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/258667 | 5 | Consider the group of $mn\times mn$ permutation matrices $\mathfrak{S}\_{mn}$ and partition each such matrix $P$ into $n^2$ blocks of $m\times m$ matrices $Q\_{i,j}$. Now, transpose each $Q\_{i,j}$ (independently) to form a new $mn\times mn$ matrix denoted $P^t$ (with an abuse of notation). Let's construct the set $U\_... | https://mathoverflow.net/users/66131 | does this set of permutations form a group? And more | For $m, n > 1$ it's never a group.
Let $\sigma \in \mathfrak{S}\_n$, $\sigma$ switches $1, m + 1$, and in fact $\sigma = \sigma^t$. Clearly, $\sigma \in U\_{mn}$. Similarly, let $\tau$ switch $1, 2$; similarly, $\tau = \tau^t$. Then $\sigma \tau \sigma$ switches $2, m+1$. This clearly is not in $U\_{mn}$.
| 10 | https://mathoverflow.net/users/44191 | 258701 | 116,740 |
https://mathoverflow.net/questions/258702 | 1 | Let $m,n$ be two positive integers and $2^{2n+1}-1\, | \, 2^{2m+1}-1$. Suppose $P\_0$ be the largest prime number such that $P\_0 \, | \, 2^{2m+1}-1$. If $P\_0 \, | \, 2^{2n+1}-1$ then is the following equation true?
$$2^{2n+1}-1 = 2^{2m+1}-1$$
| https://mathoverflow.net/users/86401 | equality of two numbers which are odd powers of 2 and satisfy a certain condition | No. $(2n+1,2m+1) = (17,51)$ is a counterexample because
$$
2^{51} - 1 = 7 \cdot 103 \cdot 2143 \cdot 11119 \cdot 131071
$$
and $131071 = 2^{17}-1$ is prime. The only other counterexample with $m \leq 100$ is $(2n+1,2m+1) = (37,111)$ with $P\_0 = (2^{37}-1)/223 = 616318177$.
P.S. **gp** code:
```
forstep(k=3,200,2,... | 3 | https://mathoverflow.net/users/14830 | 258703 | 116,741 |
https://mathoverflow.net/questions/258726 | 5 | Recently I read the following results.
(1) Zheng, F.. *Kodaira dimensions and hyperbolicity of
nonpositively curved compact K\"ahler manifolds.* Comment. Math.
Helv. 77 (2002), no. 2, 221-234.
(2) Jahnke, P.; Peternell, T.; Radloff, I.. *Some recent
developments in the classification theory of higher dimensional
ma... | https://mathoverflow.net/users/12904 | On a result of Kawamata on ampleness and nonexistence of rational curves | I suppose they might mean the basepoint-free theorem. If you definitely want a reference to Kawamata, I believe it's in his Annals paper, "The cone of curves...", but probably the standard reference is Kollár-Mori (1998).
So, the point is, if $K\_X$ is nef and big, then by the basepoint-free theorem some multiple of... | 12 | https://mathoverflow.net/users/10076 | 258728 | 116,748 |
https://mathoverflow.net/questions/258723 | 5 | I'm an undergraduate math student that learned about classical ideal gases and the associated maxwell-boltzmann distribution for particle velocities in a statistical physics course. Now, starting from this knowledge and assuming that there are no collisions between particles, I tried to show that an ideal gas maximises... | https://mathoverflow.net/users/56328 | rigorous derivation of isoperimetric inequality from ideal gas equation | For the $n$-dimensional phase space of $n$ particles on a line the isoperimetric inequality of an ideal gas was derived in: [Phase space measure concentration for an ideal gas](https://arxiv.org/abs/0811.1719) (2009).
>
> We point out that a special case of an ideal gas exhibits
> concentration of the volume of it... | 4 | https://mathoverflow.net/users/11260 | 258733 | 116,749 |
https://mathoverflow.net/questions/258732 | -1 | I am given $n$ balls. For $n$ times, I pick one of them with uniform probability and put it back after picking it.
Let $U$ be the number of balls I have never picked, so $U\in \{0,\ldots,n-1\}$. We are interested in the expected value $E\_n:= E(U)$.
What is the value of $\lim\_{n\to\infty} \frac{E\_n}{n}$, if it ex... | https://mathoverflow.net/users/8628 | Expected number of balls left out when choosing $n$ times from $n$ balls | When tossing $n$ balls uniformly and independently into $n$ bins, the distribution of the number of balls in any specific bin is binomial with parameters $n$ and $p=\frac{1}{n}$. Thus, the probability that a bin is left empty is $(1-\frac{1}{n})^n$, which tends to $\frac{1}{e}$. By linearity, the expected number of emp... | 5 | https://mathoverflow.net/users/1061 | 258736 | 116,751 |
https://mathoverflow.net/questions/258434 | 9 | A finite group $B$ is said to be a B-*group* if every primitive permutation group having a regular (transitive) subgroup isomorphic to B is $2$-transitive.
Schur proved that a cyclic group of composite order is a B-group. Wielandt showed that no group of the form $B\_1 \times \cdots \times B\_d$ with $|B\_1| = \ldot... | https://mathoverflow.net/users/7709 | Regular elementary abelian subgroups of primitive permutation groups | This is a bit long for a comment. To answer the question it would be very useful to know what groups can act primitively on a set of order $2^a$. Looking at O'Nan--Scott--Aschbacher, we see that such groups must be either affine or almost simple, or product action. Let's consider these three families one at a time.
*... | 5 | https://mathoverflow.net/users/801 | 258739 | 116,753 |
https://mathoverflow.net/questions/258737 | 2 | Let $M$ be a smooth manifold and let $\Delta \_i$ for $i=1,...,k$ be distributions of $TM$ which are integrable such that $\Delta\_i \cap \Delta \_j$ is zero distribution for $i \neq j$. Suppose $\Delta = \Delta\_1 \oplus ...\oplus \Delta \_k$. Is $\Delta$ integrable?
| https://mathoverflow.net/users/103173 | Integrability of direct sum of some integrable distributions | No. On the 3-sphere, take the usual metric and the 2-plane field $\Delta$ perpendicular to the fibers of the Hopf fibration. It splits as a sum of two line fields $\Delta\_1$ and $\Delta\_2$. I will write out the details when I have a moment.
Edit: some details. You think of $S^3$ as the unit quaternions, and so as a... | 3 | https://mathoverflow.net/users/13268 | 258740 | 116,754 |
https://mathoverflow.net/questions/120349 | 1 | I know that the only simply primitive permutation groups of degree $2p$, where $p$ is an odd prime, are $A\_5$ and $S\_5$. I want to know that: Is there a complete list of simply primitive permutation groups of degree $2p^2$, where $p$ is an odd prime?
| https://mathoverflow.net/users/27831 | simply primitive permutation groups of degree $2p^2$ | For the sake of getting this off the unanswered stack... Yes, all such actions are known.
By studying O'Nan--Scott--Aschbacher, one sees immediately that a group $S$ that has a primitive action of degree $2p^2$ must be almost simple. Now one refers to this paper:
*Cai Heng Li and Xianhua Li*, MR 3210408 [**On permu... | 7 | https://mathoverflow.net/users/801 | 258741 | 116,755 |
https://mathoverflow.net/questions/258724 | 6 | I'm trying to learn the basics of the representation theory of $p$-adic groups and I'm stuck on a few things:
Let $G$ is a connected split reductive group over a non-archimedean local field $F$, and $K=G(\mathcal{O})$, a hyperspecial maximal compact subgroup of $G(F)$. Then there are two facts that apparently go hand... | https://mathoverflow.net/users/64244 | Why are spherical representations subquotients of unramified principal series? | Statement 2. comes from the following classical fact whose proof can be found in e.g. Bushnell and Kutzko, "The admissible dual of ${\rm GL}(N)$ via compact open subgroups". This is a particular case of Proposition (4.2.3), page 147 of loc. cit. I state it in your case:
*The following sets are in natural bijection:*
... | 9 | https://mathoverflow.net/users/4767 | 258743 | 116,756 |
https://mathoverflow.net/questions/258752 | 5 | Let $M(n)$ be an $n\times n$ matrix in the variables $x\_1,\dots,x\_n$ with entries
$$M\_{i,j}(n)=\frac{x\_{\max(i,j)}}{x\_{\min(i,j)}}, \qquad 1\leq i,j\leq n.$$
I'm interested in the following:
>
> **Questions.**
>
>
> (1) Is there a neat or "closed form" evaluation for the determinant $\det M(n)$?
>
>
> (2) ... | https://mathoverflow.net/users/66131 | minimum-maximum entries matrix | Let us write $$a\_r=\frac{x\_{r+1}}{x\_r}$$ for $r=1\cdots n$.
We can then write the matrix $M(n)$ in the form
$$\begin{pmatrix} 1 & a\_1 & a\_1a\_2& \cdots & a\_1a\_2\cdots a\_{n-1} \\ a\_1 & 1 & a\_2& \cdots & a\_2\cdots a\_{n-1}\\ \vdots & \vdots & \vdots &\ddots & \vdots\\ a\_1a\_2\cdots a\_{n-1}& a\_2\cdots a\... | 10 | https://mathoverflow.net/users/41644 | 258755 | 116,763 |
https://mathoverflow.net/questions/258672 | 9 | (A) It is an old, outstanding problem to show that there are infinitely many real quadratic fields with class number one.
(B) On the other hand, Weber's class number problem (for $p=2$) asks to show that the degree $2^n$ cyclotomic extension $F\_n=\mathbb Q(\cos(2\pi/2^{n+2}))$ of $\mathbb Q$ has class number one, fo... | https://mathoverflow.net/users/48554 | Weber's class number problem and real quadratic fields of class number one | This statement is most certainly nonsense; I guess what she meant to write was that in order to prove the existence of infinitely many number fields with class number $1$ it is sufficient to prove $h(F\_n) = 1$ for infinitely many (and therefore for all) $n$.
| 8 | https://mathoverflow.net/users/3503 | 258757 | 116,764 |
https://mathoverflow.net/questions/258510 | 6 | $\DeclareMathOperator\End{End}$Let $T\_n$ be the full transformation semigroup/monoid of $[n]=\{1,\dots,n\}$. Let $\End(T\_n)$ be the set of [endomorphisms][1] of $T\_n$. Then, $\# T\_n=n^n$ and
$$\# \End(T\_n)=n!\left[1+\sum\_{m=1}^n\sum\_{k=0}^{\lfloor\frac{m-1}2\rfloor}\sum\_{r=1}^{m-2k}\frac{m^{n-m}r^{m-k-r}}{2^k(n... | https://mathoverflow.net/users/66131 | are endomorphisms "small" compared to the full transformations? | Looking at the exponential series, we have $\frac{r^k}{k!}\leq e^r$. Hence we have
$$
\frac{r^{m-k-r}}{(m-2k-r)!r!k!} = \frac{1}{r!}\frac{r^k}{k!}\frac{r^{m-2k-r}}{(m-2k-r)!} \leq\frac{e^{2r}}{r!}=\mathcal{O}(1).
$$
From this we obtain
$$
\#End(T\_n)\leq Cn! n^3\max\_m\frac{m^{n-m}}{(n-m)!} \leq Cn! n^3\max\_m\frac{m^{... | 2 | https://mathoverflow.net/users/37555 | 258766 | 116,767 |
https://mathoverflow.net/questions/258777 | 9 | I asked this question on Math Stack Exchange some time ago and a similar question recently appeared regarding $L^1$ instead [see here](https://math.stackexchange.com/questions/2072103/integration-of-power-series-when-is-the-function-in-l-1) This has prompted me to bring it to this community in the hopes of getting an a... | https://mathoverflow.net/users/20460 | When is an analytic function in $L^2(\Bbb R)$? | 1. Since you ask about "real analytic functions", should it be $f:\mathbb{R}\to \mathbb{R}$ instead of $f:\mathbb{R}\to \mathbb{C}$?
2. I think analyticity is a "local" concept according to its definition.
3. Because of 2, $f$ may not have a global power series expansion. For example, $1/(1+x^2)$ is real analytic over ... | 6 | https://mathoverflow.net/users/37087 | 258780 | 116,773 |
https://mathoverflow.net/questions/258808 | 4 | I asked this question on the stack exchange, and after no answers and the recommendation of someone else, I am posting it here on MO. I am looking for an example of two ideals $I$ and $J$ in a noetherian ring that are both primary to $p$, but their sum is not. There are a few things that I do know: If $p$ is maximal, t... | https://mathoverflow.net/users/103203 | $I,J$ are $p$-primary ideals, but $I+J$ is not | Let $R$ be the commutative ring $k[x,y,z]$. Let $I$ be the ideal generated by the regular sequence $(x^2,y)$. Let $J$ be the ideal generated by the regular sequence $(x^2,y-xz)$. Then both $R/I$ and $R/J$ are Cohen-Macaulay, hence unmixed. The prime $\mathfrak{p}=\langle x,y \rangle$ is the unique associated prime of b... | 10 | https://mathoverflow.net/users/13265 | 258813 | 116,791 |
https://mathoverflow.net/questions/257782 | 4 | Let $M:=U(n)/T^n$ be a complete flag variety, where $U(n)$ is an unitary group and $T^n \simeq (S^1)^n$ consists of its diagonal matrices. I have heard the following construction of a symplectic moment map $\mu: M \to t^\*$. Let $B=\operatorname(b\_1, \ldots, b\_n)$ be a diagonal matrix with different $b\_i \in \mathbb... | https://mathoverflow.net/users/43639 | Moment map for complete flags variety | The form on the orbit of coadjoint action is called Kirillov-Kostant-Souriau form (thank to Friedrich Knop for reference!), it descends from a form $\omega\_\alpha: T^\*\_\alpha \mathfrak g^\* \times T^\*\_\alpha \mathfrak g^\*=\mathfrak g \times \mathfrak g \to \mathbb C$ that in a point $\alpha \in \mathfrak g^\*$ is... | 1 | https://mathoverflow.net/users/43639 | 258819 | 116,793 |
https://mathoverflow.net/questions/258431 | 17 | *Sorry about the title, I couldn't resist.*
It's a classic fact that, not only does $PA$ prove every true $\Sigma\_1$ sentence, but $PA$ *proves* that $PA$ proves every true $\Sigma\_1$ sentence! In particular, restricting attention to $\Sigma\_1$ sentences of the form "$PA$ proves ---", in the modal logic of $PA$-pr... | https://mathoverflow.net/users/8133 | Who first proved that we can prove that we prove things we prove? | The theorem $\Box P \Rightarrow \Box\Box P$ is due to Martin Löb and first appears in his [1955 paper "Solution of a Problem of Leon Henkin"](http://www.math.umd.edu/~laskow/Pubs/713/Lob.pdf), *J. Symb. Logic* **20** 115–118: it appears as condition (V) (page 116) in the paper in question, and whereas conditions (I)–(I... | 10 | https://mathoverflow.net/users/17064 | 258849 | 116,806 |
https://mathoverflow.net/questions/258843 | 6 | Let us say a set $X$ satisfies Property A if$$\liminf\_{n \to \infty} {{\left|X^{\le n}\right|}\over n} = 0.$$Are there recursive sets $X$ satisfying Property A that contain infinitely many incompressible strings?
| https://mathoverflow.net/users/101049 | Are there recursive sets $X$ with Property A that contain infinitely many incompressible strings? | I assume that $X$ is a set of binary strings and the binary strings are coded by natural numbers in this way: consider this enumeration of all strings:
$\lambda$, 0,1,00,01,10,11,000,.....
(in which, at first we have the string with length zero, then all strings of length one in alphabetical orde and so on).Now co... | 5 | https://mathoverflow.net/users/65878 | 258850 | 116,807 |
https://mathoverflow.net/questions/258804 | 5 | Let $I$ be an index set. Given $A\subseteq I\times 2^\omega$ and $i\in I$, set $(A)\_i = \{x\in 2^\omega: (i, x)\in A\}$.
Now let $I\times 2^\omega = \bigcup\_{n<\omega} A\_n$. How large must $I$ be to ensure that there are indices $i\neq j\in I$ and $n< \omega$ with $(A\_n)\_i \cap (A\_n)\_j$ non-meager? In particu... | https://mathoverflow.net/users/62393 | Countable partitions of Cantor space mod meager | Let $\kappa$ be least such that whenever $\{A^i\_n : n < \omega\}$, for $i < \kappa$ are coverings of $2^{\omega}$, there are $i < j < \kappa$ and $n < \omega$ such that $A^i\_n \cap A^j\_n$ is non meager.
Claim: CH implies $\kappa > \omega\_1$.
Proof: Using an Ulam matrix.
Claim: Assume MA plus not CH. Then $\ka... | 2 | https://mathoverflow.net/users/2689 | 258851 | 116,808 |
https://mathoverflow.net/questions/258847 | 5 | After having googled for several days without locating a definitive answer, I will try my luck here!
I have implemented a version of the QR algorithm to calculate Eigenvalues and hopefully Eigenvectors of a matrix $A$ of dimension $n\times n$.
In order to speed up the convergence rate i have applied a version of t... | https://mathoverflow.net/users/103230 | SOLVED: How to retrieve Eigenvectors from QR algorithm that applies shifts and deflation | Instead of dropping one row and one column, compute at each step a $(n-1)\times(n-1)$ orthogonal transformation (or $(n-k)\times(n-k)$, after $k$ deflation steps) $Q$ by working to the reduced matrix, and then apply it to the full matrix as
$$
\begin{bmatrix}
Q^\* \\& I
\end{bmatrix}
\begin{bmatrix}
H\_{11} & H\_{12}\\... | 2 | https://mathoverflow.net/users/1898 | 258853 | 116,809 |
https://mathoverflow.net/questions/258848 | 11 | In modern condensed matter physics, one is often interested in the homotopy classes of mappings from a $d$-dimensional torus $$\mathbb{T}^d=\underbrace{S^1\times\ldots \times S^1}\_d$$
(corresponding to the Brillouin zone of a $d$-dimensional system) to various topological spaces $Y$, for example the Grassmannians $$O... | https://mathoverflow.net/users/103233 | Relation between the homotopy classes of maps on a torus, and maps on a sphere | One case in which you can establish a simple relationship is when $Y$ is a loop space. Suppose that $Y\simeq \Omega Z=\mbox{map}\_\*(S^1, Z)$. Then there is a bijection $[{\mathbb T}^d, Y]\_\*\cong [\Sigma{\mathbb T}^d, Z]\_\*$. On the other hand, there is an equivalence $$\Sigma{\mathbb T}^d \simeq \bigvee\_{i\ge 1} \... | 13 | https://mathoverflow.net/users/6668 | 258854 | 116,810 |
https://mathoverflow.net/questions/258858 | 4 | Suppose $a\_n > 0$ is a sequence of real numbers in $l^2 \setminus l^1$. i.e. $\sum a\_n^2 < \infty$ but $\sum a\_n = \infty$.
If $B\_n$ are an infinite sequence of independent Bernoulli random variables with parameter $\frac{1}{2}$ then the infinite sum $\sum a\_n (-1)^{B\_n}$ exists almost surely and converges in $... | https://mathoverflow.net/users/4959 | Can a probability distribution from summing alternating signs have atoms? | The answer is no.
For any $T \subseteq \mathbb N$, let
$$ A(T) = \sum\_{n \in T} a\_n (-1)^{B\_n}$$
Now $A = A(T) + A(\mathbb N \backslash T)$ where $A(T)$ and $A(\mathbb N \backslash T)$ are independent, and so if $A$ had atoms both $A(T)$ and $A(\mathbb N \backslash T)$ would have atoms. Thus it suffices to find... | 5 | https://mathoverflow.net/users/13650 | 258862 | 116,814 |
https://mathoverflow.net/questions/257898 | 3 | Given the group algebra of a p-group over a field of characteristic p. Can the 2-periodic indecomposable modules $M$ ($M$ with $\Omega^{2}(M)=M$) be classified? I am not experienced much with modular representation theory, but my feeling is that apart from the tame case, there might not exist many such modules.
| https://mathoverflow.net/users/61949 | 2-periodic modules over p-group algebras | This question is addressed by the paper *Wild categories of periodic modules* (Illinois J. Math. 32 (1988), no.3, 557-561) by Jon Carlson and Alfredo Jones. As suggested by the title, the answer to your question is that the $2$-periodic modules can usually not be classified. In particular, the authors consider the very... | 1 | https://mathoverflow.net/users/11791 | 258865 | 116,815 |
https://mathoverflow.net/questions/258864 | 3 | I'm trying to learn about eigenvarieties and I am confused as to whether or not the spectral variety is affinoid.
It seems to me that being the zero locus of a power series should make you affinoid, but I cannot find this stated anywhere, so I'm a bit confused.
| https://mathoverflow.net/users/103238 | Are Fredholm hypersurfaces affinoid? | No they're not in general affinoid. The problem is that the zero locus of the power series is computed within a space which is almost never affinoid -- for example in the modular curve case the ambient space would be the product of an open disc (not affinoid) and affine 1-space minus the origin (not affinoid either). T... | 4 | https://mathoverflow.net/users/1384 | 258866 | 116,816 |
https://mathoverflow.net/questions/258678 | 3 | Here is the context in which I heard it. If $G$ is a reductive group over a non-archimedean local field $F$, then let $K$ be the hyperspecial maximal compact subgroup $G(\mathcal{O})$. For example, if $G=SO(n)$, and assuming that $G$ is split, then there is a natural choice of $SO(n)(\mathcal{O})$. What exactly does th... | https://mathoverflow.net/users/64244 | What is a "split $SO(n)$"? | In fact, to properly understand what is $SO(n)(\mathcal{O})$, you need to know what are split groups over a ring. A summary of this story goes as follows:
Given a ring $R$, and given a finite Dynkin diagram (<https://en.wikipedia.org/wiki/Dynkin_diagram>), you have a unique simply connected (resp. adjoint) absolutely... | 10 | https://mathoverflow.net/users/47722 | 258867 | 116,817 |
https://mathoverflow.net/questions/258838 | 2 | Suppose we have some i.i.d. sequence $(X\_n)\_{n\in \mathbb{N}}$, and define $Y\_n :=n+X\_n$. What is the probability that $Y\_n$ is increasing from $n$ to $m$?
In more detail, we fix some $n<m \in \mathbb{N}$, and we want to know the probability:
$$\mathbb{P}\{\forall r \in\{n,\dots,m-1\}: Y\_r \leq Y\_{r+1}\} =$$
$... | https://mathoverflow.net/users/83088 | Probability that n+X_n is increasing for an i.i.d. sequence | We can compute these probabilities symbolically for some distributions using Mathematica. Letting $k=m-n$, the code is just
```
f[dist_, k_] := Integrate[(f[dist, k-1] /. w->v) (f[dist, 1] /. u->v) PDF[dist,v],
{v, -Infinity, Infinity}]
f[dist_, 1] = Boole[1 + w > u]
prob[dist_... | 3 | https://mathoverflow.net/users/nan | 258884 | 116,824 |
https://mathoverflow.net/questions/258105 | 15 | The setup is as in [this question](https://mathoverflow.net/questions/89762/bounding-the-commutator-a-b-in-terms-of-the-numerical-radius?rq=1):
Given a norm $N$ over ${\bf M}\_n(\mathbb C)$, it is a natural question to find the best constant $C\_N$ such that
$$N([A,B])\le C\_N N(A)N(B),\qquad\forall A,B\in{\bf M}\_n... | https://mathoverflow.net/users/29783 | Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$ | A somewhat more general setting, namely, finding the best constant $C\_{p,q,r}$ in
\begin{equation\*}
\|AB-BA\|\_p \le C\_{p,q,r}\|A\|\_q\|B\|\_r,
\end{equation\*}
for Schatten $p$,$q$,$r$-norms, is studied in [this paper.](http://www.sciencedirect.com/science/article/pii/S0024379510003356)
*EDIT (1st Mar'17).* See ... | 9 | https://mathoverflow.net/users/8430 | 258885 | 116,825 |
https://mathoverflow.net/questions/258878 | 4 | Assume we have a Gaussian measure $\mu$ supported on a Banach space $X$. Can we always find a Hilbert space $H$ embedded in $X$ sch that $\mu$ is also supported on $H$?
| https://mathoverflow.net/users/103245 | Gaussian measure on Banach space | No, not necessarily. It is shown in Examples 3.6.6 and 3.6.7 of Bogachev's *Gaussian Measures* that if $X = C([0,1])$ and $\mu$ is classical Wiener measure, then for any Hilbert space $H$ embedded in $X$, we have $\mu(H) =0$, so that $\mu$ is not supported on $H$.
| 2 | https://mathoverflow.net/users/4832 | 258892 | 116,828 |
https://mathoverflow.net/questions/258904 | 0 | If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete.
Is there an infinite connected $T\_2$-space $(X,\tau)$ and a discrete subset $S\subseteq X$ such that no proper superset of $S$ is discrete?
**EDIT**: Added "infinite" in ... | https://mathoverflow.net/users/8628 | Maximal discrete subsets of connected $T_2$-spaces | No. If $S$ is a maximal discrete subset of a $T\_1$-space $X$, then every point of $S$ is isolated in $X$ (in fact, $S$ must be the set of isolated points of $X$ and it must be dense in $X$). Thus if $X$ has at least two points, it is not connected.
| 2 | https://mathoverflow.net/users/17836 | 258915 | 116,835 |
https://mathoverflow.net/questions/258919 | 0 | Let $K$ and $K'$ be number fields $K \subset K'$, and let $R$ and $R'$ be the corresponding ring of integers. Let $S = Spec\ R$ and $S' = Spec\ R'$. Suppose $X \to S$ be an arithmetic surface that is a regular scheme of dimension 2 which is projective and flat over $S$.
Is $X' = X \times\_S S'$ also regular? If not ... | https://mathoverflow.net/users/11392 | Regularity of schemes under base change | I don't think this will be true in general.
Say $K=\mathbf{Q}$ and $K'=\mathbf{Q}(\sqrt{2})$, and let $X\_0$ be $Spec(R')$. Then $X\_0$ is regular of dimension 1 and the map down to $S$ is projective and flat, but the base change to $S'$ is the spectrum of $\mathbf{Z}[\sqrt{2}][X]/(X^2-2)$ which is two smooth affine ... | 3 | https://mathoverflow.net/users/1384 | 258921 | 116,836 |
https://mathoverflow.net/questions/258214 | 7 | Is there a Dieudonné theory for $p$-divisible groups (or for finite flat group schemes of $p$-power order) over *non-perfect* base-fields?
| https://mathoverflow.net/users/nan | Dieudonné theory over non-perfect base fields | Johan de Jong sent me an e-mail with a reference to Jong, A.J. de. "Finite locally free group schemes in characteristic $p$ and Dieudonné modules" Inventiones mathematicae 114.1 (1993): 89-138. <https://eudml.org/doc/144143>.
| 6 | https://mathoverflow.net/users/nan | 258924 | 116,839 |
https://mathoverflow.net/questions/258829 | 4 | Let $(X,\mu )$ be a measure space. Then, $L^2(X):=L^2(X,\mu )$ is a Hilbert space in the usual way and we may view $L^{\infty}(X):=L^{\infty}(X,\mu )$ as a subalgebra of bounded operators on $L^2(X)$ via $L^{\infty}(X)\ni f\mapsto M\_f$, where $M\_f\colon L^2(X)\rightarrow L^2(X)$ is the multiplication operator defined... | https://mathoverflow.net/users/16639 | Operator topologies on $L^{\infty}(X,\mu )$ | The argument you give for the equality of the weak operator and weak-$^\*$ (or $\sigma$-weak) topologies also shows that the strong and $\sigma$-strong topologies are equal, and similarly for the strong-$^\*$ and $\sigma$ strong-$^\*$. This equality of topologies holds for all von Neumann algebras in standard form.
I... | 3 | https://mathoverflow.net/users/99234 | 258928 | 116,841 |
https://mathoverflow.net/questions/258920 | 4 | Kosinski in his book "Differential Manifolds" states:
"A closed tubular neighbourhood $E$ of a compact submanifold $M$, which is closed neighbourhood in $N$, can always bee realised as a closed disc subbundle of a tubular neighbourhood of $M$".
He proves this statement like that:
"At first we **reparametrize** i... | https://mathoverflow.net/users/84618 | Does every disc bundle come from a vector bundle? | Equip $N$ with a Riemannian metric, and prove that the normal exponential map to any compact submanifold is a diffeomorphism onto its image on some closed $\epsilon$-neighborhood of the the zero section of the normal bundle. This is similar to proving that the exponential map is a local diffeomorphism, and boils down t... | 4 | https://mathoverflow.net/users/1573 | 258935 | 116,842 |
https://mathoverflow.net/questions/258895 | 4 | Consider the square grid $$\{-n, -n+1, ..., n-1, n\} \times \{-n, -n+1, ..., n-1, n\} \subset \mathbb{Z}^2$$ As usual, connect $(i,j)$ to $(i',j')$ with an edge if $|i-i'|+|j-j'|=1$.
We randomly specify the environment as follows: At each vertex we designate one of the 4 outgoing directions to have probability 1/2, ... | https://mathoverflow.net/users/101909 | Symmetry of random walk in a random environment | Your random walk can be thought of as a isotropic diffusion with a random iid drift. This is very close to the situation studied here: <http://link.springer.com/article/10.1007/s00222-005-0477-5> (ok, they are replacing the discrete model with a continuum one, but this isn't really important and you can find previous w... | 3 | https://mathoverflow.net/users/5678 | 258938 | 116,843 |
https://mathoverflow.net/questions/258940 | 1 | If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete.
Let $\kappa$ be an infinite cardinal. Is there a connected $T\_2$-space $(X,\tau)$ and a discrete subset $S\subseteq X$ such that $|X| = |S| = \kappa$?
| https://mathoverflow.net/users/8628 | Large discrete subsets of connected $T_2$-spaces | For any cardinal $\kappa$ at least the size of the continuum, the "really long line" of length $\kappa$ is an example.
This space, let's call it $L\_\kappa$, is defined as follows. Begin with the ordinal $\kappa$ with its usual order topology. Then, for any $\alpha \in \kappa$, connect $\alpha$ and $\alpha+1$ with a ... | 3 | https://mathoverflow.net/users/70618 | 258943 | 116,845 |
https://mathoverflow.net/questions/258931 | 3 | A **smooth vector bundle** of rank $n$ is usually defined as a smooth map $p: E \longrightarrow B$ together with **a real vector space structure on each fiber** $E\_b := p^{-1}(b)$ such that:
* **(Local Triviality):** There is an open covering $\{U\_i\}\_i$ of $B$ and trivialisations $\{\phi\_i\}\_i$ where
$\phi\_i: ... | https://mathoverflow.net/users/103274 | Vector Bundle Structure | Consider the torus $T^2$, its tangent bundle $p:TT^2\rightarrow T^2$ is an $R^2$-differentiable bundle.You can also see this bundle as an $O(n)$-bundle by defining differentiable metrics on $T^2$, here $G=O(2)$ acts on $R^2$. There exist flat and non flat differentiable metrics on the torus which define non equivalent ... | 1 | https://mathoverflow.net/users/80891 | 258949 | 116,847 |
https://mathoverflow.net/questions/258222 | 3 | I have asked the following question in math.stackexchange, but I could not receive the answer. See [here](https://math.stackexchange.com/questions/2063463/solving-a-system-of-equations-involving-smooth-functions).
Suppose $h\_{i\overline{j}}$, where $1\leq i, j\leq n$, are functions defined on $\mathbb{C}^n$ such tha... | https://mathoverflow.net/users/30375 | Solving a system of equations involving smooth functions | I find a possible way to solve it, assuming some properties of polynomials which I will state in the end:
Write
$p\_j(z)=a\_j+\sqrt{-1}b\_j$ for $1\leq j\leq n$. Equating the real and imaginary parts,
we see that solving $(1)$ is equivalent to solving
$$\tag{2}
\frac{\partial a\_j}{\partial x\_i}+\frac{\partial b\_j}{... | 0 | https://mathoverflow.net/users/30375 | 258952 | 116,848 |
https://mathoverflow.net/questions/258951 | 2 | We identify the vector space tensor product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$ with $\mathbb{R}^{mn}$
Let $X$ be the space of all non zero simple tensors $X=\{a\otimes b \mid a\in \mathbb{R}^{n} \setminus \{0\}, \;b\in \mathbb{R}^{m} \setminus \{0\}\}$.
Let $\pi:\mathbb{R}^{mn}\setminus \{0\} \to \mathbb{R}P^{(m... | https://mathoverflow.net/users/36688 | Are these vector bundles, trivial bundle? | There is an evident map $\mathbb{R}P^{m-1}\times\mathbb{R}P^{n-1}\to PX$, which is easily seen to be an isomorphism. (This is called the Segre embedding.) If we pull back the tautological bundle of $PX$, we get the tensor product of the tautological bundle $M$ over $\mathbb{R}P^{m-1}$ and the tautological bundle $N$ ov... | 3 | https://mathoverflow.net/users/10366 | 258961 | 116,852 |
https://mathoverflow.net/questions/258957 | 8 | Let $\Gamma$ be a finitely generated residually finite group. For a subgroup of finite index $\Lambda<\Gamma$ let us denote by $\pi\_\Lambda:\Gamma\rightarrow \Gamma/\Lambda$ the quotient map. Is it possible to find a subgroup $H<\Gamma$ such that the restriction of $\pi\_\Lambda$ to $H$ is surjective for every finite ... | https://mathoverflow.net/users/15125 | Subgroups of residually finite groups | A general class of such groups are [maximal subgroups](https://groupprops.subwiki.org/wiki/Maximal_subgroup) $H<\Gamma$ of infinite index. Such a subgroup $H$ must surject any finite quotient of $\Gamma$. For finitely generated linear groups like $SL\_n(\mathbb{Z})$ which are not virtually solvable, maximal subgroups o... | 7 | https://mathoverflow.net/users/1345 | 258969 | 116,857 |
https://mathoverflow.net/questions/234709 | 6 | The following form of Hensel's Lemma in Algebraic Geometry is well-documented in the literature:
>
> $\textbf{Theorem 1}$: Let $R$ be an Henselian local ring with maximal ideal $\mathfrak{m}$, and let $X$ be a smooth $R$-scheme. Then $X(R)\to X(R/\mathfrak{m})$ is surjective.
>
>
>
One can for example refer to... | https://mathoverflow.net/users/47722 | Reference for Hensel's Lemma in Algebraic Geometry | The question has been answered in the comments
| 1 | https://mathoverflow.net/users/47722 | 258979 | 116,863 |
https://mathoverflow.net/questions/258874 | 7 | Let $D\subset\mathbb{C}$ be the open unit disk. Suppose $f,g,F,G:D\rightarrow\mathbb{C}$ are analytic functions linked by
$$\vert f(z)\vert^2+\vert g(z)\vert^2=\vert F(z)\vert^2+\vert G(z)\vert^2; \qquad \forall z\in D.$$
>
> **Question 1.** If $f\neq \alpha g$ and $g\neq\beta f$ for any $\alpha, \beta\in\mathbb{C... | https://mathoverflow.net/users/66131 | modulus-related analytic functions | In fact, more is true. If $f\_j$ are linearly independent and $F\_i$ are
linearly independent, and
$$\sum\_{j=1}^n|f\_j|^2=\sum\_{i=1}^m|F\_i|^2,$$
then $m=n$ and $F\_i$ are obtained from $g\_i$ by a unitary transformation.
See, for example, <https://arxiv.org/pdf/math/0007030.pdf>, section 3. This is called the "Calab... | 2 | https://mathoverflow.net/users/25510 | 258982 | 116,865 |
https://mathoverflow.net/questions/258844 | 9 | My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals on spaces of continuous functions on a locally compact space to measures. I want to do it in a different way, and would ... | https://mathoverflow.net/users/49155 | Alternative proof of a theorem of Riesz | A proof along the lines that you describe was [worked out by V.S. Sunder here](https://www.imsc.res.in/~sunder/rrt1.pdf).
There are a few different approaches that you can take to reduce the Riesz Representation Theorem to a class of simpler spaces. For compact spaces, you can either reduce to compact metric spaces l... | 5 | https://mathoverflow.net/users/99234 | 258988 | 116,867 |
https://mathoverflow.net/questions/258611 | 31 | This question is a spin-off of [this one](https://mathoverflow.net/questions/258525/how-do-i-solve-this-displaystyle-f-ef-1), in which the OP asks whether there is a solution $f:\mathbb R\to\mathbb R$ of the functional equation (not exactly an ODE) $f'=e^{f^{-1}}$, where $f^{-1}$ is the compositional inverse of $f$. Th... | https://mathoverflow.net/users/37103 | $f'=e^{f^{-1}}$, again | There is no analytic local solution at $0$ to $f'=e^{f^{-1}}$, $f(0)=0$, that is, the formal power series solution is diverging. Together with the solution given in comments by fedja, this means the actual scenario is **2.** For convenience of notation, I shall consider the equivalent equation
$$\begin{cases} g' =e^{g\... | 17 | https://mathoverflow.net/users/6101 | 258991 | 116,869 |
https://mathoverflow.net/questions/258987 | 1 | This question comes from me reading Van den Bergh's paper(page 12 middle of the proof).<https://arxiv.org/abs/math/0211064>
Let $X=Spec R$ where $R$ is a complete local ring over an algebraically closed field $k$,$x$ its unique closed point. $Y$ an $X$-scheme. Define $\hat{Y\times Y}=(Y\times Y)\times\_{X\times X} Sp... | https://mathoverflow.net/users/48616 | $Y\times_X Y$ a closed subscheme of $\hat{Y\times Y}$? | Note that we can equivalently describe $Y \times\_X Y$ as $(Y \times Y) \times\_{X \times X} X$, where $X \to X \times X$ is the diagonal map $\Delta\_X$ (this can easily be seen from the functor of points point of view, or by a quick diagram-theoretic argument). Note that $\Delta\_X$ is a closed immersion since $X$ is... | 4 | https://mathoverflow.net/users/82179 | 258992 | 116,870 |
https://mathoverflow.net/questions/258815 | 1 | I am reading a book ***"Fourier Series and Integrals" by Dym & McKean***.
There is an exercise (Page 106):
>
> **Exercise:** *Check that if $f$ is a real, even, summable function and
> if $f(0+)$ and $f(0-)$ exist, then either $f(0-) =f(0+)$ or $\hat f(\gamma)$ changes sign
> infinitely often as $|\gamma| \to... | https://mathoverflow.net/users/76453 | Fourier transform either changes sign infinitely often far out or is continuous at $x=0$ | The exercise is stated in Dym and McKean with a mistake. The correct statement is
If $f$ is real, even, the finite limits $f(x\pm 0)$ exist for all $x$,
and $\hat{f}$ does not change sign for $|x|>A$, then $f$ is continuous at all points.
The statement is in the paper of M. Kac (1938) to which Dym and McKean refer... | 10 | https://mathoverflow.net/users/25510 | 258997 | 116,872 |
https://mathoverflow.net/questions/258914 | 28 |
>
> **Question.** Is the polynomial $x^{2k+1} - 7x^2 + 1$ irreducible over $\mathbb{Q}$ for every positive integer $k$?
>
>
>
It is irreducible for all positive integers $k \leq 800$.
| https://mathoverflow.net/users/38889 | Is $x^{2k+1} - 7x^2 + 1$ irreducible? | Here is a proof, based on a trick that can be used to prove that
$x^n + x + 1$ is irreducible when $n \not\equiv 2 \bmod 3$.
We work with Laurent polynomials in $R = \mathbb Z[x,x^{-1}]$; note that
$R$ has unit group $R^\times = \pm x^{\mathbb Z}$.
We observe that for $f \in R$, the sum of the squares of the coeffici... | 43 | https://mathoverflow.net/users/21146 | 259001 | 116,874 |
https://mathoverflow.net/questions/258999 | 1 | What would be the probability density function (pdf) of the complex random variable given below?
$$Z = \sum\_{i=1}^{M}{x\_{i}^{\*}y\_{i}}$$
where $x\_i, y\_i$ are independent r.v.'s with $\mathcal{CN}(0,c)$.
| https://mathoverflow.net/users/103291 | What is the probability density function (pdf) of the dot product of M complex normal random variables? | The joint characteristic function $\Psi(\omega\_1,\omega\_2)$ of the real and imaginary parts of $Z=z\_1+iz\_2$ is derived in: [Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications](http://ieeexplore.ieee.org/document/6047542/) (2011). That publication is behind a pay wall. You can find... | 2 | https://mathoverflow.net/users/11260 | 259002 | 116,875 |
https://mathoverflow.net/questions/259016 | 0 | Is there a category $\mathcal{A}$ of topological spaces, saturated with respect to homeomorphism relation, which is maximal with respect to the following property?
For every $X,Y \in \mathcal{A}$, $X$ is homeomorphic to $Y$ if and only if $X$ can be embedded in $Y$ and $Y$ can be embedded in $X$.
| https://mathoverflow.net/users/36688 | A category of topological spaces with certain anti symmetric property | One way of answering is just by applying Zorn's lemma.
By the Cantor-Schroeder-Bernstein theorem, we really only have to work one cardinality at a time; that is, for each cardinal $\kappa$, find a maximal class/category $C\_\kappa$ of spaces of cardinality $\kappa$ with this property. Once that is done, then take th... | 4 | https://mathoverflow.net/users/2926 | 259019 | 116,881 |
https://mathoverflow.net/questions/259017 | 1 | Are there two non homeomorphic continua $X,Y$ such that $X $ can be embedded in $Y$ **but there is no** topological space $Z$ with $$X<Z<Y.$$
The later relation means that $Z$ is homeomorphic neither to $X$ nor to $Y$ but $Z$ contains a copy of $X$ and can be embedded in $Y$.
What about if we remove the metrizabili... | https://mathoverflow.net/users/36688 | Two consecutive continua | If you really allow $Z$ to be any topological space, then no such example exists. If $X$ embeds as a proper subset of $Y$, then for any $y \in Y \setminus X$, we have open neighborhoods separating $X$ and $y$ (since $X$ is compact and $Y$ is Hausdorff). So if we let $Z = X \cup \{y\}$, then the inclusion of $X$ into $Z... | 4 | https://mathoverflow.net/users/4832 | 259025 | 116,885 |
https://mathoverflow.net/questions/259027 | 1 | Let $K$ be a compact set in $\Bbb C$. Let $\mathcal P(K)$ be the closed algebra generated by polynomials on $K$ and $\mathcal R(K)$ the closed algebra generated by rational functions without poles in $K$.
It is clear that $\mathcal P(K)\subset\mathcal R(K)$. If $\Bbb C\backslash K$ is a connected set, then [Runge's a... | https://mathoverflow.net/users/80191 | $\mathcal P(K)=\mathcal R(K)$ iff $\Bbb C\backslash K$ is connected | The following is not an entirely satisfactory answer but I thought I would put it down as something to be improved later. (I don't remember the context in which GRA stated this fact, so perhaps he had in mind some argument based on nearby parts of the book.)
It is a theorem that in a unital Banach algebra $A$ the spe... | 3 | https://mathoverflow.net/users/763 | 259029 | 116,886 |
https://mathoverflow.net/questions/258950 | 5 | I have been looking for a couple of days for the answer to this question to no avail. Let me define what $\Sigma$-injective is.
>
> Let $R$ be a unital, not necessarily commutative ring. A left $R$-module $I$ is called $\Sigma$-injective if an arbitrary direct sum of copies of itself is again injective.
>
>
>
... | https://mathoverflow.net/users/17353 | When is every injective module $\Sigma$-injective? | A result of Faith and Walker (page 205 in C. Faith, E. A. Walker, *Direct-sum representations of injective modules,* J. Algebra 5 (1967), 203-221) answers your question:
>
> If each injective left $R$-module is $\Sigma$-injective, then $R$ is left noetherian.
>
>
>
(In fact, a sufficient condition for $R$ to b... | 2 | https://mathoverflow.net/users/11025 | 259039 | 116,888 |
https://mathoverflow.net/questions/259034 | 2 | I am finding the following first order estimate.
>
> **Question.** As $y\rightarrow\infty$,
> $$\sum\_{n=1}^{\infty}\frac{\log n}n\,\arctan\frac{y}n\,\,
> \sim\,\,\frac{\pi}4\log^2y.$$
> Is it true?
>
>
>
| https://mathoverflow.net/users/66131 | asymptotic estimate for log-tan sum | Yes, it is true.
Your sum has the same asymptotics as the integral $\int\_1^\infty \frac{\log x}{x}\arctan{\frac{y}x}dx$ by standard arguments (the integrated function is decreasing for $x>e$, say, and each specific summand is bounded, this is quite enough).
Next, we denote $x=y/z$ to get the integral $\int\_0^{y}\... | 6 | https://mathoverflow.net/users/4312 | 259040 | 116,889 |
https://mathoverflow.net/questions/259043 | 7 | Let us consider the space $L^2:=L^2(\mathbb{R}^n,\mathbb{C})$ and the associated scalar product $S(f,g):=\int f \overline g$. In distribution theory, we have a situation where we have to deal with two different identifications (which makes things a little bit tricky) :
* if we identify $L^2$ with its antidual, an el... | https://mathoverflow.net/users/103312 | On the definition of Hilbert spaces and real structures on Hilbert spaces | Good question. It does seem like most naturally occurring complex Hilbert spaces come equipped with a natural real structure. Incidentally, I pointed out in [this paper](https://arxiv.org/abs/math/9908119) that the same is true of Hilbert modules over C\*-algebras, and in this case you can use the natural involution to... | 6 | https://mathoverflow.net/users/23141 | 259047 | 116,891 |
https://mathoverflow.net/questions/259015 | 2 | **Edit:** According to comment of " Fedor Petrov", I revise my question
Are there two compact convex subsets $X,Y$ of a Banach space with the following property?
>
> They are not homeomorphic spaces but $X$ can be embedded in $Y$ and $Y $ can be embedded in $X$?
>
>
>
| https://mathoverflow.net/users/36688 | The anti-symmetric property of the collection of all compact convex sets of a Banach space | If I understand you correctly, the question is answered in the book Bessaga-Pelczynski, "Selected topics in infinite-dimensional topology", Warsaw, 1975. Combine Theorem 3.1 and Proposition 3.1 in that book (on page 100). The answer is: all such infinite-dimensional sets are homeomorphic to the Hilbert cube. (As for fi... | 4 | https://mathoverflow.net/users/37822 | 259051 | 116,893 |
https://mathoverflow.net/questions/259054 | 14 |
>
> Is there a natural geometric generalization of the [winding number](https://en.wikipedia.org/wiki/Winding_number) to higher dimensions?
>
>
>
I know it primarily as an important and useful index for closed, plane curves
(e.g., the Jordan Curve Theorem),
and for its role in Cauchy's theorem integrating holomo... | https://mathoverflow.net/users/6094 | Generalization of winding number to higher dimensions | This is a very naive answer which I am sure you already considered, but isn't the most obvious generalization just given by the topological degree (<https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping>)?
The winding number of $f:S^1\rightarrow \mathbb{R}^2$ around $p$ is just the degree of the composition of... | 28 | https://mathoverflow.net/users/103319 | 259055 | 116,894 |
https://mathoverflow.net/questions/259049 | 10 | The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral
$$\int\_0^{\infty}dz\prod\_{k=1}^{\infty}\cos\frac{z}{2^k}$$
into fun once you realize $\prod\_k\cos\frac{z}{2^k}=\frac{\sin z}z$, because then it's well-known that $\int\_0^{\infty}\frac{\sin z}zdz=\frac{\pi}2$.
I've found the followi... | https://mathoverflow.net/users/66131 | This is not a dyadic cosine-product | It's not *quite* $\pi/4$ . . .
Using the same formula
$\prod\_{m=1}^\infty \cos(x/2^m) = \frac{\sin x}{x} = \text{sinc}\,x$,
we write the integrand as
$$
\prod\_{n=1}^\infty \text{sinc}\,\frac{2z}{2n-1},
$$
and then the integrals
$$
I\_N := \int\_0^\infty dz \prod\_{n=1}^N \text{sinc}\,\frac{2z}{2n-1}
$$
of the parti... | 24 | https://mathoverflow.net/users/14830 | 259060 | 116,897 |
https://mathoverflow.net/questions/259069 | 3 | I know the definition of intersection multiplicity in algebraic geometry. However, I think it is possible to define it for some sort of non-algebraic functions such as $y=\sin x$.
>
> How to define this?
>
>
>
| https://mathoverflow.net/users/nan | Intersection multiplicity in the non-algebraic case | If we work in the category of holomorphic functions, then we can give the following definition, that generalises to the complex-analytic setting the classical intersection multiplicity used in algebraic geometry.
>
> **Definition.** Let $f$, $g$ be two holomorphic functions defined in a neighborhood of a point $\b... | 7 | https://mathoverflow.net/users/7460 | 259088 | 116,904 |
https://mathoverflow.net/questions/259028 | 1 | Suppose $A = [A\_1, A\_2, \ldots, A\_n]$ is a matrix, and each $A\_i$ is a column-wise sub-matrix of $A$. Given a vector $v \in \mathbb{R}^n$, a bound of $\| \sum\_{i=1}^n A\_i v\_i \|\_2$ (spectral norm) in terms of $\|A\|\_2$ (spectral norm) and $\|v\|\_2$ (Euclidean norm) is desired. A naive approach is
\begin{align... | https://mathoverflow.net/users/103308 | tighter bound on the sum of sub-matrices | $$\left\|\sum A\_iv\_i\right\|=\sup\_{\|x\|=1} \left\|\sum A\_iv\_ix\right\|\leq \sup\_{\|x\_i\|=|v\_i|} \left\|\sum A\_ix\_i\right\|=\sup\_{\|x\_i\|=|v\_i|} \left\|A \begin{pmatrix} x\_1\\ \vdots\end{pmatrix}\right\|\\\leq \sup\_{\|x\_i\|=|v\_i|} \|A\| \left\|\begin{pmatrix} x\_1\\ \vdots\end{pmatrix}\right\|= \|A\| \... | 1 | https://mathoverflow.net/users/97148 | 259095 | 116,906 |
https://mathoverflow.net/questions/259100 | 13 | Suppose $V=L$ + reasonable hypotheses (e.g. "ZFC has a countable transitive model"). Call a countable ordinal $\alpha$ *memorable* if for some countable $\beta$, $\alpha$ is definable *without parameters* in every $L\_\gamma$ with $\beta<\gamma<\omega\_1$.
My question is:
>
> Are there uncountably many memorable ... | https://mathoverflow.net/users/8133 | Memorable ordinals | It is a very nice question, but unfortunately, the answer is no.
**Theorem.** There are only countably many memorable ordinals.
**Proof.** Let $\delta$ be a countable ordinal with $L\_\delta\prec
L\_{\omega\_1}$. I claim that every memorable ordinal is less than
$\delta$.
To see this, suppose that $\alpha$ is me... | 14 | https://mathoverflow.net/users/1946 | 259105 | 116,908 |
https://mathoverflow.net/questions/259076 | 2 | This question is mainly a reference request about the order of a Brauer class on a smooth projective variety over $\mathbb{C}$. Namely, let $X$ be a smooth complex projective variety and $\alpha$ be a Brauer class on $X$.
I have read in a few papers on the ArXiv that the order of $\alpha$ divides the rank of any $\al... | https://mathoverflow.net/users/37214 | Twisted line bundles Brauer class | OK so I think I understand what I missed from the beginning. If I take a particular $\alpha$-twisted sheaf, say $F$, then the object $F$ itself, as a twisted sheaf, **depends on** the Cech cocycle I use to represent $\alpha$. In particular, the Cech cocycle used to define $\mathcal{O}\_{Y/X}(1)$ on a Severi Brauer vari... | -1 | https://mathoverflow.net/users/37214 | 259107 | 116,909 |
https://mathoverflow.net/questions/198744 | 6 | The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in the quiver representation. We say that a quiver variety **satisfies Kirwan surjectivity** if these classes generate the co... | https://mathoverflow.net/users/66 | For which quiver varieties is Kirwan surjectivity known? | As an update, Kevin McGerty and Thomas Nevins prove Kirwan surjectivity for Nakajima quiver varieties in [this recent paper](https://arxiv.org/abs/1610.08121).
| 4 | https://mathoverflow.net/users/2384 | 259110 | 116,910 |
https://mathoverflow.net/questions/259147 | 2 | For any set $X$ set $[X]^2 = \big\{\{x,y\}: x,y \in X, x\neq y\big\}$.
Suppose $G=(V,E)$ is a simple, undirected graph, let $v^\* \notin V$. We let $a(G)$ be the minimal number of edges that we need to attach $v^\*$ to the vertices of $G$ such that the chromatic number increases, formally: $$a(G) = \min\{|Z| : Z\subse... | https://mathoverflow.net/users/8628 | Vertex-adding number | The vertex-adding number can be arbitrarily large compared to the chromatic number. To see this consider a long odd cycle, $C\_{2k+1}$. Then $\chi(C\_{2k+1})=3$, but $a(C\_{2k+1})=2k+1$.
Note that $a(C\_{2k+1})=2k+1$ because for any proper subset $U$ of $V(C\_{2k+1})$, there is a $3$-colouring of $C\_{2k+1}$ that on... | 6 | https://mathoverflow.net/users/2233 | 259153 | 116,915 |
https://mathoverflow.net/questions/259155 | 15 | As far as I know, in modern physics we assume that the underlying field of work is the field of real numbers (or complex numbers). Imagine one second that we make a crazy assumption and suggest that the fundamental equations of physics can be expressed with $p$-adic numbers. What could be really rewritten formally? Doe... | https://mathoverflow.net/users/103287 | $p$-adic numbers in physics | For an overview of applications of p-adic numbers in physics I would refer to the [Wikipedia](https://en.wikipedia.org/wiki/P-adic_quantum_mechanics) and [Physics.stackexchange](http://physics.stackexchange.com/questions/15252/p-adic-quantum-mechanic) links, and to this [nLab entry.](https://ncatlab.org/nlab/show/p-adi... | 28 | https://mathoverflow.net/users/11260 | 259156 | 116,916 |
https://mathoverflow.net/questions/258824 | 4 | Starting from a weakly compact cardinal $\kappa$ one can use Neeman's side condition forcing to obtain the tree property at $\aleph\_2$. Also It can be obtained using other side condition constructions such as Velickovic's method. By the way in the final model $\kappa$ is $\aleph\_2$ and $2^{\aleph\_0}=\aleph\_2$, I ma... | https://mathoverflow.net/users/38866 | Side condition forcing and tree property | I think the answer is no, but there are some technical issues that I don't see how to solve at the moment. Let me instead 1) use Krueger's side condition forcing instead of Neeman's; and 2) work with regular embeddings (a.k.a. complete embeddings) in the other direction, instead of projections. Some remarks about these... | 1 | https://mathoverflow.net/users/26319 | 259174 | 116,922 |
https://mathoverflow.net/questions/259170 | 2 | I am trying to prove an inequality that seems to be intuitively true, however I cannot arrive at a rigorous argument.
Consider a sequence of i.i.d random variables $X\_1,X\_2,....$, that take values in $[0,\infty)$ such that $\mathbb{E}[X\_i] = \mu$. Let $\beta\_1 > \beta\_2 \geq 0$. Suppose,
\begin{align\*}
\mathbb... | https://mathoverflow.net/users/51716 | Probability of an atypical event given a typical event | I think the inequality you want is false. Consider random variables $X\_i$ taking two values: 1 and 100 with probability one half each. Set $\beta\_2=2$ and $\beta\_1=101$. Now conditioning on $S\_2$ being small actually makes $S\_1$ larger.
| 2 | https://mathoverflow.net/users/11054 | 259175 | 116,923 |
https://mathoverflow.net/questions/259145 | 1 | Implicit in the technique of forcing is the following relative consistency result:
>
> If $\mathfrak M$$\vDash$$T$, and therefore $T$ is consistent (where $\mathfrak M$ is the ground model) then if $\mathfrak M$$[$$G$$]$$\vDash$$T^{'}$, $T^{'}$ is consistent (since I am assuming $T$ and $T^{'}$ are first-order theo... | https://mathoverflow.net/users/20597 | Forcing the consistency of $ZF$ from a fragment of $ZF$ | Leaving aside the issues around consistency (which I don't really follow - see Andreas' comments), it seems to me that the mathematical question you're asking is:
>
> If $M\models ZF-Inf$, can there be a class forcing extension $M[G]$ of $M$ which satisfies full $ZF$?
>
>
>
(Note that $ZF$ and $ZF-Inf+Inf$ are... | 2 | https://mathoverflow.net/users/8133 | 259179 | 116,924 |
https://mathoverflow.net/questions/259178 | 2 | Let $G$ be a finite group, $N$ its normal subgroup, and $\pi:N\to{\mathcal B}(X)$ a unitary representation of $N$ in a Hilbert space $X$. Consider the induced representation $\pi':G\to{\mathcal B}(L\_2(F,X))$, where $F=G/N$.
Let us extend the representations $\pi$ and $\pi'$ to the group algebras:
$$
\pi:{\mathbb C... | https://mathoverflow.net/users/18943 | Does the induced representation preserve norm? | No. Let $G=S\_3$ generated by 3-cycle $x$ and involution $y$. Let $N=A\_3$. Let $ x$ act on $X=\mathbb C$ by multiplication by the third root of unity $\omega$. Let $a=1+ix$. Then $\pi(a)$ has norm $|1+i\omega|$ but $\pi'(a)y=(1+i\omega^2)y$ showing $\pi'(a)$ has larger norm.
| 3 | https://mathoverflow.net/users/37103 | 259182 | 116,925 |
https://mathoverflow.net/questions/259180 | 1 | Let $(a\_{i})$ be an increasing sequence of positive integers given by a linear recurrence $a\_{i+n}=c\_{n}a\_{i+n-1}+\dots +c\_{1}a\_{i}$ with $c\_{i}\in\{-1,0,1\}$ and $a\_{i}=2^{i}$ for $i=1,\dots n$ such that the characteristic polynomial $p(x)=-x^{n}+c\_{n}x^{n-1}+\dots +c\_{1}x^{0}$ has a unique dominating real r... | https://mathoverflow.net/users/23542 | A question on linear recurrence | I don't know what "dominating" means so perhaps this isn't a counterexample, but how about $-p(x)=x^5 - x^4 - x^3 - x^2 + x + 1=(x^2-x-1)(x^3-1)$ (which has a unique real root greater than 1), with $a\_1,...,a\_5=1,2,3,5,8$ and $s\_1=s\_2=-1$ and $s\_3=1$ giving $\sum s\_ix^i=x^3-x^2-x$?
| 3 | https://mathoverflow.net/users/1384 | 259183 | 116,926 |
https://mathoverflow.net/questions/259176 | -2 | does Oppenheim conjecture hold for specific quadratic forms? or for generic quadratic forms with a set of measure 1.
for example can we find $x,y,z \in \mathbb{Z}$ with
$$|x^2 + y^2 - \sqrt{3} z^2| < 10^{-6}$$
is that implied by Oppenheim conjecture?
where are elementary expositions of proof? what is the current s... | https://mathoverflow.net/users/1358 | Find $x,y,z \in \mathbb{Z}$ with $|x^2 + y^2 - \sqrt{3} z^2| < 10^{-6}$ | $x=6627,y=314048,z=238678$ is an answer to the question in the title but I'm sure that the real question is something I don't really understand.
| 18 | https://mathoverflow.net/users/1384 | 259185 | 116,928 |
https://mathoverflow.net/questions/258566 | 3 | Suppose you and I are playing a two-player zero-sum game repeatedly. There is a payoff matrix $(A\_{ij})$, and if I play action $i$ and you play action $j$ then I receive $A\_{ij}$ and you receive $-A\_{ij}$. I am not clever enough to compute my best strategy (my maximin strategy), but I can just manage to compute a be... | https://mathoverflow.net/users/20598 | Iterative approximate solutions to games | The keywords I was looking for were "fictitious play" and "Brown--Robinson process". The short answer seems to be that it works, sometimes. For more information see <https://en.wikipedia.org/wiki/Fictitious_play>.
| 1 | https://mathoverflow.net/users/20598 | 259189 | 116,929 |
https://mathoverflow.net/questions/257426 | 9 | I recently got into Lickorish's paper [Prime knots and tangles](http://www.ams.org/journals/tran/1981-267-01/S0002-9947-1981-0621991-2/S0002-9947-1981-0621991-2.pdf) and a question, which I didn't have the first time I read it, naturally emerged.
The Thurston-Perelman Geometrization Theorem asserts that given a comp... | https://mathoverflow.net/users/58934 | On the geometrization of double branched covers | As Ian Agol mentioned in his comment, the OP's question can be thought of in terms of the Orbifold Theorem. There are two (contemporaneous and) independent proofs of the Orbifold Theorem:
*Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff*, MR 1778789 [**Three-dimensional orbifolds and cone-manifolds**](http:/... | 7 | https://mathoverflow.net/users/27453 | 259191 | 116,930 |
https://mathoverflow.net/questions/259106 | 18 | The group $SL\_n \times SL\_n \times SL\_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\sum\_{i=1}^n e\_i \otimes e\_i \otimes e\_i \in \mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$$ is known to be G... | https://mathoverflow.net/users/18060 | Explicit invariant of tensors nonvanishing on the diagonal | Let me start with some remarks about the classical symbolic method (without which one cannot understand 19th century invariant theory)
and multisymmetric functions.
I will use an example first. Take four series of three variables $a=(a\_1,a\_2,a\_3)$, $b=(b\_1,b\_2,b\_3)$,
$c=(c\_1,c\_2,c\_3)$ and $d=(d\_1,d\_2,d\_3)$.... | 9 | https://mathoverflow.net/users/7410 | 259195 | 116,932 |
https://mathoverflow.net/questions/259138 | 6 | Does any one prepared a list of errata for Linear algebraic groups by Springer.
I could not find any in Google search.
First typo that i came across is in page 6, Regular functions and ringed spaces:
>
> If $U $ and $V$ are open subsets and $U\subset V$, restriction defines a $k$ - algebra homomorphism $\mathca... | https://mathoverflow.net/users/nan | Errata for Linear algebraic groups by Springer | The answer to your specific question is that it's really stated backwards. More generally, your question about lists of errata comes up fairly often here and is hard to answer in detail. It's a legitimate question to ask when looking at relatively advanced books in mathematics. (Maybe a special tag is needed?) But unfo... | 13 | https://mathoverflow.net/users/4231 | 259198 | 116,934 |
https://mathoverflow.net/questions/259217 | 9 | Landau proved that the mean density of integers of the form $a^2+b^2$ up to $x$ is $K\frac{x}{\sqrt{\log x}} (1+o(1))$, where $K$ is an explicit constant. One proof is based on the fact that a prime $p$ is of this form iff $p\neq 3\mod 4$, and so we can express the generating function for these integers in terms of the... | https://mathoverflow.net/users/2042 | What is the density of integers of the form $a^2+nb^2$? | Yes there is quite a bit in the literature on this problem. Apparently it was first solved by Bernays in his 1912 PhD thesis under Landau. The density is as expected (namely proportional to $x/\sqrt{\log x}.$)
A modern paper on the topic is:
Brink, Moree, Osburn - Principal forms $X^2+nY^2$ representing many intege... | 11 | https://mathoverflow.net/users/5101 | 259220 | 116,941 |
https://mathoverflow.net/questions/259164 | 1 | Let $G$ be a finite group with $n$ elements and let $\mathbb{Q}[x\_1,\cdots,x\_n]^G = \mathbb{Q}[g\_1,\cdots,g\_m]$. Then there exist polynomials $s\_j \in \mathbb{Q}[y\_1,\cdots,y\_m]$ for $j=1,\cdots,n$ such that
$e\_j(x\_1,\cdots,x\_n) = s\_j(g\_1(x\_1,\cdots,x\_n), \cdots, g\_m(x\_1,\cdots,x\_n))$ for $j=1,\cdots,n... | https://mathoverflow.net/users/nan | A question about the invariants of a finite group | I think it's irreducible in $R$ as you suggest. Here's a sketch which I think works. If the polynomial factored in a non-trivial way, then because of the $t^n$ term the factors must have degree less than $n$ in $t$ (consider the factorization in $\mathbb{Q}(y\_1,y\_2,\ldots,y\_n)[t]$; note also that we can assume that ... | 1 | https://mathoverflow.net/users/1384 | 259222 | 116,942 |
https://mathoverflow.net/questions/259214 | 4 | I am now trying to construct a $Z\_2\times Z\_2$-cover over $\mathbf{P}^n$. From the paper of Pardini, we need line bundles $L\_1$, $L\_2$, $L\_3$ and divisors $D\_1$, $D\_2$, $D\_3$ which satisfies the following 6 relations.
$$2L\_i\equiv D\_j+D\_k\quad \textrm{and} \quad L\_i+L\_j\equiv L\_k+D\_k$$
However, the p... | https://mathoverflow.net/users/75699 | equations for a bidouble cover | As abx note in the comments, you miswrote the equations. Still the singular point remains.
The point is that the "reduced" data works well algebraically, as indeed you can deduce $L\_3$ from the other data. But if you try to "reduce" analogously the equations you are eliminating some variables and that correspond ge... | 4 | https://mathoverflow.net/users/46104 | 259224 | 116,943 |
https://mathoverflow.net/questions/259225 | 3 | Let us work in ZFC set theory.
1: We name "Very very weak universe (VVWU)" a set u such that if a and b are two member sets of u, then every function between a and b is also a member set of u; The empty set, and the singleton of the empty set are such sets. Do we know a description of VVWU's ?
2: We name "Very weak un... | https://mathoverflow.net/users/30395 | Very weak notions of Universes in ZFC | Unfortunately, I think that this universe concept will not be very useful, since although you add all functions from $a$ to $b$, you haven't ensured that the *elements* of $a$ or $b$ are in the universe, and so you won't be able to evaluate those functions while living inside that universe. That is, if you try to use t... | 9 | https://mathoverflow.net/users/1946 | 259228 | 116,944 |
https://mathoverflow.net/questions/186900 | 5 | For any cardinal $\kappa$, let $K\_\kappa$ denote the complete graph on $\kappa$. We consider the following statements:
(H) If $G$ is a graph and $\chi(G) = \kappa$ then $K\_\kappa$ is a minor of $G$.
(WH) If $G$ is a graph and there is no graph homomorphism $c: G\to K\_\kappa$, then $K\_\kappa$ is a minor of $G$.
... | https://mathoverflow.net/users/8628 | Does the weak Hadwiger conjecture imply the Hadwiger conjecture? | Well, it is possible that both (WH) and (H) are true, in which case (WH) implies (H).
If on the other hand, you are asking if there is a short proof of (H) assuming (WH), then the answer is **no**. For example, Hadwiger's Conjecture for $n=5$ says
>
> If $G$ has no $K\_5$-minor, then $\chi(G) \leq 4$,
>
>
> ... | 6 | https://mathoverflow.net/users/2233 | 259247 | 116,947 |
https://mathoverflow.net/questions/259255 | 3 | Let $(X,\mathcal O\_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O\_X$-modules, one has the *complex of morphisms* \begin{equation} \operatorname{Hom}^\bullet(\mathcal F, \mathcal G),\end{equation}
an... | https://mathoverflow.net/users/20883 | K-injective (also known as hoinjective) complexes of sheaves of modules | Let $j:U\hookrightarrow X$ be the inclusion of an open subset. Then $\mathcal H om\_{\mathcal O\_X}^\bullet (\mathcal A, \mathcal I)(U) = \operatorname{Hom}^\bullet\_{\mathcal O\_U} (j^\ast\mathcal A,j^\ast\mathcal I)=\operatorname{Hom}^\bullet\_{\mathcal O\_X}(j\_!j^\ast\mathcal A,\mathcal I)$. The functors $j\_!$ and... | 3 | https://mathoverflow.net/users/35353 | 259257 | 116,949 |
https://mathoverflow.net/questions/259244 | 3 | Let $p=2^{2m+1}-2^{m+1}+1$ be a prime number, where $m\geq1$ is an integer. Under which conditions can we say that $(kp+1) \mid 2^{4m+2}$, where $k\geq 1$ is a positive integer?
| https://mathoverflow.net/users/30252 | Conditions of solution | As Gerhard says, $m=1$ works giving $p=5$ and $k=3$. I claim that this is the only time it happens.
Let $m\geq2$ be an integer, and set $p=2^{2m+1}-2^{m+1}+1$. I don't care if $p$ is prime or not, but note at this point that $p\geq25$ is odd.
Let's try and work out the multiplicative order of 2 mod $p$. Surprisingl... | 5 | https://mathoverflow.net/users/1384 | 259259 | 116,950 |
https://mathoverflow.net/questions/259243 | 1 | Let $A$ be a Banach algebra, $I$ be a closed two-sided ideal in $A$, and $J$ be a closed two-sided ideal in $I$ such that there is no ideal between $I$ and $J$. Can we see $dim(\frac{I}{J})<\infty$?
Y. DOMAR in "On the ideal structure of certain Banach algebras", proves a lemma like this question with the following d... | https://mathoverflow.net/users/27066 | Dimension of quotient of ideals | If we allow $A$ to be noncommutative then there are well-known natural counterexamples. For instance one can look at $A=B(E)$ for various Banach spaces $E$. I imagine that there should be counterexamples of the form $A=L^1(G)$ but I do not know this for sure.
There are "silly" commutative counterexamples obtained by ... | 4 | https://mathoverflow.net/users/763 | 259266 | 116,953 |
https://mathoverflow.net/questions/259212 | 4 | This solves [this post](https://math.stackexchange.com/questions/2090523/on-x3y3-z3-the-dixonian-elliptic-functions-and-the-borwein-cubic-theta-f) and is also related to this [MO post](https://mathoverflow.net/questions/259013/can-we-use-the-rogers-ramanujan-cfrac-to-parameterize-the-fermat-quintic-x5y) by involving $\... | https://mathoverflow.net/users/12905 | On $x^k+y^k=1$ and the Dixonian elliptic functions | Define the generalized trigonometic functions (discussion of these functions is given in this [answer](https://mathoverflow.net/a/227030/82588))
$$
z=\int\_0^{\sin\_{pr}z}\frac{dt}{\sqrt[p]{1-t^r}},\qquad \cos\_{pr}z=\sqrt[r]{1-(\sin\_{pr}z)^r},\qquad \frac{1}{p}=1-\frac{1}{r}.
$$
It was shown by Edmunds,Gurka, and Lan... | 3 | https://mathoverflow.net/users/82588 | 259267 | 116,954 |
https://mathoverflow.net/questions/259270 | 2 | Let $G$ be a graph with $m$ edges and $n$ vertices. For a fixed integer $s \leq n$, what lower bound can be shown on the number of independent sets with $s$ vertices?
Letting $d$ denote the average degree $d = 2 m/n$, then when $G$ is a union of $d-1$-cliques it has $$
\frac{n (n - (d+1)) \dots (n - (d+1) (s-1))}{s!}... | https://mathoverflow.net/users/9896 | A bound on coefficient of independence polynomial | Yes. If you take the complement of your graph, then you get the clique density problem, which was solved recently: <https://arxiv.org/abs/1212.2454>
| 1 | https://mathoverflow.net/users/955 | 259276 | 116,957 |
https://mathoverflow.net/questions/259285 | 0 | Given a Dirichlet series $$\phi(s)=\sum\_{n\ge1}\frac{a\_n}{n^s}$$
let $\sigma\_{\text{conv}}\in\bar{\mathbb{R}}$ its abscissa of convergence, then we know that $\phi(s)$ is holomorphic on the half-plan $\mathrm{Re}(s)>\sigma\_{\text{conv}},$ then if we denot by $\sigma\_{\text{hol}}$ the abscissa of holomorphy of $... | https://mathoverflow.net/users/95654 | What is the relationship between the abscissa of holomorphy and abscissa of convergence of a Dirichlet series | The answer to your question is no, the counterexample is $L(s, \chi) = \sum\_{n = 1}^{\infty} \chi(n) n^{-s}$ with $\chi$ any non-principal character. $L(s,\chi)$ can be continued to an entire function thus $\sigma\_{\text{hol}} = -\infty$, while $\sigma\_{\text{abs}} = 1$ (abscissa of absolute convergence) and $\sigma... | 9 | https://mathoverflow.net/users/102997 | 259288 | 116,960 |
https://mathoverflow.net/questions/255283 | 2 | I have had a lot of questions lately about root systems and rationality questions. Several people have referred me to the article *Groupes Reductifs* by Borel and Tits, which I am slowly reading. In the meantime, I had a question about relative roots for which I hope there is a simple explanation.
$G$ is a connected ... | https://mathoverflow.net/users/38145 | Two definitions of restricted roots | As far as I can tell, there isn't anything fancy you need to prove the equivalence of these definitions beyond basic facts about rational representations of tori. Let $T$ be a torus, $S$ a closed subgroup, and $\pi: T \rightarrow \textrm{GL}(V)$ a rational representation. Then $V$ breaks up into a direct sum
$$V = \b... | 2 | https://mathoverflow.net/users/38145 | 259289 | 116,961 |
https://mathoverflow.net/questions/259264 | 3 | I'm still trying to learn projective representation theory, so please bear with me. Thanks to YCor and David Speyer for pointing out the issues with my previous formulation of this question. Hopefully this is better.
Below my field will always be $\mathbb{C}$.
Is there a classification of finite groups G with th... | https://mathoverflow.net/users/22763 | Groups with one simple projective representation | Previous answer was broken, here is what I can salvage.
To be clear, I am taking a projective representation to be a map $G \to PGL(V)$ with $\dim V \geq 2$. I call it irreducible if the corresponding representation $\tilde{G} \to GL(V)$, where $\tilde{G}$ is a central extension of $G$, is irreducible. I call two pro... | 5 | https://mathoverflow.net/users/297 | 259291 | 116,963 |
https://mathoverflow.net/questions/259284 | 1 | I'm looking to find a 2-approximation algorithm (pseudocode) for the minimum maximal matching problem. I tried to find one but I did not manage. I want to use it to implement a program in java. Can anyone help in finding pseudocode for this problem?
| https://mathoverflow.net/users/103429 | 2-approximation algorithm for Minimum Maximal Matching (MMM) problem | There is an easy $2$-approximation algorithm for finding a minimum size maximal matching. Simply find *any* maximal matching. Note that a maximal matching $M$ can be found greedily. Initialize $M=\emptyset$. Add any edge $xy$ to $M$, and in $G -x-y$ search for another edge to add and recurse.
Let $m$ be the size of ... | 3 | https://mathoverflow.net/users/2233 | 259292 | 116,964 |
https://mathoverflow.net/questions/259281 | 4 | Let $K/\mathbb{Q}\_p$ be a finite field extension and $\mathcal{O}\_K\subseteq K$ be its ring of integral elements. Let also $G/\mathcal{O}\_K$ be a finite flat $p$-group scheme that is also an $\mathcal{O}\_E$-module where $E/\mathbb{Q}\_p$ is a finite field extension and $\mathcal{O}\_E\subseteq E$ is its ring of int... | https://mathoverflow.net/users/86022 | Serre tensor construction on finite flat group schemes | Thinking about this more, it might be easier than you think. You don't seem to even care about the actions of $\mathcal{O}\_E$ or $\mathcal{O}\_{E'}$ in your question so you should consider what happens if you just forget them. Maybe $G'$ is complicated as an $\mathcal{O}\_{E'}$-group scheme, but if you just consider $... | 3 | https://mathoverflow.net/users/1384 | 259306 | 116,970 |
https://mathoverflow.net/questions/259227 | 1 | Let $X$ and $Y$ be closed Riemannian manifolds and $f,g\colon X\to Y$ two $C^1$-mappings.
Assume that for every $x\in X$ the points $f(x)$ and $g(x)$ can be joined by a unique shortest geodesic of $Y$. Denote this geodesic by $c\_x$. Moreover denote by
$P^{c\_x}\colon T\_{f(x)}Y \to T\_{g(x)}Y$
the parallel tran... | https://mathoverflow.net/users/103407 | Parallel transport in Riemannian manifold induces bounded mapping of vector bundles | Counterexample: Let $X=S^1$ and $Y$ defined below. The norm of $P^{c\_{t+\epsilon}} - P^{c\_{t}}$ is given, essentially, by (Ambrose-Singer) the norm of the parallel transport minus identity operator along the closed curve $s\_{t,\epsilon}$ formed by $(f(\tau), [t\leq \tau \leq {t+\epsilon}]), {c\_{t+\epsilon}}, (g(\ta... | 1 | https://mathoverflow.net/users/1988 | 259310 | 116,971 |
https://mathoverflow.net/questions/259317 | 3 | Let $X$ be a compact countable Hausdorff space. By Sierpinski-Mazurkiewicz Theorem we know that $X$ is a compact countable ordinal, i.e.
$$
X \simeq \omega ^{\alpha} \cdot n + 1
$$
where $\alpha$ is countable and $n \ge 1$ an integer.
My question is: What do we know about $\mathcal{C}(X)$?
Obviously it is a commuta... | https://mathoverflow.net/users/18089 | What can be said about the algebra of continuous functions on compact countable ordinals? | Let $X$ be a compact Hausdorff space. Then $C(X)$ has separable dual if and only if $X$ is countable. In this case, the dual is isomorphic to $\ell\_1$.
In the class of compact metric space, $C(X)$ has countable Szlenk index if and only if $X$ is countable.
| 4 | https://mathoverflow.net/users/15129 | 259323 | 116,975 |
https://mathoverflow.net/questions/259256 | 5 | Suppose I have a finite CW complex $X$ with $p$-skeleton $X^{(p)}$.
* Let $\varphi\_f \colon S^p \to X^{(p)}$ be part of the attaching map of a $(p+1)$-cell $f$.
* Let $\Phi\_e \colon D^p \to X^{(p)}$ be part of the attaching map of a $p$-cell $e$ and let $q\_e \colon X^{(p)} \to S^p$ be the map that collapses every... | https://mathoverflow.net/users/3995 | attaching maps in CW complexes | This is essentially Lemma 4.10 of Hatcher's book, which is the key step in proving the cellular approximation theorem. It shows that you can homotope $\varphi\_f$ so that there is an open set $U$ in your $p$-cell $e$ such that $\varphi\_f$ is piecewise linear on the inverse image of $U$. Then a general element of $U$ h... | 3 | https://mathoverflow.net/users/1310 | 259334 | 116,980 |
https://mathoverflow.net/questions/259325 | 3 | Motivated [by this post](https://mathoverflow.net/questions/259251/a-consecutive-resolution-of-continum-algebras-to-a-simple-continum-algebra) we give the following definition:
**Definition:** A pair of unital rings $(R,S)$ is called a consecutive pair of rings if they do not have non trivial idempotent and satisfy t... | https://mathoverflow.net/users/36688 | Is the pair $(C([0 \;1]),\mathbb{C})$ a consecutive pair? | The answers to the question [prime ideals in C([0,1])](https://mathoverflow.net/questions/35793/prime-ideals-in-c0-1) explain why the answer is no, assuming that Yemon Choi was mistaken in his answer in believing that you are only interesting in continuous homomorphims.
In particular it is explained that there are n... | 3 | https://mathoverflow.net/users/345 | 259340 | 116,981 |
https://mathoverflow.net/questions/259013 | 28 | Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven.
>
> **I.** $p=2$
>
>
>
We have,
$$\left(\frac{8}{\alpha^8+8}\right)^2+\left(\frac{\beta^8}{\beta^8+32}\right)^2=1\tag1$... | https://mathoverflow.net/users/12905 | Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$? | (*Courtesy of a comment by Nemo who suggested Huber's paper.*)
>
> **Part I.** $x^5+y^5 = 1$
>
>
>
In "[*A Theory of Theta Functions to the Quintic base*](https://arxiv.org/pdf/1304.0684.pdf)",
Tim Huber defines four theta functions which can be ultimately expressed in terms of the [Rogers-Ramanujan identitie... | 6 | https://mathoverflow.net/users/12905 | 259342 | 116,983 |
https://mathoverflow.net/questions/259303 | 2 | I would like to understand how the Schwartz kernel theorem works for some more difficult cases and therefore would like to discuss an example from scratch:
Let the Dirichlet Laplacian on the half-line $-\Delta:H\_0^1((0,\infty))\cap H^2((0,\infty)) \rightarrow L^2((0,\infty))$ be given. Then $f(-\Delta)$, for $f \in ... | https://mathoverflow.net/users/103442 | Schwartz kernel theorem | By the Helffer-Sjöstrand formula,
$$
f(-\Delta\_D) = \frac1\pi \int\_{\mathbb C} (\partial\_{\bar z} \tilde
f)(z) (-\Delta\_D -z)^{-1}\,L(dz),
$$
where $L(dz)$ is the Lebesgue measure and $\tilde f$ is an almost-analytic extension of $f$. Now the resolvent $(-\Delta\_D-z)^{-1}$ for $z\in\mathbb C\setminus[0,\infty)$ ... | 3 | https://mathoverflow.net/users/69194 | 259357 | 116,989 |
https://mathoverflow.net/questions/259358 | 4 | I am new here and I hope I don't make any mistakes in asking this question here.
Let $n > k$ be integers, and $\epsilon < 1/2$ be an arbitrarily small constant. What is an upper bound on the maximum size of a family $\{A\_1,\ldots,A\_N\}$ of subsets of $[n]$ such that:
* $|A\_i| = k$
* $|A\_i \cap A\_j| \le (1-\eps... | https://mathoverflow.net/users/103464 | An upper bound on the families of large k-uniform sets with small pairwise intersection | Small pairwise intersection is eqhuivalent to large symmetric difference. So you are asking about constant weight error correcting codes. You can move to complenents and assume that k is at most n/2. There is large literature about it.
| 4 | https://mathoverflow.net/users/1532 | 259360 | 116,990 |
https://mathoverflow.net/questions/259345 | 3 | For a subset $S\subset[0,1]$ with $0<|S|<1$ ($|S|$ is the Lebesgue measure of $S$) we define the multiplicity function of order $n$ $m\_{n,S}:[0,1] \rightarrow \{0,1,\ldots,n\}$ in the following way: at point $x$ the function $m\_{n,S}(x)$ is the number of elements in the set $S\cap (x+\frac{1}{n}\mathbb{Z})$.
It is... | https://mathoverflow.net/users/16040 | Nowhere dense set with high multiplicity | Yes. First fix an increasing sequence of natural numbers $(n\_k)$ such that $\frac{k}{n\_k} \to 0$ as $k \to \infty$, $\alpha = \sum\_{i=1}^\infty \frac{1}{n\_i} < 1$, and $\beta\_k = n\_k\cdot \sum\_{i=k+1}^\infty \frac{1}{n\_i} \to 0$ as $k \to \infty$. Let $(x\_k)$ be a countable dense subset of $[0,1]$ and let $S$ ... | 0 | https://mathoverflow.net/users/23141 | 259363 | 116,991 |
https://mathoverflow.net/questions/259366 | 0 | Let $\Omega$ a bounded open set of $R^n$, $\omega$ a non-empty open subset of $\Omega$ and
$\chi\_{\omega} : L^2(\Omega) \longrightarrow L^2(\omega)$
be the restriction operator to $\omega$, while
$\chi\_{\omega}^\*$ denotes the adjoint operator of $\chi\_{\omega}$ and given by
\begin{equation}\label{v1}
(\chi\_... | https://mathoverflow.net/users/95551 | Composition of a negative operator and a positive one | No, of course not. Even if $A$ is bounded, this would imply that $i\_\omega A$ is self-adjoint. So you only have a chance if $A$ and $i\_\omega$ commute.
| 0 | https://mathoverflow.net/users/23141 | 259369 | 116,993 |
https://mathoverflow.net/questions/259233 | 2 | We have an unknown $m\times n$ matrix $X=(x\_{ij})\_{i=1,j=1}^{m,n}$. Assume we are given measurements of the differences
$$x\_{i,j+1}-x\_{i,j}$$
and
$$x\_{i+1,j}-x\_{i,j}$$ for all $(i,j)\in \{1,\dots,m\}\times\{1,\dots,n\}$ where $x\_{i,n+1}=x\_{i,1}$ and $x\_{m+1,j}=x\_{1,j}$. Assume the measurements are in... | https://mathoverflow.net/users/100927 | Reconstruct matrix given all differences of neighbors | Given
* $\mathrm R \in \mathbb R^{m \times n}$, whose $m$ rows are the differences of neighboring rows of $\mathrm X \in \mathbb R^{m \times n}$
* $\mathrm C \in \mathbb R^{m \times n}$, whose $n$ columns are the differences of neighboring columns of $\mathrm X \in \mathbb R^{m \times n}$
we would like to determine... | 2 | https://mathoverflow.net/users/91764 | 259386 | 117,000 |
https://mathoverflow.net/questions/241809 | 8 | Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the [Dedekind eta function](http://mathworld.wolfram.com/DedekindEtaFunction.html) identity $\eta(-\frac{1}{z})=\sqrt{-iz}\eta(z)$
and one of the most fundamental pr... | https://mathoverflow.net/users/54532 | Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$ | Let $R(q)$ be the Rogers-Ramanujan continued fraction
$$
R(q):=\frac{q^{1/5}}{1+}\frac{q^1}{1+}\frac{q^2}{1+}\frac{q^3}{1+}\ldots,|q|<1
$$
Let also for $r>0$
$$
Y=Y(r):=R(e^{-2\pi\sqrt{r}})^{-5}-11-R(e^{-2\pi\sqrt{r}})^5
$$
It is easy to show someone that
$$
Y\left(\frac{r}{5}\right)Y\left(\frac{1}{5r}\right)=125, : (... | 2 | https://mathoverflow.net/users/88851 | 259389 | 117,002 |
https://mathoverflow.net/questions/259347 | 3 | Can someone explain the relationship between reduced and unreduced parametrized homology theories in the parametrized setting à la May-Sigurdsson with maps to a reference space $B$?. Is it just a copy of the "classical cohomology theory" on the reference space $B$. How do you go between reduced and unreduced theories u... | https://mathoverflow.net/users/21985 | Reduced Vs unreduced cohomology in the parametrized setting. | If $(X,p)$ is a space over $B$, its unreduced cohomology is the same as the reduced cohomology of $(X,p)\_+ = (X\sqcup B,p,\sigma)$, $\sigma$ being the section taking $B$ to the disjoint copy of $B$.
But I suspect you're interested in the relationship between the reduced and unreduced cohomologies of a given ex-space... | 2 | https://mathoverflow.net/users/58888 | 259392 | 117,003 |
https://mathoverflow.net/questions/259390 | 9 | Is there an example of a smooth vector field $v$ on $S^3$ such that $v$ preserves a volume form and $v$ is not a Reeb vector field?
Recall that $v$ is a *Reeb vector field* if there exists a contact $1$-form $\alpha$ such that $\alpha(v)=1$ and $v$ belongs to the kernel of $d\alpha$.
| https://mathoverflow.net/users/13441 | Reeb flows on $S^3$ versus volume preserving flows | I took the time to give a clean version of the answer and to eliminate the dependence on really hard theorems.
**Construction of geodesible, volume-preserving flows on $S^3$ that are not Reeb flows for any contact form.**
Ingredients:
1. A contact form $\alpha$ whose Reeb vector field $X$ has no nonconstant inte... | 10 | https://mathoverflow.net/users/21123 | 259405 | 117,008 |
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