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https://mathoverflow.net/questions/177946 | 2 | I have an optimization problem in the form of
[\begin{array}{l}
\mathop {{\rm{Minimize}}}\limits\_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - }}Tr\left( {{\bf{AX}}} \right)\\
{\rm{Subject to Tr}}\left( {\bf{X}} \right){\rm{ - }}{P\_{th}} \le 0\\
& \,\,\,\,\,\,... | https://mathoverflow.net/users/56824 | Is first term of my cost function convex? | Your objective is convex, and can be written in conic form (to be precise, a mixed SDP and SOCP). By cyclicity of the trace, and since $X$ is Hermitian
$$ Tr(AX^2)=Tr(XAX^H)=Tr(XF(XF)^H) $$
Now, the square root of this term is nothing but the Euclidean norm of $vec(XF)$, which can be written in second-order conic form
... | 2 | https://mathoverflow.net/users/39129 | 177953 | 89,563 |
https://mathoverflow.net/questions/177944 | 6 | Let $D\in \mathbb{R}^{n\times n}$ denote a lower triangular matrix. With $\|\cdot\|$ denoting the spectral matrix norm, is there an estimate like
$$
\|D\| \leq C\|D+D^T\|,
$$
where $C>0$ is independent of the dimension $n\in\mathbb{N}$ and $D$?
| https://mathoverflow.net/users/50562 | Norm of the upper triangular part of symmetric matrix | No. The best constant is $C\_n\sim \ln n$. See, for example, [this paper.](https://www.jstor.org/stable/2589115) In particular, for the lower bound, see example 3.3 of the paper.
| 4 | https://mathoverflow.net/users/48839 | 177956 | 89,564 |
https://mathoverflow.net/questions/177852 | 2 | I am working on a physical problem, where I need to compute the "reduced plethysm" that is all the irreducibles characterised by the Young tableaux of 2 columns or less. The plethysm problem I want to solve is the following,
$$\mathbb s\_{(3)}[\mathbb s\_{1^p}] \text{ which can be written as } s\_{(3)}[e\_p],$$
whe... | https://mathoverflow.net/users/56778 | Is the reduced plethysm (restricted to 2-columns in Young tableaux) of this Schur funtion known $\mathbb S_{3^1}(\mathbb S_{1^p})$? | This follows on from Dan Peterson's answer. For $r,s,a \in \mathbb{N}\_0$, let $N\_{r,s}(a)$ be the number of partitions of $a$ whose Young diagram fits into a box with $r$ rows and $s$ columns. By the Cayley–Sylvester formula, the multiplicity of $s\_{(rs-a,a)}$ in $s\_{(r)} \circ s\_{(s)}$ is $N\_{r,s}(a) - N\_{r,s}(... | 3 | https://mathoverflow.net/users/7709 | 177957 | 89,565 |
https://mathoverflow.net/questions/177952 | 18 | Let $q$ be a prime power and $k>1$ a positive integer. For what values of $k$ and $q$ is the number $(q^k-1)/(q-1)$ a perfect square, that is the square of another integer? Is the number of such perfect squares finite?
Note that $(q^k-1)/(q-1)$ is the number of points in a finite projective space of dimension $k-1$.... | https://mathoverflow.net/users/56827 | When is $(q^k-1)/(q-1)$ a perfect square? | The equation
$$ \frac{x^k-1}{x-1}=y^m$$
is known as the Nagell-Ljunggren equation. It is conjectured that for $x\geq 2$, $y\geq 2$, $k\geq 3$, $m\geq 2$, the only solutions are
$$ \frac{3^5-1}{3-1}=11^2,\qquad \frac{7^4-1}{7-1}=20^2,\qquad \frac{18^3-1}{18-1}=7^3.$$
For $m=2$, the equation was solved by Ljunggren (Nor... | 32 | https://mathoverflow.net/users/11919 | 177958 | 89,566 |
https://mathoverflow.net/questions/177954 | 1 | I am wondering if there is a general solution for this ODE
$\ddot X +2\gamma \alpha \dot X + (\alpha+S(t)) X = \beta $
the dot represents time derivative, and $\gamma>1$, so it is in the over-damped regime.
It is a form of parametric oscillator, but I am wondering if there may exist a general solution for any fun... | https://mathoverflow.net/users/47467 | Solution of General Parametric Oscillator | No, I'm fairly confident there is no known general solution. Even in some very simple cases, such as $S(t) = t + t^3$, $\beta = 0$, $\alpha = 0$, I don't think the solutions can be written in closed form (Maple 18 doesn't find such a form).
| 2 | https://mathoverflow.net/users/13650 | 177964 | 89,568 |
https://mathoverflow.net/questions/177930 | 9 | Let $W$ be an infinite word over a finite alphabet $\{1,\dots,n\}$ and $k$ a positive integer. An easy application of Van der Waerden's theorem implies the existence of $k$ disjoint and consecutive intervals in $W$ such that the sum of letters in each interval are equal (**Edit**. See Barber's answer).
On the other h... | https://mathoverflow.net/users/51663 | A stronger version of Van der Waerden's theorem? | Your claim is not true.
Two consecutive blocks having the same number of occurrences of each letter (such as the English word "reappear") form what is called an "abelian square". A google search will easily produce a large literature on this subject. The best result (in terms of alphabet size) is due to the Finnish m... | 18 | https://mathoverflow.net/users/44797 | 177969 | 89,570 |
https://mathoverflow.net/questions/177962 | 6 | Let K be a finite field extension of $\mathbb{C}(t)$. Then $K$ is isomorphic to the field of meromorphic functions on a compact Riemann surface $X$ with genius $g$. By an argument similar to the proof of Douady's theorem for $\mathbb{P}^1(\mathbb{C})$ ( cf. chapter3 of Szamuly's Galois groups and fundamental groups) on... | https://mathoverflow.net/users/51663 | Comparison of finite field extensions of $\mathbb{C}(t)$ | For any such $K$, we can recover the subfield of constants $\mathbb{C}\subset K$ as the elements of $K$ that have $n$th roots for all $n$. Indeed, if a rational function on a curve has roots of all orders, it must have valuation $0$ at every point and hence be constant. Thus any isomorphism between two such fields $K$ ... | 5 | https://mathoverflow.net/users/75 | 177972 | 89,571 |
https://mathoverflow.net/questions/177934 | 4 | Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}\_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us assume that
$E=\mathbf{C}/\Lambda\_{\tau}$, where $\Lambda\_{\tau}=\mathbf{Z}+\tau\mathbf{Z}$ and $\tau$ is in the Poinc... | https://mathoverflow.net/users/11765 | Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve | Among $E\_1, E\_2, \Gamma, \Delta$ we have the intersections:
$E\_1 \cdot E\_1 = E\_2\cdot E\_2 = \Delta \cdot \Delta =\Gamma \cdot \Gamma =0 $
$E\_1 \cdot E\_2 = E\_1 \cdot \Delta = E\_2 \cdot \Delta =1$
$E\_1 \cdot \Gamma = 1$
$E\_2 \cdot \Gamma = D$
$\Delta \cdot \Gamma = 1+D$.
Presumably only the last c... | 2 | https://mathoverflow.net/users/18060 | 177976 | 89,572 |
https://mathoverflow.net/questions/177989 | 8 | Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H\_i(G, \mathbb{Q})$, or equivalently $H\_i(BG, \mathbb{Q})$ (topological homology of the classifying space) be infinite-dimensional? Can they be nonzero for infinitely many $i$?
For any finitely p... | https://mathoverflow.net/users/7108 | Is there a finitely presented group with infinite homology over $\mathbb{Q}$? | Thompson's group F is an example. It's finitely presented and, according to [this](http://www.math.cornell.edu/~kbrown/papers/homology.pdf) paper of Ken Brown, the integral homology is free abelian of rank 2 in every positive dimension.
| 18 | https://mathoverflow.net/users/1463 | 177992 | 89,576 |
https://mathoverflow.net/questions/177975 | 1 | Is there some relationship between the $p$-adic regulators of isogenous curves over $\mathbb{Q}$? I've done some computations and their ratio seems to be related (equivalent in all calculations so far) to the ratio of their modular degree.
There doesn't seem to be too much literature on this. Or perhaps it's a just a... | https://mathoverflow.net/users/44551 | $p$-adic Regulators | Whether you take the $p$-adic or the real regulator won't make a difference for this question.
Let $\varphi :E\to E'$ be an isogeny defined over a number field $K$. Consider the map $\varphi\_K : E(K)\to E'(K)$. Then
$$ \frac{\operatorname{Reg}(E')}{(\# E'(K)\_{\mathrm{tors}} )^2 } \cdot \frac{\# \operatorname{coker... | 2 | https://mathoverflow.net/users/5015 | 177997 | 89,578 |
https://mathoverflow.net/questions/177965 | 10 | I'm looking for simple and reasonably tight bounds on the k-th moment of the Binomial distribution $B(n,p)$, namely, $E[B(n,p)^k]$. I'm interested in the case when k is large (say on the order of $\sqrt{n}$) and in exact bounds (not asymptotic).
Using a relatively simple calculation based on the concentration of the bi... | https://mathoverflow.net/users/56835 | Bounds on the moments of the binomial distribution | For any $\beta>0$, $$\mathbb{E}B(n,p)^k\leq k!\beta^{-k}\mathbb{E}e^{\beta B(n,p)}= k!\beta^{-k}(1-p+pe^{\beta})^n.$$
Now you can plug various $\beta$, e.g. $\beta=\frac{k}{np}$ which yields
$$\mathbb{E}B(n,p)^k\leq (np)^k k!k^{-k}\left((1-p)+pe^{\frac{k}{pn}}\right)^n.$$
I assume that you got your estimate by elabor... | 6 | https://mathoverflow.net/users/56624 | 177998 | 89,579 |
https://mathoverflow.net/questions/178001 | 8 | Let $X$ be a projective variety. Why is any surjective morphism from $X$ to itself finite?
| https://mathoverflow.net/users/43951 | Surjective morphism from $X$ to itself is finite | Let $f$ be your surjective endomorphism; it is generically finite, say of degree $d$. Then $f\_\*:H^\*(X,\mathbb{Q})\rightarrow H^\*(X,\mathbb{Q})$ is surjective, because $\ f\_\*f^\*=d.\mathrm{Id}\ $ (use $\mathbb{Q}\_\ell$ instead of $\mathbb{Q}$ in characteristic $p$). Therefore $f\_\*$ is bijective. But then $f$ ca... | 10 | https://mathoverflow.net/users/40297 | 178006 | 89,582 |
https://mathoverflow.net/questions/177990 | 7 | An FC-group is a group in which every element has a finite conjugacy class. A group G is said to be maximally almost periodic if there is an injective homomorphism from G into a compact Hausdorff group.
My question is whether every discrete countable FC-group G is maximally almost periodic.
In general an FC-group ... | https://mathoverflow.net/users/1243 | Are countable FC-groups maximally almost periodic? | Here's a counterexample. Consider a non-abelian finite, 2-step nilpotent group $F$. Let $Z$ be its derived subgroup. Define $G$ as the quotient of the direct sum $F^{(\mathbf{N})}=\bigoplus F\_n$ of copies of $F$ indexed by $\mathbf{N}$ by identification of all copies of $Z$. Thus, writing $H=F/Z$, the group $G$ lies i... | 7 | https://mathoverflow.net/users/14094 | 178009 | 89,583 |
https://mathoverflow.net/questions/178030 | 3 | Let $X,Y$ be finite etale $T$ schemes for some scheme $T$ (assume the maps $X\rightarrow T,Y\rightarrow T$ are surjective). Then the sheaf $\mathcal{Hom}\_T(X,Y)$ on $\text{Sch}/T$ (with the etale topology) sending
$$(U\rightarrow T)\mapsto\text{Hom}\_U(X\_U,Y\_U)$$
is representable, since on each component of $T$ it i... | https://mathoverflow.net/users/15242 | If $T$ is an $S$ scheme, and $X,Y$ are $T$-schemes, then do the $S$-valued points of $\mathcal{Hom}_T(X,Y)$ have an interpretation? | This might be a disappointing kind of answer and maybe there are better ways, but what is straightforward is this: morphisms $V\to H$ over $S$ are in one-to-one correspondence with pairs $(p,\varphi)$ where $p:V\to T$ is a $V$-valued point of $T$ over the $V$-valued point $V\to S$ of $S$ and $\varphi:p^\*X\to p^\*Y$ is... | 3 | https://mathoverflow.net/users/41291 | 178037 | 89,594 |
https://mathoverflow.net/questions/106606 | 7 | I have a proof that given a partition $\lambda=(\lambda\_1,\dots,\lambda\_l)$
then the number of semi-standard Young tableaux (SSYT) of shape $\lambda$ with entries in $1,2,\dots, n$ is given by
$$\frac{1}{1!2!\cdots (n-1)!} \prod\_{1\leq i\lt j\leq n} (\lambda\_i-i)-(\lambda\_j-j).$$
(We define $\lambda\_j:=0$ if $j... | https://mathoverflow.net/users/1056 | New formula for counting semi-standard Young tableaux? | So, to answer this question, the formula is known. It is also (now) on the wikipedia page about Schur polynomials, so the next person will find it easily.
| 6 | https://mathoverflow.net/users/1056 | 178050 | 89,600 |
https://mathoverflow.net/questions/178020 | 7 | Is it possible, in ZFC, to find an $\omega\_1$-branching tree $(T,\leq)$ of size and height $\omega\_1$ such that whenever $T$ is partitioned into countably many sets $T=\bigcup\_{n<\omega} T\_n$ one of the sets $T\_n$ contains a subset $S$ which is again an $\omega\_1$-branching tree of size and height $\omega\_1$
(wi... | https://mathoverflow.net/users/25700 | Partitioning $\omega_1$-branching trees of size and height $\omega_1$ | The existence of such trees is independent of ZFC.
On one hand, if $CH$ holds then $\omega\_1^{<\omega\_1}$ (and in fact - any $\sigma$-closed $\omega\_1$-branching tree) cannot be partitioned into $\aleph\_0$ many subsets, each does not contain an $\omega\_1$-branching tree.
*Proof*:
Assume otherwise and let $T\... | 12 | https://mathoverflow.net/users/41953 | 178054 | 89,601 |
https://mathoverflow.net/questions/178048 | 1 | Let $n \in \mathbb{N}$, $K$ a (nonarchimedean) local field, $\overline{K}$ its algebraic closure. Take a compact subgroup $G \leq \text{GL}\_n(\overline{K})$. Must there be a finite extension $F$ of $K$ such that $G \leq \text{GL}\_n(F)$?
What if in addition $G$ is profinite? can that make a difference?
Does this g... | https://mathoverflow.net/users/38889 | Compact subgroups of linear groups over nonarchimedean fields | Here's an argument. It works under the assumption that $K$ is a complete normed field and has countably many finite extension up to $K$-isomorphism. This holds in particular when $K$ has finitely many extensions in each given degree, e.g., $p$-adic fields.
Since given a finite extension it has finitely many embeddin... | 1 | https://mathoverflow.net/users/14094 | 178062 | 89,606 |
https://mathoverflow.net/questions/178073 | 12 | Consider the following two foliations of torus:
1)The Kronecker foliation with slope $\sqrt{2}$
2)The Kronecker foliation with slope $\pi$
As I learn from the literature, these two foliations are not topological equivalent. The proof is that the K theory of their corresponding $C^{\*}$ algebras are not isomorphic... | https://mathoverflow.net/users/36688 | "The" kronecker foliation or "a" kronecker foliation? | Just what the doctor ordered, a proof using diffeology instead of $\mathrm C^\*$-algebras:
MR0799609 Donato, Paul(F-CNRS-T); Iglésias, Patrick(F-CNRS-T)
**Exemples de groupes difféologiques: flots irrationnels sur le tore.** (French. English summary) [Examples of diffeological groups: irrational flows on the torus] ... | 8 | https://mathoverflow.net/users/19276 | 178074 | 89,611 |
https://mathoverflow.net/questions/178088 | 13 | Inspired by some other questions, ([this](https://mathoverflow.net/questions/147326/unique-domino-tiling) and [this](https://mathoverflow.net/questions/162020/which-region-in-the-plane-with-a-given-area-has-the-most-domino-tilings)),
I wonder what numbers $n$ there are that satisfy
$$p(n)=\text{there is no region tha... | https://mathoverflow.net/users/1056 | What exact number of domino tilings cannot be realizable? | To follow up on the answer of dhy, there is a simple construction that works with simply connected regions:
Take a $k\times k$ square with $k\geq 2$. Remove from the bottom left corner a staircase region with rows of $1,2,3,\ldots k-2$ cells, and from the top right corner a (suitably rotated) staircase region with ro... | 17 | https://mathoverflow.net/users/730 | 178100 | 89,619 |
https://mathoverflow.net/questions/178104 | 44 | Years ago I read about a topologist who was to enter the states as an immigrant and was asked a question about his profession. He indicated he was a topologist, but as this was not included on the officer's list, he wanted to check him in as a "mathematician", which was on his list.
But, the topologist refused being ... | https://mathoverflow.net/users/56890 | A topologist is not a mathematician - a small question | It seems to me you are referring to Egbert Rudolf van Kampen (but the problem at the immigration office was quickly solved by a phone call to the university, the Johns Hopkins I believe). The story is told by Mark Kac, in his book *Enigmas of Chance: An Autobiography*.
**rmk** The point of Mark Kac's enjoyable anecd... | 53 | https://mathoverflow.net/users/6101 | 178106 | 89,622 |
https://mathoverflow.net/questions/178094 | 10 | A number field $K$ is said to have a *power basis* if there is an $\alpha \in K$ such that the full ring of integers $O\_K$ is the $\mathbb{Z}$-linear span of $1,\alpha,\alpha^2,\ldots,\alpha^{\deg{K}-1}$. In other words, $O\_K = \mathbb{Z}[\alpha]$; another term for this is monogenic. This happens for example for the ... | https://mathoverflow.net/users/26522 | Can there be a power basis for a totally real field of high degree? | [Kedlaya](http://arxiv.org/abs/1103.5728) proves that there are infinitely many irreducible integer polynomials of every degree with real roots and square free discriminant; if $P$ has square free discriminant then $\mathbb{Z}[x]/P(x)$ is integrally closed. More precisely, he constructs $c N^{1/(n-1)}$ different fields... | 7 | https://mathoverflow.net/users/297 | 178108 | 89,623 |
https://mathoverflow.net/questions/165255 | 2 | Theorem 2.5 in Conca and Herzog, Castelnuovo-Mumford regularity of products of ideals <http://arxiv.org/abs/math/0210065>, says that if $R$ is a polynomial ring over a field $k$, $I$ a homogeneous ideal of $R$ such that $\dim R /I \le 1$ and $M$ is a finitely generated, graded $R$-module, then $reg(IM) \le reg(M) + reg... | https://mathoverflow.net/users/32906 | Theorem 2.5 in "Castelnuovo-Mumford regularity of products of ideals" by Conca & Herzog | The point is that if the base field is infinite (as one can assume after a field extension) a general enough linear form is almost regular. This follows from the so called prime avoidance.
| 3 | https://mathoverflow.net/users/48585 | 178111 | 89,626 |
https://mathoverflow.net/questions/39640 | 12 | The Free Burnside group $G=B(2,665)=\langle a,b|g^{665} \rangle$ is infinite, by the work of Adyan and Novikov. Furthermore, the centralizer of any nonidentity element in $G$ is finite cyclic, and so the group is an i.c.c. group and the associated left group von Neumann algebra $LG$ is a type $II\_{1}$ factor. It is a ... | https://mathoverflow.net/users/6269 | Do Burnside Group Factors have Gamma? | The answer is no, the group von Neumann algebra of the Burnside group $B(2,665)$ does not have property Gamma. In fact, $B(2,665)$ is not inner amenable.
To see this, first note that, as pointed out in the question, $B(2,665)$ is nonamenable, and the centralizer of every non-identity element of $B(2,665)$ is finite.... | 5 | https://mathoverflow.net/users/1243 | 178114 | 89,627 |
https://mathoverflow.net/questions/178112 | 8 | Is it possible, starting from any large cardinal assumption, to find a *countably closed* forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash\_\mathbb{P} "\kappa = \lambda^+$ and $\lambda$ is singular"?
| https://mathoverflow.net/users/11145 | Inaccessible becomes successor of singular | No. The following theorem is from a work in progress by Yair Hayut and myself.
>
> **Theorem.** If $\Bbb P$ is a proper forcing, and it changes the cofinality of $\kappa$ to $\mu>\omega$, then $\Bbb P$ adds a surjection from $\mu$ onto $\kappa$.
>
>
>
Now suppose that you had such countably closed $\Bbb P$, it... | 11 | https://mathoverflow.net/users/7206 | 178115 | 89,628 |
https://mathoverflow.net/questions/178052 | 6 | Can a (higher) local field have uncountably many finite (seperable) extensions?
| https://mathoverflow.net/users/38889 | Finite extension of local fields | Let $k$ be any field and $K=k((x,y))$ the fraction field of the ring $k[[x,y]]$ of formal Laurent series in two variables. Then Harbater and Stevenson proved that the absolute Galois group of $K=k((x,y))$ is quasi-free of rank equal the cardinality of $m={\rm card}(K)>\aleph\_0$. In particular, for any nontrivial finit... | 4 | https://mathoverflow.net/users/2042 | 178119 | 89,630 |
https://mathoverflow.net/questions/177571 | 10 | Let $M$ be **any** $n\times n$ matrix.
We define the usual cofactors: $C\_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write the determinant of $M$ using Laplace expansion along column $p$ as:
$$\det M = \sum\_{q=1}^n M\_{q,p} C\_{q,p}$$
... | https://mathoverflow.net/users/23829 | Factor a sum of products of cofactors | This is a proof that $\det M$ divides $W\_{n,p}(k)$.
I can assume that the entries of $M$ are variables of polynomial ring $\mathbb{C}[M\_{11},\ldots,M\_{nn}]$; then, it suffices to prove that points from the (irreducible) variety $\det M=0$ satisfy $W\_{n,p}(k)=0$.
Indeed, let $M'$ be a generic point on this variety... | 2 | https://mathoverflow.net/users/nan | 178126 | 89,633 |
https://mathoverflow.net/questions/176737 | 9 | [Newton basin fractals](https://en.wikipedia.org/wiki/Newton_basin) are visualizations of the Julia sets of functions of the form:
$$f\_p(z) = z - p(z)/p'(z)$$
where $p$ is a complex polynomial. My question is:
>
> When is the Julia set, $J(f\_p)$, continuously determined by the roots of $p$, in the sense of th... | https://mathoverflow.net/users/16518 | When is a Newton basin fractal continuously determined by the roots of its polynomial? | The question of when the Julia set of a function depends continuously on the parameters is well-studied. For the case of polynomials, see Douady, *Does a Julia set depend continuously on the Polynomial?* Proceedings of Symposia in Applied Mathematics, 49, 91–135, 1994.
This implies that the answer for your question i... | 7 | https://mathoverflow.net/users/3651 | 178148 | 89,638 |
https://mathoverflow.net/questions/178147 | 2 | I guess that $\lnot$SH + $\lnot$CH is consistent, but I have not found this question discussed anywhere. Is there any relatively simple model of $\lnot$SH + $\lnot$CH?
| https://mathoverflow.net/users/41274 | Is not SH + not CH consistent? | Recall that adding a single Cohen real adds a Suslin tree, and does not change the value of the continuum [1] [2]. So by taking any model of $\lnot\sf CH$ and adding a single Cohen real, we have a Suslin tree, and therefore $\lnot\sf SH$ as well.
(This makes the Solovay-Tennenbaum theorem all the more magical, since ... | 14 | https://mathoverflow.net/users/7206 | 178149 | 89,639 |
https://mathoverflow.net/questions/176029 | 6 | I have seen this claim on the Wikipedia page for the [Yang-Mills Millenium problem](https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap) by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I think [his](http://arxiv.org/abs/1005.3779) s... | https://mathoverflow.net/users/19520 | Some explanation about Dynin's formalism | The paper is currently (and will be at least for a few days) under discussion at
<http://www.physicsoverflow.org/21786/energy-mass-spectrum-yang-mills-bosons-infinite-and-discrete>
| 8 | https://mathoverflow.net/users/56920 | 178155 | 89,641 |
https://mathoverflow.net/questions/178122 | 11 | What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are there interesting facts about this category?
I am particularly interested in the special case of the category of posets ... | https://mathoverflow.net/users/55915 | The category of categories and adjunctions | You will have to make an arbitrary choice for the direction of morphisms: is the left adjoint "forward" or "backward"? To prevent that, you can add involutions. The resulting category $\mathbf{InvAdj}$ of involutive categories and adjunctions, that I'll define below, has a lot of interesting structure. It is a [dagger ... | 5 | https://mathoverflow.net/users/10368 | 178160 | 89,645 |
https://mathoverflow.net/questions/178165 | 1 | Does there exist a category C which such that there is no functor $F:C \rightarrow D$ with $D\not\cong C$ which has a left (or right) adjoint?
| https://mathoverflow.net/users/36886 | Category which has no non-trivial adjoint functors | The empty category trivially satisfies this (there are no functors at all from a nonempty category to the empty category), but no other such category exists. Let $A$ be any category with a terminal object $1$, and consider the projection $C\times A \to C$. This has a right adjoint $C\to C\times A$ given by $c\mapsto (c... | 11 | https://mathoverflow.net/users/75 | 178171 | 89,647 |
https://mathoverflow.net/questions/178102 | 3 | Given a countable discrete group $G$ with [Kazhdan's property (T)](http://en.wikipedia.org/wiki/Kazhdan%27s_property_%28T%29), consider $\mathbb{C}G$ or $l^2(G)$ as a left $G$-module, then we can consider the [group cohomology](http://en.wikipedia.org/wiki/Group_cohomology#H2),
Is it known that $H^n(G, l^2(G))=0, H^n... | https://mathoverflow.net/users/9305 | vanishing higher cohomology group for property T group? | If you are interested in cohomology with coefficients in the group ring $\mathbb CG$, then you should know the Bieri-Eckmann theory of duality for a group of finite cohomological dimension. The group ring is a bimodule, with commuting left and right actions of the group. Taking cohomology uses up one of these actions, ... | 5 | https://mathoverflow.net/users/4639 | 178181 | 89,653 |
https://mathoverflow.net/questions/178177 | 4 | Assuming P=BQP (ie we have polynomial time algorithms to solve all BQP problems) can we use it to prove that P=NP?
The argument is that since we have the Grover's algorithm which can solve NP complete problems with a quadratic speedup and since we have assumed that P=BQP, we can apply the Grovers algorithm repeatedly... | https://mathoverflow.net/users/51073 | What impact would P=BQP have on NP? | For what it’s worth, [Fortnow and Rogers](http://www.sciencedirect.com/science/article/pii/S0022000099916513) constructed an oracle relative to which P = BQP, but the polynomial hierarchy does not collapse (hence in particular P ≠ NP).
| 13 | https://mathoverflow.net/users/12705 | 178183 | 89,655 |
https://mathoverflow.net/questions/178170 | 2 | Let $n \in \mathbb{N}$, $K$ a finite field. Denote by $K[[x]]$ the (profinite) ring of formal power series over $K$. Note that $\text{GL}\_n(K[[x]])$ is a profinite group.
Is every closed subgroup of $\text{GL}\_n(K[[x]])$ (topologically) finitely generated?
An analogy is that for evey prime number $p$, every close... | https://mathoverflow.net/users/38889 | Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated? | This question has a negative answer is many respects. Firstly, there are simple constructions in the commutative case. Namely, the additive group $K[\![x]\!]$ is an infinite dimensional $\mathbb F\_p$-vector space and thus not finitely generated. Less simply, the multiplicative group $K[\![x]\!]^\times$ has infinitely ... | 12 | https://mathoverflow.net/users/52824 | 178188 | 89,657 |
https://mathoverflow.net/questions/178191 | 1 | Let $C\_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C\_0^{m,n}$ to be the stalk at $0$ of the sheaf of continuous $\mathbb{R}^n$ valued functions on $\mathbb{R}^m$. I'm trying to g... | https://mathoverflow.net/users/2622 | Continuous real function on germs | Do you mean that you give $C\_0^{m,n}$ the quotient topology? Then I think the answer is that there are no other continuous functions, on the grounds that $C\_0^{m,n}$ is badly non-Hausdorff: two germs that have the same value at $0$ belong to the same open sets. So factoring out that equivalence is effectively the sam... | 4 | https://mathoverflow.net/users/23141 | 178198 | 89,661 |
https://mathoverflow.net/questions/178192 | 6 | I have the following data from chemical kinetics research to fit the parameters of ordinary differential equations:
$$
\left[
\begin{array}{ccccccc}
\text{No.}& t & y\_1(t)&y\_2(t) & y\_3(t) & y\_4(t) & y\_5(t)\\
1&30.0000 & 9.1300 & 0.0931 & 0.0899 & 0.1000 & 0.0000 \\
2&60.0000 & 8.9300 & 0.1270 & 0.1230 & 0.2270... | https://mathoverflow.net/users/43296 | How to fit the parameters of differential equations with known data? | This kind of problem is known in the literature as a nonlinear state-space system identification. Several algorithms have been proposed in the literature to solve these problems. I think a good starting point would be (1) and the references therein, in particular the works of L. Ljung.
As far I know, in general if yo... | 4 | https://mathoverflow.net/users/22389 | 178201 | 89,663 |
https://mathoverflow.net/questions/178193 | 6 | For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X\_i X\_j f$$
for some orthonormal frame $\{X\_i\}$ of Lie vector fields?
Essentially, can we chose a left-invariant metric and orth... | https://mathoverflow.net/users/56938 | Laplace-Beltrami operator on a Lie group | As José Figueroa-O'Farrill pointed out in his comment above, the condition $\alpha\_{i j}^{~~~j}$ is equivalent to unimodularity. So we can write the Laplace-Beltrami operator in this form iff the Lie group is unimodular.
| 1 | https://mathoverflow.net/users/56938 | 178204 | 89,666 |
https://mathoverflow.net/questions/178211 | 14 | Suppose that $S\subseteq\mathbb{Z}[i]$ has the following properties:
For convenience, let $A\_n = \{z : z\in\mathbb{Z}[i], \text{Nm}(z)\le n\}$
$$\limsup\_{n\rightarrow\infty} \frac{|S\cap A\_n|}{|A\_n|} > 0$$
Then is it the case that $S$ contains arbitrarily long arithmetic progressions.
| https://mathoverflow.net/users/40983 | Szemeredi's theorem in the Gaussian integers | This is true, and follows from one-dimensional Szemeredi.
Fix $\delta > 0$ such that $|S\cap A\_n| \geq \delta |A\_n|$ infinitely often.
Let $r = \lfloor \sqrt n \rfloor$, so $A\_n$ is contained in
the square $S\_r: \{x+iy \in {\bf Z}[i] \colon |x|,|y| < r\}$.
Define the additive homomorphism $h\_r: {\bf Z}[i] \right... | 26 | https://mathoverflow.net/users/14830 | 178212 | 89,671 |
https://mathoverflow.net/questions/178150 | 8 | I'm very confused about some contradicatory statements, and I hope someone can help me clarify this.
Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for $\Gamma$ can be constructed as global sections of a sheaf $\mathcal{G}\_k$ on the modular curve $X(\Gamma)$. If $\Gamma$ con... | https://mathoverflow.net/users/15899 | modular forms, invertible sheaves, and quotients | You have to be a bit careful what "vanish" means in this context. For $k = 2 \bmod 4$, at an elliptic point of order 4, "vanishing as a section of the sheaf" and "vanishing as a function on the upper half-plane" aren't the same thing; it's easy to check that $E\_4$ is a local basis of the sections of $\mathcal{G}\_k$ i... | 5 | https://mathoverflow.net/users/2481 | 178214 | 89,672 |
https://mathoverflow.net/questions/178221 | 1 | Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E\_1 \times E\_2$ by the product $T\_1 \times T\_2$ of two translations by $2$-torsion points. Then $A$ admits two natural projections to $E\_i':=E\_i/T\_i$, and the fibers are $E\_j'$ with $i \neq j$.
Is the group of di... | https://mathoverflow.net/users/27125 | Divisors on an abelian surface | The fiber of the projection to $E'\_i$, say $F\_i$, is isomorphic to $E\_j$ ($j\neq i$), not to $E'\_j$. We have indeed $(F\_1.F\_2)=2$, and $F\_1,F\_2$ generate the group of divisors on $A$ up to numerical equivalence, because $E\_1$ and $E\_2$ generate the analogous group on $E\_1\times E\_2$.
| 3 | https://mathoverflow.net/users/40297 | 178224 | 89,677 |
https://mathoverflow.net/questions/178232 | 11 | We know, thanks to A. Weil, that such a function is rational, and the numerator has all of its zeros on the circle $z \overline{z} = q,$ where $q$ is the order of the field. The question is: can *every* polynomial in $\mathbb{Z}[x]$ with the zeros on the circle arise in this way? Is there some obvious and/or conjectura... | https://mathoverflow.net/users/11142 | Inverse problem for zeta functions of curves over finite fields | Tate and Honda show that almost all polynomials like that are the L-function of an abelian variety over the finite field. The problem with curves is much harder and it's open (for genus g>2). One necessary condition is that the expression giving the number of points on the curve ($q+1$ minus sum of zeros) has to be pos... | 18 | https://mathoverflow.net/users/2290 | 178233 | 89,680 |
https://mathoverflow.net/questions/178229 | 4 | In algebraic topology, it is a theorem of Stasheff that every A-$\infty$ space has the homotopy type of a loop space.
>
> **Question:** Is this true in homotopy type theory?
>
>
>
Let me be a little more precise. Let $X$ be a type. Assume that we have $e : X$ and $ m : X \times X \to X$ together with the foll... | https://mathoverflow.net/users/4002 | Delooping in homotopy type theory | The definition you gave is not of an $A\_\infty$-space, but just of $A\_1$-space. As Charles noted, these two classes of spaces are very different in general. For example, there are also higher isomorphisms similar to the ones in MacLane's pentagon identity for monoidal categories, and relations between them, ad infini... | 12 | https://mathoverflow.net/users/10605 | 178246 | 89,683 |
https://mathoverflow.net/questions/178239 | -1 | In the paper "Sato-Tate Distributions and Galois Endomorphism Modules in Genus 2" (arxiv: <http://arxiv.org/abs/1110.6638>), the authors use the singular homology $H\_1(A\_\mathbb{C}^{top},\mathbb{Q})$ ($A$ is an abelian variety). I have seen this construction other places, but most recently here. I am not that well ve... | https://mathoverflow.net/users/56793 | Interpretation of $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ | $A\_\mathbb C$ is not defined in the finite field case because you can't base change from a finite field to $\mathbb C$.
$H^1( A\_\mathbb C^{top})$ is a vector space whose dimension is $2g$. One can define $g$ in terms of the Tate module if you'd like, (or even more simply using the fact that the number of $\ell$-tor... | 2 | https://mathoverflow.net/users/18060 | 178248 | 89,684 |
https://mathoverflow.net/questions/178205 | 5 | Let $(X,\Delta)$ be a log pair (we assume the coefficients of $\Delta\leq 1$, but could be negative rationals), and I use the definitions in the book of "Birational Geometry of Algbebraic varieties" (Page 56, Def. 2.34; Page 58, Def. 2.37) for "terminal, canonical, klt, plt, lc, dlt" singularities.
My first question... | https://mathoverflow.net/users/29730 | Which singularities of log pairs do not depend on the resolution? | I guess I can take a stab at this. Certainly we are talking about *log* resolutions, ie that $\pi : Y \to X$ is proper and birational, $Y$ is regular, and $\text{exc}(pi) \cup \pi^{-1}\_\* \Delta$ has simple normal crossings. Then:
* *KLT* is independent of the log resolution
* *LC* is independent of the log resoluti... | 7 | https://mathoverflow.net/users/3521 | 178255 | 89,686 |
https://mathoverflow.net/questions/178262 | 1 | We know that, in a Riemannian manifold, the geodesic distance between a point O and a point P, when we fix O, is a function of P that is $C^\infty$ everywhere on a local neighborhood, except in P=O. If we consider the squarred function, we have $C^\infty$ everywhere.
(There is a topic on that subject)
Now, in Finsl... | https://mathoverflow.net/users/56819 | Derivability properties of the distance function in a Finsler Manifold | I believe that this is discussed in Zhongmin Shen's book "Lectures on Finsler Geometry". The result, if I remember correctly, is that the square of the geodesic distance function is generally *not* smooth (i.e., infinitely differentiable) at the origin in the Finsler case (as opposed to the Riemannian case), though it ... | 4 | https://mathoverflow.net/users/13972 | 178263 | 89,689 |
https://mathoverflow.net/questions/178219 | 6 | I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper (<http://arxiv.org/abs/math/0010059>)
I ran a search after the words "neck" and "stretch" and they do not appear in the paper, I guess the theory i... | https://mathoverflow.net/users/14105 | Reference Request: "Neck Stretching Procedure" (In Symplectic Field Theory) | Neck-stretching is a deformation of an almost complex structure in a neighbourhood of a hypersurface. In the Eliashberg-Givental-Hofer paper, neck-stretching is called "splitting along a contact submanifold" (see Section 1.3) - the description of the almost complex structure is sketched at the end of Section 1.4.
A m... | 10 | https://mathoverflow.net/users/10839 | 178271 | 89,694 |
https://mathoverflow.net/questions/178268 | 3 | Let $\mathbb R[x,y]\_+$ denote the set of positive polynomials in two variables. My problem can be stated as follows:
>
> Does there exist a countable set $M\subseteq \mathbb R[x,y]\_+$
> such that every $f\in \mathbb R[x,y]\_+$
> can be represented as $$ f=\sum\_{i=1}^n p\_i^2\cdot h\_i,\ h\_i\in M,\ p\_i\in\m... | https://mathoverflow.net/users/12081 | Is $\mathbb R[x,y]_+$ countably generated as a quadratic module? | Suppose that a countable set $M$ as required exists. (As you say, $M$ cannot be finite - in particular it cannot be empty.) Let $d$ be the minimal degree of all polynomials in $M$ and set $M\_d \subset M$ be the subset of elements of minimal degree. Since $M$ is not empty, $d\neq 0$, and the zero-set of any polynomial ... | 5 | https://mathoverflow.net/users/8176 | 178278 | 89,696 |
https://mathoverflow.net/questions/171782 | 25 | Let $n$ be a positive integer and $\prec$ an arbitray total order on $\{1,\dots,n\}$. I associate to this order a vector $v$ with one coordinate for every pair $(i,j)$ s.t. $1\leq i\neq j \leq n$, by this definition:
$$v\_{ij} = \left\{\begin{array}{cc}1 & i\prec j \\ 0 & i\succ j\end{array}\right. $$
This vector sat... | https://mathoverflow.net/users/51663 | Convex hull of total orders | I claim that this is false for $n=6$. I find it convenient to shift the variables to $w\_{ij} = 2 v\_{ij} -1$. So the inequalities are
$$-1 \leq w\_{ij} \leq 1 \quad (1)$$
$$w\_{ij} + w\_{ji}=0 \quad (2)$$
$$-1 \leq w\_{ij} + w\_{jk} + w\_{k i} \leq 1. \quad (3)$$
Consider the point $x\_{12} = x\_{34} = x\_{56} = 1$,... | 16 | https://mathoverflow.net/users/297 | 178288 | 89,700 |
https://mathoverflow.net/questions/178290 | 1 | In Weil's short note entitled "*On the Riemann hypothesis in function-fields*"
he mentions the notion of the **complementary correspondence** associated to a given correspondence $T:C\rightarrow C$ where $C$ is an algebraic curve. He gives a reference to one of Severi's papers for the definition, which I did not bother... | https://mathoverflow.net/users/11765 | On Severi's definition of the complementary correspondence | Given a correspondence $T \subset C \times C$ one defines an endomorphism of the Jacobian of $C$ by letting $T(P)$ be the divisor on $C$ cut out in $T$ by $\{P\}\times C$, extending by linearity and then taking linear equivalence classes. So the ring of endomorphisms is a quotient of the ring of correspondences. I thin... | 1 | https://mathoverflow.net/users/2290 | 178294 | 89,701 |
https://mathoverflow.net/questions/178296 | 19 | I have seen that [Euclid's Elements](https://en.wikipedia.org/wiki/Euclid%27s_Elements) was written 300 BC and first set in type in 1482. Are there scans of that old versions available?
How were formulas / images added to the books created with printing presses?
Can / could formulas / images also be printed automatic... | https://mathoverflow.net/users/36477 | How were formulas / images added to books in post-printing-press / pre-digital times? | 
[Euclid's Elements of Geometry](http://www.sciencephoto.com/media/527243/view) (1482)
In the 1455 Gutenberg bible the illustr... | 25 | https://mathoverflow.net/users/11260 | 178297 | 89,702 |
https://mathoverflow.net/questions/178113 | 29 | In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U\_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of $U\_q\mathfrak g$) is equivalent to the category $Rep\_k(\widetilde{L\mathfrak g})$
of level $k$ integrable highest weight modules ... | https://mathoverflow.net/users/5690 | What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps? | Here is my understanding of the situation. Question 1: yes, the equivalence is known in all cases. Kazhdan-Lusztig work does have some limitations on the level,
so Finkelberg's approach is not applicable. However the categories in question are
easy enough to work with explicitly: for instance $E\_8$ at level 1 has just... | 15 | https://mathoverflow.net/users/4158 | 178304 | 89,705 |
https://mathoverflow.net/questions/178302 | 1 | Assume that $H$ is a separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)>1$ with the following property?:
Every densely defined operator $A:D(A)\to D(A),\;D(A)\subset H$ with $p(A)=0$ is necessarily a bounded operator on $H$.
That is the polynomial-operator equation $p(A)=0$ has only... | https://mathoverflow.net/users/36688 | A question on unbounded operators | I do not think so.
**Observation:** Without loss of generality, $p(x)$ can be taken to be monic (constant multiples won't affect either $p(A) = 0$ or boundedness).
**Case 1:** $p$ is degree $2$.
By the above reduction, $p(x) = (x - \lambda)(x - \mu)$ for some $\lambda$ and $\mu$ in $\mathbb{C}$ (since $A$ clear... | 2 | https://mathoverflow.net/users/56229 | 178310 | 89,710 |
https://mathoverflow.net/questions/178314 | 6 | It seems to me that in slight paraphrase the central result of the article
* Marco Porta, Liran Shaul, Amnon Yekutieli, *On the Homology of Completion and Torsion* ([arXiv:1010.4386](http://arxiv.org/abs/1010.4386))
(theorems 6.11 and 6.12) means that for $\mathfrak{a} \subset A$ a suitably nice ideal inside a comm... | https://mathoverflow.net/users/381 | AdicCompletion$\dashv$Torsion adjunction on spectra? | If I understand what you are asking, then yes. p-completion of p-local spectra is $X \mapsto F(M, X)$, where $M=$ fiber of $S\to S\mathbb{Q}$, while the p-torsion approximation is $X\mapsto X\wedge M$.
The same story holds for any "smashing" localization. **Added.** A "smashing localization" $L$ gives a map of spect... | 5 | https://mathoverflow.net/users/437 | 178316 | 89,713 |
https://mathoverflow.net/questions/178323 | 1 | Let $(X\_t)\_{t \in \mathbb{N}}$ be a real-valued martingale that is bounded, i.e.,
there are $a, b \in \mathbb{R}$ such that $a \leq X\_t \leq b$ for all $t$.
Define the *path length* $L$ of $(X\_t)\_{t \in \mathbb{N}}$ as the distance the martingale covers,
$$
L := \sum\_{t=0}^\infty | X\_t - X\_{t+1} |.
$$
My qu... | https://mathoverflow.net/users/57014 | Bounded martingales of infinite path length | If you have a random walk on $\{0,\ldots,N\}$, it's known that the expected
number of steps to hit the boundary starting from the middle is something like $N^2/4$. That is: you travel a distance of $N^2/4$ before hitting the boundary.
You can exploit this to produce a martingale where the expected path length is inf... | 3 | https://mathoverflow.net/users/11054 | 178325 | 89,717 |
https://mathoverflow.net/questions/178286 | 9 | Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd perfect number?
I have asked the same question in [MSE](https://math.stackexchange.com/questions/882559), but did not get any answers. I was wondering if anybody here has anything to share regarding this.
Thank you!
| https://mathoverflow.net/users/10365 | Is there an odd integer $x < 105$ for which it is known that $x \nmid N$, if $N$ is an odd perfect number? | No, such a result would be a major breakthrough regarding our knowledge on odd perfect numbers.
A few years ago there was some confusion, since due to careless reading and citing of the article "Every odd perfect number has a prime factor which exceeds $10^6$" by Cohen and Hagis the impression arose that they had pr... | 11 | https://mathoverflow.net/users/37555 | 178343 | 89,724 |
https://mathoverflow.net/questions/178143 | 18 | Just to be concrete, consider the digits to be binary. Hasse showed that among all the primes, only a fraction of $17/24 < 1$ divide a number of the form $2^n+1$. As a result, the integers that divide a number with just two non-zero digits have zero density.
On the other hand, since $A + A = \mathbb{F}\_p$ for a typ... | https://mathoverflow.net/users/26522 | How many integers divide a number that involves just three non-zero digits? | For an integer $n$, call a prime divisor $p$ good, if the multiplicative order of $2$ modulo $p$ exceeds $p^{2/5}$, $p^2\nmid n$, and $(p-1, \varphi(n/p))<p^{1/5}$.
If $p$ is a good prime divisor of $n$, then the multiplicative order of $2^{\varphi(n/p)}$ is $>p^{1/5}$, by the sum-product results of Bourgain there exis... | 7 | https://mathoverflow.net/users/37555 | 178348 | 89,726 |
https://mathoverflow.net/questions/177025 | 1 | How would one find the maximal $n$ such that there exists an $n$-subset $S$ of $\mathbb{Z}^+$ such that $\forall A\subseteq S, \sum\_{a\in A}a$ is either a perfect square or a perfect cube, or can one go arbitrarily large. This is the case if one loosens the restrictions to any perfect power. Also, what about the sligh... | https://mathoverflow.net/users/40983 | Subset of the integers with certain properties | Dietmann and Elsholtz (Hilbert cubes in progression-free sets and in the set of squares, Israel J. Math. 192 (2012), 59–66) have shown that if $S\subseteq[1, x]$ is a set such that all subset sums are contained in the set of squares, then $|S|\leq (8/e+o(1))(\log\log x)^2$, and if all subset sums are contained in the s... | 4 | https://mathoverflow.net/users/37555 | 178349 | 89,727 |
https://mathoverflow.net/questions/178345 | 6 | A while ago I asked [this](https://math.stackexchange.com/questions/848065/local-geodesics-in-uniquely-geodesic-spaces)
question in Math Stackexchange. Since I didn't receive an answer so far, I thought I'd ask it here.
Suppose $Y$ is a proper length space, where every pair of points $x,y\in Y$ can be joined by a *un... | https://mathoverflow.net/users/57021 | Local geodesics in uniquely geodesic spaces | Without further hypotheses it is easy to cook up a counterexample. Start with a standard torus of revolution, choose a point $p$ on the "outer" circle $\alpha$ and follow this circle until you reach the first conjugate point $q$ from $p$. It is well known that for points $x$ past $q$, the geodesic is no longer minimizi... | 7 | https://mathoverflow.net/users/28128 | 178351 | 89,728 |
https://mathoverflow.net/questions/178353 | 1 | Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$, $W \subset X \times Y$ a closed irreducible subscheme. Suppose that the natural projection map $pr\_2:W \to Y$ is surjective on the underlying topological spaces. Note that for all $x \in W$, there is a natural map of tangent spaces $\phi\_x:T\_{W,x} \to T... | https://mathoverflow.net/users/54369 | Nonreducedness of schemes and projective morphisms | No. Take for $X$ and $Y$ smooth projective curves (say), $W\subset X\times Y$ a smooth curve such that $pr\_2$ has degree 2. Any branch point $y$ of $pr\_2$ satisfies your requirement.
| 3 | https://mathoverflow.net/users/40297 | 178357 | 89,731 |
https://mathoverflow.net/questions/178366 | 3 | Recall that a flag variety over a field $k$ is a smooth projective variety over $k$, which is a homogeneous space for some linear algebraic group.
My question concerns specialisations of flag varieties over discrete valuation rings. Namely, let $R$ be a discrete valuation ring with field of fractions $K$ and residue ... | https://mathoverflow.net/users/5101 | Specialisations of flag varieties | There is a counter-example (with $k=\mathbb{C}$) due to Pasquier and Perrin, Math. Zeitschrift 265 (2010), 589-600. The generic fiber is an orthogonal grassmannian $\mathrm{Gr}\_q(2,7)$, while the special fiber is a smooth
projective non-homogeneous ("horospherical") variety with an action of $G\_2$.
| 10 | https://mathoverflow.net/users/40297 | 178370 | 89,737 |
https://mathoverflow.net/questions/178372 | 8 | Are there known any lower and upper bounds for
$$
\sum\_{k=1}^n \frac{(-1)^{\Omega(k)}}k,
$$
where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$?
Or at least is it known if it is always positive?
| https://mathoverflow.net/users/56947 | lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$? | Let $\lambda(n) = (-1)^{\Omega(n)}$ and
\[T(x) = \sum\_{n \leq x} \frac{\lambda(n)}{n}.\]
The conjecture that $T(x) \geq 0$ for all $x \geq 1$ is called Turán's conjecture (though Turán took objection to this labelling, as he did not conjecture that it was true). As Jeremy Rouse stated, [Haselgrove](http://dx.doi.org/1... | 15 | https://mathoverflow.net/users/3803 | 178375 | 89,738 |
https://mathoverflow.net/questions/178358 | 2 | I posed this on August 8 on math.stackexchange, but there has been no response.
Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the converse true?
| https://mathoverflow.net/users/16839 | A Property of Baire Spaces | Yes. Any topological space $X$ can be decomposed as $X:=X\_0 \cup X\_1$ where $X\_0$ is an open Baire subset of $X$ and $X\_1$ is closed and meager (See e.g. [this MSE answer](https://math.stackexchange.com/a/88536/94514)). If $X$ has property A then $X\_1$ has empty interior and therefore $X$ is a Baire space.
| 4 | https://mathoverflow.net/users/17836 | 178384 | 89,742 |
https://mathoverflow.net/questions/178381 | 4 | Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$
$$
I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\}
$$
complementable (as a closed subspace) in $C(X)$ (as a Banach space)?
This is true in the case when $X\subseteq {\mathbb R}^n$ (this follows from: Stei... | https://mathoverflow.net/users/18943 | Is the ideal of functions vanishing at a set complementable in $C(X)$? | Not in general.
It's well-known in Banach space theory that the ideal $c\_0$ in $\ell^\infty$ is not complemented (see e.g. Albiac & Kalton).
By the Gelfand representation, $\ell^\infty \simeq C(\beta \mathbb{N})$ as a $C^\ast$-algebra. This maps $c\_0$ to the ideal of functions on $\beta \mathbb{N}$ that vanish on... | 7 | https://mathoverflow.net/users/22758 | 178387 | 89,743 |
https://mathoverflow.net/questions/178344 | 3 | Let $s>\frac{1}{2};$ and define a Sobolev space as follows:
$$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int\_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$
Fact: Let $m$ be an integer greater than $s+1.$ Assume that $F\in C^{m}(\mathbb R)$ and $F(0)=0.$ Then $F(f)\in H^{s}(\mathbb R)$ for all ... | https://mathoverflow.net/users/33018 | How do functions operate in a Sobolev space $H^{s}$? | In the case s>3/2, the answer looks to be given quite comprehensively as Theorem 1 (see also Remark 1) of
*Bourdaud, G., Moussai, M., and Sickel, W.
Composition operators on Lizorkin-Triebel spaces. J. Funct. Anal. 259 (2010), no. 5, 1098–1128.*
As far as I am aware, results giving necessary and sufficient conditio... | 5 | https://mathoverflow.net/users/nan | 178388 | 89,744 |
https://mathoverflow.net/questions/178391 | 4 | $\DeclareMathOperator\Sp{Sp}$Let $\Sp\_{2g}$ denote the symplectic group of $2g \times 2g$ matrices for some $g \geq 1$, and let $\Gamma(N)$ be the level-$N$ principal congruence subgroup of $\Sp\_{2g}(\mathbb{Z})$. It is already known, by certain approximation theorems for algebraic groups, that $\Sp\_{2g}(\mathbb{Z})... | https://mathoverflow.net/users/24757 | Density of $\Gamma(N)$ in $\operatorname{Sp}_{2g}(\mathbb{Z}_{\ell})$ where $\ell \nmid N$ | Yes, this also follows from the same approximation theorems.
One very concrete way of stating the strong approximation theorem for $Sp\_{2g}$ is that $Sp\_{2g}(\mathbf{Z})$ surjects onto $Sp\_{2g}(\mathbf{Z} / R \mathbf{Z})$ for every integer $R$. By taking $R = N\ell^k$ for $k \gg 0$, we can find elements of $Sp\_{2... | 9 | https://mathoverflow.net/users/2481 | 178400 | 89,748 |
https://mathoverflow.net/questions/178399 | 4 | We know for planar Brownian motion, that conformal maps composed with Brownian motion are also Brownian motion (preserve distribution).
Does it follow for higher dimensions?
I think it follows for even dimensions because by looking at pair components, the distribution will be preserved and so the joint distributio... | https://mathoverflow.net/users/40793 | Conformal invariance of Brownian motion in higher dimensions | The natural setup for talking about conformal properties of the Brownian motion is that of Riemannian manifolds (by considering the generating operator of the Brownian motion as the Laplace-Beltrami operator of the corresponding Riemannian metric). Now, the Laplacian is conformal in dimension 2 only (in higher dimensio... | 5 | https://mathoverflow.net/users/8588 | 178409 | 89,753 |
https://mathoverflow.net/questions/178354 | 8 | The most general way I can formulate my question is the following:
Question 1: *Given a Gorenstein quotient ring $S$ of a polynomial ring over a field $K$, can one construct a (topological) space $X$ such that the (even degree part of the) singular cohomology ring of $X$ with coefficients in $K$ is isomorphic to $S$... | https://mathoverflow.net/users/6989 | Constructing a space with prescribed cohomology ring | I did not understand exactly your question. What is important here, is the ring $k$. If $k=\mathbf{Z}$ then the problem you are asking for is hard, if $k$ is any commutative ring then the problem is very hard. If $k$ is a field of characteristic 0 or $\mathbf{F}\_{p}$ then there is always a solution. When the character... | 3 | https://mathoverflow.net/users/21369 | 178410 | 89,754 |
https://mathoverflow.net/questions/178131 | 2 | Let $G\_0$ be a finite group and $G\_j$ a Schur cover of $G\_{j-1}$ for $j=1,2,3\ldots$. Is $G\_2$ equal to $G\_1$? If not, will the sequence stop after finite steps in general?
| https://mathoverflow.net/users/56827 | Successive Schur covers | Let me expand on Geoff Robinson's comment. First, for $G$ perfect, there is a unique Schur cover $R$, and $R$ is perfect. As Geoff indicated, $R$ is its own Schur cover. To see this, let $S$ be a Schur cover for $R$. Then $S$ has a central subgroup $Z$ with $S/Z$ isomorphic to $R$, and thus $S$ has a normal subgroup $Y... | 7 | https://mathoverflow.net/users/9694 | 178411 | 89,755 |
https://mathoverflow.net/questions/178412 | 1 | Ok, here is what I am attempting to find an answer to:
I draw M uniformly random subsets of size K from the set of numbers $\Omega=\{1, \dots, N\}$ (where uniformly random means that each unique subset of $\Omega$ is equally likely). What is the probability that $S \leq M$ subsets share $R \leq K$ elements?
I am es... | https://mathoverflow.net/users/46629 | Probability of k overlapping subsets in N trials | To be precise: given $S \in \{2,\ldots,M\}$ and $R \in \{0,\ldots,K\}$ you want the probability $P(S,R)$ that the intersection of some $S$ of your $M$ random sets has cardinality $R$.
Well, I'll do the case $M=3$, $S=2$. Let your random sets be $X\_1, X\_2, X\_3$.
Of course you need $N + R \ge 2 K$.
$$ P(|X\_1 \ca... | 2 | https://mathoverflow.net/users/13650 | 178416 | 89,756 |
https://mathoverflow.net/questions/178422 | 2 | Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal definition in *An Introduction to Kolmogorov Complexity and Its Applications* by Ming Li and Paul M.B. Vitányi:
**Definition... | https://mathoverflow.net/users/14024 | How to define the input of computable function or Turing machine over real numbers | A good place to start learning about different representations of reals and their computability- and complexity-theoretic consequences is Weihrauch's book *Computable Analysis*.
| 4 | https://mathoverflow.net/users/4600 | 178427 | 89,761 |
https://mathoverflow.net/questions/178244 | 4 | My question arose after studying the article "John K. Beem: Conformal Changes and Geodesic Completeness". (<http://projecteuclid.org/euclid.cmp/1103899983>) One of the results there is:
>
> Let $(M,g)$ be a causal spacetime which satisfies condition $N$. If $E$ is an open subset of $M$ with compact closure $\overli... | https://mathoverflow.net/users/56972 | Subset of causal spacetime+Imprisonment Condition+Compact Closure -> Stably Causal spacetime? | *Edit:* In Beem's article on Page 180 before the introduction of condition (N) he mentions Carter's example (see also Hawking, Ellis The large scale structure of space-time, p. 195, Fig. 39). There you should get your counterexample by $E=(t\_1,t\_2)\times S^1 \times S^2$.
| 3 | https://mathoverflow.net/users/47189 | 178435 | 89,763 |
https://mathoverflow.net/questions/178434 | 2 | Let $A = \mathbb{C}[x,y,z]/(x y - z^k)$. In fact $A$ is the ring of $\mu\_k$ invariants: $A = \mathbb{C}[u,v]^{\mu\_k}$ where $g \in \mu\_k$ acts by $g(u,v) = (g u, g^{-1} v)$.
This allows one to understand vector bundles on the smooth surface $Spec\ A - (0,0,0)$ as $\mu\_k$ equivariant bundles on $Spec\ \mathbb{C}[u... | https://mathoverflow.net/users/7 | vector bundles on $\mathbb{C}[x,y,z]/(x y - z^k)$ | The only invertible sheaf on $U := \text{Spec}A \setminus \langle x,y,z\rangle$ that extends to all of $\text{Spec} A$ is the trivial invertible sheaf. As you correctly surmise, the invertible sheaves on $W$ are precisely the $\mu\_k$-linearized invertible sheaves on $V := \text{Spec}\mathbb{C}[u,v] \setminus \langle u... | 2 | https://mathoverflow.net/users/13265 | 178452 | 89,768 |
https://mathoverflow.net/questions/178455 | 0 | Let $C$ be a smooth projective curve over a field $k$ of characteristic zero and $S$ a reduced divisor on $C$ (so just a collection of points). Consider the sheaf of logarithmic differentials $\Omega^1\_C(\log S)$.
Why is it possible to find a rational section $\omega$ of $\Omega^1\_C(\log S)$ such that the residue o... | https://mathoverflow.net/users/57064 | rational sections of logarithmic differentials on a curve | I think what you mean is a *regular* section of $\Omega ^1\_C(\log S)$, that is, a rational 1-form $\omega $ which is regular outside $S$ and has a simple pole at each point of $S$. Consider the exact sequence of sheaves
$$0\rightarrow \Omega ^1\_C\rightarrow \Omega ^1\_C(\log S)\rightarrow \oplus\_{s\in S}\ k(s)\right... | 3 | https://mathoverflow.net/users/40297 | 178460 | 89,771 |
https://mathoverflow.net/questions/178456 | 4 | Suppose that on a certain coordinate system the coefficients $\Gamma^i\_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am interested in the torsionless totally symmetric case $\Gamma^i\_{jk}=\Gamma^k\_{ij}=\Gamma^i\_{kj}$, so that $\Gamma$ has $... | https://mathoverflow.net/users/40549 | Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients? | It's not hard to see that you don't *always* have uniqueness up to sign when $n\ge 4$. Just consider the case when $\Gamma^i\_{jk}$ is zero except when $i=j=k$. In this case, one has $R^i\_{jkl}=0$ for all $i$, $j$, $k$, and $l$, independent of the values of $\Gamma^i\_{ii}$, so, in dimension $n$, this gives an $n$-par... | 7 | https://mathoverflow.net/users/13972 | 178462 | 89,772 |
https://mathoverflow.net/questions/178459 | 3 | Let $f: X \to Y$ be a finite, surjective morphism of smooth, projective, irreducible varieties over $\mathbb{C}$ and let $y \in Y$.
1. Can I find a smooth curve $C \subseteq Y$ with $y \in C$ such that $f^{-1}(C)$ is again a smooth curve?
2. If not, can I find a curve $C \subseteq Y$ such that $y$ is a smooth point ... | https://mathoverflow.net/users/36563 | Preimage of smooth curves under morphism of smooth varieties | Consider the map $\mathbb P^2 \to \mathbb P^2$ with equations $(xy,x^2-y^2,z^2)$. Any smooth curve passing through the point $(0,0,1)$ has an equation that looks locally linear, and its inverse image has an equation that looks locally quadratic.
| 5 | https://mathoverflow.net/users/18060 | 178463 | 89,773 |
https://mathoverflow.net/questions/178449 | 5 | If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$).
For odd integers $n$ we have $n=\chi(K\_n) = 2\cdot\mu(K\_n) + 1$. Question: is there a non-complete graph $G$ with $\chi(G) = 2\cdot\mu(G) + 1$?
| https://mathoverflow.net/users/8628 | Matching number and chromatic number | Yes there are, but they only differ from the complete graph by adding isolated vertices. The proof goes as follows:
Let $G$ be a graph with $\chi(G) = 2\cdot\mu(G) + 1$ we will show that it is a complete graph plus some isolated vertices. Consider the vertices of a maximal matching: $x\_1, x\_2, \ldots , x\_{2\mu(G)... | 9 | https://mathoverflow.net/users/38267 | 178465 | 89,775 |
https://mathoverflow.net/questions/178466 | 6 | Let $O(2,\mathbb{Z}\_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}\_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for should be the whole group. Is it really so or I am confused? On the other hand the even stabilizer $O^+$ should have inde... | https://mathoverflow.net/users/4096 | orthogonal group in characteristic 2 | Assuming that, by $\mathbb{Z}\_2$, you mean the finite field of order $2$, then
$$ O\_2^{\pm}(q) \cong D\_{2(q\mp 1)}$$
where $D\_{2(q\mp 1)}$ is the dihedral group of order $2(q\mp 1)$. Taking $q=2$, one obtains that $O\_2^+(2)$ is cyclic of order $2$ (although not equal to $\pm I$ you suggest, because $-I=I$), while... | 5 | https://mathoverflow.net/users/801 | 178478 | 89,778 |
https://mathoverflow.net/questions/178451 | 11 | For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[x, x+0.1\sqrt{x}]$ with $P^+(n)>x^{1/2+c}$ is $\geq c\sqrt{x}$.
The proof is quite standard: You essentially need a non-... | https://mathoverflow.net/users/37555 | Integers with a large prime divisor in short intervals | See the very first paper on the subject by Ramachandra (A note on numbers with a large prime factor; JLMS 1969). Ramachandra's Theorem 1 states that there is a constant $\alpha <1/2$ such that for all large $x$ the interval $[x,x+x^{\alpha}]$ contains an integer with a prime factor larger than $x^{\frac 12+\frac{1}{13}... | 8 | https://mathoverflow.net/users/38624 | 178483 | 89,779 |
https://mathoverflow.net/questions/178485 | 2 | I'm considering the congruence in the title, *i.e.*,
$$a^p \equiv 1 \pmod{b^p},$$
where $a \ge b \ge 1$ are positive integers and $p$ is an odd prime.
For $p=3$, a brute-force computer search found many solutions with $b > 3$.
For $p=5$, there appear to be only solutions with $1 \le b \le 5$.
For $p=7$, my search... | https://mathoverflow.net/users/19844 | Does the congruence $a^p \equiv 1 \pmod{b^p}$ with prime $p \ge 5$ force $b \le p$? | There are counterexamples with $b=kp$ any multiple of $p$. Namely, for $a=1+k^pp^{p-1}$ we have
$$ a^p=1+\sum\_{j=1}^p\binom{p}{j}(k^pp^{p-1})^j\equiv 1\pmod{k^pp^p}. $$
For example, $p=5$ and $k=3$ yields the congruence $151876^5\equiv 1\pmod{15^5}$.
Let us now assume that $b$ is not a multiple of $p$. Then $a^p\equ... | 5 | https://mathoverflow.net/users/11919 | 178490 | 89,782 |
https://mathoverflow.net/questions/178254 | 8 | Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel $(\rho\_x^\mathcal{E})\_{x \in X}$ on $X$ is a *regular conditional distribution of* $\rho$ *with respect to* $\mathcal{... | https://mathoverflow.net/users/15570 | Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra? | I've found the answer - it's NO!
The paper I found addressing the question is the following:
<http://projecteuclid.org/euclid.aop/1175287757> ("0-1 Laws for Regular Conditional Probabilities")
A simple counterexample (which I've just slightly adapted from the counterexample in Example 2 of the above paper) is the... | 4 | https://mathoverflow.net/users/15570 | 178498 | 89,785 |
https://mathoverflow.net/questions/178497 | 2 | In [these notes](http://www.math.uiuc.edu/K-theory/0316/vv.pdf), page $10$, bullet $(5)$, it is stated that if $X$ is a scheme of finite type over a field $k$, then the motivic cohomology $\mathrm{H}^{p,q}(X,R)$ of $X$ over $k$, where $R$ is a ring, vanishes if $q<0$. Is there a reference available online (for free) th... | https://mathoverflow.net/users/nan | Vanishing of Motivic Cohomology | I know very little, but my impression from looking at these lecture notes is that it vanishes for non-negative p simply by using bullet (4). Bullet (4) follows from Theorem 2.9 on the previous page which says how to relate these K-groups to algebraic K-groups for smooth varieties. Using Theorem 2.9 again when both p an... | 1 | https://mathoverflow.net/users/5031 | 178513 | 89,789 |
https://mathoverflow.net/questions/178276 | 2 | Let $A$ be a $C^\*$-algebra, and $R:A\to A$ its right multilplier. Is it true that
$$
\exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad
$$
implies $A$ is unital. I know this is true if A is a weak$^\*$ dense ideal of $W^\*$-algebra. But what about the general case?
This question on [MSE](https://math.stackexcha... | https://mathoverflow.net/users/19593 | $R$ is a right multiplier and $R(a)b=a\overset{?}{\implies} A$ is unital | $A$ is unital. Something a little more general is true: if a C\*-algebra $A$ has an element $b$ that is a right divisor of all its elements, then $A$ is unital. Proof: We have $(b^\*b)^{1/4}=xb$ for some $x\in A$. So
$
(b^\*b)^{1/2}=b^\*x^\*xb\leq \|x\|^2(b^\*b).
$
The inequality $(b^\*b)^{1/2}\leq \|x\|^2(b^\*b)$ hol... | 3 | https://mathoverflow.net/users/13381 | 178514 | 89,790 |
https://mathoverflow.net/questions/178494 | 7 | Let $M$ be a complex manifold of complex dimension 2. What do we know about the set all Kähler metrics on $M$ in general and in the case of 4-torus $C^2/Z^4$?
For the case of surfaces ($dim\_C=1$), any compatible metric is Kähler and by the uniformization theorem, the answer is that every two such metrics are confor... | https://mathoverflow.net/users/19325 | All Kähler metrics on a complex manifold? | Let $M$ be a compact complex manifold. The set of Kahler metric on $M$ in a given Kahler class is parametrized by the set of positive volume forms with given integral, because (by Calabi-Yau theorem) any given positive volume form is a volume form of a Kahler metric in a given cohomology class, assuming their integrals... | 6 | https://mathoverflow.net/users/3377 | 178515 | 89,791 |
https://mathoverflow.net/questions/178518 | 6 | Why is the problem called the ten martini problem? Sounds like an interesting name for people who drink.
| https://mathoverflow.net/users/10035 | The ten martini problem - reason for name | The name was coined by Barry Simon in this 1982 [article](http://www.math.caltech.edu/SimonPapers/R26.pdf) (page 487):
>
> **The Ten Martini Problem:** *The almost Mathieu operator has a Cantor spectrum.*
>
>
> The name comes from the fact that Mark Kac\* has offered ten
> martinis to anyone who solves it. [...]... | 9 | https://mathoverflow.net/users/11260 | 178523 | 89,793 |
https://mathoverflow.net/questions/178467 | 3 | Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its [hammock localization](http://ncatlab.org/nlab/show/simplicial+localization#hammock_localization) in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:
>
> If there is an initial object (up to homotopy) in $... | https://mathoverflow.net/users/18263 | Zigzags and contractibility of categories | The answer to your new question is positive. Dwyer and Kan prove in *Calculating simplicial localizations* that the hammock localization ${\bf C}’=L^H\bf C$ is weakly equivalent to the standard simplicial localization $L\bf C$. In *Simplicial localizations of categories*, they had previously showed that $L\bf C$ and $\... | 7 | https://mathoverflow.net/users/12166 | 178524 | 89,794 |
https://mathoverflow.net/questions/178529 | 4 | My question is about (abstract) simplicial complices.
In particular, how many are they if I consider $n$ unlabelled vertices?
For example, if $n=4$, the two complices
$$
\{\varnothing, \{1\}, \{2\}, \{3\}, \{4\}, \{1, 2\}, \{3, 4\}\}
$$
and
$$
\{\varnothing, \{1\}, \{2\}, \{3\}, \{4\}, \{2, 3\}, \{1, 4\}\}
$$
are ... | https://mathoverflow.net/users/42995 | Simplicial complices on unlabelled vertices | This is <http://oeis.org/A006602> . I checked the numbers there.
The correspondence is because the maximal elements form an antichain cover.
| 4 | https://mathoverflow.net/users/9025 | 178536 | 89,797 |
https://mathoverflow.net/questions/178525 | 7 | Is it known whether Todorcevic's Open Coloring Axiom implies $2^{\aleph\_0}=\aleph\_2$?
The only consistency proofs for OCA that I know are the following:
1) PFA implies OCA (and also $2^{\aleph\_0}=\aleph\_2$).
2) Constructing a finite support ccc iteration of length $\omega\_2$ starting from a model satisfying ... | https://mathoverflow.net/users/41274 | Does OCA imply $2^{\aleph_0}=\aleph_2$? | Whether $OCA$ implies $2^{\aleph\_0}=\aleph\_2$ appears as problem 7.2 in J.T. Moore´s [The Proper Forcing Axiom](http://www.math.cornell.edu/~justin/Ftp/ICM.pdf) in the Proceedings of the ICM, 2010.
Apparently it was still open by the end of 2012 according to T. Yorioka´s [abstract](http://settheory.mathtalks.org/spe... | 7 | https://mathoverflow.net/users/17836 | 178541 | 89,799 |
https://mathoverflow.net/questions/175953 | 12 | For $p,q \in (0,\infty)$ and $s \in \mathbb{R}$, one can define certain function spaces, $B\_s^{p,q}(\mathbb{R}^n)$ and $F\_s^{p,q}(\mathbb{R}^n)$, the Besov and Triebel-Lizorkin spaces respectively. The definition of these spaces is a bit complicated, and it probably won't offer any new insight to anybody who hasn't a... | https://mathoverflow.net/users/nan | Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces | This is really a comment with possibly relevant references, but I would prefer to avoid creating an account. The additional parameter is sometimes referred to as the "microscopic parameter" in the literature (this may help in searching google and MathSciNet). Two references of interest are:
W. Sickel, L. Skrzypczak a... | 3 | https://mathoverflow.net/users/nan | 178546 | 89,802 |
https://mathoverflow.net/questions/176923 | 9 | **Background:**
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in [Symplectic Reflection Algebras](http://arxiv.org/abs/0712.1568), and Ginzburg & Etingof's list given at a workshop in 2007 [here](http://www.icms.org.uk/downloads/cher... | https://mathoverflow.net/users/33854 | Update on list of open problems for Cherednik/Symplectic Reflection Algebras | My paper with Charles classifies only aspherical values for the monomial groups $G(r,p,n)$; for most exceptional reflection groups the calculation of the aspherical locus is still open (and our technique, based on explicit norm calculations with orthogonal functions, doesn't easily generalize).
Looking only at the li... | 4 | https://mathoverflow.net/users/15933 | 178547 | 89,803 |
https://mathoverflow.net/questions/178553 | 2 | I've seen the following theorem around in various forms:
To give an object $A \in \mathcal{C}$ the structure of a $\Omega$-algebra object in $\mathcal{C}$ is equivalent to giving a lift of the contravariant hom-functor $\mathcal{C}(-,A) \colon \mathcal{C} \to Set$ to a functor $\mathcal{C} \to \Omega$.
Here $\Omega... | https://mathoverflow.net/users/57128 | Algebraic objects and lifts of their represented functors | Giving a functor $C \rightarrow \Omega$ is the same as giving an object with an $\Omega$-algebra structure in the category of functors $[C, Set]$. Having observed this the theorem follows form Yoneda and the fact that it preserves products.
| 3 | https://mathoverflow.net/users/39004 | 178556 | 89,807 |
https://mathoverflow.net/questions/178557 | 4 | Let $\{a\_n\}\_{n\in\mathbb{N}}$ be defined as $a\_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a\_k$ is prime. It is well known that there are infinitely many primes in this series,
| https://mathoverflow.net/users/40983 | Smallest prime in an arithmetic progression | This is Linnik's theorem, and the best known bound is $O(b^5)$ due to Xylouris. (This is in the Wikipedia page, and as I admitted in this year's JMM I added it to the Wikipedia page. It's in his thesis but otherwise unpublished as far as I know.)
| 10 | https://mathoverflow.net/users/6043 | 178559 | 89,808 |
https://mathoverflow.net/questions/178549 | 11 | In a recent Math Stack Exchange [question](https://math.stackexchange.com/q/887671/137524) I asked about the function $$f(z)=\sum\_{n=0}^\infty z^{2^n},$$ and was informed of its status is a canonical example of a lacunary series with natural boundary at $|z|=1$. A phenomenon observed by the accepted answer was that th... | https://mathoverflow.net/users/55904 | Distribution of zeroes of lacunary functions | In addition to the unique real zero at $z=-0.658626\ldots$, Mahler in a 1982 paper [*On the zeros of a special sequence of polynomials*, Math. Comp.] determined, to within eight decimal places, eight complex conjugate pairs of zeros of $f(z)$. This gives a total of $17$ fairly precisely located zeros. At the end of tha... | 9 | https://mathoverflow.net/users/26522 | 178561 | 89,809 |
https://mathoverflow.net/questions/178566 | 1 | The problem bothers me for a long time.
Suppose, we have two matrix $A$ and $B$, where $A$ is a $m$ by $n$ complex matrix while $B$ is a $n$ by $m$ complex matrix.
Apparently, $AB$ and $BA$ have the same non-zero eigenvalues.
However, can we predict the non-zero eigenvalues just based on the information of singular... | https://mathoverflow.net/users/57133 | The relation between eigenvalue and singular value of non-symmetric square matrix | We certainly can't predict phases: e.g. multiply $A$ by a complex number $\omega$ with $|\omega|=1$, and you don't change any of the singular values but you multiply the eigenvalues of $A$ and $AB$ by $\omega$.
Consider the case $A = I$, $B$ a unitary $n \times n$ matrix. Then the singular values of $A$, $AB$, $BA$ a... | 3 | https://mathoverflow.net/users/13650 | 178570 | 89,814 |
https://mathoverflow.net/questions/178562 | 0 | Lets say that we have n matrices of data $X\_i : i \in [1, n]$. All $X\_i$ have the same number of rows.
Their associated projection matrices are $P\_i = X\_i(X\_i^T X\_i)^{-1}X\_i^T$
Also say that we have have already found the SVDs of $X\_i = U\_iS\_iV\_i \forall i \in [1, n]$. Then I could figure out that the sin... | https://mathoverflow.net/users/29887 | Efficient way to find SVD of sum of projection matrices? | If $n$ is small, and matrix sizes for all $X\_i$ are large, then there is certainly a more efficient way of doing it.
First, note that $P\_i = U\_i U\_i^\top,$ and therefore
$$M=\begin{bmatrix}U\_1 & \ldots & U\_n\end{bmatrix}\begin{bmatrix}U\_1^\top \\ \ldots \\ U\_n^\top\end{bmatrix}.$$
If the concatenation of all b... | 2 | https://mathoverflow.net/users/52964 | 178573 | 89,816 |
https://mathoverflow.net/questions/178555 | 32 |
>
> Given two integers $a \ge b >1$, can we encode them as a unique integer $a^b + b^a$?
>
>
>
---
I [asked](https://math.stackexchange.com/questions/888009/for-integers-a-ge-b-ge-2-is-fa-b-ab-ba-injective) this question on math.SE, and after surviving a week with a bounty, it seems that this question is ha... | https://mathoverflow.net/users/54339 | For integers $a \ge b > 1$ is $f(a,b) = a^b + b^a$ injective? | I will show that if we assume the $abcd$ conjecture (which is the case $n=4$ of Browkin and Brzezinski's [$n$-conjecture](http://cr.yp.to/bib/1994/browkin.pdf) that generalizes the $abc$ conjecture), then $a^b+b^a=c^d+d^c$ has only finitely many solutions with $\{a,b\}\neq \{c,d\}$.
The $abcd$ conjecture claims that ... | 24 | https://mathoverflow.net/users/23008 | 178581 | 89,820 |
https://mathoverflow.net/questions/178576 | 9 | I have a short paper I'm working on where I prove:
*Theorem: Every graph on (2t-1) vertices with no (t+1)-clique has a vertex that is contained in every t-clique.*
By "t-clique", I mean a complete subgraph with t vertices. It's actually a lemma for the main result in the paper, but it's where most of the work is do... | https://mathoverflow.net/users/57139 | Looking for history on a theorem of clique intersections | Hajnal's Clique Collection Lemma implies your theorem. See Corollary 2.10 from this paper <http://arxiv.org/pdf/1101.4564v5.pdf> or Hajnal's original paper <http://cms.math.ca/cjm/v17/cjm1965v17.0720-0724.pdf> .
| 11 | https://mathoverflow.net/users/57145 | 178583 | 89,821 |
https://mathoverflow.net/questions/178579 | 11 | I was curious and quite clueless as to how we can equip the Grassmann Manifold with a canonical metric - I have yet to find anything upon this subject.
| https://mathoverflow.net/users/56059 | Canonical Metric on Grassmann Manifold | Since Grassmannian $Gr(n,m)=SO(n+m)/SO(n)\times SO(m)$ is a homogeneous manifold,
you can take any Riemannian metric, and average with $SO(n+m)$-action. Then you show that an $SO(n+m)$-invariant metric is unique up to a constant. This is easy, because the tangent space $T\_VGr(n,m)$ (tangent space to a plane $V\subset ... | 14 | https://mathoverflow.net/users/3377 | 178585 | 89,822 |
https://mathoverflow.net/questions/17608 | 43 | If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent
when the following statement is added to it as a new axiom?
"There exists a denumerably infinite and ordinal definable set of real numbers, not all of whose elements
are ordinal definable"
If the answer to the ab... | https://mathoverflow.net/users/4423 | A question about ordinal definable real numbers | The original problem solves in the positive: there is a model of ZFC in which there exists a countable OD (well, even lightface $\Pi^1\_2$, which is the best possible) set of reals $X$ containing no OD elements. The model (by the way, as conjectured by Ali Enayat at <http://cs.nyu.edu/pipermail/fom/2010-July/014944.htm... | 23 | https://mathoverflow.net/users/53091 | 178591 | 89,823 |
https://mathoverflow.net/questions/175803 | 7 | The fractal dimension of the 3D Apollonian packing is computed in [this paper](http://www.worldscientific.com/doi/abs/10.1142/S0218348X94000739).
In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (Hausdorff dimension of the complement) can be approximated... | https://mathoverflow.net/users/20595 | For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature? | OK, I found the reference:
For those who care, it's recently proved **in a much stronger form** by Oh and Shah in [The asymptotic distribution of circles in the orbits of Kleinian groups](http://www.ams.org/mathscinet-getitem?mr=2874933). The paper is about circle packing, but the method can be generalized to higher ... | 1 | https://mathoverflow.net/users/20595 | 178597 | 89,826 |
https://mathoverflow.net/questions/124268 | 6 | A [question](https://math.stackexchange.com/questions/326804/probability-of-two-vertices-to-be-connected-in-gn-p) I asked at math.SE without elliciting an answer.
Let $G(n,p)$ be an Erdős–Rényi graph on $n$ vertices. Is there an explicit expression for the probability $P\_{n,p}(u,v)$ that two fixed (distinct) vertice... | https://mathoverflow.net/users/30264 | Probability of two vertices to be connected in G(n,p) | Explicit expressions for the all-terminal reliabilty were established back in 1959 by Gilbert.
Gilbert, E. N. Random graphs. Ann. Math. Statist. 30 1959 1141--1144. [MR0108839](http://www.ams.org/mathscinet-getitem?mr=0108839) (21 #7551)
| 1 | https://mathoverflow.net/users/30264 | 178601 | 89,827 |
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