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/256717 | 0 | Let $a(n)$ be the $k$-ary tuple of the exponents of the prime factorization of $n$. For example,
$$a(5184)=a(2^{6}⋅3^{4})=(6, 4), a(65536)=a(2^{16})=(16).$$
Formally, let $p\_{1}^{a\_{1}}, p\_{2}^{a\_{2}}, \ldots, p\_{k}^{a\_{k}}$ be the prime factorization of a positive integer $n$, then
$$a(n) = a(p\_{1}^{a\_{1... | https://mathoverflow.net/users/102059 | Upper bound for tuple of exponents of prime factorization | It is unfortunate to use the notation $|\cdot|$ for the bit-length function, because in number theory (where this post belongs) it denotes the absolute value function. At any rate, in standard notation, you are asking if
$$ \sum\_{i=1}^k\lfloor 1+\log\_2 a\_i\rfloor\ll\log\log n. $$
The answer is no, even if we replace... | 5 | https://mathoverflow.net/users/11919 | 256721 | 115,953 |
https://mathoverflow.net/questions/256718 | 6 | Let $\mathbf{Q}^{\mathrm{ab}}$ be the maximal abelian extension of the field of rational numbers $\mathbf{Q}$. I'm interested in the following question:
Is it true that $K^{M}\_{2}(\mathbf{Q}^{\mathrm{ab}})/pK^{M}\_{2}(\mathbf{Q}^{\mathrm{ab}})=0$ for any prime $p$ ?
where $K^{M}\_{\ast}$ is Milnor $K$-theory.
| https://mathoverflow.net/users/102216 | The second Milnor $K$-theory of a field | By the Milnor-Bloch-Kato conjecture, this is equivalent to $\mathrm{Br}(\mathbf{Q}^{\mathrm{ab}})[p] = 0$ (by the Kummer sequence $1 \to \mu\_p \to \mathbf{G}\_m \to \mathbf{G}\_m \to 1$, Hilbert 90 $H^1(K,\mathbf{G}\_m) = 0$ and $H^2(K,\mathbf{G}\_m) = \mathrm{Br}(K)$). This follows from [Neukirch-Schmidt-Wingberg, Co... | 13 | https://mathoverflow.net/users/nan | 256724 | 115,955 |
https://mathoverflow.net/questions/256709 | 3 | Let $1<p<\infty$. Recall that a Banach space $X$ is $l\_{p}$-saturated if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l\_{p}$. I have a seemingly stronger notion of $l\_{p}$-saturated Banach spaces. We say that a Banach space $X$ is strongly $l\_{p}$-saturated if for every infinite-dim... | https://mathoverflow.net/users/41619 | Two questions on $l_{p}$-saturated Banach spaces | As for question 1; this is the famous [distortion problem](https://en.wikipedia.org/wiki/Distortion_problem). Your question has negative answer already for some renorming of $\ell\_p$:
>
> E. Odell, Th. Schlumprecht, The distortion problem of Hilbert space, *Geom. Funct. Anal.*, **3**, 201–207.
>
>
>
As for yo... | 4 | https://mathoverflow.net/users/15129 | 256731 | 115,958 |
https://mathoverflow.net/questions/256630 | 15 | How can a physicist understand a **2-dimensional topological field theory** as a **Frobenius algebra**? Are there some explicit examples in order to understand this relation?
The definition (e.g. on Wikipedia) of the Frobenius algebra is quite clear, it is a finite dimensional associative algebra equipped with a spec... | https://mathoverflow.net/users/102157 | Why is a Topological Field Theory equivalent to a Frobenius algebra? | As the general relation between 2d TQFT and Frobenius algebras has already been given in another answer, let me describe the Frobenius algebras occurring in the A and B-models.
1) The A-model is defined for a compact symplectic manifold $(X,\omega)$. The vector space underlying the Frobenius algebra is the cohomology... | 15 | https://mathoverflow.net/users/25309 | 256734 | 115,959 |
https://mathoverflow.net/questions/256713 | 17 | It is an old result of Schützenberger that in a free group, a basic commutator cannot be a proper power. A look at the original reference
M.-P. Schützenberger, *Sur l'équation $a^{2+n} = b^{2+m}c^{2+p}$ dans un groupe libre*, C. R. Acad. Sci. Paris 248 (1959), 2435–2436 (French).
quickly reveals that a lot of detai... | https://mathoverflow.net/users/8176 | A result of Schützenberger on commutators and powers in free groups | The case $n=2$ (originally due to Lyndon) admits a very nice geometric argument: one notes that elements $a,b,c$ with $[a,b]=c^2$ lead to a map from the surface $\Sigma\_{-1}$ of Euler characteristic -1 to a graph. Pulling back midpoints of edges, one obtains essential, two-sided, simple closed curves on $\Sigma\_{-1}$... | 18 | https://mathoverflow.net/users/1463 | 256737 | 115,961 |
https://mathoverflow.net/questions/256733 | 6 | Consider a surjective holomorphic map between two complex projective manifolds $\pi :X \rightarrow Y$. Iitaka conjectured the subadditivity of Kodaira dimensions: $\kappa(X)\geqslant\kappa(Y)+\kappa(X\_y)$ where $X\_y$ is a generic fibre. We know already this holds when $dim(X)=dim(Y)+1$ and when $\pi$ is a fibre bundl... | https://mathoverflow.net/users/nan | Additivity of Kodaira dimension for a nice fibration | There is another inequality which says the following:
>
> **Easy addition**
> (Using the same notation):
> $$
> \kappa(X)\leqslant \kappa(X\_y) + \dim Y
> $$
>
>
>
Consequently, if $Y$ is of general type, i.e., $\kappa(Y)=\dim Y$, then the subadditivity conjecture is equivalent with equality instead of ineq... | 10 | https://mathoverflow.net/users/10076 | 256739 | 115,963 |
https://mathoverflow.net/questions/256743 | 9 | If $G$ is just an ordinary set-theoretic group, then the answer to the question in the title is *yes*: the automorphisms of $G$ as a (left) $G$-set are all of the form "multiply (on the right) by an element of $G$."
I'm trying to understand the case where $G$ is a group scheme; then as I understand it the stack of $G... | https://mathoverflow.net/users/1474 | Is $G$ always the automorphism group of the trivial $G$-torsor? | The two morphisms are no $G$-morphisms. An $R$-algebra homomorphism $f : R[x]/(x^2-1) \to R[x]/(x^2-1)$ commutes with the $G$-action if and only if the diagram
$$\begin{array}{c} R[x]/(x^2-1) & \xrightarrow{f} & R[x]/(x^2-1) \\ \Delta \downarrow ~~~&& ~~~\downarrow\Delta \\ R[x]/(x^2-1) \otimes\_R R[y]/(y^2-1) & \xrigh... | 12 | https://mathoverflow.net/users/98306 | 256746 | 115,965 |
https://mathoverflow.net/questions/255957 | 5 | It is known (a theorem of Komjáth and Hajnal) that it is consistent (by adding a Cohen real to a universe where there is no Suslin line) that there exists an order type $\theta$ such that for any other order type $\eta$, there exists a bad coloring $f: [\eta]^2 \to \omega$ such that for any order preserving embedding $... | https://mathoverflow.net/users/23835 | Strongly non-Ramsey order type in polarized partition problems | We can't find such pair of bad order types, aka there indeed is some Erdös-Rado phenomenon in the polarized partitions with respect to linear orderings. Indeed given $\gamma$ number of colors and $\theta$, $\psi$ we can find large saturated dense linear orders $\alpha, \beta$ such that for any $f: \alpha\times \beta \t... | 2 | https://mathoverflow.net/users/23835 | 256753 | 115,969 |
https://mathoverflow.net/questions/256750 | 0 | Consider the (real) linear system of equations $A\mathbf{x}=\mathbf{c}$ of size $N$ as
$$
\begin{bmatrix} a\_{N}-a\_{2}& a\_{2}& 0 &\dots& 0 & -a\_{N} \\-a\_{1} & a\_{1}-a\_{3}&a\_{3}& 0 & \dots&0\\
0 & -a\_{2} & a\_{2}-a\_{4}&a\_{4}&0 & \dots \\\vdots&\ddots&\ddots&\ddots&\ddots\\ a\_1& 0& \dots& 0&-a\_{N-1}&a\_{N-1}-... | https://mathoverflow.net/users/75491 | How to show inconsistency of a linear system of equations? | For a counterexample to (1) with all $a\_i$ nonzero, consider the $N=4$ case with $a\_3 = a\_1$, noting that
$$ \left[\matrix{a\_4-a\_2 & a\_2 & 0 & -a\_4\cr
-a\_1 & 0 & a\_1 & 0\cr
0 & -a\_2 & a\_2-a\_4 & a\_4\cr a\_1 & 0 & -a\_1 & 0\cr}\right] \left[\matrix{0 \cr a\_4 \cr 0 \cr a\_2}\right] = 0$$
For a counterexamp... | 4 | https://mathoverflow.net/users/13650 | 256757 | 115,971 |
https://mathoverflow.net/questions/256758 | 2 | Given $m\in\Bbb N$ large enough is there a coprime pair $a,b$ in $[2^{m-1},2^m]$ and $m\_1,n\_1, m\_2,n\_2\in[0,m^{c'}]\cap\Bbb Z$ with each of
$$m\_1a-n\_1b,m\_2a-n\_2b\in[0,m^c]\cap\Bbb Z$$
$$m\_1n\_2\neq n\_1m\_2$$
true at some $c,c'>0$ independent of $m$?
| https://mathoverflow.net/users/nan | Existence of certain coprime numbers | What you seem to request is that the matrix $M=\begin{pmatrix}m\_1 & -n\_1 \\
m\_2 & -n\_2\end{pmatrix}$ were non-degenerate, and the vector $d:=M\binom ab$
had its coordinates of size polynomial in $m$. If this could be arranged, as
a result we would get $\binom ab=\Delta^{-1}Ad$, where $\Delta:=\det M$ is a
non-zero ... | 3 | https://mathoverflow.net/users/9924 | 256762 | 115,973 |
https://mathoverflow.net/questions/256712 | 3 | Consider the following family of functions
$$f\_n(w):=\sum\_{k=0}^{\infty}\frac{(-1)^{k-1}}{k!}(k+n)^{k-1}w^k.$$
>
> **QUESTION 1.** Does the following hold?
> $$f\_n(w)=-\frac1{n(f\_{-1}(w))^n}.$$
>
>
>
Deeper look:
>
> **QUESTION 2.** Is there a conceptual reason why the linear shift, in $n$ of $f\_n$, ... | https://mathoverflow.net/users/66131 | sequencial shift on families =flipped powers. How? | It is well-known that the Abel polynomials $p\_k(x)=x(x-ak)^{k-1}$ are
a sequence of polynomials of binomial type, i.e.,
$$ \sum\_{k\geq 0} p\_k(x)\frac{w^k}{k!} = \left( \sum\_{k\geq 0}
p\_k(1)\frac{w^k}{k!}\right)^x, $$
which explains your formula.
| 6 | https://mathoverflow.net/users/2807 | 256778 | 115,981 |
https://mathoverflow.net/questions/256745 | 1 | What is the largest cardinal consistent with $ZFC$ + $V$=$L$? The reason for the question is this: under the assumption that all of 'ordinary mathematics' (as reverse mathematics understands the term) can be interpreted in $ZFC$+ $V$=$L$ (see Simpson's paper "The Goedel Hierarchy and Reverse Mathematics", preprint, Nov... | https://mathoverflow.net/users/20597 | What is the largest cardinal consistent with $ZFC$ + $V$=$L$? | Re: ordinary mathematics, I think your assumptions are incorrect (although the question is interesting on its own, since ZFC+V=L is a natural set theory). ZFC+V=L is (most, including Simpson, would argue) more than sufficient for ordinary mathematics. But then, so is ZFC, and indeed much less; there's nothing that sing... | 5 | https://mathoverflow.net/users/8133 | 256782 | 115,983 |
https://mathoverflow.net/questions/256725 | 9 | The following are very familiar and basic items, individually.
(1) The number $a(n)$ of rectangles (parallel to axes) in an $n\times n$ square grid.
(2) The number $b(n)$ of cubes (parallel to axes) in an $n\times n\times n$ cube.
However, I could not find a reference to a direct **bijective** proof for $a(n)=b(n... | https://mathoverflow.net/users/66131 | in need of a direct combinatorial/bijective proof | Following @wojowu's suggestion, we have:
Let $h$ be the side of the inner cube, and let $(i,j,k)$ be its corner nearest the origin. Then we have $0\le i,j,k < n-h+1 \le n$.
Let us describe our rectangle by the corners $(x\_1, y\_1)$ and $(x\_2, y\_2)$ with $x\_1<x\_2$ and $y\_1<y\_2$. Then our mapping from $(i,j,k,... | 3 | https://mathoverflow.net/users/4422 | 256790 | 115,988 |
https://mathoverflow.net/questions/256751 | 1 | Let $O$ be the ring of $S$-integers in a real quadratic number field. Let $G$ be an $S$-arithmetic subgroup of $SL\_2(O)$ whose intersection with $SL\_2(\mathbb Z)$ is not of finite index in $SL\_2(\mathbb Z)$.
*Assume the action of $G$ is properly discontinuous for simplicity.*
Can the fundamental domain of the ac... | https://mathoverflow.net/users/101231 | Subgroup of $SL_2(O)$ with nice fundamental domain in complex upper half-plane | No arithmetic subgroup of $SL\_2(O\_S)$ will ever act properly on the upper half-plane. These groups are simply too big, and the orbits will not be discrete subgroups.
To see this explicitly, consider the orbit of $z \in \mathbb{H}$ under the subgroup of elements of $G$ of the form $\begin{pmatrix} 1 & x \\ 0 & 1\en... | 4 | https://mathoverflow.net/users/2481 | 256798 | 115,992 |
https://mathoverflow.net/questions/256813 | 5 | Given a set of $n$ vertices and the fact that none of them is of degree greater than $2$, how many distinct such graphs are there?
| https://mathoverflow.net/users/102269 | Number of graphs on a given set of vertices with maximum degree of $2$ | This is [OEIS A003292 Number of 4-line partitions of n decreasing across rows](https://oeis.org/A003292).
>
> a(n) is the number of unlabeled graphs on n nodes whose connected components are a path or a cycle. - Geoffrey Critzer, Nov 28 2011
>
>
>
OEIS gives generating function and references which may contain... | 5 | https://mathoverflow.net/users/12481 | 256821 | 115,999 |
https://mathoverflow.net/questions/256801 | 8 | I am interested in studying equidistribution of Hecke eigenvalues and proving statistical properties of arithmetical objects. On the road, I face the following problem: how to express sums of the form
$$\sum\_{c(\pi)<X} a\_n(\pi)$$
where $a\_n(\pi)$ is the $n$-th coefficient of the Dirichlet series defining $L(s, \... | https://mathoverflow.net/users/43737 | Reaching Hecke eigenvalues from a trace formula | Yes, this is a standard thing to do. If you want to look at traces of Hecke operators on a definite quaternion algebra, this is the same as what are known as "traces of Brandt matrices." These have been studied with trace formulas classically by Eichler, Shimizu, etc. E.g., see the notes
>
> [The basis problem for ... | 9 | https://mathoverflow.net/users/6518 | 256827 | 116,004 |
https://mathoverflow.net/questions/256810 | 3 | Let $F$ be a free group on $x,y,z$. Fix $n>1$ (I am ready to assume that $n$ is large enough). Let $\mathcal{W}$ be the set of *cyclically reduced* words $w$ in $F$ where the letter $z$ appears at least once (i.e., such that $w\notin \langle x,y \rangle$).
Let $N$ be the normal subgroup of $F$ generated by $\{w^n:w\... | https://mathoverflow.net/users/38889 | Free subgroup of a quotient | As observed by Ilya Bogdavov in a comment, the answer is obviously no since the quotient $F/N$ has exponent $n$.
| 4 | https://mathoverflow.net/users/14094 | 256837 | 116,008 |
https://mathoverflow.net/questions/256836 | 2 | In his paper "The Evaluation and Estimation of the Coefficients in the Chebyshev Series Expansion of a Function", David Elliott writes:
>
> Writing $x = \cos(\theta)$, it can easily be shown that
> $$
> \int\_{-1}^1 \frac{T\_n(x)}{\sqrt{1-x^2} \, (z-x)} \, dx
> =
> \int\_0^\pi \frac{\cos(n\theta)}{z - \cos(\theta)... | https://mathoverflow.net/users/101798 | Chebyshev coefficient of $1/(z-x)$ | The integral equals $\frac12\int\_{-\pi}^\pi \frac{e^{in\theta}}{z-\cos \theta}d\theta$. Consider the integral over the rectangle with vertices $\pm \pi$, $\pm \pi+iT$ for large $T>0$. The integrals over vertical sides cancel, since the function if $2\pi$-periodic, the integral over high horizontal side tends to 0 for ... | 2 | https://mathoverflow.net/users/4312 | 256840 | 116,009 |
https://mathoverflow.net/questions/256842 | 3 | I am trying to understand the proof of Laurent phenomenon of cluster algebras in [the book (Sergey Fomin, Lauren Williams, Andrei Zelevinsky, *Introduction to Cluster Algebras. Chapters 1-3*, arXiv:1608.05735v1)](https://arxiv.org/abs/1608.05735).
On page 45, it is said that "We see that $x\_j'$ is linear in $x\_q$ ... | https://mathoverflow.net/users/11877 | Trying to understand the proof of Laurent phenomenon of cluster algebras | You need to use the fact that $x'\_j = M x\_q + M'$, where $M$ and $M'$ are Laurent *monomials* in the remaining variables. Their observation is that if $x'\_j$ factored, it would be as $x'\_j = (P\_1 x\_q + P\_2)P\_3$, where $P\_1, P\_2, P\_3$ are Laurent *polynomials* in the remaining variables. But then $P\_1 P\_3 =... | 3 | https://mathoverflow.net/users/16002 | 256844 | 116,011 |
https://mathoverflow.net/questions/256039 | 3 | *Note:* This is a crosspost from <http://math.stackexchange.com>; the original question may be found [here](https://math.stackexchange.com/questions/2036746/comprehension-question-within-script-on-bessel-process).
I have a question regarding a script by Greg Lawler on Bessel processes:
<http://www.math.uchicago.ed... | https://mathoverflow.net/users/101850 | Comprehension question within script on Bessel process | Here is another way to arrive at the result. Set $Y\_t = X\_t^2$ (for clarity). As the OP shows,
$$
dY\_t = d \; dt + 2 \sum\_{j=1}^d W\_t^j dW\_t^j \;, \quad Y\_0 = x > 0 \;.
$$ By definition of the infinitesimal generator of $Y\_t$, we have
\begin{align\*}
L f(x) &= \lim\_{t \to 0^+} \frac{1}{t} \mathbb{E}\_x \left\{... | 1 | https://mathoverflow.net/users/64449 | 256849 | 116,012 |
https://mathoverflow.net/questions/256855 | 0 | Given $\mathscr{A}\subseteq\mathbb{N}$ an infinite set, consider its $h$-fold sumsets
$$h\mathscr{A}:=\left\{\sum\_{i=1}^{h} k\_i : k\_1,\ldots, k\_h\in \mathscr{A} \right\},$$
and let "$\simeq$" be the equivalence relation (in $\mathcal{P}(\mathbb{N})$) given by $\mathscr{A}\simeq \mathscr{B}$ if and only if $\mat... | https://mathoverflow.net/users/74026 | Sets with $\mathrm{d}(h\mathscr{A}) = 1$ but s.t. $h\mathscr{A}$ doesn't contain all large integers | Let $(n\_k)$ be a rapidly increasing sequence. Remove from the positive integers all segments $[n\_k,2n\_k-1]$ and call the remaining set $A$. Then $2n\_k$ does not lie in $A+A$, so $A+A$ is not equivalent to $\mathbb{N}$. But $A+A$ has density 1. Indeed, $A$ contains all the numbers from $2n\_k$ to $n\_{k+1}-1$, thus ... | 1 | https://mathoverflow.net/users/4312 | 256856 | 116,013 |
https://mathoverflow.net/questions/256619 | 11 | Is there an example of a strongly proper ccc forcing that is not equivalent to Cohen forcing?
| https://mathoverflow.net/users/11145 | Strongly proper ccc forcing | The answer to your question is yes, and it follows from the following paper of Koppelberg and Shelah [Subalgebras of Cohen algebras need not be Cohen](https://arxiv.org/abs/math/9610227).
In this paper, it is shown, for each $\kappa \geq \aleph\_2,$ there exists a non-Cohen complete subalgebra of $Add(\omega, \kappa)... | 7 | https://mathoverflow.net/users/11115 | 256872 | 116,018 |
https://mathoverflow.net/questions/256819 | 5 | I have a list of $m$ affine inequalities in $n$ variables of the following form
$$a\_1 x\_1 + \cdots + a\_n x\_n \leq c\_n$$
I would like to know whether there is any point on the unit sphere in $\mathbb R^n$ that satisfies all of them. Is there any easy way to check whether this is the case?
(Based on randomly ... | https://mathoverflow.net/users/nan | Intersecting a convex polytope with the unit sphere | It was [proved](https://pdfs.semanticscholar.org/1e16/eff75b69f9e85593fd0641cccb2fae7ec066.pdf) by Freund and Orlin that the problem of checking whether a polytope specified by linear inequalities is not entirely contained in a ball specified by its centre and radius is NP-complete.
And it is obvious that one can assu... | 5 | https://mathoverflow.net/users/11100 | 256881 | 116,022 |
https://mathoverflow.net/questions/256781 | 22 | It is well known that the symmetric group $S\_n$ admits presentation with
$\{(ij) \mid i\neq j\}$ as the set of generators and the following list of relations
(in every formula distinct letters denote distinct indices):
\begin{align}
(ij) =&\, (ji), \label{Sym0} \tag{S0} \\
(ij)^2 = &\, 1, \label{Sym1} \tag{S1} \\
(jk... | https://mathoverflow.net/users/68204 | A symmetric-like group and the quaternion group $Q_8$ | To simplify typing I will call these groups $G\_n$ rather than $\tilde{S}\_n$. Note that $G\_2$ is just the free product of two groups of order $2$, so is the infinite dihedral group, and I conjecture that $|G\_n| = 2^n n!$ for $n \ge 3$, and $G\_n$ is a central product of an extraspecial or symplectic type $2$-group o... | 7 | https://mathoverflow.net/users/35840 | 256894 | 116,024 |
https://mathoverflow.net/questions/256870 | 7 | I just started reading "The calculi of lambda conversion" by Church.
Church defines functions like: id x = x, and says the domain and range are understood to be as permissible as possible. Permitting even itself, id id = id
In my experience, I've always been told to specify a domain and range with the functions I'... | https://mathoverflow.net/users/47532 | Relationship of lambda calculus to the rest of math | The question you are trying to ask is "What is a [denotational semantics](https://ncatlab.org/nlab/show/denotational+semantics) for the untyped lambda calculus?"
This is a difficult problem because, as Bjorn Kjos-Hanssen said in his answer, if you try and make variables range over elements of some set $D$ you find th... | 8 | https://mathoverflow.net/users/333 | 256902 | 116,026 |
https://mathoverflow.net/questions/256901 | 2 | Let $G$ be a finite nonabelian simple group. Must $G$ admit a pair of generators $g,h$ such that $g,[g,h]$ also generate?
I've computationally verified this for the first 21 nonabelian finite simple groups, though I don't know if this should hold in general.
| https://mathoverflow.net/users/88840 | Which nonabelian simple groups admit a generating pair $g,h$ such that $g,[g,h]$ is also generating? | Yes, this is true, at least for large enough groups. It is known that a random pair of elements generate a finite simple group (with probability approaching $1,$ as size goes to infinity), and also
*Robert M. Guralnick, Martin W. Liebeck, Jan Saxl, and Aner Shalev*, MR 1707675 [**Random generation of finite simple gr... | 6 | https://mathoverflow.net/users/11142 | 256913 | 116,032 |
https://mathoverflow.net/questions/256895 | 2 | I am thinking of a quantitative (possibly based on random graph theory) or qualitative (say, based on topological ideas, e.g. Baire's theorem in the Gromov-Hausdorff metric space) information about how many finite graphs (perhaps: how many graphs on a given number of nodes $n$) are line graphs, or perhaps just *general... | https://mathoverflow.net/users/26039 | How many line graphs are there? | (Per request, post edited to discuss $G\_{n,p}$ for values other than $p=1/2.$ This is a routine line of reasoning in probabilisitic combinatorics. See the first few chapters of [Alon and Spencer](http://www.wiley.com/WileyCDA/WileyTitle/productCd-1119061954.html) for more.)
Line graphs are [claw-free](http://mathwor... | 6 | https://mathoverflow.net/users/22512 | 256917 | 116,034 |
https://mathoverflow.net/questions/256891 | 0 | I am studying the paper, "[Solutions in the large for nonlinear hyperbolic systems of equations](http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160180408/abstract)", and I'd like to know a few references for following questions :
1. This seems to be an important theorem, does this theorem he proved has got any famou... | https://mathoverflow.net/users/14414 | reference request : "Solutions in the large for nonlinear hyperbolic systems of equations" | 1. It is sometimes referred to as "Glimm's existence theorem". Though in some ways the proof is "more famous" than the theorem, and is frequently referred to as "Glimm's difference scheme".
2. Not that I know of. Glimm proves the existence of a *specific class* of weak solutions; the class is pretty tied to the scheme ... | 3 | https://mathoverflow.net/users/3948 | 256918 | 116,035 |
https://mathoverflow.net/questions/256898 | 2 | Cross-posting from math stack-exchange since it's not getting any visibility there.
I am given a function $F: \{[0, y]: y \in I\} \to \Sigma(I)$, such that $\lambda(F([0, y])) = y$, and $F([0, y]) \subseteq F([0, z])$ for $y \le z$. Here $\lambda$ is the Lebesgue measure on the unit interval $I$ and $\Sigma(I)$ denot... | https://mathoverflow.net/users/4923 | Automorphism on the unit interval compatible with a measure preserving set function | Are you sure that you are not missing an hypothesis? If I take
$$F([0,y]) = [0,y] \setminus {\bf Q}$$
there can be no such bijective $f$ because then
$$f(\{y\}) = f([0,y]) \setminus \bigcup\_{z<y} f([0,z]) =
([0,y]\setminus {\bf Q}) \setminus \bigcup ([0,z]\setminus {\bf Q}) = \{y\}$$
and thus $f$ would be the identit... | 2 | https://mathoverflow.net/users/6129 | 256919 | 116,036 |
https://mathoverflow.net/questions/256907 | 2 | How can I give a set of transitions sufficient to transform any spanning tree into any another spanning tree in a finite number of steps via spanning trees? I was wondering if someone help me.Thanks.
| https://mathoverflow.net/users/102311 | Spanning tree with sufficient transformation | I assume that by a transition you mean adding one edge and removing another. Then it seems that a simple algorithm works: Add an edge from the new tree. This will create a cycle so just remove one edge from the cycle that is not in the new tree. Such an edge must exist, since there are no cycles in the new tree. The ru... | 2 | https://mathoverflow.net/users/39187 | 256922 | 116,037 |
https://mathoverflow.net/questions/256786 | 17 | This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't.
In the category of modules over a ring $R$, the module $R$ is a projective generator. This property does not determine it uni... | https://mathoverflow.net/users/41291 | Can $\mathcal O_X$ be recognized abstract-nonsensically? | As Will points out, the best you can hope for is to recognize being a line bundle. Also, I find you're a bit vague about what category of sheaves you want work in; I'm thinking about coherent sheaves, which is an appropriate analogue for finite dimensional $R$-modules. As we'll see below, I'll also want to assume my sc... | 6 | https://mathoverflow.net/users/66 | 256930 | 116,041 |
https://mathoverflow.net/questions/256893 | 0 | M. Schechter introduced an operational quantity characterizing strictly singular operators as follows:
For an operator $T:X\rightarrow Y$, we set $$\tau(T)=\sup\_{M}\inf\_{x\in S\_{M}}\|Tx\|,$$ where $M$ represent an infinite-dimensional closed subspace of $X$.
If $A$ and $B$ are two nonempty subsets of a Banach s... | https://mathoverflow.net/users/41619 | A question on operational quantities characterizing strictly singular operators | It seems to me that one can show that $\tau(T)\leq \chi(T)$ as follows. The case $\tau(T)=0$ is clear. Let $\tau(T)>0$, $\varepsilon\in(0,\tau(T))$, and let $M\subset X$ be an infinite dimensional subspace for which $\inf\_{x\in S\_{M}}\|Tx\|\ge \tau(T)-\varepsilon$. Then $T|M$ is an isomorphism and $T(B\_M)$ contains ... | 0 | https://mathoverflow.net/users/37822 | 256936 | 116,043 |
https://mathoverflow.net/questions/251767 | 5 | There is a nice smooth retraction from $\operatorname{GL}(n,\mathbb{C})$ onto $\operatorname{U}(n)$, which can be explained using polar decomposition. There is an analogous one from $\operatorname{GL}(n, \mathbb{R})$ onto $\operatorname{O}(n)$. Both are "natural" from a Lie theoretic point of view (in a sense that shal... | https://mathoverflow.net/users/81645 | What is the largest subgroup of $GL^{+}(7,\mathbb{R})$ which smoothly retracts onto $G_2$? | There is no retraction of $\mathrm{SO}(7)$ onto $\mathrm{G}\_2$. If such a retraction $\rho:\mathrm{SO}(7) \to \mathrm{G}\_2$ existed, then the composition
$$
\mathrm{G}\_2 \hookrightarrow \mathrm{SO}(7)\ {\buildrel{\rho}\over{\rightarrow}}
\ \mathrm{G}\_2\,,
$$
would induce a composition of the homotopy group homomorp... | 12 | https://mathoverflow.net/users/13972 | 256937 | 116,044 |
https://mathoverflow.net/questions/254702 | 15 | Related to [Kahler version of Darboux's Theorem](https://mathoverflow.net/questions/254563/kahler-version-of-darbouxs-thorem)
I know a few theorems that feel like Darboux's theorem. By that, I mean some kind of geometry based around a "pointwise" condition and the existence of a tensor or something generalizing a ten... | https://mathoverflow.net/users/44191 | Darboux-like theorems | While Francois' answer is fine, as far as it goes, I think that it is important to bear in mind the history of this problem.
The original problem of 'flatness' or integrability (and more generally, equivalence) of what are now called $G$-structures was formulated by Élie Cartan in his fundamental paper *Les sous-grou... | 20 | https://mathoverflow.net/users/13972 | 256943 | 116,046 |
https://mathoverflow.net/questions/256946 | 2 | Let $(X,\tau)$ be a topological space. We say that $x\neq y\in X$ are *close* if every neighborhood of $x$ intersects every neighborhood of $y$. We associate to $(X,\tau)$ a graph $G(X,\tau)=(V,E)$ with $V = X$ and $$E=\big\{\{x,y\}: x\neq y \in X \text{ and } x,y \text{ are close}\big\}.$$
Given an infinite simple g... | https://mathoverflow.net/users/8628 | Graph associated to a topological space | I think the answer is (sadly) no : take a complete graph $K\_n = (V, E)$ and remove an edge, say between $x$ and $y$. Then assume you have such a topology $\tau$. Take any $U, W\in \tau$ such that $x\in U$, $y \in W$ and $U\cap W=\emptyset$ (two such open sets exist,otherwise $x$ would be close to $y$). Since any point... | 4 | https://mathoverflow.net/users/102343 | 256951 | 116,048 |
https://mathoverflow.net/questions/256945 | 3 | Let $f$ be a non-constant polynomial with integers coefficients, and for each prime number $p$ let $\eta\_f(p)$ be the number of zeros of $f$ modulo $p$. It is known that the average (in the natural way) of $\eta\_f(p)$ among all primes $p$ is equal to the number $r$ of irreducible factors of $f$ in $\mathbb{Q}[X]$. (O... | https://mathoverflow.net/users/nan | Asymptotic formula for the average number of zeros of a polynomial modulo p | Your statement is essentially Mertens' theorem for number fields. Apparently a reference is
M. Rosen, A generalization of Mertens’ theorem, J. Ramanujan Math.
Soc. (1) 14 (1999), 1–19
However it requires a little bit of effort to transform this into your problem. You might not be satisfied by that.
Clearly it's suf... | 4 | https://mathoverflow.net/users/18060 | 256953 | 116,050 |
https://mathoverflow.net/questions/256846 | 0 | For any metric space $X$ and $\varepsilon>0$, let $$\text{cov}(X,\varepsilon)=\min\{n\,|\,X\text{ has a cover by }n\text{ many closed }\varepsilon\text{-balls}\},$$
be the ordinary covering numbers. For any $D>0$ and $N:(0,\infty)\rightarrow \mathbb{N}$, let $$U(D,N)=\{X\,|\,X\text{ is a compact metric space, diam}(X... | https://mathoverflow.net/users/83901 | Covering numbers of uniformly bounded subsets of Gromov-Hausdorff space | Given a metric space $X$ with $\operatorname{cov}(X,\epsilon) \leq N(\epsilon)$, there exist $N(\epsilon\_1)$ points such that each point is within $\epsilon\_1$ of each of them. That set of $N(\epsilon\_1)$ points is itself a metric space $Y$, with Gromov-Hausdorff distance at most $\epsilon\_1$ to $X$. That metric sp... | 1 | https://mathoverflow.net/users/18060 | 256955 | 116,051 |
https://mathoverflow.net/questions/234223 | 15 | I am looking for explicit examples (for all positive integers $n \ge 5$) of degree $2n$ even polynomials $f(x)=h(x^2)$ over the field $\mathbb{Q}$ of rational numbers such that the Galois groups of $f(x)$ over $\mathbb{Q}$ are the Weyl groups $W(B\_n), W(D\_n)$ or their normal subgroups of small index 2 or 4. (In this ... | https://mathoverflow.net/users/9658 | Weyl Groups as Galois groups | It's possible to produce examples using congruence conditions. If there is a prime modulo which $h(x)$ is irreducible, a prime modulo which it splits into the product of a linear polynomial and an irreducible polynomial, and a prime modulo which it splits into the product of a linear polynomial and $n-2$ irreducible po... | 5 | https://mathoverflow.net/users/18060 | 256956 | 116,052 |
https://mathoverflow.net/questions/256954 | 5 | The distribution $\nu\_{\lambda}$ of the random series $\sum\pm\lambda^n$ is the infinite convolution product of $\frac12(\delta\_{-\lambda^n}+\delta\_{\lambda^n})$. This problem has been [studied extensively](http://www.math.uchicago.edu/~schlag/papers/sixty.pdf).
I am curious about the following modification:
> ... | https://mathoverflow.net/users/66131 | a modification on an infinite Bernoulli convolution | 1. If $\lambda\in(0,1/3)$, then $\mu\_\lambda$ is supported by a Cantor set of zero measure and therefore, is singular.
2. If $\lambda=1/3$, then it is the Lebesgue measure on its support.
3. If $\lambda^{-1}$ is a Pisot number, then $\mu\_\lambda$ is singular, via a standard Erdős type argument. (Its Fourier transform... | 4 | https://mathoverflow.net/users/8131 | 256968 | 116,059 |
https://mathoverflow.net/questions/256970 | 3 | Suppose $L / K$ is a cyclic extension of number fields. Is there a straightforward way to determine if a given $\alpha\in K^{\*}$ is in the image of the norm map $N\_{L/K}:L^{\*}\rightarrow K^{\*}$? Or to at least find an element of $K^{\*}$ not in the image?
For example, if $K=\mathbb{Q}(\sqrt{-3})=\mathbb{Q}(\omega... | https://mathoverflow.net/users/102357 | Determining the image of the norm map of a cyclic extension | This articles describes an algorithm: <https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/norms_jsc.pdf> (Jürgen Klüners, V. Acciaro: Computing Local Artin Maps, and Solvability of Norm Equations, J.Symb.Comput., 30, 2000, 239-252.)
| 2 | https://mathoverflow.net/users/nan | 256971 | 116,061 |
https://mathoverflow.net/questions/256571 | 2 | It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$
**My questions:**
1. Given a finite measure $\mu$ on $\Omega \subseteq \mathbb{R}$, under what conditions does the res... | https://mathoverflow.net/users/42864 | Sturm Liouville problems for non-classical orthogonal polynomials | A reference in english for Bochner's theorem is section 20.1, p.508, of the book by Mourad E.H.Ismail,
Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, 98. Cambridge University Press, Cambridge, 2009.
| 4 | https://mathoverflow.net/users/89429 | 256982 | 116,066 |
https://mathoverflow.net/questions/256506 | 4 | Everything is over an algebraically closed field of characteristic $\neq 2$. I had constructed a root datum for $\textrm{GSp}\_4$ in a less than ideal way, and I had hoped to show that it was self dual. I wanted to know if my approach was salvegeable.
I came up with the following two lemmas to help me do it.
**Lemm... | https://mathoverflow.net/users/38145 | $\textrm{GSp}_{4}^{\wedge} \cong \textrm{GSp}_4$ | Define $A=\begin{pmatrix}1& 0\\0&1\\0&0\end{pmatrix},\
B=\begin{pmatrix}2&-1\\-2&2\\-1&1\end{pmatrix}
$.
The matrices $A$ and $B$ consist of coefficents of simple roots and simple
coroots, respectively. To be explicit, each column of $A$ corresponds to a simple root vector and consists of the coefficients of the si... | 2 | https://mathoverflow.net/users/102320 | 256991 | 116,069 |
https://mathoverflow.net/questions/256987 | 6 | Let $p$ and $q$ be two distinct primes. Let $f\in \mathcal{S}\_k^{\ast}(pq,\psi)$ be a holomorphic newform of level $pq$, nebentypus $\psi$, and weight $k$, where $\psi = \chi\_p \chi\_{0(q)}$, with $\chi\_p$ a primitive character modulo $p$ and $\chi\_{0(q)}$ the principal character modulo $q$ (i.e. $\psi$ is an impri... | https://mathoverflow.net/users/102362 | Root number of the Rankin-Selberg convolution of two newforms | You need to do this via a local argument. A good reference for local components of $\mathrm{GL}\_2 \times \mathrm{GL}\_2$ automorphic representations is Gelbart and Jacquet, "[A relation between automorphic representations of $\mathrm{GL}(2)$ and $\mathrm{GL}(3)$](https://dx.doi.org/10.24033/asens.1355)". For just the ... | 6 | https://mathoverflow.net/users/3803 | 256995 | 116,070 |
https://mathoverflow.net/questions/256993 | 3 | Is
$\mathrm{sd}^2 (\Delta^n) = \mathrm{sd}^2(\partial \Delta^n) \times \Delta^1 \cup\_{\mathrm{sd}^2(\partial \Delta^n) \times \{1\}} Cone(\mathrm{sd}^2(\partial \Delta^n))$
? Here $\mathrm{sd}^2$ means the second barycentric subdivision of a simplicial set / complex, and
$Cone(\mathrm{sd}^2(\partial \Delta^n)) ... | https://mathoverflow.net/users/2362 | Geometry of the second barycentric subdivision (and Thomason-fibrant replacement) | Yes, that's true. See Remark 4.1 in [this paper](https://arxiv.org/abs/1408.2743) by Meier and Ozornova. A generalization to $k > 2$ would be interesting and I believe that everything you wrote about it is right. However, I don't know any general statement.
| 2 | https://mathoverflow.net/users/12547 | 256999 | 116,072 |
https://mathoverflow.net/questions/256998 | 4 | Short version: what can we say about subsets of $\omega\_2$ which - in a generic extension where $\omega\_2$ is the new $\omega\_1$ - contain a club?
*We could of course generalize beyond $\omega\_2$, but already the questions seem hard.*
---
**Motivating example**:
Even in much weaker theories than ZFC (alth... | https://mathoverflow.net/users/8133 | "Potentially club" filters on $\omega_2$ | First, your notation is nonstandard; when we write $\mathrm{Col}(\kappa,\lambda)$, this typically means the set of partial functions $p : \kappa \to \lambda$ of size $<\kappa$, i.e. reverse of yours.
First note the following fact:
>
> (1) If $\kappa$ is regular and $\mathbb P$ is $\kappa$-c.c., then every club su... | 7 | https://mathoverflow.net/users/11145 | 257012 | 116,075 |
https://mathoverflow.net/questions/237043 | 3 | Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I\_n$ be the $n\times n$ identity matrix, and let $A\_n$ be the $n\times n$ diagonal matrix having coefficient $1,q,...,q$ on the diagonal.
Note that $I\_n$ and $A\_n$ are always congruent over the $... | https://mathoverflow.net/users/47722 | Are those $2$ quadratic forms congruent over $\mathbb{Z}[1/q]$ | The answer to the main question is yes ! (The question has been answered by the commentators).
As a consequence, it is not possible to push the Bruck-Ryser theorem by looking at congruences over rings of the form $\mathbf{Z}[1/q]$.
| 3 | https://mathoverflow.net/users/47722 | 257013 | 116,076 |
https://mathoverflow.net/questions/256566 | 0 | I am studying some theories around FEM method in 2D, and I am trying to solve this problem from Ciarlet's book (the proof was not provided): Consider a simplex $T$ in $R^d$ with $N\_1(T) = \left\{N\_i\right\}\_{i=0}^{d}\subset P\_1^{\*}(T)$ be the Lagrange nodal variables (or nodal evaluation). By the Riesz representat... | https://mathoverflow.net/users/81537 | Dual basis of Lagrange nodal variables in $R^d$ | (failed to edit the comment-answer in 5 minutes fully)
In fact, no, I am sure there is a nice way to compute all this but now can think only of a brute force one. $\int\_{T^{hat}} \lambda\_1^2 = \int\_{T^{hat}} x\_1^2 dx = \int\_0^1 x\_1^2 V\_{d-1}(1-x\_1) dx\_1$ where $V\_k(r)$ is the volume of $k$-dimensional canon... | 0 | https://mathoverflow.net/users/97620 | 257016 | 116,077 |
https://mathoverflow.net/questions/257007 | 8 | Let $f:A\to B$ be a monoid homomorphism. Where can I find an explicit description of the its cokernel? Are there any books on this topic?
By the *cokernel* of $f$, I mean the universal arrow which postcomposes with $f$ to give the trivial homomorphism. (Sorry for not including this from the start, I just thought ther... | https://mathoverflow.net/users/69037 | What's the cokernel of a monoid homomorphism? | First of all, the construction is as for all (pointed) algebraic structures. Let $\sim$ be the congruence relation generated by $f(a) \sim 1$ for $a \in A$. Here, congruence relation means an equivalence relation on the underlying set of $B$ satisfying $b \sim b' \Rightarrow x b \sim x b' \wedge b x \sim b' x$ for all ... | 5 | https://mathoverflow.net/users/98306 | 257033 | 116,082 |
https://mathoverflow.net/questions/257025 | 1 | Suppose $H$ is some group with finite isomorphic subgroups $A$ and $B$, and isomorphism $\psi: A\rightarrow B$. Suppose that the HNN-extension $G=\langle H, t\mid a^t=\psi(b)~\forall~a\in A\rangle$ is hyperbolic. Is $H$ hyperbolic?
My thoughts on this are rather sparse. I know that the opposite is true - that if $H$ ... | https://mathoverflow.net/users/35478 | Hyperbolic HNN-extension with finite associated subgroups implies hyperbolic base group? | As Derek Holt indicates in comments, this is standard. You should do the following easy exercise.
>
> If $G = A\*\_CB$ or $G=A\*\_C$ is hyperbolic and $C$ is quasiconvex, then so is $A$ (and $B$).
>
>
>
Since finite subgroups are trivially quasiconvex, and quasiconvex subgroups are themselves hyperbolic, the r... | 2 | https://mathoverflow.net/users/1463 | 257035 | 116,084 |
https://mathoverflow.net/questions/257031 | 6 | We start with a model of $\sf ZFC$, $V$, for simplicity we can imagine that $V$ satisfies $\sf GCH$ or even $V=L$.
Let $\Bbb P$ be $\operatorname{Add}(\omega,\omega\_1)$, and let $G$ be a $V$-generic filter for $\Bbb P$. For $E\subseteq\omega\_1$, we will write $G\restriction E$ as the restriction of $G$ to the reals... | https://mathoverflow.net/users/7206 | Cohen real without a minimal support | Define $x \in 2^{\omega} \cap V[G]$ by $x(n) = $ the first bit of the $n$th Cohen real. Then for every $E \subseteq \omega\_1$, $E \in V$, we have $x \in V[G \upharpoonright E]$ iff $\omega \setminus E$ is finite. So there is no such minimal $E$ for $x$.
| 6 | https://mathoverflow.net/users/2689 | 257036 | 116,085 |
https://mathoverflow.net/questions/256785 | 6 | Let $T\_0$ be the set theory axiomatized by $ZFC^-$ (that is $ZFC$ without powerset) + every set is countable + $\mathbb{V}=\mathbb{L}$.
**Question 1:** Suppose $\phi$ is a sentence of set theory. Must there be a large cardinal axiom $A$ such that $\phi$ is decided by $T\_0$ + "there are transitive set models of $A$... | https://mathoverflow.net/users/26705 | A "Completion" of $ZFC^-$ | Without a definition of “large cardinal axiom”, I’m going to ignore Q1 and Q3.
The answers to Q2 and Q4 are negative by the following general principle. (For Q4, we take $T\_0$ to be the set of $\mathcal L\_{\mathrm{set}}$-consequences of $T\_1$.)
>
> $\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\Wit{Wit}\let\... | 4 | https://mathoverflow.net/users/12705 | 257044 | 116,086 |
https://mathoverflow.net/questions/257047 | 2 | I am given $n$ objects and for $n$ times, I pick one of them with uniform probability and put it back after picking it.
For $k\in\{1,\ldots,n\}$ let $f\_k$ denote the number of times that I have picked object number $k$. So we have $f\_k\in \{0,\ldots,n\}$ for all $k$.
We consider $M:= \max\big\{f\_k: k\in\{1,\ldot... | https://mathoverflow.net/users/8628 | Choosing $n$ times from $n$ objects | Fix some $k$. The probability that the element $1$ is chosen exactly $k$ times is ${n\choose k}\frac1{n^k}\left(1-\frac1n\right)^{n-k}\to \frac1{ek!}$ as $n\to\infty$. So $1$ is chosen at least $k$ times with probability at least, say, $\frac1{2ek!}$
Under the condition that $1$ has been chosen less than $k$ times, t... | 4 | https://mathoverflow.net/users/17581 | 257049 | 116,087 |
https://mathoverflow.net/questions/256900 | 4 | $\newcommand{\abs}[1]{\left|#1\right|}$
There is a population $O$ with a countable (finite or infinite) number of subjects. The population is colored randomly: for each subject, an unbiased coin-toss is used to decide whether the subject is colored red or green. Then, a sub-population containing $t$ subjects are sele... | https://mathoverflow.net/users/34461 | Concentration inequalities for random sets | I'll combine Ryan's answer and my elaboration of it into a single (hopefully, coherent) answer.
I'll represent Erel's random R/G coloring as the assignment of a random $\sigma\_i\in\{-1,1\}$ to each point $x\_i\in O$. Put $n:=|O|$. I'll represent the set $T$ as a function $f:O\to\{0,1\}$, where $f$ is the characteris... | 1 | https://mathoverflow.net/users/12518 | 257050 | 116,088 |
https://mathoverflow.net/questions/256967 | 3 | Suppose that $A$ is a finite dimensional unital commutative Banach algebra and $A\hat{\otimes} A$ is its Projective tensor product by itself. Define $\Delta:A\hat{\otimes} A\to A$ with
$$\Delta(\sum\_{i=1}^\infty a\_i\otimes b\_i)=\sum\_{i=1}^\infty a\_i b\_i.$$
There is $m\in A\hat{\otimes} A$ with
$$a\Delta(m)=a,\q... | https://mathoverflow.net/users/99284 | Special bound of net and inequality | I am not entirely sure you have correctly stated the logical ordering of what you want to ask.
Your question appears to be: suppose $A$ is a finite-dimensional unital CBA, and suppose it has an approximate diagonal satisfying certain conditions; does the unique diagonal element $m\in A \otimes A$ satisfy a certain no... | 1 | https://mathoverflow.net/users/763 | 257053 | 116,089 |
https://mathoverflow.net/questions/256979 | 7 | Suppose we have a dynamical system and a sequence of functions $0=f\_0\leq f\_1\leq\cdots\leq f\_k$. Define $J\_{r,\lambda}$ to be the set of points $x$ such that there are $j\_0<j\_1<\cdots<j\_r$ so that, for each $i<r$, $\sup\_n A\_n(f\_{j\_{i+1}}-f\_{j\_i})(x)>\lambda$. Using the maximal ergodic theorem, we see that... | https://mathoverflow.net/users/8991 | The Maximal Ergodic Theorem more than once | You can't do substantially better than $\|f\_k\|\_\lambda$. Here's a simple example: Consider $X=\{0,1,\ldots,2^{N}-1\}$, equipped with normalized counting measure. The transformation is $T(x)=x+1\bmod 2^{N}$. The functions are
$$
f\_i(x)=\begin{cases}
2&\text{if $x<i$;}\\
0&\text{otherwise.}
\end{cases}
$$
for $i=0,\... | 4 | https://mathoverflow.net/users/11054 | 257060 | 116,090 |
https://mathoverflow.net/questions/257046 | 3 | Suppose $U \subsetneq \mathbb{R}^d$ is open. How do I see that the distance function$$u(x) = \min\_{y \in \mathbb{R}^d \setminus U} |x - y|$$is the unique nonnegative continuous function on $\mathbb{R}^d$ that satisfies$$\begin{cases} \lim\_{r \to 0} {1\over r}((u(x) - \min\_{\partial B(x, r)} u) = 1 & \text{if }x \in ... | https://mathoverflow.net/users/98682 | Distance function is unique nonnegative continuous function on $\mathbb{R}^d$ satisfying following | Give yourself "an epsilon of room" and apply the continuity method.
It suffices to check it for $x\in U$.
For one direction, let $\epsilon>0$ and $y\notin U$ such that $d:=d(x,\mathbb{R}^n-U)=|y-x|$. Let
$A=\{t\in[0,1]:\forall s\in[0,t],u(x+s(y-x))\ge u(x)-(1+\epsilon)sd\}$.
Then $A$ is both open and closed in ... | 5 | https://mathoverflow.net/users/37103 | 257065 | 116,092 |
https://mathoverflow.net/questions/257056 | -1 | Let $k=\mathbf Q[i]$ be the field of Gaussian numbers.
I've proved the following easy lemma:
"If $x \in \mathcal{O}\_k$ (the ring of integers of $k$) and $p$ is an odd rational prime dividing the norm $N\_k(x)$, then there is a prime ideal $P$ in $\mathcal{O}\_k$ such that $P$ divides both $p\mathcal{O}\_k$ and $x\ma... | https://mathoverflow.net/users/100299 | Prime dividing norm of algebraic integer | Yes. It's sufficient to prove $(p,x)$ is not $(1)$, because then we can take any prime factor of $(p,x)$. But if $ap + bx =1 $ for $a,b \in \mathcal O\_K$, then $N(b)N(x) =N(bx) = N(1-ap) \equiv 1$ mod $p$. This is a contradiction as $N(b)$ is an integer if $N(x)$ is divisible by $p$.
| 1 | https://mathoverflow.net/users/18060 | 257066 | 116,093 |
https://mathoverflow.net/questions/257067 | 3 | The fact that the axiom of foundation doesn't imply the axiom of choice is pretty standard (the model Cohen created to prove the consistency of $\neg AC$ models the axiom of foundation as well), and it's also known that you can have both of them (in $\mathbb{L}$ for instance), or neither of them (Fraenkel's model where... | https://mathoverflow.net/users/102343 | Relation between AC and the axiom of foundation | Of course the axiom of choice is consistent with the failure of the axiom of foundation.
To get Fraenkel's model with the atoms, you usually start with a model of $\sf ZFA+AC$. You can find the relevant proofs in Jech "The Axiom of Choice" in Chapter 4.
(Note that atoms are usually obtained by weakening extensional... | 6 | https://mathoverflow.net/users/7206 | 257069 | 116,094 |
https://mathoverflow.net/questions/257074 | 7 | There is a set of notes by Lachlan from 1973 on casting priority arguments in topological language; references to these notes are few and far between, but [one source](http://www.mathcomp.leeds.ac.uk/turing2012/inc/Talks/soareChicheley.pdf) refers to them as "Topology for Priority Arguments," which for now I'll assume ... | https://mathoverflow.net/users/8133 | Lachlan on topology for priority arguments | After about my third priority argument in a graduate class, I too was interested in finding a general approach for handling them. You can find something similar from Lachlan in published form as [The priority method for the construction of recursively enumerable sets](http://link.springer.com/chapter/10.1007/BFb0066779... | 5 | https://mathoverflow.net/users/99234 | 257075 | 116,098 |
https://mathoverflow.net/questions/257085 | 13 | Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}'$ its subalgebra. Then the universal enveloping algebra $U(\mathfrak{g}')$ can be canonically embedded into $U(\mathfrak{g})$, that of $\mathfrak{g}$.
Now I'm interested in the reverse direction. Given a subalgebra $Y$ of $U(\mathfrak{g})$, under what conditions o... | https://mathoverflow.net/users/1832 | characterization of subalgebras of universal enveloping algebra coming from Lie subalgebras | In characteristic $0$, if $Y$ is a *Hopf* subalgebra of $U(\mathfrak{g})$, then $Y = U(\mathfrak{g}')$ for some Lie subalgebra $\mathfrak{g}'$ of $\mathfrak{g}$ (and conversely, every such subalgebra is a Hopf subalgebra). This is a consequence of Proposition 6.13 and Theorem 5.18 of John W. Milnor and John C. Moore's ... | 15 | https://mathoverflow.net/users/7932 | 257090 | 116,101 |
https://mathoverflow.net/questions/257014 | 4 | Under some conditions, a distribution is determined by all of its moments. Furthermore, there is a certain value of entropy for a given distribution. So my question is:
1.Can I say that its entropy contains all the information of its moments?
2.Is there any mathematical link between entropy and moments of a given... | https://mathoverflow.net/users/100893 | Mathematical links between entropy and moments of a given distribution | If you're interested in entropy of a discrete distribution, then entropy tells you nothing about the moments (unless the random variable is constant).
First, it literally tells you nothing about the expected value or the variance since all the functions $aX+b$ have the same entropy (provided $a \neq 0$). And what's w... | 1 | https://mathoverflow.net/users/22512 | 257091 | 116,102 |
https://mathoverflow.net/questions/257072 | 1 | Suppose we have a local ring $L$ (not necessarily commutative) such that $L/rad(L)$ is a division algebra (here $rad(L)$ is the Jacobson radical of $L$). We clearly have the canonical surjection $\varphi:L\to L/rad(L)$ from which we may induce the surjection $\varphi\_\ast:L[x\_1,\ldots,x\_n]\to(L/rad(L))[x\_1,\ldots,x... | https://mathoverflow.net/users/73736 | When is $rad(L)[x_1,\ldots]$ radical in $Ker(\varphi_\ast)$? | Consider $L=\mathbb{Z}\_{(2)}$ (the set of rational numbers which in reduced form have denominators coprime to $2$). This is a local ring with ${\rm rad}(L)=2L$ and $L/2L\cong \mathbb{F}\_2$. However $2L[x\_1,\ldots,x\_n]$ is not contained in the Jacobson radical of $L[x\_1,\ldots, x\_n]$. For instance, $1+2x\_1$ is no... | 1 | https://mathoverflow.net/users/3199 | 257093 | 116,103 |
https://mathoverflow.net/questions/257032 | 28 | Given a projective variety $X$, each of its embeddings $i:X\hookrightarrow \mathbb P^N$ gives rise to an integer valued Hilbert polynomial $P\_{X,i}(t)\in \mathbb Q[t]$.
These polynomials depend however on the chosen embedding $i$: for example $\mathbb P^1$ linearly embedded into $\mathbb P^2$ has Hilbert polynomial... | https://mathoverflow.net/users/450 | Which intrinsic invariants of a projective variety can you deduce from its Hilbert polynomials? | The OP encouraged me to post my comment, along with some examples, as an answer, so here goes.
>
> **Lemma.** Let $X$ be a smooth projective variety over an algebraically closed field $k$. Then there exists a numerical polynomial $p \in \mathbb Q[x\_1,\ldots,x\_r]$ such that for every (ample) divisor $H$ on $X$, th... | 24 | https://mathoverflow.net/users/82179 | 257097 | 116,105 |
https://mathoverflow.net/questions/257100 | 4 | how can one construct a finite set of points in the euclidean plane from its Voronoi Diagram and, what is the complexity of the problem?
| https://mathoverflow.net/users/31310 | Algorithm for Reconstructing Point Sites from a Voronoi Diagram | This is a well studied problem and there are a couple of algorithms, for example using linear programming. For an overview take a recent reference, for example:
[Fitting Voronoi Diagrams to Planar Tesselations](https://arxiv.org/abs/1308.5550) by Greg Aloupis, Hebert Pérez-Rosés, Guillermo Pineda-Villavicencio, Perou... | 5 | https://mathoverflow.net/users/39495 | 257110 | 116,111 |
https://mathoverflow.net/questions/204446 | 2 | I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a partially observed Ito diffusion process
\begin{align}
dX\_t&=f(t,\theta\_0,X\_t)dt+g(t,\theta\_0,X\_t)dW\_t, \\
dY\_t&=h(t... | https://mathoverflow.net/users/69603 | Recursive parameter estimation for partially observed Ito SDEs | The formulation of a likelihood function for a continuous-time stochastic process requires the choice of a reference measure. In the context of parameter estimation, the reference measure has to be parameter-independent. In the case of partially observed diffusion processes with additive observation noise, the innovati... | 3 | https://mathoverflow.net/users/69603 | 257112 | 116,112 |
https://mathoverflow.net/questions/257115 | 2 | In his paper, "Completed versus Incomplete Infinity in Arithmetic" (which can be found [here](http://web.math.princeton.edu/%7Enelson/papers/e.pdf)), the late Edward Nelson defines the notion of 'counting number' as follows:
>
> 0 is a counting number
>
>
> if $y$ is a counting number, so is $y{'}$ [ $^{'}$ is th... | https://mathoverflow.net/users/20597 | Is the statement "All numbers are counting numbers" independent of $PA$? | The statement asserting that every number is a counting number is $\forall n\ C(n)$, and this is definitely independent of PA, if PA is understood to include induction only in the usual language of arithmetic, without the predicate $C$. To see this, we can simply observe that the statement is true in the standard model... | 7 | https://mathoverflow.net/users/1946 | 257117 | 116,114 |
https://mathoverflow.net/questions/257124 | 12 | This problem arose when considering storage of cannonballs in n-dimensional pirate ships, as explained in [this MSE post](https://math.stackexchange.com/questions/2042839/solutions-to-the-diophantine-equation-xn-2yn-1-can-the-sum-of-the-first-n). [This MO question](https://mathoverflow.net/questions/39561/is-there-an-e... | https://mathoverflow.net/users/100723 | Are there any solutions to the diophantine equation $x^n-2y^n=1$ with $x>1$ and $n>2$? | Delone (1930) and Nagell (1928) showed for any nonzero integer $d$ that the equation $x^3 - dy^3 = 1$ has at most one solution in integers $(x,y)$ besides $(1,0)$, with no constraint on the signs of $x$ and $y$. In particular, since $x^3 - 2y^3 = 1$ has the integral solution $(-1,-1)$, there is no integral solution $(x... | 20 | https://mathoverflow.net/users/3272 | 257131 | 116,121 |
https://mathoverflow.net/questions/257133 | 8 | Does every connected, compact Riemann surface $\Sigma$ with boundary, $\partial \Sigma\not =\emptyset$, admit a holomorphic function (smooth on the boundary) $f:\Sigma\to\mathbb C$ whose derivative is everywhere non-zero?
| https://mathoverflow.net/users/5690 | Does every Riemann surface with boundary immerse in C? | A more general result is proven in
Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967).
As for compact surfaces with boundary, it is essentially a part of the definition that they embed holomorphically into open Riemann surfaces.
Incidentally, it is an open problem... | 15 | https://mathoverflow.net/users/21684 | 257135 | 116,122 |
https://mathoverflow.net/questions/257121 | 6 | The question is very simple and I apologize for that, but I am not an expert of this kind of problem.
Given the polynomial
$$ P(x\_1,\ldots,x\_{2n})=x\_1^2+\ldots+x\_n^2-x\_{n+1}^2-\ldots-x\_{2n}^2,$$
I would like to know if there are non trivial integer roots $(y\_1,\ldots, y\_{2n})$ such that
$$y\_1+\cdots+y\_{n}=y\_... | https://mathoverflow.net/users/45729 | Integer roots of a symmetric polynomial | Fix large $N$ and consider all $n$-tuples $(x\_1,\dots,x\_n)\in \{1,\dots,N\}^n$. There are $N^n$ such $n$-tuples, at least $N^n/n!$ tuples modulo permutations, and for them the pairs $(x\_1+\dots+x\_n,x\_1^2+\dots+x\_n^2)$ take at most $n\cdot N\cdot n\cdot N^2=n^2N^3$ possible values. Thus by pigeonhole principle som... | 17 | https://mathoverflow.net/users/4312 | 257136 | 116,123 |
https://mathoverflow.net/questions/257138 | 3 | (This is a cross-post of [this unanswered math.stackexchange question](https://math.stackexchange.com/questions/2054693/in-what-sense-is-every-element-of-h-2g-represented-by-a-free-action-on-some))
In Edmond's 1982 paper [Surface Symmetry II](https://projecteuclid.org/download/pdf_1/euclid.mmj/1029002844), at the bot... | https://mathoverflow.net/users/88840 | In what sense is every element of $H_2(G)$ "represented by a free action on some surface" | The group $\Omega\_d(X)$ being used is the following oriented bordism group. The objects are equivalence classes of closed oriented d-dimensional manifolds $N$ with a map $N \to X$. Two elements $N\_1 \to X$ and $N\_2 \to X$ are equivalent if there exists a compact oriented (d+1)-manifold $W$ with a map $W \to X$ such ... | 5 | https://mathoverflow.net/users/360 | 257144 | 116,126 |
https://mathoverflow.net/questions/257139 | 5 | (The following question arose in a joint research with Adam Przeździecki and Boaz Tsaban.)
For a $\sigma$-ideal $\mathcal{I}$ of subsets of the unit interval
$[0,1]$, define
$$\newcommand{\card}[1]{\left|#1\right|}\newcommand{\cov}{\operatorname{cov}}\newcommand{\sub}{\subset} \cov(\mathcal{I}):=\min\{\card{\math... | https://mathoverflow.net/users/91504 | Covering measure one sets by closed null sets | The answer to the first question is yes: it is always true that $\kappa\_{\mathcal E} = \mathrm{cov}(\mathcal E)$. The answer to the second question is no.
As Piotr points out, a negative answer to the second question implies a positive answer to the first. [This is because if $A \subseteq [0,1]$ is a set of measure ... | 3 | https://mathoverflow.net/users/70618 | 257152 | 116,129 |
https://mathoverflow.net/questions/257116 | 3 | For a compact semisimple Lie group $G$, what is an example of a homogeneous space of $G$ which is not a torus bundle over a [generalized flag manifold](https://en.wikipedia.org/wiki/Generalized_flag_variety) of $G$. Examples for $SU(N)$ would be of most interest.
| https://mathoverflow.net/users/42100 | Homogeneous spaces which are not torus bundle over flag manifolds | Just to summarize the common feature of both Allen's example and my example: there are many Lie subgroups $H$ of $G$ that are contained in no proper parabolic subgroup. Some examples arise from the fact that a proper subgroup of a parabolic may have a normalizer that is not contained in the subgroup; the example $H$ th... | 4 | https://mathoverflow.net/users/13265 | 257159 | 116,133 |
https://mathoverflow.net/questions/257157 | 0 | Let $[\theta\_1,\theta\_2, \dots, \theta\_N]^\mathrm{T} \, \in \mathbb{R}^N$. The angles are not all identical (on the circle), i.e. $[\theta\_1,\theta\_2, \dots, \theta\_N] \not \equiv c [1,1,\dots, 1]^\mathrm{T}\,\, \mathrm{mod}\,\, 2\pi$. Define matrices $C$ and $S$ as:
\begin{align}
\begin{split}
[C]\_{jl}&= 1 \,... | https://mathoverflow.net/users/102447 | Do these matrices have the same null space? | You must have made a mistake.
$E = -C + 2I + i S$ has entries $\exp(i (\theta\_j - \theta\_l)) = \exp(i\theta\_j) \exp(-i \theta\_l)$ and thus is the product $V V^\*$ where $V$ is the column vector with entries $\exp(i \theta\_j)$, so it has rank $1$.
Then $S = (E - \overline{E})/(2i)$ has rank at most $2$ (it wou... | 1 | https://mathoverflow.net/users/13650 | 257161 | 116,134 |
https://mathoverflow.net/questions/256711 | 13 | Just as a monoid is a category with a single object, a semigroup may be seen as a non-unital category, still with associative composition. Then an $S$-set for $S$ a semi-group can be seen as a functor from the category corresponding to $S$ into the category of sets.
One of the nice things about the functor category o... | https://mathoverflow.net/users/69037 | Category without identities? |
>
> which portions of category theory still hold without identities?
>
>
>
Almost everything. <https://arxiv.org/abs/1311.3524v1>. (The link is to v1 of the paper, because the author has removed v2 from the arXiv.)
I'm honestly astonished by the fact that this paper hasn't the impact it deserves.
| 9 | https://mathoverflow.net/users/7952 | 257174 | 116,141 |
https://mathoverflow.net/questions/257169 | 2 | Calling '$L$-function' any automorphic $L$-function belonging to the Selberg class, what are the known $L$-functions $L(s,F)$ and $L(s,G)$ of respective degrees $d$ and $d'$ such that the Rankin-Selberg convolution $L(s, F \otimes G)$ is provably (as of today, December 13 2016) an $L$-function of degree $dd'$?
| https://mathoverflow.net/users/13625 | Known degrees of L-functions F and G whose Rankin-Selberg convolution is an L-function | An automorphic L-function $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.
This includes two important cases: Hecke characters and holomorphic modular forms.
Automorphicity of Rankin-Selberg convolution is known in that range for $\mathrm{GL}(1)\times \... | 5 | https://mathoverflow.net/users/43108 | 257184 | 116,145 |
https://mathoverflow.net/questions/257165 | 9 | $\newcommand{\al}{\alpha}$
$\newcommand{\ga}{\gamma}$
$\newcommand{\e}{\epsilon}$
Let $X,Y$ be Riemannian manifolds, such that $\dim(X) > \dim(Y)$.
I am trying to prove the following statement (mentioned by Gromov in his book on metric geometry):
There is no arcwise isometry (i.e length preserving map) from $X$ t... | https://mathoverflow.net/users/46290 | There is no arcwise isometry from a high dimensional manifold into a low dimensional manifold | Your proof is correct, but you need to add words "amost everywhere" at ane more place.
We use [Rademacher's theorem](http://en.wikipedia.org/wiki/Rademacher%27s_theorem) and lemma about length of curve, which says that if a curve parametrized by length then its velocity is 1 almost everywhere, see 2.7.4 in Metric Geo... | 5 | https://mathoverflow.net/users/1441 | 257188 | 116,149 |
https://mathoverflow.net/questions/257151 | 4 | The Birman-Schwinger principle says that if $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and we consider the operator $H=-\Delta-V$ for a positive potential $V$, then, for any $\lambda>0$, the number of eigenvalues at most $-\lambda$ of this operator is the same as the number of eigenvalues at least 1 of the opera... | https://mathoverflow.net/users/102429 | Birman-Schwinger Principle | This runs into obvious technical issues. For example, $K\_0$ will not be bounded (let alone compact) even for very nice $V$. So one also has to think about what exactly one wants to prove.
[This paper](https://arxiv.org/abs/0911.2134) discusses these issues. In particular, the Birman-Schwinger principle for $\lambda=... | 5 | https://mathoverflow.net/users/48839 | 257191 | 116,150 |
https://mathoverflow.net/questions/257175 | 4 | Let $F : \mathcal{A}\rightarrow\mathcal{B}$ be an additive functor of abelian categories, such that $F$ has cohomological dimension $\le n$. Suppose $\mathcal{A}$ has enough injectives. Let $P\subset\text{Ob}(\mathcal{A})$ be the set of objects such that $R^iF(X) = 0$ for all $X\in P$ and $i\ne n$, and suppose every ob... | https://mathoverflow.net/users/88840 | Question about the proof of Prop I.7.4 in Hartshorne's Residues and Duality | This is a standard dimension shifting argument (*décalage*): split the long exact sequence
$$0 \to X \to I^0 \to I^1 \to \ldots \to I^{n-1} \to \ker(d^n) \to 0$$
into short exact sequences
$$0 \to J^i \to I^i \to J^{i+1} \to 0.$$
Here, $J^0 = X$ and $J^n = \ker(d^n)$. For each $k$, the associated long exact sequence gi... | 6 | https://mathoverflow.net/users/82179 | 257195 | 116,153 |
https://mathoverflow.net/questions/257101 | 8 | Let $G=(V,E)$ be a simple, undirected graph. A *clique cover* is a set ${\cal C}\subseteq {\cal P}(V)$ such that
1. every element of ${\cal C}$ is a clique, and
2. $\bigcup {\cal C} = V$.
We call a clique cover ${\cal C}$ *beatable* if there is a clique cover ${\cal C}\_1$ such that $$|{\cal C}\_1 - {\cal C}| < |{... | https://mathoverflow.net/users/8628 | Unbeatable clique covers | I think it is a known conjecture. The special case that there always exists a clique cover such that the union of any two of those cliques is not a clique, can be proved from Zorn's lemma (or without it), cf. my book with Totik: Problems and Theorems in Classical Set Theory, Problem 14.6.(m). This special case is actua... | 9 | https://mathoverflow.net/users/6647 | 257196 | 116,154 |
https://mathoverflow.net/questions/256965 | 2 | Let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle on a projective manifold $(X,\omega)$ of dimension $n$, where $\omega$ is a Kähler form; the holomorphic structure of $E$ defines an operator $\bar{\partial}\_E:\Omega^0(E)\to\Omega^{0,1}(E)$ and, since $\varphi:\Omega^0(E)\to\Omega^{1,0}(E)$, one defines $D^{\prime\prim... | https://mathoverflow.net/users/57030 | Semistable Higgs bundles and flat connections | Let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle on a complex, projective manifold $(X,\omega)$ of dimension $n$, where $\omega$ is a Kähler form; by hypothesis: $ch\_1(E)\cdot\omega^{n-1}=0$, $ch\_2(E)\cdot\omega^{n-2}=0$ and $\mathfrak{E}$ is semistable.
By [S] theorem 2, $\mathfrak{E}$ is the extension of stable H... | 0 | https://mathoverflow.net/users/57030 | 257206 | 116,162 |
https://mathoverflow.net/questions/255684 | 5 | A *pseudo-disk arrangement* is a collection of planar bodies whose boundaries are Jordan curves that pairwise intersect at most twice.
I would like to know if given seven points in the plane whether it is possible to find a pseudo-disk arrangement with $\binom 73$ disks, such that for each triple of the seven points th... | https://mathoverflow.net/users/955 | Is there a crossing-free planar embedding of the 2-skeleton of the 6-simplex? | I think that there is no pseudo-disk arrangement even with $\binom{5}{3}$ pseudo-disks for the 5 point case; I sketch a proof below.
Suppose for contradiction that there is such an arrangement.
Let $a\_1,\dots, a\_5$ be the points. For each closed Jordan curve $c\_{ijk}$ that surrounds $a\_i,a\_j$ and $a\_k$ we defin... | 3 | https://mathoverflow.net/users/102469 | 257212 | 116,164 |
https://mathoverflow.net/questions/257214 | 13 | Write an integer partition $\lambda\vdash n$ in two different ways:
(1) $\lambda=\lambda\_1\geq\lambda\_2\geq\lambda\_3\cdots\geq\lambda\_k\geq1$
(2) $\lambda=1^{m\_1}2^{m\_2}3^{m\_3}\cdots n^{m\_n}$ for some $m\_i\geq0$.
Denote the length of a partition, compatible with (1) and (2), by $m\_1+m\_2+\cdots+m\_n=k$.... | https://mathoverflow.net/users/66131 | an identity for a sum over partitions | Multiply the whole equality by $(-1)^n$.
First of all, notice that $k\choose m\_1,\dots,m\_k$ is the number of ways to permute the numbers $\lambda\_1,\dots,\lambda\_k$, so the left-hand side equals
$$
L=\sum\_{\lambda\_i\geq 1, \; \lambda\_1+\dots+\lambda\_k=n}
(-1)^k\prod\_{i=1}^k{n+1\choose \lambda\_i}.
$$
Denot... | 22 | https://mathoverflow.net/users/17581 | 257219 | 116,167 |
https://mathoverflow.net/questions/257181 | 4 | The [Bell numbers](http://mathworld.wolfram.com/ComplementaryBellNumber.html) $B(n)$ can be given as a sum of the (signed) [Stirling numbers of the second kind](http://mathworld.wolfram.com/ComplementaryBellNumber.html) $S(n,k)$ as $B(n)=\sum\_{k=0}^nS(n,k)$. There are also the so-called [complementary Bell numbers](ht... | https://mathoverflow.net/users/66131 | Congruence for complementary Bell numbers | The definition of the complementary Bell numbers should be
$$B\_1(n):=\sum\_{k=0}^n(-1)^kS(n,k).$$
Define the polynomials $B\_n(x)$ by
$$B\_n(x)=\sum\_{k=0}^nx^kS(n,k),$$
so $B\_1(n) = B\_n(-1)$.
These polynomials are called *Bell polynomials* or *exponential polynomials*. Christian Radoux proved the congruence
$$ B... | 9 | https://mathoverflow.net/users/10744 | 257234 | 116,176 |
https://mathoverflow.net/questions/256896 | 6 | Are there proofs of the measurability of $\omega\_1$ (under $\operatorname{AD}$) that do not use Turing degrees nor the $\Sigma\_1^1$ boundedness lemma?
I've been struggling to find an "elementary" proof of this fact. Note that I consider the proof of "Assume $\operatorname{AD}$. Then every ultrafilter is $\sigma$-c... | https://mathoverflow.net/users/102300 | $\operatorname{AD}$ and the measurability of $\omega_1$ | I'm not sure this will work for you, but there's a way to recast the Turing argument so that it avoids recursion theory; if this is the reason you want to avoid the Turing argument, this might be the way to go.
Namely, instead of working with Turing degrees, work with a coarser reducibility, which is easier to explai... | 4 | https://mathoverflow.net/users/8133 | 257241 | 116,178 |
https://mathoverflow.net/questions/257244 | 2 | Consider a semigroup $(T(t))\_{t\in\mathbb{R}^+}$ generated by a densely defined strictly positive symmetric linear operator $A: D(A) \subset X \to X$, where $X$ is a Banach space with norm $\|\cdot\|$.
Besides, we introduce a Sobolev scale $(X\_n)\_{n\in\mathbb
{Z}}$ induced by completion of $D(A^\infty)$ with respe... | https://mathoverflow.net/users/41105 | Estimate of semigroup with dual norm? | First, if $A$ is symmetric, then $X$ should be a Hilbert space, but I remain in a Banach space.
If
$$\|T(t)x\| \leq C\|A^{-1} x\|$$
holds for all $x\in X$, then using the substitution $y=A^{-1}x$, you obtain
$$\|AT(t)y\|\leq C\|y\|.$$
But by Theorem 5.3 in Section 2.5 of
*A. Pazy*, MR 710486 [**Semigroups of l... | 2 | https://mathoverflow.net/users/12898 | 257248 | 116,182 |
https://mathoverflow.net/questions/257239 | 2 | How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial differential equation?
$$\frac{\partial}{\partial t}p = \frac{k^2}2\frac{\partial^2}{\partial v^2}(vp)+\frac{\partial}{\partial... | https://mathoverflow.net/users/32660 | Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum | The given PDE (1) can be written as:
$$
\partial\_t p(t,x) = L^\* p(t,x) \tag2
$$ where $L^\*$ is the (formal) adjoint of the following operator
$$
Lf(x) = - \gamma (x - \theta) f'(x) + \frac{1}{2} k^2 x f''(x) \tag3
$$
Associated to this operator $L$ is a diffusion process which satisfies the SDE:
$$
d X = - \gamma... | 3 | https://mathoverflow.net/users/64449 | 257252 | 116,184 |
https://mathoverflow.net/questions/257081 | 7 | I had a question for anyone familiar with the proofs of the classification of reductive groups. I skipped most of the details of classification when I originally learned linear algebraic groups, and now I'm trying to go back and fill the gaps in my knowledge.
Let $G$ be a connected algebraic group over an algebraical... | https://mathoverflow.net/users/38145 | Does the classification of reductive groups follow from that of semisimple groups? | As indicated in the comments, there is no need to redo the entire classification argument when passing from "semisimple" to "reductive". But it's useful to recall some of the history. The emphasis in the Chevalley seminar 1956-58 was on achieving a uniform classification of semisimple algebraic groups over an algebraic... | 7 | https://mathoverflow.net/users/4231 | 257256 | 116,186 |
https://mathoverflow.net/questions/257263 | 1 | Here is the definition of $\xi(s,\chi)$:
$\xi(s,\chi)= \left(\frac{s(s-1)}{2} \right)^{1\_{\chi=1}} (q/\pi)^{\frac{s+a}{2}} \Gamma \left( \frac{s+a}{2} \right) L(s,\chi)$
Here is the definition of the **Hadamard product applied to $\xi(s,\chi)$:**
Since it is an entire function of order one, there exists constant... | https://mathoverflow.net/users/100898 | infinitely many non-trivial zeros for $L(s,\chi)$ using Hadamard product for $\xi(s,\chi)$ | The existence of the Hadamard product by itself doesn't show the function has any zeroes (it might just be an exponential!) -- you need a little more.
The idea is to study the logarithmic derivative, which by your two expressions above satisfies ($\delta\_\chi = 1$ for the principal character):
$$ -\frac{\xi'}{\xi}... | 11 | https://mathoverflow.net/users/327 | 257265 | 116,188 |
https://mathoverflow.net/questions/257260 | 4 | Let $\cal K$ be a 2-category, and $A$ an object therein. Is there a way to define the "slice object" $A/a$ in $\cal K$ formally (in such a way that if $K = Cat$ we recover precisely the notion of the slice category $A/a$)?
I suggest that if $\cal K$ has comma objects then $A/a = (1\_A\downarrow a)$ where $a\colon 1\t... | https://mathoverflow.net/users/7952 | Slice categories in a general 2-category | Yes, that's the right way to define "slice objects" in a 2-category. But there's no way to say what the colimit of that diagram is for a general $K$ and $A$. For instance, $A$ might not admit any "global elements" $1\to A$, so that the colimit would be the initial object. Or, if $K$ is a 1-category regarded as a 2-cate... | 5 | https://mathoverflow.net/users/49 | 257269 | 116,190 |
https://mathoverflow.net/questions/101225 | 6 | Let $\Phi(t)$ be an $n\times n$ complex matrix whose columns are (independent) solutions of the system of ordinary differential equations (ODE):
$\frac{d}{dt}y= A(t) y$, where $A(t)$ is a $t$-dependent $n\times n$ complex matrix.
Then, the Liouville formula provides a very simple relation between the determinant of $... | https://mathoverflow.net/users/23261 | A generalization of Liouville formula for the determinant of a system of ODE? | **NB:** I'll slightly rearrange this for clarity:
As I should have remarked at the beginning,
writing the fundamental solution to your system in the form
$$
y(t) = \Phi(t)\,y(0),
$$
where $\Phi(0)=I$, is only possible when you allow $\Phi(t)$ to take values in $$\mathrm{Aut}\_\mathbb{R}(\mathbb{C}^n,\mathbb{C}^n) \... | 2 | https://mathoverflow.net/users/13972 | 257274 | 116,192 |
https://mathoverflow.net/questions/257271 | 1 | This is a follow-up on a [previous question](https://mathoverflow.net/q/257239/32660). Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions.
$$
\partial\_t P = L^\* P \tag1
$$
$$L^\*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^2}+2\rho\kappa\frac{\partial^2}{\partial x\partial v}+\frac{\partial^2}... | https://mathoverflow.net/users/32660 | Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum II | *Almost a symmetric diffusion.* To see this, note that:
$$
L f(x,y) = \operatorname{Trace}\left( M(x,y) D^2 f(x,y) \right) + \mu(x,y) \bullet Df(x,y) \tag{2}
$$ where
$$
M(x,y) =\frac{1}{2} y \begin{bmatrix} 1 & \rho \kappa \\ \rho \kappa & \kappa^2 \end{bmatrix} \;, \quad \mu(x,y) = - y \begin{bmatrix} \frac{1}{2} \... | 2 | https://mathoverflow.net/users/64449 | 257277 | 116,194 |
https://mathoverflow.net/questions/257279 | 7 | The **Proper Forcing Axiom** kills CH in a particularly specific way: it implies that $2^{\aleph\_0}=\aleph\_2$. However, its impact on the continuum function above $\aleph\_0$ is much less clear. It is known, for example, that it implies the *Singular Cardinal Hypothesis*, SCH.
In [this paper](http://www.math.unt.ed... | https://mathoverflow.net/users/8133 | Getting PFA + GCH above $\omega$ | The proper forcing axiom is known to be indestructible by ${<}\aleph\_2$-directed closed forcing, and since the forcing of the GCH for uncountable cardinals admits this degree of closure (iteratively add a Cohen set to the successor cardinals that are still there), it follows that one can simply force the GCH above $\a... | 7 | https://mathoverflow.net/users/1946 | 257281 | 116,196 |
https://mathoverflow.net/questions/257249 | 2 | I wonder what could be a Kähler surface, which is a total space of principal elliptic bundle over a curve. I believe that there is a classification and that it must be pretty simple, but I cannot find it in the literature.
Fo example, though there exist a full and complicated classification of elliptic surfaces, I di... | https://mathoverflow.net/users/82309 | Principal elliptic bundles over curve with Kähler total space | A (partial) answer was given by [Donu Arapura](https://mathoverflow.net/users/4144/donu-arapura) at [Are most Kähler manifolds non-projective?](https://mathoverflow.net/questions/257147/are-most-k%C3%A4hler-manifolds-non-projective)
>
> Let $C$ be a smooth projective curve of genus $g>0$, $\Gamma =\pi\_1(C)$, and $... | 0 | https://mathoverflow.net/users/82309 | 257286 | 116,197 |
https://mathoverflow.net/questions/257231 | 7 | The Grothendieck ring of complex varieties $K(Var\_\mathbb C)$ is the free abelian group generated by isomorphism classes $[X]$ of $\mathbb C$-varieties, modulo the scissor relation $[X]=[Z]+[X\setminus Z]$ for every closed subvariety $Z\subset X$. The product is given by $[X]\cdot [Y]=[X\times\_\mathbb CY]$. There is ... | https://mathoverflow.net/users/97902 | What is the motivic class of a quotient stack? | Yes: if $G$ is a special group, then $[X/G] = [X]/[G]$ in the Grothendieck group of stacks. This is the analogue of the fact that if $X \to E$ is a $G$-torsor over an algebraic variety (and $G$ is still special) then $[X] = [E]\cdot [G]$ in the Grothendieck group of varieties. In the category of stacks, the map $X \to ... | 7 | https://mathoverflow.net/users/1310 | 257287 | 116,198 |
https://mathoverflow.net/questions/257273 | 0 | Suppose $X$ a normal project variety over complex numbers, $D$ is an integral Cartier divisor on $X$, and $L$ is a line bundle on $X$. Suppose one know the natural map $H^0(X, L) \to H^0(D, L|\_D)$ is **surjective**, I want to know if the following statement holds:
For any effective divisor $G\_D \in |L|\_D|$, is the... | https://mathoverflow.net/users/29730 | Restriction of a linear system to a divisor | Any effective divisor in $|L\vert\_D|$ is the zero locus of a global section of $L\vert\_D$. If $s\_D$ is such a section, by the surjectivity assumption there is a global section $s$ of $L$ on $X$ that restricts to $s\_D$. If $G$ is its zero locus then $G\vert\_D = G\_D$. So, what you ask is a tautology.
| 4 | https://mathoverflow.net/users/4428 | 257296 | 116,201 |
https://mathoverflow.net/questions/257266 | 6 | These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".
We concern ourselves with questions of the Picard variety $P$, and its dimension, of a complete nonsingular surface $F$ over an algebraically closed ground field $k$ of arbitrary characteristic. Let $\mathfrak{o}$ be the sheaf... | https://mathoverflow.net/users/nan | Intuition behind results in Mumford's "Lectures on curves on an algebraic surface", I | I think I can provide some intuition for (A), both in characteristic $0$ and $p > 0$. What follows below is more or less a proof, but with a lot of omissions (and hopefully not too many lies...).
I believe that everything I state works for all (geometrically) integral projective $k$-schemes. For simplicity, let's ass... | 7 | https://mathoverflow.net/users/82179 | 257298 | 116,202 |
https://mathoverflow.net/questions/257257 | 14 | Let $S$ be the set of injective sequences in $\mathbb{R}$:
$$S = \{s: \mathbb{N} \rightarrow \mathbb{R}: s(m) \neq s(n) \text{ if }m \neq n\}.$$
Consider $S$ with the topology of pointwise convergence, and $C(S,S)$ the associated continuous functions on $S$. For any sequence $s$ in $S$, let $\text{ran}(s)$ be the corre... | https://mathoverflow.net/users/nan | Given a sequence of reals, we can find a dense sequence avoiding it, but can we find one continuously? | I think there is not such an $f:S\to S$. Consider the sequence $x^t\in S$ continuously depending on $t\in[0,1]$, such that $x\_0^t=-t$ and $x\_n^t=1/n $ for all $n\ge1$. Since $f(x^1)$ is dense, for some index, say $17$, we have $-1 <f\_{17}(x^1)<0$. Therefore, for $t=1$, we have $$-t=x^t\_0<f\_{17}(x^t)<x^t\_n=1/n$$ f... | 6 | https://mathoverflow.net/users/6101 | 257299 | 116,203 |
https://mathoverflow.net/questions/257275 | 6 | First, I'd like to understand what the compact open subgroups of $H(\mathbb{Q}\_p)$ are, where $H$ is an inner form of $GL\_n$ over $\mathbb{Q}\_p$.
Second, I'd like to know where I can read about this for other reductive groups.
Any pointers would be greatly appreciated. Thanks!
| https://mathoverflow.net/users/97316 | Reference Request: Compact subgroups of p-adic Reductive Groups | You get compact subgroups by taking compact-open subgroups of algebraic subgroups. My understanding is that they are more-or-less all compact subs, e.g. any compact subgroup $H$ should have a finite index subgroup of this form.
Uri Bader's reference to Pink's 1998 paper is a good start. Pink proves this sort of rigid... | 1 | https://mathoverflow.net/users/5301 | 257301 | 116,204 |
https://mathoverflow.net/questions/257309 | 6 | It is known that Modal Logic can be interpreted in First-Order logic via [Standard translation](https://en.wikipedia.org/wiki/Standard_translation). However, this translation needs a unary predicate for every propositional variable. It is also known that without these propositional variables certain formulas in ML can'... | https://mathoverflow.net/users/46003 | Modal vs First-Order Logic on finite models | On p. 30–31 of Van Benthem’s *Notes on modal definability* (Notre Dame Journal of Formal Logic 30 (1988), #1, pp. 20–35), you can find a (brief!) sketch of a proof that the modal formula
$$(\Diamond\Diamond\top\land\Box(\Diamond\top\to\Diamond p))\to\Diamond(\Diamond\top\land\Box p)$$
is not FO-definable on finite fram... | 9 | https://mathoverflow.net/users/12705 | 257314 | 116,210 |
https://mathoverflow.net/questions/257048 | 21 | Is anything known about the maps out of an Eilenberg Mac-lane Space $K(G,n)$?
Obviously I'm interested in extensions of Miller's resolution of the Sullivan conjecture, that $Map\_\*(K(G,1),X)\simeq\ast$ for $G$ a discrete, locally finite group and $X$ a connected, finite complex. On the other hand Gray has shown that... | https://mathoverflow.net/users/54788 | Maps out of Eilenberg-Mac Lane Spaces | This question was totally answered by Alex Zabrodsky, right after Haynes Miller proved the Sullivan conjecture. See the paper: "On phantom maps and a theorem of H. Miller", Israel J. Math. 58 (1987), 129-143.
In summary, all maps are phantom, and all the homotopy groups of the space of maps can be determined from rat... | 21 | https://mathoverflow.net/users/102519 | 257318 | 116,211 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.