reductions to get to the essential point, along with sentences crammed with dense arguments. Though he is almost always right on target, it can be frustrating to unpack such p... | 9 | https://mathoverflow.net/users/4231 | 111988 | 64,151 |
https://mathoverflow.net/questions/111965 | 37 | The title says pretty much what I want. Of course, the abelian categories should contain at least one nonzero object.
In particular, is there an abelian category containing only one nonzero object? On the one hand, this is equivalent to construct a ring which is the endomorphism of the nonzero object. On the other h... | https://mathoverflow.net/users/27976 | Are there any nontrivial abelian categories with only finitely many objects? | Take the category of (at most) countable-dimensional vector spaces over your favourite field. Then take the quotient by the Serre subcategory of finite-dimensional vector spaces. (And take a skeletal subcategory so that it strictly has only two objects.)
Then this is an abelian category with only one non-zero object,... | 57 | https://mathoverflow.net/users/22989 | 111989 | 64,152 |
https://mathoverflow.net/questions/111868 | 12 | I once read a statement (not memorized precisely) that a certain physics quantity between two states of charge $d\_1$ and $d\_2$ respectively could be computed by running over the states of charge $d\_1+d\_2$ which is the extension of the original two states. Therefore we need to consider some Hall algebras on a moduli... | https://mathoverflow.net/users/3478 | The use of Hall algebras in physics | In supersymmetric field theories and string theories there are special states called BPS
states which are annihilated by some of the supercharges and whose mass is determined
in terms of their charges by the supersymmetry algebra. The study of these states
and how they behave as various moduli are varied has been one o... | 8 | https://mathoverflow.net/users/10475 | 111991 | 64,153 |
https://mathoverflow.net/questions/32412 | 25 | Suppose that $n=\prod\_{i=1}^{k} p\_i^{e\_i}$ and $m=\prod\_{i=1}^{l} q\_i^{f\_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e\_1 \le e\_2 \cdots \le e\_k$, and $f\_1 \le f\_2 \le \cdots \le f\_l$. Then if $k=l$ and $e\_i=f\_i$ for all $i$, we say that $n$ and $m$ ... | https://mathoverflow.net/users/1355 | Question on consecutive integers with similar prime factorizations | I'm coming into this late, but am wondering why no one seems to have mentioned the results of Goldston, Graham, Pintz and Yildirim:
<http://arxiv.org/pdf/0803.2636.pdf>
In particular, their Theorem 4 answers the OPs first question in the affirmative.
| 14 | https://mathoverflow.net/users/27984 | 111992 | 64,154 |
https://mathoverflow.net/questions/111994 | 6 | Can anyone tell me whether or not it is true that for all odd primes p the multiplicative order of 2 modulo p is strictly less than the multiplicative order of 2 modulo p^2 ? What are some good references regarding this problem ? Thank you
| https://mathoverflow.net/users/27986 | multiplicative order of 2 mod p | It is well-known that there are primes $p$ such that $2^{p-1} \equiv 1$ (mod $p^{2}$), a question which arises in connection with Fermat's Last Theorem. For such a prime $p,$ let $e$ be the smallest positive integer such that $p$ divides $2^{e}-1,$ and write (as we may) $p-1 = ed$ with $d$ an integer. Then we see easil... | 10 | https://mathoverflow.net/users/14450 | 111997 | 64,155 |
https://mathoverflow.net/questions/112007 | 15 | Proving things suspected to be true in anabelian geometry is usually very hard. Maybe it is easier to disprove things suspected to be false?
In particular, I am interested in false generalizations of the main conjecture to higher dimensions.
>
> Is there a pair of non-isomorphic varieties, $X$ and $Y$, which one ... | https://mathoverflow.net/users/18060 | What conjectures in anabelian geometry are false? | Grothendieck, in his letter to Faltings, suggested that the moduli space of abelian varieties (quote: "I would assume that the same should hold for the multiplicities of moduli of polarized abelian varieties") would be anabelian. Ihara and Nakamura showed that this was not the case, as the anabelian recipe for the auto... | 12 | https://mathoverflow.net/users/2290 | 112009 | 64,157 |
https://mathoverflow.net/questions/111950 | 12 | Let $T$ be a locally compact abelian (LCA) group. For any other LCA group $G$, let
$\hom(G,T)$ be the set of continuous homomorphisms $G\to T$. With the compact-open
topology, $\hom(G,T)$ is certainly a topological group, but is not in general locally
compact, even if $T$ is compact. In any case there is an obvious ... | https://mathoverflow.net/users/6856 | When does Pontryagin duality generalize? | I gave an answer to the second question already on math.stackexchange at <https://math.stackexchange.com/questions/124379/why-unitary-characters-for-the-dual-group-in-pontryagin-duality-if-g-is-not-co>, but I'll put it here too: if $\alpha\_G$ is an isomorphism of topological groups for all locally compact abelian grou... | 14 | https://mathoverflow.net/users/3272 | 112012 | 64,159 |
https://mathoverflow.net/questions/112015 | 9 | 1. Who first recognized that the torus supports a flat structure?
2. Who first characterized the moduli space of flat structures on the torus?
3. Who first recognized that the closed, orientable genus 2 supports a hyperbolic structure?
4. Who first thought of a geometrized surface in terms of the property that for any ... | https://mathoverflow.net/users/683 | The history of the geometrization of closed surfaces | I believe that on the first two questions, there is no answer (so they are not well-posed:-)
The reason is that the flat structure on a torus and the moduli spaces of tori were very
well understood long before the notions of "flat", "structure", "moduli space" and even "tori"
were introduced. Mathematicians just did no... | 13 | https://mathoverflow.net/users/25510 | 112020 | 64,163 |
https://mathoverflow.net/questions/111984 | 2 | Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and reduce all the entries of $M$ mod $4$. Is there a nice clean description of all the finite types of such mod $4$ reductions o... | https://mathoverflow.net/users/12909 | Gram matrix modulo 4 | I use the 2-adic decompositions for various tasks. I can't say I know what would be useful for you, but let me call your attention to page 141, Lemma 4.3. This refers back to Lemma 5.2 on page 123, (paraphrase) if $d(g) = u^2 d(f)$ for some unit $u$ and
$$ g\_{ij} \equiv f\_{ij} \pmod {2^{ v\_2(d(f)) + 2}}, $$
then $f,... | 1 | https://mathoverflow.net/users/3324 | 112022 | 64,164 |
https://mathoverflow.net/questions/112003 | 0 | Is it possible that an element of a sylow basis of a finite solvable group G lies in the frattini subgroup of G?(i.e could an element of a sylow basis of a finite solvable group G be a non-generator subgroup?)
| https://mathoverflow.net/users/27962 | sylow basis of finite solvable groups | You should tell the reader what a Sylow basis is. Assuming that it s a set of pairwise permutable Sylow subgroups of $G,$ one for each prime divisor, the answer is no. For if $P\_{1}$ is a Sylow $p\_{1}$-subgroup of $G$ belonging to a basis, then deleting $P\_{1}$from the generating set given by the Sylow basis leaves ... | 1 | https://mathoverflow.net/users/14450 | 112023 | 64,165 |
https://mathoverflow.net/questions/111321 | 20 | Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$:
1. $f(z)$ is bounded when $\Re z>1+\delta$
2. $f(z)$ is unbounded when $\Re z=1$
3. $f(z)$ grows polynomially in vertical strips, ie for all $\sigma$ there is $C\_\sigma$ so that $|f(\sigma+i t)|\ll|t|^{C\_\sigma}$
4. $f(z)$ ... | https://mathoverflow.net/users/9849 | Provable zero-free region for any entire function that analytically is similar to zeta(s) | OK, shameless cheating, as promised.
Part 1. Let's start with something.
We need a function bounded in $\Re z>1$ and growing not too fast on each vertical line whose zeroes are somewhere on the left. The first thing that comes to mind is $1$. No zeroes anywhere in sight, beautiful control on vertical lines. All tha... | 14 | https://mathoverflow.net/users/1131 | 112031 | 64,168 |
https://mathoverflow.net/questions/112026 | 2 | This is probably well-known, but... Define the $n$-dimensional hypercube graph $H\_n$ as having for vertices the integers between 0 and $2^n-1$, and edges between integers differing by a power of 2. The characteristic polynomial of $H\_n$ is then $\prod\_{k=0}^n(x-n+2k)^{\frac {n!}{k!(n-k)!}}$, i.e. $(x-3)(x-1)^3(x+1)^... | https://mathoverflow.net/users/17164 | Characteristic polynomial of hypercube graph | View the vertices as elements of $\mathbb{Z}^n$. If $a\in\mathbb{Z}^n$, define a function $f\_a$ on the vertices by
$$
f\_a(x) = (-1)^{a^Tx}.
$$
This function is an eigenvectors and if $a$ has weight $w$, the eigenvalue is $n-2w$. I can make this look more combinatorial by viewing vertices (and $a$) as subsets of $\{1... | 5 | https://mathoverflow.net/users/1266 | 112035 | 64,170 |
https://mathoverflow.net/questions/112021 | 4 | Let $n$ be a positive integer.
The $n$ by $n$ Fourier matrix may be defined as follows:
$$
F^{\*} = (1/\sqrt{n}) (w^{(i-1)(j-1)})
$$
where
$$
w = e^{2 i \pi /n}
$$
is the complex $n$-th root of unity with smaller positive argument
and $\*$ means transpose -conjugate.
It is well known that $F$ is diagonalizab... | https://mathoverflow.net/users/11016 | Eigenvectors of the Fourier matrix | This has a little number-theoretic content, having to do with real-valued *characters* modulo $n=2k+1$. For example, for $n=p$ an odd prime number, there are exactly two such functions (up to scalar multiples), the function that is $1$ for non-zero-mod-$p$ inputs, and the quadratic character $\chi$ mod $p$, which is $\... | 3 | https://mathoverflow.net/users/15629 | 112036 | 64,171 |
https://mathoverflow.net/questions/112029 | 19 | Is it possible to mathematically prove that the speed of gravitational waves in general relativity equals the speed of light, without linearizing the Einstein field equations? The approach via the linearization of the EFE's, which is used in many books on relativity, does not seem to provide an exact proof that disturb... | https://mathoverflow.net/users/26077 | The speed of gravitational waves in general relativity | What is the speed of a wave in a non-linear theory? Answering before considering your question is important, because that answer will tell you where to look for your answer.
A useful notion is that of *domain of dependence* (see for example a decent book on GR for a detailed discussion, e.g., Wald or Hawking & Ellis)... | 29 | https://mathoverflow.net/users/2622 | 112037 | 64,172 |
https://mathoverflow.net/questions/112024 | 8 | Let's assume we start with Chern--Weil theory in the following form:
>
> Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l\_n(\mathbb C)$ connection and from this connection compute a closed differential $2k$-form (from the curvature of the connection) wh... | https://mathoverflow.net/users/35353 | How does one go from Chern--Weil to cohomology classes on BGL(n,C)? | Expanding on Will Sawin's comment:
Every finite CW complex is homotopy equivalent to a finite simplicial complex (by an approximation argument) and a simplicial complex is homotopy equivalent to a manifold (take a regular neighborhood of the natural geometric realization in $\mathbb R^{\#\text{vertices}}$). The natur... | 4 | https://mathoverflow.net/users/35353 | 112038 | 64,173 |
https://mathoverflow.net/questions/112039 | 4 | In this question, a "graph" and a "subgraph" will always be a loop-free multigraph (finite, undirected, unweighted).
*Motivation*: Given a graph in which each vertex has even degree, we can find a cycle (possibly a 2-cycle). If we delete this subgraph, we preserve the vertex degrees modulo $2$. Thus, we can repeatedl... | https://mathoverflow.net/users/2264 | Does a graph with all vertex degrees divisible by 3 contain a cubic subgraph? | Counterexample:
[alt text http://www.freeimagehosting.net/newuploads/8m77r.png](http://www.freeimagehosting.net/newuploads/8m77r.png)
| 14 | https://mathoverflow.net/users/5513 | 112043 | 64,176 |
https://mathoverflow.net/questions/112050 | 0 | $x\_{n+1}=4x\_n(1-x\_n)$ I already proved that for $x\_n\subset [0,1]$, $x\_n=sin^2(2\pi y\_n)$
with $y\_{n+1}=\begin{cases}2y\_n & 0 \le y\_n < 0.5 \\vee 2y\_n -1 & 0.5 \le y\_n < 1 \end{cases}$
Now I would like to prove that for an arbitrary number $m\in\mathbb N$ there exists an $x\in [0,1]$ of the recursion wit... | https://mathoverflow.net/users/28002 | Logistic map periodic point | Essentially you have proved that the logistic map is conjugate to the doubling map $Tx=2x\bmod 1$. Now, $T$ is in turn conjugate to the shift map $\sigma:\Sigma\to\Sigma$, where $\Sigma$ is the space of infinite 0-1 words.
More precisely, if
$$
\pi(w\_1,w\_2,\dots)=\sum\_{n=1}^\infty w\_n2^{-n},
$$
then you have
$$... | 2 | https://mathoverflow.net/users/8131 | 112054 | 64,179 |
https://mathoverflow.net/questions/111849 | 11 | I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.
1. For the algebraic cobordism theory $MGL$ and a smooth variety $X$ over a (perfect?) field is it true that $MGL^{2n+i,n}(X)=0$ for any $n\in \mathbb{Z},i... | https://mathoverflow.net/users/2191 | The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$? | (1) is true if $char(k)=0$. This follows from a combination of results. First of all, it is true over any field that the spectrum $MGL$ is connective, which means that
$$MGL^{p,q}(X)=0$$
if $p>q+dim(X)$, $X\in Sm/k$ [1, Cor. 2.9]. (Slightly more is true: for any $p\geq q+dim(X)$, the orientation map $MGL\to H\mathb... | 14 | https://mathoverflow.net/users/20233 | 112064 | 64,182 |
https://mathoverflow.net/questions/112059 | 8 | The Novikov--Boone Theorem, which is perhaps the archetypal local unsolvability result in group theory, states existence of a finitely presented group whose word problem is recursively unsolvable. Recall that the *word problem* asks whether there exists an algorithm to decide whether an arbitrary word in generators of ... | https://mathoverflow.net/users/2051 | The equality problem between conjugate group elements | If C is a right-coset Hg then the equality problem in C is equivalent to the word problem in H. The word problem in H can be solvable even if the word problem in G is unsolvable, as long as the membership problem for H is unsolvable.
For instance, take a group G' with unsolvable word problem and set $G= G'\times H$ w... | 7 | https://mathoverflow.net/users/1463 | 112065 | 64,183 |
https://mathoverflow.net/questions/48461 | 12 | There have been many results on the first sign change of $\pi(x)-{\mathrm{li}}(x)$: among others, Lehman, te Riele, Bays & Hudson, Demichael, Chao & Plymen, and most recently Saouter & Demichel. These provide upper bounds (as well as lower bounds on some region in which $\pi(x)>{\mathrm{li}}(x)$).
Could these methods... | https://mathoverflow.net/users/6043 | Analytic lower bounds on the first sign change of pi(x) - li(x)? | There are some explicit results in the work S. B. Stechkin, A. Yu. Popov, “The asymptotic distribution of prime numbers on the average”, Uspekhi Mat. Nauk, 51:6(312) (1996), 21–88
| 4 | https://mathoverflow.net/users/5712 | 112068 | 64,185 |
https://mathoverflow.net/questions/112079 | 11 | I find many references to Gödel's pairing function on ordinals but I have not found a definition. What is it?
| https://mathoverflow.net/users/38783 | What is Gödel's pairing function on ordinals? | Define an order on pairs of ordinals $(\alpha,\beta)$ by ordering first by maximum, then by first coordinate, then by second coordinate. That is, one pair preceeds another if the maximum is smaller, or they have the same maximum and the first coordinate is smaller, or they have the same maximum and first coordinate, bu... | 18 | https://mathoverflow.net/users/1946 | 112081 | 64,191 |
https://mathoverflow.net/questions/112086 | 14 | As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho\_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf B &= 0, &\text{No-name law (no monopoles),}\\\
\nabla \times \mathbf E &= - \partial\_t \mathbf B, &\text{Faraday's law,}\\\
\nab... | https://mathoverflow.net/users/5295 | Regularity of the Maxwell equations | There are a few, not many, books on hyperbolic equations. You might have a look to that of S. Benzoni-Gavage and myself: *Multi-dimensional hyperbolic partial differential equations. First order systems and applications*, Oxford Mathematical Monographs, Oxford University Press (2007).
A basic fact of hyperbolic syste... | 19 | https://mathoverflow.net/users/8799 | 112089 | 64,197 |
https://mathoverflow.net/questions/112074 | 2 | Given m & n, we have to find out the number of possible matrices of order m\*n with the property that A(i,j) can be either 0 or 1 and that no contiguous sub-matrix of both length > 1 & breadth > 1 should have same entries i.e. all of its cells shouldn't be 0 or 1. For example if m = 2 & n = 2, the answer is 14: Total p... | https://mathoverflow.net/users/27991 | How many matrices are possible for the given arrangement? | Let $a\_n$ be the number of $2 \times n$ -matrices avoiding constant 2\*2-submatrices.
Then
$$a\_n = \frac{2^{-n} \left(4 \left(17+4 \sqrt{17}\right)
\left(3+\sqrt{17}\right)^n+\left(\sqrt{17}-17\right)
\left(\sqrt{17}-3\right)^n e^{i \pi n}\right)}{17
\left(3+\sqrt{17}\right)}$$
This should be fairly straightf... | 5 | https://mathoverflow.net/users/1056 | 112093 | 64,199 |
https://mathoverflow.net/questions/112101 | 7 | I was wondering what the cardinality of $\omega\uparrow^\omega\omega$ is, with $\uparrow$ being Knuth's up-arrow notation. I ask this purely out of curiosity; after finding out about set theory I feel like a child with a new toy. I'm not on the same level as everyone else on this site, and everything I've learned about... | https://mathoverflow.net/users/28016 | Cardinality of $\omega\uparrow^\omega\omega$ | The [Knuth arrow notation](http://en.wikipedia.org/wiki/Knuth's_up-arrow_notation#Definition) is most often defined only on the natural numbers, but the central idea of it
can be easily extended to the ordinals, for example as follows:
$$\alpha\uparrow^0\beta=\alpha\beta$$
$$\alpha\uparrow^\eta 0=1\qquad\text{for }\... | 14 | https://mathoverflow.net/users/1946 | 112103 | 64,203 |
https://mathoverflow.net/questions/112091 | 5 | Let $(u\_1, \ldots, u\_n)$ and $(v\_1, \ldots, v\_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u\_1 \cdots u\_n]$, and similarly for the second basis. Prove that the first basis can be continuously transformed into the second one, while rem... | https://mathoverflow.net/users/27742 | Continuous change of basis (and on the definition of determinant) | Basically what you're asking in the first question is whether $SL\_n({\mathbb R})$ is pathwise connected. Using the polar decomposition, this follows from the fact that $SO\_n({\mathbb R})$ is pathwise connected. That, in turn, follows from the properties of Givens rotations.
| 6 | https://mathoverflow.net/users/13650 | 112106 | 64,204 |
https://mathoverflow.net/questions/112113 | 2 | Hello,
At for instance, <http://en.wikibooks.org/wiki/Famous_Theorems_of_Mathematics/Number_Theory/Totient_Function#Sum_of_integers_relatively_prime_to_and_less_than_or_equal_to_n>,
there is a closed form for the integers relatively prime and less than an integer n, given by
$\displaystyle\sum\_{1\leq n\leq k ,gcd(n,... | https://mathoverflow.net/users/28023 | Sum of integers squared relatively prime to and less than n ??? | It's always a good idea to plug the first few terms into the OEIS. For 1,1,5,10,30,26, this leads to <http://oeis.org/A053818> from which there's a reference to an exercise in Apostol's *Introduction to Analytic Number Theory* deriving the formula
$${1\over3}n^2\varphi(n) + {n\over6}\prod\_{p|n}(1-p)$$
**Added 11/1... | 15 | https://mathoverflow.net/users/15837 | 112115 | 64,207 |
https://mathoverflow.net/questions/112127 | 17 | In the Wikipedia article about [descriptive set theory](https://en.wikipedia.org/wiki/Descriptive_set_theory) I read that $\mathbb{R}$ (with its usual topology) is a Polish space, and that every Polish space
1) can be obtained as a continuous image of the Baire space $\mathcal{N}$
2) can be obtained as the image o... | https://mathoverflow.net/users/4721 | The reals as continuous image of the irrationals | For any irrational number $x$, let $f(x)$ be the real number arising from the integer part of $x$, together with every other digit of the rest of the expansion of $x$.
This is surjective, since one may interleave the digits of any real $y$ with any nonrepeating pattern, and thereby find an irrational $x$ with $f(x)=... | 28 | https://mathoverflow.net/users/1946 | 112130 | 64,214 |
https://mathoverflow.net/questions/112128 | 0 | Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic subsets of $X$. Can I say something general about the relation between the two following expressions?
$$\int\_X f(x) g(... | https://mathoverflow.net/users/26798 | Integral inequality | 1. Your first integral is $|A|$ (volume), and similarly the other integrals.
Cauchy-Schwarz inequality implies $|A\cap B|^2\leq |A||B|$ and nothing else can be said.
2. Your inequality with 3 sets is not true. Take $A,B,C$ independent. That is the volume of
each product set is equal to the product of the volumes. Such ... | 2 | https://mathoverflow.net/users/25510 | 112134 | 64,217 |
https://mathoverflow.net/questions/112125 | 5 | This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!
Suppose $f(x)=\sum\_{k\in\mathbb{Z}}c\_ke^{ikx}$, whereby we mean pointwise convergence. What properties must $f(x)$ then satisfy? Clearly c... | https://mathoverflow.net/users/28028 | Weakest assumption for pointwise convergence of Fourier series | The function must be integrable in a certain sense defined by Denjoy and others. Here is an interesting [survey paper](http://dx.doi.org/10.1090/S0002-9904-1955-09853-7) on the subject:
>
> One of the problems in the theory of trigonometric series
> $$\frac12a\_0+\sum\_{n=1}^\infty(a\_n\cos nx+b\_n\sin nx)\tag{1.1... | 7 | https://mathoverflow.net/users/19276 | 112138 | 64,219 |
https://mathoverflow.net/questions/112139 | 2 | The title pretty much says it all. What is the cardinality of $G$, the group of all functions $f: \mathbb{R} \to \mathbb{R}$ such that $\forall x,y\in \mathbb{R} \left( x>y\Rightarrow f(x)>f(y)\right)$? Obviously $G$ is a subgroup of the symmetric group on $\mathbb{R}$, which has cardinality $\beth\_2$, and $G$ is also... | https://mathoverflow.net/users/28033 | Cardinality of group of order-preserving functions from R to R | Such a function can have only countably many points of discontinuity, since any discontinuity will be a jump discontinuity and hence the range will skip over an interval unique to that point, and so we can associate each point of discontinuity with a distinct rational number, meaning there are only countably many.
S... | 10 | https://mathoverflow.net/users/1946 | 112140 | 64,220 |
https://mathoverflow.net/questions/112153 | 0 | Hello everyone my question is:
$Question:$ Consider a function $f:X \rightarrow \mathbf R$ where $X$ is a convex subset of $\mathbf{R}^n$. The convex envelope of $f$ over $X$ is defined as the pointwise supremum of convex under-estimator of $f$, denoted as $g$. I want to ask if the minimum of $f$ over $X$ is the same... | https://mathoverflow.net/users/20929 | Is minimum of convex envelope the same as minimum of the original function? | If $g(x)\leqslant f(x)-\varepsilon$ for some $x$ in $X$ such that $f(x)$ is the minimum of $f$ on $X$, then the function $h=\max\{g,f(x)\}$ is a convex under-estimator of $f$ such that $h\geqslant g$ everywhere and $h\gt g$ around $x$. This contradicts the definition of $g$.
| 3 | https://mathoverflow.net/users/4661 | 112155 | 64,229 |
https://mathoverflow.net/questions/112096 | 5 | I have read in the paper of Meigniez "Submersions, fibrations and bundles"
that a smooth surjective submersion $f: E \rightarrow B$ whose fibers are all diffeomorphic to $\mathbb{R}^{n}$ is locally trivial, i.e. a fiber bundle (corollay 31).
1) Is there a counterexample for $f$ not a submersion, which is not a fiber... | https://mathoverflow.net/users/10758 | Questions about local triviality | Here is a counter-example to Q1. Consider a Reeb-like foliation on the annulus $E=S^1\times\mathbb R$. Namely fix a point $p\in S^1$, let $I=S^1\setminus \{p\}$ and fix a smooth function $h:I\to\mathbb R$ which tends to $+\infty$ at both ends of the interval $I$. One leaf of the foliation is $\{p\}\times\mathbb R$, and... | 6 | https://mathoverflow.net/users/4354 | 112159 | 64,230 |
https://mathoverflow.net/questions/112158 | 1 | Let $X$ be a projective complex manifold. Is it true that any coherent sheaf $\mathcal{E}$ on $X$ whose Hilbert polynomial is a constant has a surjective morphsim $\mathcal{O}\_X\rightarrow \mathcal{M}$? Or dually is it true that any coherent sheaf $\mathcal{F}$ whose Hilbert polynomial is different from that of $\math... | https://mathoverflow.net/users/28040 | Surjective and injective criteria via Hilbert polynomials | No. For the first question, let $X$ be a point and $\mathcal M=\mathbb C^2$ (vector space) or any $X$ and $\mathcal M$ be a vector space of dimension $\ge 2$ supported only at $x$.
For the second question, consider a Riemann surface $X$ and $\mathcal F$ a line bundle. Then the Hilbert polynomial of $\mathcal F$ is t... | 1 | https://mathoverflow.net/users/3485 | 112160 | 64,231 |
https://mathoverflow.net/questions/112157 | 0 | Is it true that, given a space $X$ and a probability measure $\mu$ on it, given some sets $A, B \subset X$ and a finite number of disjoint sets $C\_{\sigma}$ such that $\bigcup\_{\sigma} C\_{\sigma} =X$, the following inequality holds,
$$\mu (A \cap B ) = \sum\_{\sigma}\mu(A \cap C\_{\sigma}\cap B) \leq \sum\_{\sigma... | https://mathoverflow.net/users/26798 | Relation between measure of sets | The answer is no. Consider $B\subset A$ with $\mu(B)$ very small, $C\_1=A\setminus B$ and $C\_2=X\setminus C\_1$. Then the r.h.s. equals $\mu(B)^2/\mu(A)\mu(C\_2)$ which can be less than $\mu(B)$ since $\mu(A)$ and $\mu(C\_2)$ can be grater than $1/2$ and $\mu(B)$ less than $1/10$.
**Added later.** Under the assumpti... | 6 | https://mathoverflow.net/users/4354 | 112161 | 64,232 |
https://mathoverflow.net/questions/112171 | 2 | Does there exist a positive integer N such that for all odd primes p the multiplicative order of 2 mod p is strictly less than the multiplicative order of 2 mod p^N ? Once again, references would be greatly appreciated as this pertains to ongoing graduate research.
| https://mathoverflow.net/users/27986 | multiplicative order of 2 mod p^N | I don't think this is known. You might want to have a look at:
A. Granville, Refining the conditions on the Fermat quotient,
Mathematical Proceedings of the Cambridge Philosophical Society, 98 (1985) 5-8.
You might be able to to say more assuming the ABC conjecture, which may or may not have been proved.
| 1 | https://mathoverflow.net/users/2290 | 112174 | 64,240 |
https://mathoverflow.net/questions/112152 | 4 | I was asking myself about some necessary and/or sufficient conditions for a function to be bandlimited (i.e. its Fourier transform is zero t residing out of $[-B,B]$ for some $B>0$).
Of course, if a function is bounded (timelimited), it cannot be bandlimited.
But for a non-bounded function, how can we tell if it's band... | https://mathoverflow.net/users/28037 | Space of Bandlimited Functions | There is a necessary and sufficient condition for a function $f$ of the real variable
to be "band-limited". It is called the Wiener-Paley theorem. $f$ must be a restriction on the real
line of an entire function of exponential type. Of course, here one implicitly assumes that
$f$ belongs to an appropriate space which p... | 10 | https://mathoverflow.net/users/25510 | 112178 | 64,243 |
https://mathoverflow.net/questions/112180 | 4 | The following apparently elementary question came out of a somewhat naive attempt to
prove that every distribution $u\in \mathscr D'(\mathbb R^2)$ with $\partial\_1 u=\partial\_2 u =0$ is a constant function (this can be reduced to $\mathscr C^1$-functions by convolution with an approximate identity and for $\mathscr C... | https://mathoverflow.net/users/21051 | Which test functions are the divergence of a vector field? | If you use use the Euclidean metric on $\mathbb{R}^2$ and apply Hodge duality, your question is about $H^2\_c(\mathbb{R}^2)$, the De Rham cohomology with compact supports on $\mathbb{R}^2$.
Bott & Tu, *Differential forms in Algebraic Topology*, Theorem 4.7 and Corollary 4.7.1 give $H\_c^2(\mathbb{R}^2) = \mathbb{R}$,... | 4 | https://mathoverflow.net/users/2622 | 112184 | 64,246 |
https://mathoverflow.net/questions/112181 | 1 | Let $K$ be a compact of the plane of Lebesgues measure 0 and $\Omega$ a domain containing $K$. Denote by $E$ the vector space of functions that are holomorphic on $\Omega - K$.
I'm interested in knowing whether we can define a $\overline{\partial}$ operator on $E$ in the sense of distributions . The problem is, elem... | https://mathoverflow.net/users/19189 | About principal values and Wirtinger derivative | You probably want $L$ to be continuous with respect to the topology of $E$, not just $C(z)$ And $E$ is equipped with its usual topology: uniform convergence on compact subsets of
$\Omega\backslash K$.
If this is so, the extension is impossible. Even if your $K$ is one point.
In this case a function $f\in E$ expands i... | 2 | https://mathoverflow.net/users/25510 | 112187 | 64,249 |
https://mathoverflow.net/questions/112154 | 12 | For the rank $8$ elliptic curve with a-invariants $(0, 0, 1, -23737, 960366)$
sage 5.3 reports analytic rank $4$ in about 2.4 hours.
Almost sure this a bug, so I am interested what other CAS say on the matter of
the analytic rank. Currently testing pari's *ellanalyticrank()* but I have the impression
sage is several ... | https://mathoverflow.net/users/12481 | What CASes say about the analytic rank of rank 8 elliptic curve '457532830151317a1' | Did you read the Sage's documentation on analytic\_rank()?
```
Return an integer that is *probably* the analytic rank of this
elliptic curve. If leading_coefficient is "True" (only implemented
for PARI), return a tuple (rank, lead) where lead is the value of
the first non-zero derivative of the L-functi... | 15 | https://mathoverflow.net/users/11100 | 112200 | 64,254 |
https://mathoverflow.net/questions/112198 | 5 | Let $G$ be a finite group which has a cyclic maximal subgroup. Is $G$ solvable?
| https://mathoverflow.net/users/27962 | solvable groups | Yes. Let $M$ be the cyclic maximal subgroup (actually the proof works for an Abelian maximal subgroup). We may suppose by induction that $M$ contains no non-trivial normal subgroup of $G.$ Then for each non-identity subgroup $X$ of $M,$ we have $M = N\_{G}(X),$ as $M$ is maximal and $X \lhd M.$ It follows easily that $... | 17 | https://mathoverflow.net/users/14450 | 112202 | 64,255 |
https://mathoverflow.net/questions/112137 | 13 | I'm guessing the answer to this question is well-known:
Suppose that $Y:C \to P$ and $F:C \to D$ are functors with $D$ cocomplete. Then one can define the point-wise Kan extension $\mathbf{Lan}\_Y\left(F\right).$ Under what conditions does $\mathbf{Lan}\_Y\left(F\right)$ preserve colimits? Notice that if $C=P$ and $Y... | https://mathoverflow.net/users/4528 | When do Kan extensions preserve limits/colimits? | The pointwise left Kan extension of F along Y is a coend of functors $Lan\_{Y}(F) = \int^{x}P(Yx,-).Fx$ where each functor $P(Yx,-).Fx$ is the composite of the representable $P(Yx,-):P \to Set$ and the copower functor $(-.Fx):Set \to D$. As a coend (colimit) of the $P(Yx,-).Fx$ the left Kan extension preserves any coli... | 23 | https://mathoverflow.net/users/17696 | 112206 | 64,258 |
https://mathoverflow.net/questions/112173 | 2 | We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it.
Here Jacobi PDE system is
$a\_1+b\_1\frac{\partial h\_1}{\partial x\_1}-c\_1\frac{\partial h\_1}{\partial x\_2}-d\_1\frac{\p... | https://mathoverflow.net/users/nan | Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation? | Thanks for the clarification; I wasn't familiar with this terminology. I assume that the coefficients $a\_i$, $b\_i$, $c\_i$, $d\_i$, $e\_i$, and $f\_i$ are specified functions of $x\_1,x\_2,h\_1,h\_2$. (Let me know if this is not correct.)
Geometrically, what you have is a pair of $2$-forms on a $4$-dimensional mani... | 4 | https://mathoverflow.net/users/13972 | 112208 | 64,260 |
https://mathoverflow.net/questions/112211 | 2 | In analyzing the spherical pendulum the cotangent space of the sphere is defined as
$ T^\*S^2 = \lbrace (q,p) \in \mathbb{R}^3 \times \mathbb{R}^3; |q| = 1, q \cdot p = 0 \rbrace$
my problem with this is that I see the right-hand side of the equation as a set of points, whereas I see the left-hand side as a set of... | https://mathoverflow.net/users/28053 | Cotangent space of the sphere | What you have written down to me seems to be the tangent bundle, not the cotangent bundle (though they are isomorphic, I suppose).
Consider the projection from your space (let's call it $X$) to the first copy of $\mathbb{R}^3$. This has image exactly those $q \in \mathbb{R}^3$ such that $|q| = 1$ i.e. $S^2$. The fibr... | 2 | https://mathoverflow.net/users/1703 | 112213 | 64,264 |
https://mathoverflow.net/questions/112205 | 4 | What is the best available less or more modern introduction to the subject?
| https://mathoverflow.net/users/4312 | T. Carleman's method on eigenvalues asymptotics | All approaches I am familiar with are based on some Tauberian theorem: one obtains informations about various weighted counts of eigenvalues and then one removes the weights via some Tauberian theorem.
There are three implementations of this strategy that I know of.
1. One based on the heat equation. For Laplace op... | 6 | https://mathoverflow.net/users/20302 | 112216 | 64,266 |
https://mathoverflow.net/questions/112179 | 9 | Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes.
Let us denote the category of stacks over $\mathcal{C}$ by $Stacks(\mathcal{C})$. This is naturally a full subcategory of the category $Pre\_{grpds}(\mathcal{... | https://mathoverflow.net/users/28048 | stackification commutes with finite limits? | The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online: [Tag04Y1](https://stacks.math.columbia.edu/tag/04Y1) in the Stacks Project. I quote:
>
> Lemma 8.4. Let $C$ be a site. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of fibred categories over $C$. In this ca... | 9 | https://mathoverflow.net/users/11682 | 112220 | 64,268 |
https://mathoverflow.net/questions/112199 | 16 | In the most general context, the Picard-Lindelöf theorem (aka Cauchy-Lipschitz in French) asserts the existence of a maximal solution for $\dot{x}(t) = f(t,x(t))$, i.e. of a solution $x(t)$ defined on a interval $I$ such that there exist no other solution whose restriction to $I$ coincide with $x$. The usual proofs of ... | https://mathoverflow.net/users/17164 | Differential equations and axiom of choice | At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)
Any IVP admits an upper and ... | 11 | https://mathoverflow.net/users/6101 | 112225 | 64,271 |
https://mathoverflow.net/questions/112190 | 4 | Let $X$ be a smooth and proper variety over $\mathbb{C}$. Let $F$ be an $\mathbb{A}^1$ family of $\mathbb{G}\_m$ gerbes over $X$. Suppose the fibers over every point away from 0 in $\mathbb{A}^1$ are all isomorphic. Must the fiber over zero be isomorphic to the general fiber $F\_t$ ? This is certainly true for line bun... | https://mathoverflow.net/users/6986 | family of gerbes over smooth and proper algebraic varieties | By a family of gerbes you mean, I suppose, a gerbe over $X \times \mathbb A^{1}$. In any case, it has a class in $\mathrm H^0(\mathbb A^1, \mathrm R^2 \mathrm{pr}\_{2\*}\mathbb G\_{\rm m})$. Since $\mathrm R^2 \mathrm{pr}\_{2\*}\mathbb G\_{\rm m}$ is a torsion sheaf, this class comes from $\mathrm H^0(\mathbb A^1, \mat... | 5 | https://mathoverflow.net/users/4790 | 112229 | 64,274 |
https://mathoverflow.net/questions/112233 | 8 | My question is:
I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space and Yang-Mills Moduli space?
My question is a reference request.
My question is based on the following observation: Fo... | https://mathoverflow.net/users/25054 | Relation of SW and Donaldson Invariant | This was conjectured by Witten in his paper in his paper [Monopoles and four-manifolds](http://arxiv.org/abs/hep-th/9411102). The conjecture says that if $X$ has Seiberg-Witten simple type (meaning that $SW\_X(\mathfrak{s})$ is nonzero only when the moduli space associated to $X$ is 0-dimensional) and satisfies some mi... | 14 | https://mathoverflow.net/users/428 | 112235 | 64,276 |
https://mathoverflow.net/questions/112239 | 5 | It is well-known that any ideal in a Dedekind domain can be generated by at most two elements. However, already for Noetherian domains of dimension 2, it is easy to construct examples of ideals that require arbitrarily large finite numbers of elements to generate.
Nevertheless, I believe I can prove the following the... | https://mathoverflow.net/users/2106 | Radical generation of ideals in Noetherian rings | This is a classical result. See, for instance the book of Iyengar, Leuschke, Leikin, Miller, Miller, Singh, Walther, [*Twenty-Four Hours of Local Cohomology*](http://www.cmi.univ-mrs.fr/masters/master2mf/lib/exe/fetch.php?media=local_cohomology_hypertext.pdf), Remark 9.14.
| 8 | https://mathoverflow.net/users/23950 | 112240 | 64,277 |
https://mathoverflow.net/questions/112214 | 2 | Consider the divided-power ring $A := \mathbb Z \langle x\_1, \ldots, x\_n \rangle$ consisting of $\mathbb Z$-linear combinations of divided-power monomials of the form $x\_1^{(a\_1)} \cdots x\_n^{(a\_n)}$; this can be defined as the subring of the polynomial ring $\mathbb Q[ x\_1, \ldots, x\_n ]$ which is generated as... | https://mathoverflow.net/users/1528 | On the divided power ring over the integers | If we let $k = \mathbb{F}\_2$, then there are only three polynomials of total degree 1 in the variables $x$ and $y$, namely $x,y,x+y$. The space of degree 3 elements in the divided power ring over $k$ is 4 dimensional, spanned by $x^{(3)}, x^{(2)}y, xy^{(2)}, y^{(3)}$. Unless I have made a mistake, I am forced to concl... | 3 | https://mathoverflow.net/users/121 | 112246 | 64,280 |
https://mathoverflow.net/questions/112231 | 16 | I tried to find some paper published in Notices of the American Mathematical Society 1975
By E. Calabi, On manifolds with non-negative Ricci curvature II
Notices of the American Mathematical Society 22 1975 A205
But when I searched at mathscinet and click Calabi, there is no such a paper listed under his nam... | https://mathoverflow.net/users/1190 | Is "Notices of the American Mathematical Society" available before 1995? | As Andreas Blass and others surmised, this is indeed an abstract. It was for a 10-minute talk at the Annual Meeting in January, 1975. Here is the entire "paper":
>
> **On Manifolds with nonnegative Ricci curvature II**
>
>
> Let $M$ be an $n$-dimensional
> Riemannian manifold with nonnegative
> Ricci curvature.... | 27 | https://mathoverflow.net/users/15837 | 112247 | 64,281 |
https://mathoverflow.net/questions/112217 | 2 | Suppose $\pi:X\to Y$ is a flat projective morphism with connected fibers. Suppose $Y$ is a smooth projective variety ($X$ and $Y$ are over $\mathbb C$). Let $E$ be a locally free sheaf on $X$ such that its restriction to fibers of $\pi$ is trivial. Suppose finally that for each fiber $F$ of $\pi$ the higher cohomology ... | https://mathoverflow.net/users/13441 | Locally free sheaves trivial on fibers of a flat projective morphism | One possible approach to this is to divide the problem into two and figure out what conditions are needed for the two parts.
1
-
First assume that $\pi\_\*E$ is locally free of the same rank as $E$ (say $r$). By adjointness of $\pi^\*$ and $\pi\_\*$ we have a natural map
$$
\nu:\pi^\*\pi\_\*E\to E
$$
**If** $E|\_F$... | 6 | https://mathoverflow.net/users/10076 | 112252 | 64,282 |
https://mathoverflow.net/questions/112271 | 0 | Can you exhibit an example of a noncontractible domain in R^n with the d-th cohomology groups trivial for all d greater or equal to 1?
Thank you
| https://mathoverflow.net/users/27960 | Noncontractible domain with trivial cohomology | There are a million examples. You should google "acyclic space". Here is one: if you remove a point from a homology sphere you get a manifold whose cohomology is trivial in positive degrees. Take a tubular neighbourhood of it in some $\mathbf R^n$ and you get an open domain.
| 4 | https://mathoverflow.net/users/1310 | 112274 | 64,290 |
https://mathoverflow.net/questions/112269 | 2 | Let $X, Y$ be two metric spaces and $f$ be a continuous bijection (i.e. one-to-one map) from $X$ to $Y$. Let $E$ be a $G\_{\delta}$ subset of $X$. I want to know weather the image $f(E)$ is also a $G\_{\delta}$ subset of $Y$. Thanks!
| https://mathoverflow.net/users/28067 | On the image of a G_\delta set under a continuous bijection | Yes, if $X$ is $\sigma$-compact: $X\smallsetminus E$ is $F\_\sigma$, hence a countable union of compact sets. Since a continuous image of a compact set is compact (and therefore closed, in a Hausdorff space), $f(X\smallsetminus E)$ is also $F\_\sigma$, hence $f(E)$ is $G\_\delta$.
EDIT: While ljjpfx seems to be happy... | 13 | https://mathoverflow.net/users/12705 | 112277 | 64,292 |
https://mathoverflow.net/questions/112268 | 1 | Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form
$q: \Pi \oplus \Pi \rightarrow R $
$\big( \alpha ,\beta \big) \longrightarrow \alpha \wedge \beta =q \big( \alpha , \beta \big) \Omega $
where $\alpha , \beta \in \Pi$ . We say $ \Pi \subset \Lambda^{2}\big(V^{ \star... | https://mathoverflow.net/users/nan | if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic? | No. Note that $V$ has dimension $4$. The maximum dimension of an elliptic subspace of $\Lambda^2(V^\ast)$ is $3$, so if $\Pi\_1$ and $\Pi\_2$ don't intersect, then $\Pi\_1\oplus \Pi\_2$ is *never* elliptic.
To see why this is so, let $\Omega\in \Lambda^4(V^\ast)$ be a volume form and define a symmetric bilinear form ... | 2 | https://mathoverflow.net/users/13972 | 112283 | 64,295 |
https://mathoverflow.net/questions/112249 | 0 | Hi,
Given C>0. Let $f,g,h$ be $L^2$ functions such that $f,g,h$ have a compact and finite-measure support, and $f\*f(x)=g\*g(x)=h\*h(x)=f\*g(x)=g\*h(x)=f\*h(x)=0$ (where $\*$ is the convolution) for all $|x| > C$.
Does that imply $f(x)=g(x)=h(x)=0$ for all $x$?
What is I won't require $f\*f=g\*g=h\*h=0$?
What if ... | https://mathoverflow.net/users/28037 | Convolutional Equation | I don't know why you say "compact and finite measure". Every compact set has finite measure.
Fourier transform of a function with compact support is an entire function.
Fourier transform of a convolution is the square of the Fourier transform. So
EACH of the equations $f\*f=0$ or $f\*g=0$ already implies that $f=0$ and... | 3 | https://mathoverflow.net/users/25510 | 112285 | 64,296 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.