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https://mathoverflow.net/questions/178602 | 1 | I work on a bounded domain in $\mathbb{R}^n$. Let $u \in H^1(0,T;H^{-1})\cap L^2(0,T;H^1)$ be a solution of the heat equation:
$$\langle u', v \rangle + \int \nabla u \nabla v = 0$$
for each test function $v$. The solution has the property
$$\int\_\Omega u(t) = 0$$
for each $t$.
Is it possible to conclude from this... | https://mathoverflow.net/users/57159 | Weak solution of a heat equation is zero? | I think the answer to your question is 'no' since the function $e^{-t}\sin (x)$ solves the heat equation on $\Omega=(-1,1)\times \mathbb R\_+$ but the integral is zero by symmetry. If you want an example where $n>1$, just add a dummy variable.
| 4 | https://mathoverflow.net/users/57161 | 178604 | 89,828 |
https://mathoverflow.net/questions/178603 | 1 | This, in view of the analogies between CM points on Shimura curves and torsion points on elliptic curves, is a sequel to an earlier question I had asked: [The torsion point count in higher dimension](https://mathoverflow.net/questions/121202/the-torsion-point-count-in-higher-dimension) .
Mazur proved that the number... | https://mathoverflow.net/users/26522 | The special point count on Shimura varieties | For the $j$-line, a special point corresponds to a CM-elliptic curve, so to an order in an imaginary quadratic field. As you are looking at the degree of the point, this corresponds to the degree of the Hilbert class field of the order, which can be estimated using the Brauer-Siegel formula in terms of the discriminant... | 4 | https://mathoverflow.net/users/2290 | 178605 | 89,829 |
https://mathoverflow.net/questions/178564 | 4 | Do there exist Jordan analytic curves $J$ in the complex plane $C$, other than circles, with the following property:
There exists a harmonic function $u$ in the unbounded component of $C\backslash J$,
$u(z)=0$ and $|\mathrm{grad}\, u(z)|=1$ for $z\in J$.
If we impose mild restrictions on $u$ near $\infty$, for exa... | https://mathoverflow.net/users/25510 | A free boundary problem | Sorry, I solved it few hours after posting.
Consider two functions
$$\phi(z)=z+\frac{2a^2}{z}-\frac{a^4}{3z^3},$$
and
$$v(z)=(1+a^4)\log|z|-\frac{a^2}{2}\Re(z^2-z^{-2}).$$
They have the following properties.
(i) When $a>0$ is sufficiently small, $\phi$ is
univalent in $\Delta=\{ z:|z|\geq 1\}$, and $\phi'(z)\neq 0$... | 2 | https://mathoverflow.net/users/25510 | 178611 | 89,831 |
https://mathoverflow.net/questions/178609 | 6 | Suppose $\mathbb{P}$ is a definable proper forcing (for instance Sacks forcing). Let $\alpha$ be some ordinal. Let $\mathbb{P}\_\alpha$ be the countable support iteration of $\mathbb{P}$ of length $\alpha$.
It is well known that $\mathbb{P}\_\alpha$ is also proper. Hence for any countable elementary structure $M \pr... | https://mathoverflow.net/users/43354 | Iteration of Proper Forcing and Support of Master Conditions | I may be missing something here, because if I've got the question right then the definability doesn't come in to play:
The answer is yes, and the reason is implicit in the proof that $\mathbb{P}\_\alpha$ is proper.
Shelah's proof of Theorem 3.2 on page 109 of Proper and Improper forcing mentions this. In particular, ... | 7 | https://mathoverflow.net/users/18128 | 178614 | 89,832 |
https://mathoverflow.net/questions/178586 | 5 | Let $q$ be a power of prime $p$, and $n, m, k$ positive integers such that $mk=2n$ and $2\leq m<2n$. Let $\mathrm{Sp}(2n,q)$ be the symplectic group of dimension $2n$ over $\mathrm{GF}(q)$ and $\mathrm{SL}(m,q^k)$ the special linear group of dimension $m$ over $\mathrm{GF}(q^k)$. For what values of $m,k$ does $\mathrm{... | https://mathoverflow.net/users/56827 | Special linear groups contained in symplectic groups | The question is this:
>
> When can $SL(m,q^k)$ be a subgroup of $Sp(2n,q)$ with $mk=2n$?
>
>
>
As you point out, this is possible if $m=2$. There are many particular cases that can be ruled out by order considerations - using Zsigmondy primes. However to give a complete answer, one should observe that if such ... | 5 | https://mathoverflow.net/users/801 | 178617 | 89,833 |
https://mathoverflow.net/questions/178634 | 0 | Let H be an infinite dimensional and separable Hilbert Space. Do there exist disjoint, closed and bounded subsets A,B of H which satisfy the following conditions? (1) Each of A,B is convex and has a non-empty interior with respect to H. (2) Given any positive real number e, there is a point of A and a point of B whose ... | https://mathoverflow.net/users/4423 | Two questions about convex subsets of Hilbert Space | No. Convex closed subsets are weakly closed, and since $A,B$ are also assumed bounded, we can make $x\_n,y\_n$ converge weakly to limits $x\in A$, $y\in B$ on a subsequence if $x\_n\in A$, $y\_n\in B$. If now also $\|x\_n -y\_n\|\to 0$, then $x=y\in A\cap B$.
| 3 | https://mathoverflow.net/users/48839 | 178637 | 89,840 |
https://mathoverflow.net/questions/178632 | 2 | Let $G$ be an affine algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra. Given $X\in\mathfrak{g}$, it has (relative to $G$) its Jordan-Chevalley decomposition $X=s+n$. Since the decomposition is preserved by morphisms of affine groups... | https://mathoverflow.net/users/56901 | Global centralizers in Jordan-Chevalley decomposition in bad characteristic | The answer is affirmative for $G$ any smooth affine $k$-group. We may assume $p = {\rm{char}}(k) > 0$. By 14.26 in Borel's textbook on algebraic groups, $n$ is in the Lie algebra of a unipotent subgroup $U$ of $G$. The $p$-operator $Y \mapsto Y^{[p]}$ on the Lie algebra of a smooth affine group (or more generally group... | 4 | https://mathoverflow.net/users/54268 | 178644 | 89,842 |
https://mathoverflow.net/questions/178607 | 4 | I'm currently reading the paper <http://arxiv.org/abs/0908.2473> and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a sequence $(X\_t^h)\_{t \geq 0}$ of processes such that $X^h\_0=0$ and show that for $0<s<t$ and $p \geq 1$
$$ \operatorname{... | https://mathoverflow.net/users/56931 | Can't figure out "standard application" of the Garsia-Rodemich-Rumsey Lemma | Let $p \ge 1$ and $\alpha >p^{-1}$. From the GRR inequality, there exists a constant $C\_{\alpha,p} >0$ such that for any continuous function $f$ on $[0,T]$, and for all $t,s \in [0,T]$ one has:
$
|f(t)-f(s)|^p \le C\_{\alpha,p} |t-s|^{\alpha p-1} \int\_0^T \int\_0^T \frac{ |f(x)-f(y)|^p}{ |x-y|^{\alpha p+1}} dx dy.
... | 3 | https://mathoverflow.net/users/48356 | 178646 | 89,844 |
https://mathoverflow.net/questions/178621 | 6 | Let $V$ be a finite-dimensional vector space with an inner-product $(,)$ and let $C\subset V$ be a cone in $V$. Let $C^\vee$ denote the dual of $C$ with respect to $(,)$, i.e., the set of vector $v\in V$ having non-negative intersection with all of $C$. Of course there is a natural identification of $V$ with its dual $... | https://mathoverflow.net/users/56505 | When does a cone contain its dual cone? | The cone $C$ contains its dual cone if and only if for every $x\in\partial C$ there exists $y\in C$ so that $(x,y)\leq0$.
(This is quite obvious in two dimensions.)
Let $S\subset V$ denote the unit sphere.
The cone $C$ is uniquely determined by $A=C\cap S$ and similarly $C^V$ by $B=C^V\cap S$, so let us restrict our ... | 3 | https://mathoverflow.net/users/55893 | 178648 | 89,846 |
https://mathoverflow.net/questions/178630 | 4 | Given a planar graph (no loops, no multiple edge), is it always possible to perform edge contractions\* in order to obtain a graph $T$ which has no loops, and if one ignores parallel edges, $T$ is a tree?
\*delete the edge and identify its two ends. This may create loops and multiple edges in the process.
Is there ... | https://mathoverflow.net/users/18974 | Contracting a planar graph to a (multiply-edged)-tree | A counterexample is the planar dual to the [Tutte graph](http://en.wikipedia.org/wiki/Tutte_graph). Most of the credit goes to David Eppstein (see the comments below)
Let $G$ be the dual to the Tutte graph. Then $G$ is a planar triangulation. Towards a contradiction, suppose that $G$ has a set of edges $C$ such that ... | 3 | https://mathoverflow.net/users/2233 | 178650 | 89,847 |
https://mathoverflow.net/questions/178319 | 3 | I asked this initially in [math.stackexchange](https://math.stackexchange.com/questions/894399/identities-that-connect-antipode-with-multiplication-and-comultiplication):
The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities:
$$
S\otimes S\circ \Delta=\sigma\circ\Delta\circ S
... | https://mathoverflow.net/users/18943 | Identities that connect antipode with multiplication and comultiplication | This also holds in an arbitrary braided monoidal category and is not hard to see. See for example Majid: Foundations of quantum group theory. He gives a graphical calculus proof of this in Figure 9.14.
Strictly speaking, you need to turn this proof upside down (i.e. dualize) to get the corresponding identity for the co... | 4 | https://mathoverflow.net/users/33854 | 178651 | 89,848 |
https://mathoverflow.net/questions/178635 | 3 | Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t.
$$f(x)\ll |\psi(x) - x|$$
where $\psi$ is the Chebyshev function.
| https://mathoverflow.net/users/40983 | Lower bounds on the error term of the prime number theorem | Littlewood proved that
$$
\psi(x)-x = \Omega\_{\pm}(x^{\frac 12} \log \log \log x);
$$
here $\Omega\_{\pm}$ means that the LHS infinitely often gets as large as a positive constant times the RHS, and also infinitely often as small as a negative constant times the RHS. Montgomery conjectured that
$$
\limsup\_{x\to... | 16 | https://mathoverflow.net/users/38624 | 178652 | 89,849 |
https://mathoverflow.net/questions/178481 | 8 | Let $\mathbb{G}\_a$ be the additive group over an algebraically closed field $k$ of any characteristic. Let $X \to Y$ be a $\mathbb{G}\_a$-torsor of $k$-schemes (of finite type - in case that is relevant). Suppose that $X$ is quasi-affine. Is it true that $Y$ is quasi-affine?
EDIT: As David pointed out below, one sho... | https://mathoverflow.net/users/13302 | Quasi-affineness of the base of a $\mathbb{G}_a$-torsor | It's false! Take $\mathbb{A}^3$ with coordinates $(x,y,z)$. Blow up the origin, and let $E$ be the exceptional divisor. Delete the line $E \cap \{ z=0 \}$. The resulting quasi-projective variety will be our $Y$. Since $Y$ is quasi-projective, it is separated. $Y$ is an example of a quasi-projective variety with no comp... | 3 | https://mathoverflow.net/users/297 | 178657 | 89,851 |
https://mathoverflow.net/questions/178654 | 8 | I have a straightforward question. Let $f$ be a holomorphic cusp form of weight $k$, level $N$, and nebentypus $\chi$ that is new in the sense of Atkin-Lehner theory. Write its Fourier expansion at $\infty$ as
$$f(z) = \sum\_{n=1}^\infty \lambda\_f(n)n^{(k-1)/2}e(nz)$$
and form the $L$-function $L(f,s)$ by the Dirichle... | https://mathoverflow.net/users/37110 | Characterizing the newforms s.t. the associated symmetric square $L$-function has a pole | Yes, your understanding is correct. Here's a little bit more detail. If $f$ is a CM form, $f$ is associated to a Hecke Grössencharacter $\xi$. If $k \geq 2$, then $\xi$ is associated to an imaginary quadratic field and then $\xi$ is a homomorphism $\xi : I(\Lambda) \to \mathbb{C}^{\times}$ is a homomorphism from the gr... | 7 | https://mathoverflow.net/users/48142 | 178658 | 89,852 |
https://mathoverflow.net/questions/178295 | 6 | Let $M(n)$ be the vector space of $n\times n$ matrices over a local non-archimedian field $K$.
Let $\mathcal S$ denote the space of locally constant compactly supported functions
on $M(n)$. Similarly, let $\mathcal S ^0$ denote the space of locally constant compactly supported functions
on the group $GL(n)$ (which is a... | https://mathoverflow.net/users/3891 | A question on non-archimedian Fourier transform | Actually, the answer is indeed negative and the explanation is very simple. Namely, for $n=2$ let $\phi$ be the delta-function of the space of matrices whose second row is zero (considered as a distribution). Then its Fourier transform is the delta-function of matrices whose first column is zero. Hence this is a distri... | 5 | https://mathoverflow.net/users/3891 | 178661 | 89,854 |
https://mathoverflow.net/questions/178666 | 6 | For every natural number n, let:
* Gn be the number of distinct group structures with at most n elements;
* An be the number of distinct abelian group structures wit at most n elements;
* Sn be the number of distinct solvable group structures with at most n elements.
Question 1: Is there a known limit for the quoti... | https://mathoverflow.net/users/30395 | Asymptotic density of finite abelian and solvable groups | The number of abelian groups of order at most $n$ is $O(n)$, whereas if $n=2^k$, the number of class $2$ nilpotent groups of order $n$ is $2^{(2/27)k^3+O(k^{8/3})}=n^{\Omega(\log^2n)}$ by a result of Sims, hence the answer to question 1 is $0$. It is conjectured that the global asymptotic density of $2$-groups of nilpo... | 8 | https://mathoverflow.net/users/12705 | 178669 | 89,858 |
https://mathoverflow.net/questions/178677 | 7 | For many times, I find people working on schemes over DVRs, and prove theorems on such setting. For example, my latest experience is the "semi-stable reduction theorem" by Kempf, Knudsen, Mumford and Saint-Donat: they proved semi-stable reduction theorem first over $\mathbb{C}$, then turned to the situation over DVR.
... | https://mathoverflow.net/users/29730 | Why considering schemes over discrete valuation rings? | Well, varieties and schemes typically come in families. If you accept that, then it's natural to consider the simplest case where the base is one dimensional and regular. This includes geometers smooth curves, and things like $Spec\mathbb{Z}$ or $Spec \mathbb{Z}\_p$ of interest to number theorists. If you want to conce... | 9 | https://mathoverflow.net/users/4144 | 178679 | 89,859 |
https://mathoverflow.net/questions/178089 | 2 | The following question kept me wondering for some time:
Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive semidefinite (hence not necessarily invertible) with $\text{trace}(B)\neq0$, prove that
$\text{trace}\big\{C^{-1/2}BC^{−1/2}... | https://mathoverflow.net/users/56884 | An inequality involving traces and matrix inversions | The last inequality is not true. For example, take $n=2$, $A, C$ to be identity matrix. It suffices to compare $\text{trace}\big\{(I+B^{-1})^{−1}\big\}$ and $\text{trace}\big\{\big(I+\frac{2}{\text{trace}(B)}I\big)^{−1}\big\}$. Now take $B=diag(1,2)$. You find that $\text{trace}\big\{(I+B^{-1})^{−1}\big\}=7/6<6/5=\text... | 1 | https://mathoverflow.net/users/54458 | 178687 | 89,860 |
https://mathoverflow.net/questions/178578 | 2 | There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:
Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex domain, $\phi$ a plurisubharmonic function on $\Omega$, and $P = \{z \in \Omega:z\_n = 0\}$. If $f$ is a holomorphic function... | https://mathoverflow.net/users/1106 | Does the "Ohsawa-Takegoshi theorem without bounds" have a name? | This is often called Cartan-Serre theorem. Actually Serre's part is that the restriction map $\mathcal O(\Omega)\to\mathcal O(\Omega\cap P)$ is surjective if all cohomology groups $H^{0,p}(\Omega)$ vanish. Now Cartan "B" theorem says that this is the case if the domain $\Omega$ is pseudoconvex. Hence the combination of... | 2 | https://mathoverflow.net/users/57201 | 178697 | 89,864 |
https://mathoverflow.net/questions/178137 | 8 | *[Cross-posted from [Math.SE](https://math.stackexchange.com/questions/880738/gaps-between-roots-of-trigonometric-polynomials) because I got no responses there.]*
Given a polynomial in $e^{\mathrm{i}k t}$ of the form
$$ p(t) = \sum\_{-n\leq k\leq n} c\_k e^{\mathrm{i}k t} $$
with $\bar c\_{-k} = c\_k$, is there a goo... | https://mathoverflow.net/users/10423 | Gaps between roots of trigonometric polynomials | The technical term for what you want to do is *root isolation* or *root bracketing*. One way to approach this to find the minimal distance between the roots, like you are suggesting, and also a large enough bounded interval to contain all the roots. This idea was in fact used early on in the history of root isolation f... | 6 | https://mathoverflow.net/users/2622 | 178713 | 89,870 |
https://mathoverflow.net/questions/178665 | 2 | Let $X$ be a smooth connected projective curve of genus $g$ over an algebraically closed field.
Let $E$ be a vector bundle on $X$ of rank $n$.
Is it true that there exists a constand $N(g,n)$ such that each such $E$ has a filtration
$$
0\subset E\_1\subset ...\subset E\_n=E
$$
by subbundles with $rank(E\_i)=i$, such ... | https://mathoverflow.net/users/3891 | FIltrations on a vector bundle on a curve | According to lemma 4 of
M.F. Atiyah, *Vector bundles over an elliptic curve*, Proc. London Math. Soc. (3) 7 (1957), 414-452.
you could take $N(g,n) = - 2 g$.
| 4 | https://mathoverflow.net/users/4707 | 178717 | 89,872 |
https://mathoverflow.net/questions/178724 | 5 | ### Short Version (the question)
Let $\text{Cat}$ be the category of (small) categories and $\text{sSet}$ the category of simplicial sets. There is a functor $\Gamma:\text{sSet}\to \text{Cat}$ that takes every simplicial set to its [category of simplices](http://ncatlab.org/nlab/show/category+of+simplices). This func... | https://mathoverflow.net/users/50409 | Does the right adjoint of the category of simplices functor is "homotopicaly inverse" to the category of simplices functor? | This indeed true and is discussed in depth by Latch, Thomason and Wilson in a paper called *Simplicial Sets from Categories*. Your question is answered in Corollary 4.7 and relies on Theorem 4.1, the main result of the paper which says that $N \to T$ is indeed a weak equivalence. The argument can also be adapted to giv... | 5 | https://mathoverflow.net/users/12547 | 178725 | 89,876 |
https://mathoverflow.net/questions/178714 | 11 | Let $f$ be a function from the positive integers to the real numbers (or some ring...). Let
$$(\star) \quad F(n) = \sum\_{n\_1 \leq \cdots \leq n\_j\atop n\_1 + \cdots + n\_j = n} f(n\_1) \cdots f(n\_j), $$
for each positive integer $n$, where the sum runs over all the partitions of $n$, i.e., $n\_1 \leq \cdots \leq n... | https://mathoverflow.net/users/nan | Faster formula to compute sum over partitions | The identity $np(n) = \sum\_{m=1}^n p(n-m)\sigma(m)$, where $\sigma(m)$ is the sum of divisors of $n$ generalizes to this setting. The proof I sketched [here](https://mathoverflow.net/questions/127000/partitions-sum-of-divisors-identity) shows that
$$ nF(n) = \sum\_{r=1}^n F(n-r) g(r) $$
where
$$ g(r) = \sum\_{m ... | 12 | https://mathoverflow.net/users/7709 | 178726 | 89,877 |
https://mathoverflow.net/questions/178596 | 2 | suppose $x=\Delta$, $y=M \Phi \Delta$, where $\Delta\in N\times 1$, $M^T=M \in N \times N$ and $\Phi^T=\Phi \in N \times N$. Define $Z=xy^T+yx^T$. It is known from [the answer](https://mathoverflow.net/a/177685/32660) to [my previous question](https://mathoverflow.net/q/177683/32660) that $Z$ has two eigenvalues, one i... | https://mathoverflow.net/users/42511 | Asymptotic property of a quadratic form | Not sure if I understand your question correctly.
Denote $A=M\Phi$. Then for $x$ nonzero, $$\frac{y^Tx+\sqrt{x^Txy^Ty}}{\|x\|^2}=\frac{x^TA^Tx}{x^Tx}+\sqrt{\frac{x^TA^TAx}{x^Tx}}.$$
As $A$ is a finite dimensional matrix, $\frac{x^TA^Tx}{x^Tx}+\sqrt{\frac{x^TA^TAx}{x^Tx}}$ is bounded above by a constant, (the nume... | 1 | https://mathoverflow.net/users/54458 | 178733 | 89,879 |
https://mathoverflow.net/questions/178737 | 1 | Let $u$ be solution of $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial n} = 0$ on $\partial \Omega$.
Is it true that if $f \in L^{\infty}(\Omega)$ then $u \in W^{2,\infty}(\Omega)$?
(Assuming a 'nice' boundary of course.)
I think that I already found such a result in quite an old book, but I currently d... | https://mathoverflow.net/users/57234 | W^{2,∞} regularity of solutions of Poisson's equation if the right hand side is in L^{∞} | It is true that $f\in W^{k,2}$ implies that $u\in W^{k+2,2}$ (see theorem 4 [here](https://www.math.ualberta.ca/~xinweiyu/527.1.11f/lec14.pdf)), but this is not true if you replace the power 2 with $\infty$ (see remark 9 [here](https://www.math.ualberta.ca/~xinweiyu/527.1.11f/lec11.pdf)).
I think the boundary conditi... | 4 | https://mathoverflow.net/users/55893 | 178739 | 89,880 |
https://mathoverflow.net/questions/178738 | 0 | Take two posets $A, B$ (partially ordered sets). Now consider these posets to be categories $Cat(A), Cat(B)$ respectively. Consider a map from $A$ to $B$, $f: A \rightarrow B$. This can be seen as a functor $F : Cat(A) \rightarrow Cat(B)$. Under what conditions on the posets $A, B$ and the map $f$, will the functor $F$... | https://mathoverflow.net/users/10007 | a dcpo seen as a category: when does a dcpo map induce a functor with an adjoint? | *I would like to say that if f is Scott-continuous, then F will have an adjoint. Is this true?*
No.
If $A$ and $B$ are complete (semi)lattices then $f:A\to B$ has a right adjoint $f\dashv g$ iff $f$ preserves all joins, and then $g:B\to A$ that preserves all meets, and *vice versa*.
We could call this the **(Spec... | 3 | https://mathoverflow.net/users/2733 | 178740 | 89,881 |
https://mathoverflow.net/questions/178769 | 9 | I asked this question on [MSE](https://math.stackexchange.com/questions/732854/integer-valued-polynomial-through-some-known-points) about 5 months ago, but, even after offering a bounty, I didn't receive any answer, I hope this question isn't too easy for MO.
If we have a set of points $(x\_i,y\_i)$ with rational coo... | https://mathoverflow.net/users/49381 | Integer valued polynomial through some points with rational coordinates | No. The integer-valued polynomials have a basis (over $\mathbb{Z}$) given by the Newton polynomials
$$\displaystyle {x \choose n} = \frac{x (x - 1)\dots(x - (n-1))}{n!}$$
and as such if
$$\displaystyle f(x) = \sum a\_n {x \choose n}, a\_n \in \mathbb{Z}$$
is an integer-valued polynomial, then, for example, $... | 22 | https://mathoverflow.net/users/290 | 178772 | 89,892 |
https://mathoverflow.net/questions/178394 | 1 | One version of the Szemeredi-Trotter theorem states the following:
Given a set of $L$ lines in the plane, the number of points incident to at least $k$ lines is bounded above by a constant times $L/k + L^2/k^3$.
This version of the theorem can be found here in notes written by Adam Hesterberg for Larry Guth's poly... | https://mathoverflow.net/users/57043 | Is there a "Bipartite" Szemeredi-Trotter theorem? | I will describe a construction in the dual setting.
Take a $(k+1)\times n/k$ size grid of points.
Color the top row red, the rest blue, so we have $n$ red points and $k$ blue points.
The number of lines that contain one point of each row is $\Theta(n^2/k^3)$.
So translating it back to your problem, this gives a con... | 0 | https://mathoverflow.net/users/955 | 178775 | 89,893 |
https://mathoverflow.net/questions/178781 | 0 | I'm new to mathoverflow but hopefully anyone here can point me in the right direction.
The problem is as follows, imagine you have 4 beads, lets give them numbers 1,2,3,4. Now I want the unique `necklaces` without rotation, mirror nor duplication of the beads. So 1234 is the same as 3412 and 4321.
How do you call t... | https://mathoverflow.net/users/57250 | Terminology for beads/necklace/bracelet problem | I don't know the name for this set, but it seems straightforward to generate it. In each equivalence class there is a unique representative starting with "1", and such that either the position of "2" is less than $\frac{N+2}{2}$, or it is equal to $\frac{N+2}{2}$ and the position of "3" is less than $\frac{N+2}{2}$. Yo... | 1 | https://mathoverflow.net/users/56624 | 178785 | 89,899 |
https://mathoverflow.net/questions/178787 | 4 | It is well-known that, assuming the axiom of choice, there are unbounded linear maps defined not only on a dense subset but on all of Hilbert space. Is it possible that such a map is invertible?
| https://mathoverflow.net/users/56920 | Invertible unbounded linear maps defined on a Hilbert space | Yes.
Consider the space $E=\ell^2(\mathbb N)$.
($0\notin\mathbb N$.)
It has an orthonormal Hilbert basis $A=\{e\_i;i\in\mathbb N\}$, where $e\_i(n)=\delta\_{in}$.
Extend this to a Hamel basis $B$ of $E$ (this is where the axiom of choice comes in).
Then define $L:E\to E$ by $L(e\_i)=ie\_i$ and $L(x)=x$ for $x\in B\setm... | 5 | https://mathoverflow.net/users/55893 | 178788 | 89,901 |
https://mathoverflow.net/questions/177029 | 2 | Let $R = \mathbb{C}[x\_1,…,x\_n]$, let $J\subset R$ be a graded ideal, and consider the initial monomial ideal $\operatorname{in}(J)$ with respect to some term order. Suppose that we are given a linear subspace $V \subset \mathbb{C}\{x\_1,\ldots,x\_n\}$ such that $R/J$ and $R/\operatorname{in}(J)$ are both free of fini... | https://mathoverflow.net/users/10273 | Degenerations and spanning monomials | The answer is no.
Let $R = \mathbb{C}[x,y,z]$, $J = \langle y+z, (x+y)z\rangle$, $V = \mathbb{C}\{x+y+z\}$, and $S = \{1,x\}$. Then $\operatorname{in}(J) = \langle y, xz\rangle$, and $S$ is a $\operatorname{Sym}(V)$-basis for $R/\operatorname{in}(J)$. The algebra $R/J$ is also free over $\operatorname{Sym}(V)$, but $... | 1 | https://mathoverflow.net/users/10273 | 178792 | 89,902 |
https://mathoverflow.net/questions/178768 | 5 | In the paper **"Triangulations of Surfaces"** Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes each boundary curve to have exactly one end point and constructed a flow (Hatcher flow) to show it is contractible. ... | https://mathoverflow.net/users/9485 | A query about Hatcher flow on arc complex | The flow consists of a sequence of surgeries using one fixed oriented arc $\alpha$ to cut (and isotope) all other arcs $\beta$ to remove one point of $\alpha\cap\beta$ at a time. Each surgery cuts one arc into two arcs. Say $\beta$ is cut into $\beta'$ and $\beta''$. It can happen that one of these two arcs, say $\beta... | 10 | https://mathoverflow.net/users/23571 | 178793 | 89,903 |
https://mathoverflow.net/questions/178640 | 14 | I had asked a [question](https://math.stackexchange.com/questions/896596/learning-roadmap-to-topological-quantum-field-theories-from-a-mathematics-perspe) on math.stackexchange but did not receive any answers. I hope that this question is appropriate for this website as it is about an advanced subject. Hence I am posti... | https://mathoverflow.net/users/40386 | Learning roadmap to TQFT from a mathematics perspective | A good place to start is in low dimensions and there a good introduction can be found in
J. Kock, 2003, Frobenius Algebras and 2-D Topological Quantum Field Theories, number 59 in London Mathematical Society Student Texts, Cambridge U.P., Cambridge.
There are also old notes of Quinn that take a very neat homotopy t... | 6 | https://mathoverflow.net/users/3502 | 178797 | 89,905 |
https://mathoverflow.net/questions/178790 | 20 | A conjecture by the late Romanian mathematician Alexandru Lupas.
Posted in [sci.math](http://mathforum.org/kb/message.jspa?messageID=3704516) in 2005, but no proof was found.
Physicist [Alan Sokal](http://www.physics.nyu.edu/faculty/sokal/) just reminded me of it, saying it was related to something he is working on.
... | https://mathoverflow.net/users/454 | A conjectured formula for Apéry numbers | Here as an approach.
It is known that:
$P\_{n}(x) = \sum\_{k=0}^{n}\binom{n}{k}\binom{-n-1}{k}\left(\frac{1-x}{2}\right)^{k}$ (see for example <http://en.wikipedia.org/wiki/Legendre_polynomials>)
Therefore these expressions are equal
$$
\sum\_{k=0}^{n}\binom{n}{k}\binom{n+k}{k}\sum\_{j=0}^{k}\binom{k}{j}^{3}=\sum\_{k... | 23 | https://mathoverflow.net/users/50901 | 178799 | 89,906 |
https://mathoverflow.net/questions/178778 | 21 | The statement that surjective maps are epimorphisms in the category of sets can be shown in a constructive way.
What about the inverse?
Is it possible to show that every epimorphism in the category of sets is surjective without reverting to a proof by contradiction / negation?
| https://mathoverflow.net/users/1261 | In the category of sets epimorphisms are surjective - Constructive Proof? | **Theorem:** Every epi is surjective.
*Proof.*
Let $h : A \to B$ be an epimorphism. We define maps $f, g : B \to \mathcal{P}(B)$ by
\begin{align\*}
f(b) &= \{b\} \cap \mathrm{im}(h)\\
g(b) &= \{b\}
\end{align\*}
where we recall that $\mathrm{im}(h) = \{b \in B \mid \exists a \in A \,.\, h(a) = b\}$.
For every $a ... | 22 | https://mathoverflow.net/users/1176 | 178804 | 89,909 |
https://mathoverflow.net/questions/178747 | 5 | How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical relation signs; I thus disregard solutions that appeal to selectors as the epsilon operator. My motivation is to extend an i... | https://mathoverflow.net/users/37385 | How short can we state the Axiom of Choice? | The following paper by Kurt Maes is focused on a version of the question at hand here, namely, finding an equivalent formulation of AC in the language of set theory using the fewest number of quantifiers, rather than merely the shortest length. In his main result, Maes finds a 5-quantifier assertion equivalent
to the a... | 20 | https://mathoverflow.net/users/1946 | 178808 | 89,910 |
https://mathoverflow.net/questions/178762 | 1 | Let $Q$ be a anisotropic quadric of dimension $d$ over $k$.
We work in the category of effective Chow-Motives over $k$.
Let $T$ be the Tate-Motive.
For a motive $M$ we write $M(l)$ for its $l$-th Tate-Twist.
Assume we have a motivic decomposition $M(Q) = A \oplus B$.
Assume further that over the algebraic closure of ... | https://mathoverflow.net/users/51251 | Dimension of binary motives of a quadric | The binary motive theorem states that a binary summand of the motive of a quadric has dimension $2^r-1$ for some $r$. This should be Theorem 6.1 in:
O. Izhboldin and A. Vishik. Quadratic forms with absolutely maximal splitting. Contemp. Math. 272 (2000), 103-125.
| 4 | https://mathoverflow.net/users/50846 | 178809 | 89,911 |
https://mathoverflow.net/questions/178811 | 9 | There seems to be a lot of work on the upper bound for the [irrationality measure](http://mathworld.wolfram.com/IrrationalityMeasure.html) of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. [Looks like](https://mathoverflow.net/questions/134066/numbers-with-known-finite-irratio... | https://mathoverflow.net/users/51484 | Lower bound on the irrationality measure of $\pi$ | The expected value is $2$, simply because almost all real numbers have irrationality measure $2$. Note that this is "almost all" in the sense of measure theory. In the sense of topological category, it's the other way around: the set of reals of measure $\infty$ is [residual](http://en.wikipedia.org/wiki/Meagre_set). A... | 12 | https://mathoverflow.net/users/17064 | 178818 | 89,917 |
https://mathoverflow.net/questions/178027 | 2 | It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} \Xi(t)^{-1}$ solves the Euler Poincaré equations:
$\frac{d}{dt}\frac{\partial \ell}{\partial \xi} = -ad^\*\_{\xi} \left(\frac{\p... | https://mathoverflow.net/users/41654 | Euler-Poincaré equations with constraints | Generally no. The Euler-Poincaré equation derives all of its structure from the Lie bracket, and an arbitrary constraint does not respect this structure. The most simple counter-example is probably a subspace constraint. Let $V \subset \mathfrak{g}$ be a subspace which is not a sub-algebra. This subspace generates a co... | 4 | https://mathoverflow.net/users/16852 | 178825 | 89,919 |
https://mathoverflow.net/questions/178759 | 1 | Given two spectra $A$ and $B$, the set $[A,B]$ of homotopy classes of maps from $A$ to $B$ forms an abelian group. Can the dual abelian group $\text{Hom}([A,B],\mathbb{Q}/\mathbb{Z})$ be expressed as a group of homotopy classes of maps between spectra?
The *Brown-Comenetz dual* $I\_{\mathbb{Q}/\mathbb{Z}}E$ of a spec... | https://mathoverflow.net/users/51107 | pontryagin dual and maps between spectra | I think this is the closest thing there is to what you are looking for. Let $G$ be any injective (i.e., divisible) abelian group. Then there is a spectrum $I\_G$ representing the cohomology theory $X\mapsto \operatorname{Hom}(\pi^s\_\*(X),G)$. For any $A$ and $B$, we can then describe the dual $\operatorname{Hom}([A,B]... | 5 | https://mathoverflow.net/users/75 | 178826 | 89,920 |
https://mathoverflow.net/questions/178761 | 6 | Is it true that given a smooth manifold M (with or without boundary), a "generic" metric g on M does not possess any non-trivial (non-constant) first integral for the geodesic flow induced by g on the unit sphere bundle?
What if we restrict ourselves to considering the first integrals of the geodesic flow which is a ... | https://mathoverflow.net/users/57241 | Generic absence of non-trivial first integrals of geodesic flows | In the case the integrals are polynomial in momenta, a generic metric does not poses those (except trivial integrals such as the energy and polynomial functions of the energy). This is a local statement.
The formal theorem, which also explains the notion ``generic'' in this case, could be formulated here as following... | 4 | https://mathoverflow.net/users/14515 | 178836 | 89,925 |
https://mathoverflow.net/questions/178830 | 2 | Ok, I am reading Fillipov book on discontinuous right hand side differential equations (the red book).
He states the next lemma:
"
Let the function $f(t,x)$ satisfy the Caratheodory conditions and let the function $x(t)$ for $a\leq t \leq b$ be measurable.
Then the composite function $f(x(t),t)$ is summable."
Where... | https://mathoverflow.net/users/13904 | Caratheodory equations | You can find that in §10, XVIII of Walter,W.: Ordinary Differential Equations, volume 182 of Graduate Texts in Mathematics. Springer,
New York, 1998. Translated from the sixth German (1996) edition by Russell Thompson, Readings in Mathematics.
| 3 | https://mathoverflow.net/users/47189 | 178837 | 89,926 |
https://mathoverflow.net/questions/178531 | 8 | If $R$ is a (commutative) ring and $P$ is a projective $R$-module, then every localization of $P$ at a prime of $R$ is free by Kaplansky's theorem, and has a well-defined rank. If these ranks are all finite, must $P$ be finitely generated?
The answer is no: one can take $R=k[x\_1,x\_2,\dots]/(x\_1^2-x\_1,x\_2^2-x\_2... | https://mathoverflow.net/users/1474 | Is a projective module of constant finite rank finitely generated? | For any commutative ring $R$ with 1, any projective $R$-module of constant finite rank is finitely generated. This is the content of Exercises I.2.13 and I.2.14 of Weibel's $K$-book.
The argument goes as follows (all the relevant hints are in Weibel's book), I hope I did not introduce any tacit finiteness assumptions.... | 7 | https://mathoverflow.net/users/50846 | 178842 | 89,928 |
https://mathoverflow.net/questions/178794 | 5 | I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, *i.e.* at the coordinate axes. The topological description of this blow up is
$$
\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)\_1\sim(x^{-1},y^{-1})\_2\right)\_{x,... | https://mathoverflow.net/users/57252 | Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$ | Here is a different way of computing the cohomology ring (which starts directly from the fan). The space is constructed by gluing to copies of $\mathbb{C}^2$ along an inclusion of $(\mathbb{C}^\times)^2$. The Mayer-Vietoris sequence associated to that would look like
$$
\cdots\to H^i(X\_\Sigma)\to H^i(\operatorname{pt}... | 8 | https://mathoverflow.net/users/50846 | 178847 | 89,930 |
https://mathoverflow.net/questions/178845 | 6 | Let $H$ be a separable Hilbert space and $A$ is an invertible bounded operator on $H$. Can we approximate $A$ with an invertible operator $B$ such that $sp(B)$ is a countable set?
**Motivation:**
If the answer is yes, this would give's us an alternative proof of connected ness of $GL(H)$. This alternative proof is ... | https://mathoverflow.net/users/36688 | Invertible operator with countable spectrum | I think the answer is "no"; here is a sketch of an argument though I will have to go back and check the details. First, there is nothing special about invertibility here; if every invertible operator is approximable by operators with countable spectrum, then every operator is, just by translation. I claim the unilatera... | 6 | https://mathoverflow.net/users/13360 | 178858 | 89,936 |
https://mathoverflow.net/questions/178802 | 1 | This is my question:
Given a projective noetherian scheme $X$, the structural sheaf $\mathcal{O}\_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable locally free sheaves (with fixed Hilbert polynomial) is parametrized by a subset $S$ of the moduli space $M$ of stable... | https://mathoverflow.net/users/54269 | Open subset of the moduli space of stable sheaves on a noetherian scheme | I will answer the second question.
Let $X$ be any smooth projective surface and $P \in X$ a point. Let $F = O\_X \oplus I\_P$. Then $F$ is torsion free but not locally free. But its general deformation is locally free. The reason is that $\dim Ext^1(F,F) = 3$, so a generic deformation of $F$ should erase the singular... | 1 | https://mathoverflow.net/users/4428 | 178874 | 89,943 |
https://mathoverflow.net/questions/178659 | 18 | Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't rational to be decomposable in the way the square knot is decomposable. But there are more, and I'd like to see a list of the s... | https://mathoverflow.net/users/5010 | Classification of tangles? | As Sam Nead points out, there is a relationship between spatial graphs and tangles obtained by drilling out one edge of a spatial graph having two vertices or drilling out the vertex of a spatial graph having a single vertex. There have been a few people working on building tables of spatial graphs. Here are a few refe... | 4 | https://mathoverflow.net/users/30679 | 178880 | 89,945 |
https://mathoverflow.net/questions/178872 | 13 | If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that
$$ \pi\_0(\mathcal{F}) = \mathbb{Z}\, , $$
i.e. the connected components are classified by the index of the Fredholm operator.
But what is about higher homotopy groups? What is k... | https://mathoverflow.net/users/56995 | Homotopy groups of Fredholm operators | EDIT (much later): My answer below is not quite correct, as pointed out by Dan Ramras, because I ignored the importance of base points in the definition of $\pi\_n$. See Ramras' answer for the needed corrections.
---
It is a theorem (due I think to Atiyah) that $\mathcal{F}$ is the classifying space for the topol... | 21 | https://mathoverflow.net/users/4362 | 178884 | 89,949 |
https://mathoverflow.net/questions/178875 | 1 | In the paper [FB] ([ArXiv](http://arxiv.org/abs/0805.0407), [J. Phys. A](http://iopscience.iop.org/1751-8121/41/37/372001)), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of the minimum in a suitable asymptot... | https://mathoverflow.net/users/30264 | Minimum of Random Energy Model (REM) with logarithmically correlated potential | The Bessel-function distribution you quote is the convolution of two [Gumbel distributions](https://en.wikipedia.org/wiki/Gumbel_distribution). This form for the extreme value statistics was rigorously derived for a broad class of stochastic processes by Subag and Zeitouni, [Freezing and decorated Poisson point process... | 3 | https://mathoverflow.net/users/11260 | 178892 | 89,952 |
https://mathoverflow.net/questions/178831 | 15 | (this is basically a repost of [a question](https://math.stackexchange.com/q/526100/19661) I asked at M.SE last year)
Is there an explicit real algebraic number (such that we can write its minimal polynomial and a rational isolating interval) that cannot be expressed as a combination of the constants $\pi, e,$ intege... | https://mathoverflow.net/users/9550 | Is there an algebraic number that cannot be expressed using only elementary functions? | I addressed this exact question in my *American Mathematical Monthly* paper,
[What is a closed-form number?](http://alum.mit.edu/www/tchow/closedform.pdf) Corollary 1 in that paper states that if [Schanuel's conjecture](http://en.wikipedia.org/wiki/Schanuel%27s_conjecture) holds, then the EL numbers (i.e., the numbers ... | 23 | https://mathoverflow.net/users/3106 | 178893 | 89,953 |
https://mathoverflow.net/questions/178879 | 3 | This question is related to [this](https://mathoverflow.net/questions/19079/what-is-the-status-of-the-gauss-circle-problem) and [this](https://mathoverflow.net/questions/53388/recent-results-on-the-gauss-circle-problem "this") ones. The Gauss Circle problem asks for the number $N(r)$ of integer points within a sphere o... | https://mathoverflow.net/users/44293 | Practical use of estimates for the Gauss Circle Problem | I wouldn't expect any of these bounds to give really superb estimates of the last time your ratio achieves a certain record because $E(r)$ and your ratio undergo rapid fluctuations. Also, the exciting places are unlikely to happen at integers. The local minima of $E(r)$ occur at the places where $r=\sqrt{a^2+b^2}.$ The... | 3 | https://mathoverflow.net/users/8008 | 178907 | 89,959 |
https://mathoverflow.net/questions/178906 | 6 | If $p \geq 5$ is a prime, are there any integers $x, y, z > p$ such that
$(x, y) = 1$
and
$$x^{p} - 4y^{p} = z^{2}$$
| https://mathoverflow.net/users/51615 | The Diophantine equation $x^p - 4y^p = z^2$ | See Theorem 1.2 of the paper by [Bennett and Skinner](https://www.math.ubc.ca/~bennett/BS.pdf), which settles the problem for $p\ge 7$ (take there $C=1$ and $\alpha\_0=2$). Note that the Bennett-Skinner results are more general. (Earlier work of [Darmon and Granville](http://www.math.mcgill.ca/darmon/pub/Articles/Resea... | 9 | https://mathoverflow.net/users/38624 | 178909 | 89,961 |
https://mathoverflow.net/questions/105835 | 7 | Has anyone defined what it means for a modular operad to be Koszul, or what the Koszul dual of a modular operad is? In particular, is it meaningful to say that a modular operad is quadratic? Merkulov, Markl and Shadrin (Wheeled PROPs, graph complexes and the master equation) give a definition for wheeled properads, whi... | https://mathoverflow.net/users/1310 | Koszul duality for modular operads | Here are two answers to this question based on different interpretations of the meaning of ``Koszul.''
One reasonable way to define things is that Koszul duality depends not only on algebraic data (operad, modular operad, algebra, whatever) but also on a presentation of the algebraic data. This is both elegant and a ... | 5 | https://mathoverflow.net/users/3075 | 178911 | 89,963 |
https://mathoverflow.net/questions/178473 | 9 | The dependent choice principle ${\rm DC}\_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there is a function $f:\kappa\to S$ such that for every $\alpha<\kappa$, $f\upharpoonright\alpha R f(\alpha)$. The axiom of cho... | https://mathoverflow.net/users/5984 | Relationship between fragments of the axiom of choice and the dependent choice principles | The idea is to mimic the permutation models as given in Jech. One can then ask, "Well, in Jech he chooses some set of objects in the full universe, and shows it has a support. But in forcing we don't have a simple access to names like that, since they might not be "sufficiently determined" for us to collect them into a... | 7 | https://mathoverflow.net/users/7206 | 178914 | 89,965 |
https://mathoverflow.net/questions/178916 | 4 | On page 4 of [this paper](http://www.math.uvic.ca/~ibrahim/Ibra-Maj-Mas-cpam.pdf) of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to
$\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), \partial\_tu(0,x)=g(x)\end{array}\right.$
enjoy the estimate
$||u||\_{L\_T^4(C^{1/4}(... | https://mathoverflow.net/users/57302 | Besov Characterization of Strichartz Estimate. | Look at Figure 3 in the paper by Ginibre-Velo, point $C\_2$ is the estimate of $\|u\|\_{L^4\dot B^{1/4}\_{\infty,2}}$ in terms of the $L^1L^2$ norm of $F$, and one has $\dot B^{1/4}\_{\infty,2}\hookrightarrow \dot B^{1/4}\_{\infty,\infty}$.
| 2 | https://mathoverflow.net/users/7294 | 178921 | 89,968 |
https://mathoverflow.net/questions/178444 | 8 | Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^\*$-algebraist, I prefer to define $X^{\mathcal U}$ as the compact Hausdorff space satisfying
\begin{align\*}
C(X^{\mathcal U},\mathbb C) &\... | https://mathoverflow.net/users/22052 | Ultracoproducts and Cartesian products | Aaron, the answer is big **no** if $X$ and $Y$ are infinite (no metrasibility assumptions required to show that the answer is always negative).
Indeed, $C((X\times Y)^{\mathcal{U}})$ is a [Grothendieck space](http://en.wikipedia.org/wiki/Grothendieck_space) as an ultraproduct of $\mathscr{L}\_\infty$-spaces over a c... | 7 | https://mathoverflow.net/users/15129 | 178923 | 89,969 |
https://mathoverflow.net/questions/178922 | 0 | $ZF$ define membership by conditions demanding the existence of some constructable right-side-terms $M $ ($x \in M$). Is it meaningsful to ask for a categorical axiom system here? Shouldn't it be prooved, that any two relations $\epsilon$ and $\varepsilon$ which satisfying the axioms allways fullfill: $\forall x \foral... | https://mathoverflow.net/users/57255 | Is any axiom system for sets categorical? | Of course not, since you can always take a permutation of $M$ and redefine $\in$ as the transport of structure defined by the permutation.
But even if you mean "up to isomorphism", the answer is still negative. Since given a model of set theory, by usual compactness+Lowenheim-Skolem arguments we can produce a non-iso... | 3 | https://mathoverflow.net/users/7206 | 178926 | 89,971 |
https://mathoverflow.net/questions/163811 | 17 | With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K\_i(R)$ are completely known *for all* $i\geq 0$?
Here, by "trivial examples" I mean for instance examples like $R = \mathbb{F}\_q[t]$, or other examples that can easily be deduced from Quillen'... | https://mathoverflow.net/users/1437 | Besides F_q, for which rings R is K_i(R) completely known? | For commutative rings, all the K-theory computations I am aware of are discussed in Chapter VI of Weibel's $K$-book (actually with the exception of finite fields).
**Function rings:** I would say that for smooth affine curves over finite fields, all the $K$-groups have been computed "essentially": Theorems VI.6.4 an... | 12 | https://mathoverflow.net/users/50846 | 178943 | 89,978 |
https://mathoverflow.net/questions/178939 | 1 | Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the Hilbert scheme of $X$. Thus $C$ defines the point $[C]\in Hilb(X)$.
In my situation one may assume that $X$ and $C$ are smoo... | https://mathoverflow.net/users/16183 | Hilbert scheme of an infinitesimal neighborhood of a subvariety | That already fails when $n$ equals $2$, $X$ equals $\mathbb{P}^3$ and $C$ is a line in $\mathbb{P}^3$. Choose homogeneous coordinates $[y\_0,y\_1,y\_2,y\_3]$ on $\mathbb{P}^3$ so that $C$ is $Z(y\_2,y\_3)$. Then an affine neighborhood $U$ of $[C]$ in $\text{Hilb}(X)$ is the affine $4$-space with affine coordinates $(a\... | 2 | https://mathoverflow.net/users/13265 | 178949 | 89,980 |
https://mathoverflow.net/questions/178341 | 0 | The residual life time distribution of a random variable $X$ with distribution function $F$ is given by the formula
\begin{equation}R(t)=P[X\_\text{res}\leq t] = 1-\frac{1}{\mathbb{E}[X]}\int\_{y=0}^\infty(1-F(t+y))\text{d}y. \end{equation}
However this formule only works if $\mathbb{E}[X]<\infty$. What happens to the ... | https://mathoverflow.net/users/57020 | Residual lifetime of heavy-tailed random variable | A partial answer is that, when $X$ is not integrable, $\hat X\_x$ the residual lifetime of $\min\{X,x\}$ converges to infinity in distribution when $x\to\infty$. Since the residual lifetime $\hat X$ of $X$ should logically be bounded below by $\hat X\_x$ for every $x$, this remark points to a "simply cease to exist" ve... | 0 | https://mathoverflow.net/users/4661 | 178950 | 89,981 |
https://mathoverflow.net/questions/178952 | 6 | Suppose that $X$ and $Y$ are Banach spaces. Is $\{f\in B(X,Y):f\ \text{has a left inverse}\}$ an open subset of $B(X,Y)$?
| https://mathoverflow.net/users/23240 | Left invertible operators of $B(X,Y)$ | Yes.
Let $U=\{A\in B(X,Y);\text{there is }B\in B(Y,X)\text{ so that }BA=I\_X\}$.
(You probably want to have the left inverse in $B(Y,X)$. Otherwise you can get counterexamples from compact injections.)
Take any $A\in U$.
Then there is $B\in B(Y,X)$ so that $BA=I\_X$.
Suppose $C\in B(X,Y)$ satisfies $\|C\|\leq\frac12\... | 11 | https://mathoverflow.net/users/55893 | 178957 | 89,983 |
https://mathoverflow.net/questions/178951 | 5 | Is it true that there exists $n \in \mathbb{N}$ with arbitrarily many prime factors such that $n$ divides $F\_n$, where $F\_n$ represents the n-th **Fibonacci number**?
| https://mathoverflow.net/users/57067 | Arbitrarily large $n$ divides $F_n$ | Note that (see [here](http://oeis.org/A023172)):
* If $a$ and $b$ are in your sequence, then so is $\text{lcm}(a,b)$;
* If $n$ is in the sequence, then so is $F\_n$.
Now take any $n>12$ that belongs to the sequence, then $a\_1=\text{lcm}(n,F\_n)$, $a\_2=\text{lcm}(n,F\_n,F\_{F\_n})$, and so on. Each of $n$, $F\_n$,... | 14 | https://mathoverflow.net/users/nan | 178959 | 89,984 |
https://mathoverflow.net/questions/178958 | 2 | Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$.
>
> **Question :** Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in $A$, we have $\deg\_\mathcal{L} C\leq r$ ?
>
>
>
Thanks a lot.
| https://mathoverflow.net/users/57319 | Degree of a smooth curve in an abelian variety | Well, this is surely false for *elliptic* curves, i.e. when $g=1$.
What is true is that, for any $t \in \mathbb{N}$, the number of elliptic curves in $A$ with $\deg \_{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exist elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.
T... | 4 | https://mathoverflow.net/users/7460 | 178961 | 89,986 |
https://mathoverflow.net/questions/178903 | -1 | Do we have some good examples of local subrings of number fields which are not valuation rings?
Do we have an easy criterion for determining whether a local subring of a number field is a valuation ring? There are certainly many such criteria on Wikipedia and Atiyah-Macdonald. What I want to ask for is a criterion wh... | https://mathoverflow.net/users/2701 | When is a local subring of a number field a valuation ring? | A localization of a Dedekind ring at a (non-zero) prime is a Dedekind ring with a unique non-zero prime, hence finitely-many, hence is a principal ideal domain with a unique non-zero prime...
| 1 | https://mathoverflow.net/users/15629 | 178962 | 89,987 |
https://mathoverflow.net/questions/178320 | 10 |
>
> UPDATE: Because I was hoping that state the question as concisely as
> possible, the original post did not include a precise definition of
> arithmetic 3-manifold only a reference to Maclachlan and Reid's book where it is
> defined. However, this lead to an ambiguity that caused conflicting
> answers. With th... | https://mathoverflow.net/users/27453 | Heegaard genera of arithmetic 3-manifolds | The answer is no, for trivial reasons: if a hyperbolic 3-manifold fibers over $S^1$ with fiber of genus $g$, then the Heegaard genus is bounded by $2g+1$. However, cyclic covers of this manifold dual to the fiber will retain this property, so there are infinitely many manifolds of bounded Heegaard genus.
One may obtain... | 12 | https://mathoverflow.net/users/1345 | 178969 | 89,991 |
https://mathoverflow.net/questions/178970 | 0 | Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J\_{r\_1},\cdots,J\_{r\_s})\in M\_n(K)$ where $J\_k$ is the nilpotent Jordan block of dimension $k$ and with $r\_1\geq\cdots\geq r\_s$. Let $S(A)=\{P^{-1}AP|P\in GL\_n(K)\}$ be the simila... | https://mathoverflow.net/users/9091 | Dimension of a similarity class | There is an explicit formula. Let $r'\_1\geq \ldots \geq r'\_t$ be the transpose partition of $[r\_1,\ldots ,r\_s]$ (obtained by exchanging rows and columns in the Young diagram). Then $\ d(A)=n^2-\sum r'\_i{}^2$ . See for instance Collingwood-McGovern, *Nilpotent orbits in semisimple Lie algebras*, Cor. 7.2.4.
| 4 | https://mathoverflow.net/users/40297 | 178976 | 89,993 |
https://mathoverflow.net/questions/178964 | 6 | Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a *discrete* normal distribution over $\mathbb{Z}$, obtained by sampling from $\mathcal{N}(0, \sigma^2)$ and rounding the result to the ne... | https://mathoverflow.net/users/57321 | Estimating the variance of a discrete normal distribution | Using the Poisson summation formula, I find that the variance is
$$ \sigma^2 + \dfrac{1}{12} + \sum\_{k=1}^\infty (-1)^k e^{-2\sigma^2 k^2 \pi^2} (4 \sigma^2 + 1/(\pi^2 k^2)) $$
If $\sigma$ is not too small, the series converges quite rapidly.
| 6 | https://mathoverflow.net/users/13650 | 178981 | 89,996 |
https://mathoverflow.net/questions/178999 | 4 | I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint").
I know that all the eigenvalues of matrices in this class have (algebraic) multiplicity one or two. Some very interesting phenomena happens when all the eigenvalues have mul... | https://mathoverflow.net/users/20838 | Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two? | Assuming it is feasible to compute the characteristic polynomial, $p(x)$, in your situation, (which can certainly be done in principle if you know the matrix entries) there is a simple strategy. Given the information you already have, all roots have multiplicity two if and only if ${\rm gcd}(p(x),p^{\prime}(x))$ has de... | 9 | https://mathoverflow.net/users/14450 | 179008 | 90,009 |
https://mathoverflow.net/questions/179017 | 28 | Fix a field $k$. The Grothendieck ring $K\_0(\mathrm{Var}\_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor relation. The notation suggests that there might be higher $K$-groups $K\_i(\mathrm{Var}\_k)$ as well, but naive attempt ... | https://mathoverflow.net/users/57335 | Is there a higher Grothendieck ring of varieties? | Torsten Ekedahl proposed a definition of higher Grothendieck groups of varieties. Unfortunately it seems that he never wrote anything down on this topic before passing away. Torsten had quite a large number of unfinished mathematical manuscripts and projects. I don't know what happened to them, although surely someone ... | 34 | https://mathoverflow.net/users/1310 | 179022 | 90,012 |
https://mathoverflow.net/questions/179006 | 1 | Let $V$ be a vector space over $\mathbb{C}$.
Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^\*$. If the coordinate ring $\mathbb{C}[X]$ is arithmetically Cohen-Macaulay (aCM) is the same true for $\mathbb{C}[X^\vee]$?
An example where thi... | https://mathoverflow.net/users/17386 | Does projective duality preserve arithmetic-Cohen-Macaulay-ness? | Not true. Start from a variety $Y\subset \mathbb{P}V^\*$ which is *not* aCM -- for instance a quartic rational curve in $\mathbb{P}^3$. In most cases, in particular in the example, its dual $X:=Y^\vee$ is a hypersurface in $\mathbb{P}V$, hence is aCM. But $X^\vee=Y$ is not.
| 1 | https://mathoverflow.net/users/40297 | 179023 | 90,013 |
https://mathoverflow.net/questions/179021 | 2 | Let $F$ be a number field and $v$ a finite place of $F$.
Let $\chi\_v$ be a unramified unitary character of $F\_v$.
Then we define local L-function $L\_v(s,\chi\_v):=\frac{1}{1-\chi\_v(\omega)q^{-s}}$ where $\omega$ is a uniformizer of $F\_v$ for $s>1$.
Then I know that it has a unique meromorphic continuation to... | https://mathoverflow.net/users/29422 | On the pole of local L-function | No, you are not right. The local $L$-function $L(s,\chi\_v)=(1-\chi\_v(\omega)q^{-s})^{-1}$ has infinitely many poles, namely the solutions of the equation $q^s=\chi\_v(\omega)$. All these poles are simple of course. Also, it has no zeros. The local $L$-function can be more complicated, but it is usually a product of t... | 8 | https://mathoverflow.net/users/11919 | 179030 | 90,017 |
https://mathoverflow.net/questions/179035 | 3 | Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. **My question is whether this identity is true, known, and whether it has a direct proof.** It might also happen that this identity is simpler than I think.
Let $d... | https://mathoverflow.net/users/16183 | An identity for elementary symmetric functions | Viete formula gives
$\sum\_{k=0}^{d} t^k e\_{d-k}\left(\alpha\_0,\ldots,\alpha\_{d-1}\right)=(t+\alpha\_1)\ldots(t+\alpha\_d).$
Therefore (substituing $t=-(x-d)^2$ and $\alpha\_j=(x-j)^2$), LHS is equal to
$\prod\_{j=0}^{d-1} \left(-(x-d)^2+(x-j)^2\right)=
\prod\_{j=0}^{d-1} (2x-j-d)(d-j)=d!\prod\_{k=d}^{2d-1} (2x-k... | 14 | https://mathoverflow.net/users/49822 | 179036 | 90,020 |
https://mathoverflow.net/questions/179050 | 6 | It appears that a variation on [Bulgarian solitare](http://en.wikipedia.org/wiki/Bulgarian_solitaire) has a fixed point regardless
of the starting $n$.
For example, let $n=69$, and consider this partition:
$$
(8,8,7,7,5,5,5,5,5,4,3,3,2,2)
$$
In Bulgarian solitare, $1$ would be removed from each "stack/pile" to form a... | https://mathoverflow.net/users/6094 | A variation on Bulgarian solitare | Yes, that is what happens if there are at least $k$ stacks at each step.
It is quite clear that all stacks will be eventually bounded by $k$, since either the highest number of cards or the number of stacks with the highest number of cards reduces at every iteration if there is a stack of strictly more than $k$ cards... | 4 | https://mathoverflow.net/users/55893 | 179053 | 90,025 |
https://mathoverflow.net/questions/178955 | 4 | I am studying the paper "Laminations, trees, and irreducible automorphisms of free groups" of Bestvina, Feighn and Handel. But I found a note in the paper "Stabilisers of $\mathbb{R}$-trees with isometric $F\_n$ actions" of Kapovich and Lustig that the proof of the proposition 2.6 of the first paper has a gap.
The stat... | https://mathoverflow.net/users/42001 | Stabiliser of the lamination of a free group - Invariant subgraphs | The gap is easily fixable in the context of the paper. Let me explain the fix after first explaining the critique of Kapovich and Lustig.
The first paragraph of the proof starts by choosing a leaf $\ell$ of $\Lambda$, and choosing a proper $h$-invariant subgraph $H\_0$, and then noting that $\ell$ is a concatenation... | 8 | https://mathoverflow.net/users/20787 | 179059 | 90,028 |
https://mathoverflow.net/questions/178967 | 5 | Let $X$ be an arithmetic surface over $\mathbb{Z}$, that is we have $\pi: X\rightarrow Spec(\mathbb{Z})$, $X$ is integral, two-dimensional and regular and $\pi$ is projective and flat.
What is known about the Brauer group of $X$?
Is there a relation between the Brauer group of $X$ and the Brauer group of the gener... | https://mathoverflow.net/users/43247 | What is known about the Brauer group of an arithmetic surface? | As Jason says you should read "Dix Exposes". Part 3 of "Le Groupe de Brauer III" uses a result of Artin to show that $R^i\pi\_\*G\_m =0$ for $i\geq2$ for surfaces such as you are interested in. The issue, of course, is dealing with $p$ torsion when the residue field of a valuation ring has characteristic $p$. So the Le... | 7 | https://mathoverflow.net/users/20674 | 179071 | 90,031 |
https://mathoverflow.net/questions/178947 | 3 | I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial bundles, i.e., where the Chern classes as well as Atiyah-Rees-$\alpha$-invariant are trivial.
A precise question woul... | https://mathoverflow.net/users/50846 | Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$ | As I mentioned earlier, Peskine (and possibly Kollar too) asked whether given a family of smooth curves in 3-space with general member a complete intersection, is the special member also a complete intersection. To the best of my knowledge, the answer is not known (over complex numbers). Under the above hypothesis, it ... | 5 | https://mathoverflow.net/users/9502 | 179072 | 90,032 |
https://mathoverflow.net/questions/179052 | 1 | I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface.
It is a statement about the representability of some functor. It is stated on page 108 and says the following. Let $k$ be a field (does anything change if we consider a noetherian scheme instead?) Let $G$ be a scheme o... | https://mathoverflow.net/users/57325 | Is Mumford's statement about the representability of some functor wrong? | The proof works *slightly* differently; the $G\_0$ one needs may be not the one you name but a **retract** of it. In other words, the map you ask about may be nonsurjective but it has a retraction.
The assumption can be equivalently formulated as follows: for any $\alpha:h\_S\to B$, the pullback of $i:A\hookrightarr... | 3 | https://mathoverflow.net/users/41291 | 179076 | 90,033 |
https://mathoverflow.net/questions/179031 | 5 | I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this:
$G(s) = e^{a(s-1)^2}=\sum s^np(n)$
I need first to do Maclaurin expansion of the exponential and then get the $n$th order term for $p(n)$.
My first thinking was it would be simple to calculate the derivatives. Bu... | https://mathoverflow.net/users/57344 | An interesting calculation of derivative | **Robert Israel** noted that $p(n)$ is a $e^a$ times a polynomial of degree $n$
in $a$ with a zero of order $\lceil n/2 \rceil$ at $a=0$.
We express this polynomial in terms of a
[Hermite
polynomial](http://en.wikipedia.org/wiki/Hermite_polynomials#Generating_function) $H\_n$ evaluated at an imaginary argument $\alpha ... | 10 | https://mathoverflow.net/users/14830 | 179090 | 90,034 |
https://mathoverflow.net/questions/179070 | 6 | Is every pseudo-automorphism (self-birational map which does not contract any hypersurface) of a smooth Fano variety of Picard rank $1$ equal to a biregular automorphism?
Remark: For $\mathbb{P}^n$, the answer is yes, and easy: every birational map of degree $>1$ contracts a hypersurface, given by its Jacobian. Same ... | https://mathoverflow.net/users/23758 | Pseudo-automorphisms on Fano varieties | You don't need your variety, say $X$, to be Fano, only $\mathrm{Pic}(X)=\mathbb{Z}$. A pseudo-automorphism $u$ of $X$ induces an automorphism of $\mathrm{Pic}(X)$, which must be the identity. Let $L$ be a very ample line bundle on $X$; since $u^\*L\cong L$, $u$ induces an automorphism of $H^0(X,L)$ (here you use Hartog... | 13 | https://mathoverflow.net/users/40297 | 179091 | 90,035 |
https://mathoverflow.net/questions/179048 | 2 | This is a cross-post from [Math.SE](https://math.stackexchange.com/questions/893974/odd-form-of-controlling-derivatves) since the question got nothing (but upvotes) even after offering a decent bounty. If it is too trivial or in other ways not suited for this site, please let me know and I'll delete it.
In Muscalu, S... | https://mathoverflow.net/users/45119 | Controling mixed derivatives | As stated, the inequality indeed fails. But the final estimate (11.27) in the book is essentially correct. From the product rule the derivative
$$ (\frac{d}{dt})^n (t^z (1 - \psi(t\tau)) \chi(t) $$
contains three types of terms:
1. A term where all derivatives fall on the weight $t^z$. This gives the first term i... | 1 | https://mathoverflow.net/users/3948 | 179093 | 90,036 |
https://mathoverflow.net/questions/179094 | 3 | Given von Neumann equation
$$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$
If we know that $[H, \rho(0)] \neq 0$, how do we prove that the solution will fluctuate forever. (stabilization means the fluctuation approaching to zero, but the derivative can still be large, so here, we want to... | https://mathoverflow.net/users/48827 | Proof the solution of von Neumann equation will fluctuate forever if Hamiltonian and initial density matrix does not commute | In the eigenbasis of $H$ the density matrix has elements
$$\rho\_{nm}(t)=\rho\_{nm}(0)\exp[i(E\_m-E\_n)t]$$
where $E\_n$ are the eigenvalues of $H$; since you assume that $\rho(0)$ does not commute with $H$, there are nonzero matrix elements $\rho\_{nm}(0)$ with $n\neq m$. So these matrix elements will oscillate fo... | 0 | https://mathoverflow.net/users/11260 | 179099 | 90,039 |
https://mathoverflow.net/questions/179079 | 5 | There is quite some reference on aperiodicity of the edge-type of Wang Tile. But I could not yet find aperiodic corner type of Wang Tiles... Could someone provide me some instances (better with reference) of aperiodic Wang Tile of corner-type? Thank you:)
| https://mathoverflow.net/users/40780 | Aperiodic set of corner Wang Tile | So I read the answer to [your previous question](https://mathoverflow.net/questions/164400/relationship-between-corner-tile-and-edge-tile-of-wang-tile). I believe it's correct. This gives a recipe to translate any edge-type Wang tile to a corner-type Wang tile. If you start with an aperiodic set of edge-type Wang tiles... | 3 | https://mathoverflow.net/users/11054 | 179100 | 90,040 |
https://mathoverflow.net/questions/179110 | 4 | If $A(t)$ is an analytic, Hermitian matrix-valued function of a real variable $t$, then it is known that there are analytic functions $\lambda\_i(t)$ and $x\_i(t)$ corresponding to the eigenvalues and eigenvectors of $A(t)$. My question is: what if $A$ is an analytic function of two or more real variables? Does anyone ... | https://mathoverflow.net/users/54756 | Eigendecomposition of analytic Hermitian matrix-valued functions of several variables | See assertions (L) and (M) of the main theorem of
* Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011). [(pdf)](http://www.mat.univie.ac.at/~michor/DC-perturb.pdf)
Essentially it says, ... | 4 | https://mathoverflow.net/users/26935 | 179115 | 90,041 |
https://mathoverflow.net/questions/179095 | 2 | I have a 400x132 rectangular matrix with only 0 and 1.
I am looking for the linear combinations of the columns of the matrix that sum to 0.
For example C1 + C2 - C3 = 0.
I want to find the linear combinations with as few columns as possible. So for example, I don't want to find the 2 combinations: C1 + C2 - C4 = 0 an... | https://mathoverflow.net/users/57373 | Find base of kernel with as many 0 as possible | It sounds like you're describing an instance of the "sparse null basis problem" – this was the title of what (I believe is) the seminal work on this problem, by Coleman and Pothen. You can find their 1984 tech report at <http://ecommons.library.cornell.edu/handle/1813/6438>, and it may be helpful to search for work tha... | 2 | https://mathoverflow.net/users/2502 | 179116 | 90,042 |
https://mathoverflow.net/questions/178360 | 2 | This is a continuation of a question asked by me previously with some added hypothesis. Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$, $W \subset X \times Y$ a closed irreducible subscheme. Suppose that the natural projection map $pr\_2:W \to Y$ is surjective on the underlying topological spaces, and *... | https://mathoverflow.net/users/54369 | Non-reducedness of schemes and projective morphisms(revisited) | This is not true. Let $X$ be $\mathbb{A}^1$ with coordinate $s$. Let $\mathbb{A}^2$ have coordinates $t$ and $u$. Let $Y\subset \mathbb{A}^2$ be the irreducible, reduced closed subset with defining equation $u^2 - t^2(t+1) = 0$, i.e., $Y$ is a nodal cubic curve. Let $W\subset X\times Y$ be the closed subset with defini... | 1 | https://mathoverflow.net/users/13265 | 179118 | 90,043 |
https://mathoverflow.net/questions/143498 | 3 | I would like to bound from above the expression
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}
$$
for $x>y>0$. By plotting the above expression I have found that it should hold
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}< y^{\alpha-\beta}
$$
for $\alpha <... | https://mathoverflow.net/users/35445 | Estimate of a ratio of two incomplete gamma functions | In fact it seems to be a consequence of the Cauchy theorem from calculus. Really, by it and a formula for derivative of incomplete gamma function (cf. Wiki for example) we evaluate
$\frac{f(x)-f(y)}{g(x)-g(y)}=\frac{f'(c)}{g'(c)}$
with some intermediate c, $0<y<c<x$
and then by the above mentioned formula
$\Gamma^{'... | 2 | https://mathoverflow.net/users/49208 | 179129 | 90,048 |
https://mathoverflow.net/questions/179098 | 1 | I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations (which are second order).
I know how to derive the Euler Poincare equations (which are first order) for the geodesics. ... | https://mathoverflow.net/users/41654 | Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton | There is no easy way to do this, since you have to solve an ODE between the E-L equations and the E-P equations. For numerical solutions you can minimize the functional
$$
F(X,g\_0,g\_1) = \int\_0^1 \|X(t)\| dt + \epsilon d(g(1).g\_0 - g\_1)
$$
with respect to $X:[0,1]\to \mathfrak g$, where
$$
\partial\_t g(t) = X(t)... | 1 | https://mathoverflow.net/users/26935 | 179130 | 90,049 |
https://mathoverflow.net/questions/179117 | 4 | I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is transversal to a stratification for which our sheaf is smooth). I would like a reference for any of the following settings:
*... | https://mathoverflow.net/users/2095 | Nearby cycles and specialisation - properties | For the first item, section 3 of M. Saito's *Modules de Hodge Polarizables* is pretty thorough. He works in the filtered $D$-module setting, but I suppose you can ignore that aspect. For the constructible setting, perhaps you can look at Dimca's *Sheaves in Topology*.
| 3 | https://mathoverflow.net/users/4144 | 179132 | 90,051 |
https://mathoverflow.net/questions/178318 | 17 | A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: <http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem>
Are there any other proofs of this, or generalizations of it?
| https://mathoverflow.net/users/40983 | Proofs of the Chevalley-Warning Theorem | I am working on a book-length manusript, *Around the Chevalley-Warning Theorem*. A complete answer to your question is estimated at about 150 pages!
In terms of what exists at the moment, here are two papers. Both of them make connections between the classical results of Chevalley and Warning and modern polynomial me... | 28 | https://mathoverflow.net/users/1149 | 179143 | 90,055 |
https://mathoverflow.net/questions/179144 | 6 | The title of my Q. says it all:
**QUESTION**: Who defined and who coined: *module*?
Would it be Emmy Noether?
**EDIT** In view of @anon's and KConrad's answers, and as it could have been expected, the situation is a bit complex. Thus it looks that while Dedekind coined *module*, the final notion was defined b... | https://mathoverflow.net/users/8385 | Who defined and who coined "module"? | From: <http://jeff560.tripod.com/m.html>
MODULE. A JSTOR search found the English term in E. T. Bell’s “Successive Generalizations in the Theory of Numbers,” American Mathematical Monthly, 34, (1927), 55-75. Bell was describing the work of Dedekind, basing his account on Dedekind’s French article, “Sur la Théorie des... | 11 | https://mathoverflow.net/users/57398 | 179147 | 90,057 |
https://mathoverflow.net/questions/179138 | 1 | From one of my previous question [Aperiodic set of corner Wang Tile](https://mathoverflow.net/questions/179079/aperiodic-set-of-corner-wang-tile) (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type aperiodic Wang Tiles. But what I am more inte... | https://mathoverflow.net/users/40780 | A possible minimal aperiodic set of corner Wang Tile | You could look at the corner types that actually arise in the aperiodic edge-tilings you mention and thereby find small collections of corner-tiles with analogous aperiodic tilings, using the method in [my answer to your previous question](https://mathoverflow.net/a/164405/1946), which converts edge-tilings to corner-t... | 3 | https://mathoverflow.net/users/1946 | 179152 | 90,060 |
https://mathoverflow.net/questions/179112 | 15 | Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index subgroups of type F. For example, $SL(n,\mathbb{Z})$, mapping class groups, automorphism groups of free groups, etc.
I... | https://mathoverflow.net/users/57385 | Torsion-free group that is not of type F but is virtually of type F | It is conjectured that no such example exists. It is conjectured that $G$ is of type $F$, but it is hard to prove a group is type $F$ without explicitly exhibiting a classifying space. The class of groups of type $FP$ is a well-behaved proxy. Moreover, it is conjectured that every finitely presented group of type $FP$ ... | 11 | https://mathoverflow.net/users/4639 | 179154 | 90,061 |
https://mathoverflow.net/questions/179114 | 5 | Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the Grassmannian $G$ is $\mathrm{det} Q$ and is isomorphic to the pullback of the very ample line bundle of the projective space (in ... | https://mathoverflow.net/users/54369 | Plucker embedding and tautological/universal quotient bundle | Your claim descends directly from the definition of the Plucker embedding.
Let $G(n,h)$ be the Grassmannian parametrizing $h$-linear subspaces of $\mathbb{P}^n = \mathbb{P}(V)$. The fiber of the universal bundle $\mathcal{S}$ over $[L]\in G(n,h)$ is the vector space $W$, where $L = \mathbb{P}(W)$.
Now, the Plucker... | 6 | https://mathoverflow.net/users/14514 | 179155 | 90,062 |
https://mathoverflow.net/questions/179122 | 9 | The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda\_n = -(2\pi/L)^2 n^2$.
On the other hand the Laplacian of a non-compact manifold may be continuous. For example on $\mathbb{R}$ the spectrum of the second... | https://mathoverflow.net/users/7631 | Behavior of the spectrum of the Laplacian under pointed smooth convergence | This is only a partial answer since it concerns only compact manifolds. There is a result due to T. Shioya <http://projecteuclid.org/download/pdf_1/euclid.jmsj/1213023969> a special case of which says that if a sequence of compact $n$-dimensional Riemannian manifolds $M\_i$ converges to another compact n-dimensional Ri... | 6 | https://mathoverflow.net/users/16183 | 179166 | 90,067 |
https://mathoverflow.net/questions/179024 | 3 | I'm trying to prove the convergence of Matrix factorization.
The problem is described below.
$|X-WH|^2 + |H|\_2^2 +|W|\_2^2$.
My optimization steps are using Alternating least squares which update H with fixing W and update W with fixing H.
Although I can prove the convergence of subproblems(e.x. update H with fix... | https://mathoverflow.net/users/57338 | The convergence of Matrix factorization | Your problem is a special case of **Alternating Minimization**, on which many papers have been written.
One of the clearest (for your nonconvex problem) is this classic paper, which pays particular attention to the two block case: L. Grippo, M. Sciandrone, *On the convergence of the block nonlinear Gauss–Seidel metho... | 3 | https://mathoverflow.net/users/8430 | 179167 | 90,068 |
https://mathoverflow.net/questions/177066 | 2 | Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec\_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:
<http://ac.els-cdn.com/S0022404908002387/1-s2.0-S0022404908002387-main.pdf?_tid=120cfede-1405-11e4-91e8-00000aab0f6c&acdnat=1406297453_6fca2d4de380c88... | https://mathoverflow.net/users/nan | Singularities of secant varieties of rational normal curves | The following is a consequence of Theorem 1 in "A. Bertram, *Moduli of Rank-$2$ Vector Bundles, Theta divisors, and the geometry of curves in projective space*, J. Differential Geom. 35, 1992, 429-469."
Let $C\subset\mathbb{P}^{2h}$ be a degree $2h$ rational normal curve. Consider the following sequence of blow-ups:
... | 2 | https://mathoverflow.net/users/14514 | 179191 | 90,075 |
https://mathoverflow.net/questions/179141 | 19 | Who was first to use symbols GL, SL, O, SO, U, SU, Sp and their projective versions, and how did this notation become standard?
The notation appears in fairly modern form in Weyl's "The Classical Groups" except that he uses $O^+$ for SO. I know that Weyl coined the term "symplectic group", so the notation Sp certainl... | https://mathoverflow.net/users/33757 | Who originated the standard symbols for Lie groups GL, SL, SU, etc.? | It's hard to provide definitive confirmation of Weyl's role, but his 1939 book was highly influential in all further developments. It's important to realize that notation (and terminology) in mathematics tend to evolve over time, and what are now considered "standard" symbols for the various linear/Lie groups didn't al... | 12 | https://mathoverflow.net/users/4231 | 179193 | 90,076 |
https://mathoverflow.net/questions/179204 | 10 | I am looking for a reference/proof that shows that the Morava $K$-theory spectra, $K(n)$ are not $E\_{\infty}$ ring spectra. I suspect that this should be a calculation but I can't quite get it right.
Thank you as always.
| https://mathoverflow.net/users/24706 | Morava $K(n)$'s are not $E_{\infty}$ | The assertion is Lemma 5.6.4 in Rognes's "Galois extensions of structured ring spectra" available on the [arXiv](http://arxiv.org/abs/math/0502183). In fact, the $K(n)$ spectra do not even admit $E\_2$-algebra structures. The reason is that the free $E\_2$-algebra with $p = 0$ is known to be (by a theorem of Hopkins-Ma... | 19 | https://mathoverflow.net/users/344 | 179213 | 90,084 |
https://mathoverflow.net/questions/179202 | 3 | These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant ([see this question for further details](https://math.stackexchange.com/questions/902107/is-there-an-easier-way-to-find-the-natural-integration-constant)).
The first one is based on New... | https://mathoverflow.net/users/10059 | Prove that these two definitions of "natural" integration constant coincide when both converge | If $f$ is an $L^2$ function, with Fourier transform $\hat{f}$, then the identity you're trying to prove is
$$\sum\_{m = 0}^\infty \binom{-1}{m} \sum\_{k = 0}^m \binom{m}{k} (-1)^{m - k} (-i\xi)^k \hat{f}(\xi) = \frac{1}{-i\xi} \hat{f}(\xi).$$
In other words, you want to show that
$$\sum\_{m = 0}^\infty \binom{-1}{m} \s... | 6 | https://mathoverflow.net/users/1096 | 179218 | 90,087 |
https://mathoverflow.net/questions/179220 | 2 | Has anyone ever considered something like the following separation axiom?
$(\*)$ For any pair of open sets $O$ and $N$, there exist disjoint open sets containing $O\setminus N$ and $N\setminus O$.
Any $T\_1$ space that is not Hausdorff will fail $(\*)$ but I would be interested to know of any Hausdorff spaces faili... | https://mathoverflow.net/users/38085 | Separating Differences of Open Sets | Note that condition $(\*)$ is equivalent to "every open subspace is normal" and it is a standard exercise to show that spaces with this property are exactly the completely normal ($T\_5$) spaces.
| 2 | https://mathoverflow.net/users/17836 | 179223 | 90,089 |
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