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 |
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https://mathoverflow.net/questions/235542 | 3 | Given a $2\times2$ matrix, which entries are functions in the complex plane $$\hat{A}(z)=\left(\begin{array}{cc}a(z)&b(z)\\c(z)&d(z)\end{array}\right)$$Where $a(z),b(z),c(z)$ and $d(z)$ are functions in the complex plane. I would like to obtain two new matrices, denoted as $\hat{\phi}\_+(z)$ and $\hat{\phi}\_-(z)$, suc... | https://mathoverflow.net/users/88906 | Wiener-Hopf factorization of matrices | Pointers to the literature on matrix-Wiener-Hopf factorization can be found in [A brief historical perspective of the Wiener-Hopf technique:](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.426.46&rep=rep1&type=pdf)
>
> Matrix Wiener-Hopf kernels are fundamentally distinct from their
> scalar counterparts... | 2 | https://mathoverflow.net/users/11260 | 235573 | 109,110 |
https://mathoverflow.net/questions/235549 | 3 | Define the upper uniform density of a set $A\subset\mathbb{Z}$ to be
$$
D^+(A)=\lim\_{r\rightarrow\infty}\sup\_{a\in\mathbb{R}}\frac{|A\cap[a,a+r)|}{r}
$$
Fix an arbitrary permutation of the integers $\omega:\mathbb{Z}\rightarrow\mathbb{Z}$ ( i.e. $\omega$ is a bijection) and let $\varepsilon>0$.
Does there exist... | https://mathoverflow.net/users/16040 | Density of permutation of syndetic sets of integers | Yes. Let $\mathbb{Z} = \bigcup\_{i=1}^\infty I\_i$ be a decomposition of $\mathbb{Z}$ into disjoint intervals of length $n$. We will find $A\subset\mathbb{Z}$ such that $|A\cap I\_i| = 1$ for each $i$ and such that $|\omega(A)\cap I\_i|\leq 1$ for each $i$. Thus $A$ is syndetic and $D^+(\omega(A)) \leq 1/n$.
Define a... | 0 | https://mathoverflow.net/users/20598 | 235578 | 109,112 |
https://mathoverflow.net/questions/235597 | 8 | Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety.
Is it true that the obvious map
$$
\mathfrak g\to \Gamma (T\mathcal B)
$$
from $\mathfrak g$ to the Lie algebra of globally defined algebraic vector fields on $\mathcal B$ is an isomor... | https://mathoverflow.net/users/5690 | Symmetries of the flag variety | Yes, this is true. In fact, the homogeneous spaces $G/P$ such that $\mathfrak{g}\rightarrow \Gamma (T\_{G/P})$ is *not* an isomorphism have been classified (see e.g. M. Demazure, Inventiones math. 39, 179-186 (1977)): they are the odd-dimensional projective spaces, the Grassmannian of linear subspaces of maximal dimens... | 7 | https://mathoverflow.net/users/40297 | 235605 | 109,116 |
https://mathoverflow.net/questions/234769 | 11 | [Veblen $\phi$ functions](https://en.wikipedia.org/wiki/Veblen_function) extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon\_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't repeat the definitions, which are on Wikipedia). For a reason that isn't en... | https://mathoverflow.net/users/17064 | Is there a modern account of Veblen functions of *several* variables? | After some digging around, I found the following paper:
Kurt Schütte, “Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen”, *Math. Ann.* **127** (1954), 15–32 (MR0060556)
The author describes Veblen functions (under a slightly different notation, which he calls “bracket symbol”) in a very clear way... | 5 | https://mathoverflow.net/users/17064 | 235623 | 109,122 |
https://mathoverflow.net/questions/235635 | 2 | A while back I came across orbifolds, in particular the quotients $SU(2)/U(1)\cong S^2$, $SU(3)/(SU(2)\times U(1)\cong \mathbb{C}P^2$ and $SU(3)/(U(1)\times U(1))$. The way I needed them, was as an embedding in the corresponding Lie-algebra $\mathfrak{su}(n)$.
To be a bit more precise: Consider first the $SU(2)$ cas... | https://mathoverflow.net/users/90068 | Projections of orbifolds | Yes, it is already known.
The story perhaps starts with:
**Kostant's convexity theorem.** Let $G/K$ be a symmetric space of compact type,
$\mathfrak g=\mathfrak k +\mathfrak p$ the decomposition into the eigenspaces of the involution, $\mathfrak a$ a maximal Abelian subspace of $\mathfrak p$ and $W$ the restrict... | 2 | https://mathoverflow.net/users/15155 | 235637 | 109,127 |
https://mathoverflow.net/questions/235587 | 2 | Let $S^n$ be the $n$-sphere. If $n=2k+1$ is odd, then we can identify $S^n$ as a subset of $\mathbb{C}^{k+1}$. We define the $S^1$ action on $S^n$ by multiplication, namely
$$ \Psi \colon S^1 \times S^n \to S^n, \ (c, (z\_0, \dots , z\_k)) \mapsto (cz\_0, \dots cz\_k).$$
If we endow $S^n$ with the standard-metric $g... | https://mathoverflow.net/users/75382 | isometric action on the $n$-sphere | The actions $\Psi$ and $\Theta$ commute and hence $\Theta$ maps $\Psi$ orbits into $\Psi$ orbits. Since $\Theta$ acts by $g$-isometries, it follows that it also preserves the splitting of the tangent space into $\Psi$-direction and its ortho-complement. You modify $g$ only in the $\Psi$-direction so it's clear that $\T... | 3 | https://mathoverflow.net/users/6818 | 235639 | 109,128 |
https://mathoverflow.net/questions/235591 | 6 | Recall a stationary subset $S$ of a regular cardinal $\kappa$ is fat when for every $\alpha < \kappa$, and every club $C$, there is a closed set of order type $\alpha$ contained in $S \cap C$. It is a result of Stavi, proved [here](http://www.ams.org/mathscinet-getitem?mr=716625), that:
(1) For every regular cardinal... | https://mathoverflow.net/users/11145 | Fat stationary sets | Suppose that $\lambda$ is a singular cardinal, $\square\_\lambda$ holds, and $2^\lambda=\lambda^+$. Then:
1. There exists a partition of $\lambda^+$ into $\lambda^+$ many pairwise disjoint fat stationary sets.
2. There exists a family of $2^{(\lambda^+)}$ pairwise almost-disjoint fat stationary sets.
Both clauses f... | 8 | https://mathoverflow.net/users/20033 | 235642 | 109,129 |
https://mathoverflow.net/questions/235620 | 11 | I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to learn algebraic geometry, since I find the modern language used there is category theory (topos, sheaves, schemes,...). He... | https://mathoverflow.net/users/82229 | How much do I need to learn algebraic geometry to understand arithmetics over number fields | Well if you want to count rational points on varieties than you probably want to know what abelian varieties are, and general type varieties, and Fano varieties, and K3 surfaces, and what Azumaya algebras are, and so on to understand the main conjectures and theorems of the subject. You should probably understand how s... | 25 | https://mathoverflow.net/users/18060 | 235644 | 109,130 |
https://mathoverflow.net/questions/235652 | 6 | Does there exist any finite dimensional irreducible rep. of Euclidean or Poincare group in which translation and rotation both act nontrivially?
Let me firstly clarify my question. For example, we obviously have a faithful rep. of Poincare group, $\begin{pmatrix} \Lambda & x \\ 0 & 1 \end{pmatrix}$, where $\Lambda$ ... | https://mathoverflow.net/users/43941 | Does there exist finite dimensional irreducible representation of Euclidean or Poincare group in which translation and rotation both act nontrivially? | The answer is "No".
You don't specify which kind of representation you have in mind. I assume these are finite dimensional complex representations. Thus you ask about (continuous) homomorphisms $G\to \text{GL}\_n(\mathbb{C})$. In the groups $G$ you consider you have a non trivial normal nilpotent subgroup $N$, the gr... | 12 | https://mathoverflow.net/users/89334 | 235657 | 109,135 |
https://mathoverflow.net/questions/235654 | 2 | I have two questions on the second dual of $C[0,1]$:
R. D. Mauldin ([[1](http://www.math.unt.edu/~mauldin/papers/no5.pdf)]) proved that: For a given bounded linear functional
$T: C[0,1]^\*\to \mathbb{C}$ there is a bounded function $\psi$ defined on $B$, the set of all Borel subsets of $[0,1]$, with $T(\mu)=\int \ps... | https://mathoverflow.net/users/84390 | On the second dual of $C[0,1]$ | It is easy to see that $\int \psi d\delta\_t$ where $\delta\_t$ is the Dirac mass at $t$ is $\psi(\{t\})$
So you are starting from a $T \in C([0,1])^{\*\*}$ and you are attaching to it the function $t \mapsto T(\delta\_t)$. This has absolutely nothing to do with Mauldin's result, so I'm not sure why you are mentionin... | 1 | https://mathoverflow.net/users/22131 | 235662 | 109,136 |
https://mathoverflow.net/questions/235614 | 3 | Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?
| https://mathoverflow.net/users/90061 | Is there a closed form expression for $E(X e^{-\mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? | Split ?
$$E(Xe^{-\mu\sqrt{X}})=E(Xe^{-\sqrt{\mu^2X}})\leq E(Xe^{-\mu^2X}\mathbf{1}\_{\mu^2X\leq 1})+E(X\mathbf{1}\_{\mu^2X\geq 1})$$
| 0 | https://mathoverflow.net/users/89993 | 235669 | 109,137 |
https://mathoverflow.net/questions/226512 | 6 | Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold:
>
> (F1) $f(\mathbf N) = 1$;
>
>
> (F2) $f(X) \le f(Y)$ whenever $X \subseteq Y$;
>
>
> (F3) $f(X \cup Y) \le f(X) + f(Y)$;
... | https://mathoverflow.net/users/16537 | Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weakly additive? | I think that the answer is **No**.
The space $\mathcal B(\mathcal P(\mathbf N), \mathbf R)$ with pointwise convergence is a [locally convex space](https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space). (It can be considered a subspace of $\mathbf R^{\mathcal P(\mathbf N)}$.)
The set $\mathscr U$ is... | 2 | https://mathoverflow.net/users/8250 | 235672 | 109,139 |
https://mathoverflow.net/questions/143312 | 4 | I'm looking for references for the rate of escape and return probability for a group of intermediate growth.
Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then (via isoperimetry, the only reference I have for this is Woess' book) one gets that the return probability $P\_{2n}(e,e) \prec... | https://mathoverflow.net/users/18974 | Estimates for simple random walks in groups of intermediate growth | It was pointed out to me that Lemma 5.1 in [this](https://eudml.org/doc/116198) paper of Erschler (Critical constants for recurrence of random walks on $G$-spaces. Ann. Inst. Fourier (Grenoble), 55(2):493--509, 2005) gives bounds on speed and entropy in terms of volume (the bound on entropy is implicit in the proof). N... | 1 | https://mathoverflow.net/users/18974 | 235679 | 109,142 |
https://mathoverflow.net/questions/235683 | 6 | Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X\_j\} \in Vect(M)$ be a finite "basis" of some integrable subbundle of $TM$ (meaning that their locally linearly independent and closed under l... | https://mathoverflow.net/users/22810 | Vector fields, diffeomorphism subgroups and lie group actions | Every orbit of a torus is a torus, since every orbit of a Lie group action is a homogeneous space of the Lie group.
| 7 | https://mathoverflow.net/users/13268 | 235685 | 109,144 |
https://mathoverflow.net/questions/235374 | 16 | An *ultrafilter ornament* is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter.
Let $\mathfrak{ufo}$ be the minimal cardinality of
an ultrafilter ornament.
I arrived at this definition back in 2008, while teaching Ramsey theory at the Weizmann Institute of Science, b... | https://mathoverflow.net/users/2415 | $\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum? | This invariant is known for Boolean algebras in general as pseudo-altitude. That it is $\omega\_1$ is proved in van Douwen's chapter in the Boolean algebra handbook 11.1 and 12.7, in the more general form that this holds for any weakly countably complete BA.
| 18 | https://mathoverflow.net/users/90095 | 235690 | 109,147 |
https://mathoverflow.net/questions/118385 | 3 | Hello,
I am interested in what is known about anisotropic Sobolev spaces, by which I mean spaces of functions satisfying
$ \| f \|\_p < \infty, \|Df \|\_q < \infty, $
where $p \ne q$ (as opposed to the alternate usage signifying that a different Sobolev exponent is imposed on normal versus tangential derivatives)... | https://mathoverflow.net/users/25490 | Reference request: Anisotropic Sobolev spaces | You can find this in two of my following papers
MR3296206 Reviewed Han, Qi Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions. Monatsh. Math. 176 (2015), no. 1, 107–141. (Reviewer: Dimitri Mugnai) 35J66 (35B09 35B33 35J20 35J91 46E22 46E35)
<http://www.scienc... | 2 | https://mathoverflow.net/users/90097 | 235695 | 109,149 |
https://mathoverflow.net/questions/235700 | 2 | Often I want to define a structure on a set $S$ which is like a poset, but lacks the antisymmetry condition: i.e., one is allowed both $a\succeq b$ and $a \preceq b$ for $a, b$ different elements of my set. One way to say this is "a category structure with underlying set $S$ which is equivalent to a poset" (where pairs... | https://mathoverflow.net/users/7108 | What word can I use for a poset with equivalences | These are called [preorders](https://ncatlab.org/nlab/show/preorder). (The nLab also suggests the term "proset" but I think this is terrible; "proset" should mean a pro-object in sets.) They're the same thing as categories enriched over truth values.
| 7 | https://mathoverflow.net/users/290 | 235701 | 109,150 |
https://mathoverflow.net/questions/235702 | 2 | Some Maass form can be written ($K\_{iR}$ is the K-Bessel function):
$$f(x+iy)=\sum\_{n \ne 0}^{\infty} a\_n \sqrt{y} \;K\_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$
with the $a\_n$ multiplicative, but inversly if I fix multiplicative coefficients, for example $a\_n=\frac{\chi(n)}{\sqrt{|n|}}$ can I expect the function de... | https://mathoverflow.net/users/38290 | Maass form properties and their fourier coefficients | A few things.
* All Maass forms can be written in the Bessel form that you mention. It is deduced from the growth condition of the Fourier coefficients of an arbitrary Maass forms.
* If you want to twist a Maass form $f$ by a primitive Dirichlet character $\chi$, the natural object to consider is the twisted L-functi... | 3 | https://mathoverflow.net/users/43108 | 235712 | 109,155 |
https://mathoverflow.net/questions/235709 | 5 | Assume you have a smooth projective variety $X$ over the complex numbers, a smooth prime divisor $D$ on it, and a torsion free coherent sheaf $E$ on $X$ of rank $r>0$. Let $E|\_{2D}:=E\otimes\_{\mathcal{O}\_X}\mathcal{O}\_{2D}$, where $\mathcal{O}\_{2D}:=\mathcal{O}\_{X}/\mathcal{I}\_{D}^2$, be the restriction of $E$ t... | https://mathoverflow.net/users/4721 | Is locally freeness of a sheaf (of fixed rank) around a divisor detectable from a first order neighbourhood? | Let $X$ be a connected reduced noetherian scheme and $\mathscr F$ a coherent sheaf on $X$. Let $\varrho(x)=\dim\_{\kappa(x)}\mathscr F\_x\otimes \kappa(x)$ where $x\in X$ is a point and $\kappa(x)$ is the residue field at $x$.
Using Nakayama's lemma you can prove the following:
The function $\varrho$ is upper semi... | 6 | https://mathoverflow.net/users/10076 | 235718 | 109,158 |
https://mathoverflow.net/questions/235474 | 5 | Let $X$ be a smooth cubic surface in $\mathbb{P}^3$. It is a classical theorem of Cayley and Salmon that $X$ contains exactly 27 lines over an algebraically closed field.
In 2002, Heath-Brown proved in his paper "The density of rational points on curves and surfaces" that if $F$ is a binary form of degree $d \geq 3$... | https://mathoverflow.net/users/10898 | A question regarding lines on a cubic surface | To each automorphism of the binary form there correspond actually $d$ lines, not only one. In case $d = 3$, since there are always 6 automorphisms, we get $6\cdot 3 = 18$ lines, which summed to the 9 lines coming from roots give a total of 27.
This is very well explained in [this article](http://arxiv.org/abs/math/06... | 2 | https://mathoverflow.net/users/43951 | 235719 | 109,159 |
https://mathoverflow.net/questions/235724 | 6 | Is there a name for the following process?
Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x\_1,\dots,x\_k$ from $f$. Then, I let $x^\*$ be the sample for which $f(x\_i)$ is the largest, i.e. $x^\* = \arg \max\_i f(x\_i)$. My question is, ... | https://mathoverflow.net/users/70190 | Choosing a sample based on where the density function is highest | If your $k$ samples are taken with replacement, then your method is a *Bootstrap estimate of the mode*. In the limit of a large number of such samples, the Bootstrap estimate coincides with the true mode.
[Bootstrapping](https://en.wikipedia.org/wiki/Bootstrapping_(statistics)) is useful when seeking the variance of ... | 0 | https://mathoverflow.net/users/89654 | 235731 | 109,164 |
https://mathoverflow.net/questions/235387 | 1 | Is there a finite non-abelian $2$-group $G$ without non-trivial elementary abelian direct factor and of order $2^9$ satisfying the following condition: $$Z(G) \cap Z(\Phi(G))= \langle \prod\_{i=1}^{2^d} a^{x\_i} \;|\; a\in Z(\Phi(G)) \rangle,$$ where $\{x\_1,\dots,x\_{2^d} \}$ is a right transversal of $\Phi(G)$ in $G$... | https://mathoverflow.net/users/19075 | Cohomologically trivial $G$-modules | The answer is no. I did a computer search through the $10494213$ groups of order $512$ (which you could have done yourself) and found that the only group that satisfies the condition is the elementary abelian group.
I did successfully confirm that there are $10$ examples of order $2^8$, in addition to the elementary ... | 5 | https://mathoverflow.net/users/35840 | 235743 | 109,167 |
https://mathoverflow.net/questions/235708 | 4 | *(this question is about a particular aspect of [a previous question](https://mathoverflow.net/q/229825/22606), which was not duly stressed)*
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^\*M
$$
be the $(2n-1)$-dimensional manifold of tangent hyperplanes to $M$.
>
> QUES... | https://mathoverflow.net/users/22606 | Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian? | I also believe that the answer to your question is yes, but I think that things are more complicated than indicated in the answer by @Ben\_McKay and in your partial solution. The point is that while you have an exact sequence of the form claimed in your answer, requiring that the two arrows are metric does not pin down... | 3 | https://mathoverflow.net/users/64141 | 235746 | 109,168 |
https://mathoverflow.net/questions/235749 | 6 | What is the fundamental difference between Poisson Point Process and Binomial Point Process?
I am evaluating a solution in a Binomial Point Process setup. If I want to evaluate that in a Poisson Point Process setup, what all issues need to be considered?
| https://mathoverflow.net/users/61400 | Fundamental difference between Poisson Point Process and Binomial Point Process | 
This figure (copied from these [notes](http://ahvaz.ist.unomaha.edu/azad/temp/sac/07-baddeley-point-process-poisson-coverage-sensor-simulation.pdf)) serves to illustrate the difference between a binomial and a Poisson point process. Shown are $n=100$ points randomly... | 5 | https://mathoverflow.net/users/11260 | 235760 | 109,173 |
https://mathoverflow.net/questions/235729 | 4 | Since $C[0,1]^{\*}$ is an abstract $L$-space, $C[0,1]^{\*}$ is order isometric to $L\_{1}(\mu)$ for some measure $\mu$.
My question is: the measure $\mu$ can be choosen to be a finite positive measure?
Thank you!
| https://mathoverflow.net/users/41619 | The dual space of $C[0,1]$ | In an $L^1(\mu)$ space, we can tell when two elements are disjoint (that is, have disjoint support):
$f\_1, f\_2$ are disjoint if and only if $\|f\_1\pm f\_2\| = \|f\_1\|+\|f\_2\|$.
I claim that if $L\_1(\mu)$ is isometric to $C[0,1]^\*$, then $\mu$ is not sigma-finite. To do this, it suffices to exhibit an uncountab... | 6 | https://mathoverflow.net/users/454 | 235766 | 109,176 |
https://mathoverflow.net/questions/235666 | 15 | The lack of a suitable resolution of singularities comes up often in work on étale cohomology from the 1960s and 70s, And I think even the latest version of Milne's lecture notes says "It is likely that de Jong’s resolution theorem (Smoothness, semi-stability and alterations. Inst. Hautes Etudes Sci. Publ. Math. No. 83... | https://mathoverflow.net/users/38783 | Resolution of singularities in étale cohomology | I am turning Giulia's comment into a CW answer. At over 400 pages, I think that "Travaux de Gabber", published in Astérisque, ought to count as a "comprehensive treatment of [some aspects of ] étale cohomology" using the method of alterations. The style is very SGA, and indeed there is a nonempty intersection among the... | 6 | https://mathoverflow.net/users/4144 | 235771 | 109,177 |
https://mathoverflow.net/questions/176264 | 17 | Let $G/K$ be a symmetric space. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition,
with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant
differential operators in this case is commutative, and "polynomial conjecture"
states that it is isomorphic to $S(\mathfrak{p}... | https://mathoverflow.net/users/9833 | Is the Duflo polynomial conjecture open? | As far as I know, Duflo's conjecture is still open.
Let me make several remarks:
1. Duflo's conjecture actually says that the algebra of invariant differential operators on a symetric space is isomorphic to the $\mathfrak k$-invariant part of $S(\mathfrak g)/(h-\chi(h),h\in\mathfrak k)$, where $\chi$ is the charact... | 3 | https://mathoverflow.net/users/7031 | 235772 | 109,178 |
https://mathoverflow.net/questions/235739 | 9 |
---
Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph\_0}=\aleph\_2=\kappa.$
It is known that in the extension there are no special $\aleph\_2$-Aronszajn trees but there are $\aleph\_2$-Aronszajn trees.
>
> **Question 1.** Is ... | https://mathoverflow.net/users/11115 | Two questions about higher Souslin trees | **About question 1:** If $\kappa$ is not weakly compact in $L$, then there is an $\aleph\_2$-Suslin tree in $L[G]$.
In $L$ there is a $\kappa$-Suslin tree, $T$, and since Mitchell's forcing is $\kappa$-Knaster, it cannot add an antichain of cardinality $\kappa$ to $T$: If $\dot{\mathcal{A}}$ is a name for unbounded ... | 6 | https://mathoverflow.net/users/41953 | 235779 | 109,179 |
https://mathoverflow.net/questions/235647 | 14 | For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of $G$(whenever it exists) is cohomological. I am trying to understand what information do we exactly obtain when we know ... | https://mathoverflow.net/users/90075 | Philosophy behind cohomological representations | The real place of an automorphic representation is an irreducible unitary representation $\pi$ of $G(\mathbb R)$. A fundamental basic case is when $\pi$ has the same infinitesimal character as a finite dimensional representation (also known as regular and integral infinitesimal character). By a result of Susana Salaman... | 11 | https://mathoverflow.net/users/6030 | 235782 | 109,180 |
https://mathoverflow.net/questions/235814 | 3 | Is there a general simple theorem for the third cohomology of cyclic groups $H\_3(\mathbb{Z}\_n, U(1))= ?$. In particular, I am interested in finding $H\_3(\mathbb{Z}\_8, U(1))$. I know the answer can be found using GAP, but I wanted a formal theorem, and also I don't have access to the HAP package of GAP, which is use... | https://mathoverflow.net/users/nan | Third (co-) homology of Cyclic groups | For cyclic groups, the modified cohomology groups are periodic with period 2. In particular, there are isomorphisms $$H\_3(\mathbb Z/n\mathbb Z,U(1))=\hat H^{-4}(\mathbb Z/n\mathbb Z,U(1))\cong \hat H^0(\mathbb Z/n\mathbb Z,U(1))=U(1)^{\mathbb Z/n\mathbb Z}/N(U(1))=U(1)/n=0$$
Here $N$ is the norm map $x\mapsto \sum\_... | 6 | https://mathoverflow.net/users/76105 | 235816 | 109,187 |
https://mathoverflow.net/questions/235800 | 31 | There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book [Isoperimetric Inequalties in Mathematical Physics](http://rads.stackoverflow.com/amzn/click/0691079889)) which is as follows:
"Of all $n$-gons of a fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue."
Surprisingly, ... | https://mathoverflow.net/users/48438 | A long-lasting conjecture of Pólya & Szegő | I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals.
There have been some numerics as well as some refined inequalities for triangles. [This article](https://www.math.tecnico.ulisboa.pt/~pa... | 12 | https://mathoverflow.net/users/1540 | 235830 | 109,190 |
https://mathoverflow.net/questions/235827 | 58 | Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f\*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits\_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
| https://mathoverflow.net/users/87890 | Square root of dirac delta function | So I guess my initial intuition was wrong; there is enough "room at infinity" to concoct such a function $f$. The key lemma is
>
> **Lemma.** Let $m\_1,m\_2,m\_3,\dots$ be an enumeration of the integers. Then there exists an increasing sequence $0 = f\_0 \leq f\_1 \leq f\_2 \leq \dots$ of finitely supported functio... | 48 | https://mathoverflow.net/users/766 | 235836 | 109,191 |
https://mathoverflow.net/questions/235840 | 15 | It is well known that $S^n$ admits an H-space structure if and only if $n=0,1,3,7$. I'm interested in whether there are other suspensions $\Sigma X$ that admit H-space structures:
**Question 1** For which $X$ (not a sphere) is $\Sigma X$ an H-space? And what about $\Sigma X$ that are *associative* H-spaces?
My moti... | https://mathoverflow.net/users/2004 | H-space structures on non-sphere suspensions? | If $Y$ is a connected CW-complex of finite type which is both an H-space and a co-H-space, then $Y$ has the homotopy type of $S^1$, $S^3$, $S^7$ or a point. This is a result of Robert West:
*Robert W. West*, [**$H$-spaces which are co-$H$-spaces**](http://www.ams.org/mathscinet-getitem?mr=285006), *Proc. Amer. Math. ... | 24 | https://mathoverflow.net/users/8103 | 235843 | 109,192 |
https://mathoverflow.net/questions/235758 | 21 | I would like to understand what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$ and $pq \neq 0$. My working definition of $Out$ is as follows:
Let us denote by $Aut(G)$ the automorphism group of a Lie group $G$. I take the inner-automorphism group $Inn(G)$ of $G$ to be all elements $... | https://mathoverflow.net/users/66688 | Automorphism group of real orthogonal Lie groups | Let's first address your comment in response to Igor Rivin's answer: why don't we find this topic addressed in textbooks on Lie groups? Beyond the definite (= compact) case, disconnectedness issues become more complicated and your question is thereby very much informed by the theory of linear algebraic groups $G$ over ... | 23 | https://mathoverflow.net/users/81332 | 235850 | 109,193 |
https://mathoverflow.net/questions/235849 | 7 | Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l\_p$ norm ($p\geq 2$).
Can we prove that the following problems are equivalent:
$$\max\_{\|x\|\_p=\|y\|\_p=1} \left| \langle x, Ay\rangle \right|$$
and
$$\max\_{\|x\|\_p=1} \left| \langle x, Ax\rangle \right| $$
Can the result be generalized... | https://mathoverflow.net/users/90183 | Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$? | This is false for every $p>2$. Take $A=\left( \begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right)$. Then the maximum of $|\vec{x}^T A \vec{x}|$ on $|x|\_p=1$ is $1$, achieved on the coordinate axes. (The proof is an easy computation with Lagrange multipliers.) But the maximum of $\vec{x}^T A \vec{y}$ is $2^{1... | 7 | https://mathoverflow.net/users/297 | 235854 | 109,196 |
https://mathoverflow.net/questions/235809 | 3 | I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula
>
> $$O\_\gamma(f) = \int\_G f(x^{-1}\gamma x)dx = \Theta\_\pi(\gamma) dim(\pi)^{-1}$$
>
>
>
where $f$ is a suitably normalized matrix coefficient ... | https://mathoverflow.net/users/43737 | Orbital integral for matrix coefficients | You said in comments that you are OK with using orthogonality of matrix coefficients, but let's suppose you changed your mind. Let $\langle\cdot, \cdot\rangle$ be a $G$-invariant pairing on the space $V$ of $\pi$, and let $\mathrm dg$ be the Haar measure on $G$ normalised to give it total mass $1$. What follows is all ... | 5 | https://mathoverflow.net/users/2383 | 235855 | 109,197 |
https://mathoverflow.net/questions/235863 | 0 | Let $l$ be a prime number, $n\in \mathbb{Z}$. Is it true that any finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-module has a finite (left) resolution by free finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-modules?
I am not an expert in the field, so the question might not be on the research level. Sorry about that.
| https://mathoverflow.net/users/16183 | Projective resolutions of torsion modules | No. In fact, this is as far from true as possible: a finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-module has a finite free resolution iff it is free. To see this, note that a finitely generated $\mathbb{Z}/l^n\mathbb{Z}$ is a direct sum of modules of the form $\mathbb{Z}/l^m\mathbb{Z}$ for $m\leq n$ (this follows from... | 3 | https://mathoverflow.net/users/75 | 235865 | 109,201 |
https://mathoverflow.net/questions/235867 | 3 | We know that the Riemann zeta function can be generalized to [multivariate zeta functions](https://en.wikipedia.org/wiki/Multiple_zeta_function).
Is there a multivariate analog of the Weil conjectures?
| https://mathoverflow.net/users/10035 | Weil Conjectures Analog for Multivariate Zeta Functions | The Weil conjectures have to do with local zeta-function over finite fields.
The Riemann zeta function and the multiple zeta functions are defined over $\mathbb{Q}$
So Weil conjectures are simply the wrong type of conjecture to make about this kind of L-functions. On the other hand, the Weil conjectures were modele... | 6 | https://mathoverflow.net/users/43108 | 235872 | 109,202 |
https://mathoverflow.net/questions/235612 | 9 | Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line bundle decomposition $E = \bigoplus\_{\alpha}{L\_\alpha}$.
1. Let $S(E)$ the pointed sphere bundle obtained by fiberwise co... | https://mathoverflow.net/users/22810 | Closed formulas for topological K-theory? | Put $u\_\alpha=[\mathbb{C}]-[L\_\alpha]$.
It is a standard fact, known as the projective bundle theorem, that
$$K(\mathbb{P}(E))=K(X)[t]/\prod\_{\alpha}(t-u\_\alpha)$$
One can express $Fl(E)$ as the top of a tower in which each level is the projective bundle associated to a vector bundle over the level below. Using thi... | 11 | https://mathoverflow.net/users/10366 | 235876 | 109,203 |
https://mathoverflow.net/questions/235858 | 1 | I am trying to solve a 4th order nonlinear PDE for a real function $u(x,y)$ of two variables. It is too complicated to reproduce here but it exhibits the following two very nice properties:
1) if $u(x,y)$ is a solution, then $f(u(x,y))$ is also a solution **for any function** $f$.
2) if we think of the equation in ... | https://mathoverflow.net/users/7154 | Nonlinear PDE for a 2D foliation | It's easy to derive a *third*-order (nonlinear) differential equation for $u(x,y)$ that satisfies your conditions (1) and (2): Namely, set $\theta(x,y) = \arctan\bigl(u\_y(x,y)/u\_x(x,y)\bigr)$ and then consider
$$
\theta\_{xx} + \theta\_{yy} = 0.
$$
When one expresss this equation explicitly in terms of the partial de... | 2 | https://mathoverflow.net/users/13972 | 235887 | 109,206 |
https://mathoverflow.net/questions/235896 | 5 | I'm interested in the existence and properties of an analogue version of $L$ for models of ZF$^-$ (ZF without the power set axiom), which for simplicity I'll call $L^-$. By "analogue" I mean the least inner model of a set theory universe $V$ (same ordinals and transitive).
1. Assuming $V$ is a model of ZF$^-$, does a... | https://mathoverflow.net/users/59012 | Least inner model of ZF without power set axiom | Your intuition is correct, for we have $L^-=L$.
Inside any model $V$ of $\text{ZF}^-$, we can still build $L$. You don't need the power set axiom to construct $L$. And furthermore, the resulting inner model $L$ will satisfy $\text{ZF}^-$.
From this, it follows that $L^-=L$, since if $W$ is a transitive model of
$\... | 5 | https://mathoverflow.net/users/1946 | 235902 | 109,211 |
https://mathoverflow.net/questions/235856 | 10 | Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}\_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}\_0(M)$ denotes the identity component of $\operatorname{Homeo}(M)$). More generally, we could let $M$ be compact with bou... | https://mathoverflow.net/users/35353 | Elements of infinite order in the topological mapping class group | Let me suppose that $M$ is a closed manifold of dimension $d$, and let $\varphi : M \to M$ be a diffeomorphism / homeomorphism which is homotopic to the identity, and choose such a homotopy $h\_t$. The mapping torus $X\_\varphi$ of $\varphi$ is a smooth / topological manifold fibering over $S^1$, and our choice of homo... | 4 | https://mathoverflow.net/users/318 | 235904 | 109,213 |
https://mathoverflow.net/questions/235869 | 7 | [System U](https://en.wikipedia.org/wiki/System_U) is an inconsistent PTS in that one has a term of type $\bot = \forall p\colon \ast \ldotp p$, and such a term is explicitly constructed in Hurkens' [A Simplification of Girard's Paradox](https://www.cs.cmu.edu/~kw/scans/hurkens95tlca.pdf).
One-sorted circular PTS $\l... | https://mathoverflow.net/users/89916 | Easier Girard's paradox in a circular pure type system (PTS) | I feel like most of my posts on mathoverflow and cstheory.stackexchange consist of this answer, but the most perspicuous (in my opinion) proof of inconsistency of U and $\*:\*$ is a construction by Alexandre Miquel, given in his [phd dissertation](https://www.fing.edu.uy/~amiquel/publis/these.pdf). Tragically, it is in... | 8 | https://mathoverflow.net/users/36103 | 235908 | 109,216 |
https://mathoverflow.net/questions/234433 | 4 | Let $F$ be the field of **real** algebraic numbers. Is it true that the positive multiplicative group $(F\_{pos}^\*,\cdot,1)$ is isomorphic to the additive group $(F,+,0)$ (as abstract groups, not topological or ordered groups)?
Note that there cannot be any such continuous (or monotone) isomorphism
(i.e $F$ is not [... | https://mathoverflow.net/users/46290 | Are the positive multiplicative group and the additive group of the field of real algebraic numbers isomorphic? | To prevent the question from being "unanswered"...
Both groups are abelian torsionfree divisible groups, so they are vector spaces over $\mathbb{Q}$ and completely determined by their dimension. Both are countable, so their dimensions are countable (finite or infinite). To show they are isomorphic as abelian groups i... | 10 | https://mathoverflow.net/users/3959 | 235912 | 109,219 |
https://mathoverflow.net/questions/235901 | 2 | I'm reading Nomizu & Sasaki's "*Affine Differential Geometry: Geometry of Affine Immersions*" and I'm having some trouble with Proposition 1.4.
I have an immersed surface in $M \hookrightarrow \mathbb R^{3}$. Assume that we have a volume form, say $\omega$ on $\mathbb R^{3}$. This could, for example, be the determina... | https://mathoverflow.net/users/44642 | Proof about affine connections | The formulas all look correct. With the definition of $\nabla\_X\theta$, one gets
\begin{align\*}
[\nabla\_X\theta](Y,Z)&=X[\theta(Y,Z)]-\theta(\nabla\_XY,Z)-\theta(Y,\nabla\_XZ)\\
&=X[\omega(Y,Z,\xi)]-\omega(D\_XY,Z,\xi)-\omega(Y,D\_XZ,\xi)-\omega(Y,Z,D\_X\xi)+\tau(X)\theta(Y,Z)\\
&=[D\_X\omega](Y,Z,\xi)+\tau(X)\the... | 1 | https://mathoverflow.net/users/70808 | 235918 | 109,220 |
https://mathoverflow.net/questions/235607 | 5 | Let $X\subseteq \mathbb{A}^n$ be an affine variety.
The local ring of $X$ at $p\in X$, given by $\mathcal{O}\_{X,p}=\{f\in k(X):f \text{ regular at } p\}$ is noetherian because it is a localization of $k[X]$.
If $U\subseteq X$ is open, let $\mathcal{O}\_X(U)=\bigcap\_{p\in U}\mathcal{O}\_{X,p}$. Is this ring noethe... | https://mathoverflow.net/users/37103 | Structure sheaf of affine variety consists of noetherian rings (again) | I am just reposting my comment as an answer.
Vakil shows in [this note](http://math.stanford.edu/~vakil/files/nonfg.pdf), §3 that you can modify the construction you refer to to get an affine example. For completeness, I'll write down the example here.
Let $E$ be an elliptic curve over a field $k$, let $N$ be a deg... | 4 | https://mathoverflow.net/users/33088 | 235924 | 109,224 |
https://mathoverflow.net/questions/235926 | 8 | (This question is a variant of an unanswered question at [math.stackexchange](https://math.stackexchange.com/questions/1350331/justification-of-surface-area-integral).)
[The Definition section of Wikipedia's article on surface area](https://en.wikipedia.org/wiki/Surface_area#Definition) currently starts as follows:
... | https://mathoverflow.net/users/4600 | Characterizing surface area | I would add some kind of monotonicity.
Say if there is a distance nonexpanding map between surfaces $f\colon S\to S'$ then
$$\mathop{\rm area} S\ge \mathop{\rm area} S'.$$
**P.S.** Instead of all distance nonexpanding maps one can take only orthogonal projections to planes.
| 4 | https://mathoverflow.net/users/1441 | 235930 | 109,226 |
https://mathoverflow.net/questions/235948 | 11 | Reference for Y. Manin's idea of "algebraic geometry over the symmetric monoidal model category of motives."
Has been sugested to me that this was made in a Manin's letter. There is an escaned copy?
Some work in this direction has been made in the thesis of Spitzweck, but I refer to first references of Manin.
| https://mathoverflow.net/users/83957 | Reference for Manin's idea on algebraic geometry over the symmetric monoidal model category of Motives | The only reference ever given for this is
>
> Y. Manin, *Letter*, March 2000
>
>
>
I don't think this letter (from Manin to Bertrand Toen, I presume) has ever been made public, and definitely it hasn't been published. But you might want to try asking Toen for a copy.
As you known, some of the math involved w... | 10 | https://mathoverflow.net/users/43108 | 235950 | 109,227 |
https://mathoverflow.net/questions/235949 | 10 | Freiling's [axiom of symmetry](https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry) states that if you assign to each real number $x$ a countable set $A\_x\subset\mathbb{R}$, then there should be two reals $x,y$ for which $x\notin A\_y$ and $y\notin A\_x$.
This principle turns out to be equivalent to the fa... | https://mathoverflow.net/users/1946 | What is the optimal size in the finite axiom of symmetry? | Given $k$, let $X=\{1,2,\dots,2k\}$. We must have $1$ in $A\_1$, else we could take $x=y=1$. We may assume $A\_1=\{1,2,\dots,k\}$. So $r$ is not in $A\_1$ for $r=k+1,k+2,\dots,2k$. So we must have $1$ in $A\_r$ for $r=k+1,k+2,\dots,2k$. That is, $1$ must be in $k+1$ of the sets $A\_r$ (since it's also in $A\_1$). But t... | 8 | https://mathoverflow.net/users/3684 | 235951 | 109,228 |
https://mathoverflow.net/questions/235937 | 3 | First, some background: recently in learning more about functional programming I saw one use for coproducts that surprised me a little bit: A function $f: A \rightarrow B \coprod C$ may result when considering a computation that starts with an $a \in A$ and results with an element of $B$ unless "something exceptional h... | https://mathoverflow.net/users/10110 | Coproducts and "Error Conditions" in Math vs CS | This interpretation of $C$ as a place to report failures doesn't really have teeth until you see how it interacts with function composition, which is the following. Suppose $f : X \to Y \coprod C$ and $g : Y \to Z \coprod C$ are two "functions with $C$-valued failures." Then I claim that there is a meaningful way to co... | 8 | https://mathoverflow.net/users/290 | 235955 | 109,230 |
https://mathoverflow.net/questions/235945 | 1 | Let $\mathbb H = \mathbf H \otimes\_{\mathbf R} \mathbf C$ be the tensor product of the quaternions with $\mathbf C$, and let $\mathcal J\_3(\mathbb H)$ denote the set of $\mathbb H$-hermitian $3 \times 3$ matrices (a $15$-dimensional vector space over $\mathbf C$). I have read that $\mathcal J\_3(\mathbb H)$ is isomor... | https://mathoverflow.net/users/nan | Jordan algebra of $3 \times 3$ quaternionic hermitian matrices | Choose an isomorphism $\iota: \mathbf{H}\otimes \mathbf{C} \simeq M\_2(\mathbf{C})$. For example, one such map is given by $a+bi+cj+dk \mapsto \left(\begin{array}{cc} a+ ib & c+id \\-c+id &a-ib\end{array}\right).$ A way to remember this is is $z + wj \mapsto \left(\begin{array}{cc} z & w \\-w^\* &z^\*\end{array}\right)... | 4 | https://mathoverflow.net/users/25514 | 235959 | 109,234 |
https://mathoverflow.net/questions/235941 | 4 | Is there a general study of the symmetries of tilings on surfaces?
Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. But I found nothing about general surfaces, except for the cylinder which is an easy consequence from the result above.
I ... | https://mathoverflow.net/users/49443 | Classification of symmetries of tilings in surfaces? | As the OP points out, it is natural to study tilings from the viewpoint of symmetries acting on a space. From this perspecitve, finite subgroups of $SO(3)$ lead to tilings of $\mathbb{S}^2$ and the wallpaper groups yield tilings of $\mathbb{R}^2$.
Along the same lines, Chapter 7 of **Farb, Benson, and Dan Margalit. A... | 1 | https://mathoverflow.net/users/27453 | 235968 | 109,240 |
https://mathoverflow.net/questions/235970 | 1 | Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K\_X +D = 0$ necessary for the existence of Ricci-flat metric?
| https://mathoverflow.net/users/86428 | A necessary condition for existence of Ricci flat metric on pair (X,D) | **No**, the condition $K\_X +D = 0$ is not necessary for the Ricci-flat case, when $D$ is singular. For instance,consider $X=\overline X\setminus D$ and take $\overline X=\mathbb CP^n$ and define the divisor $D:=\{p(z)\prod\_{i=0}^{n-1}z\_i=0\}$, where
$$f(z)=z\_0^{m-1}z\_n+P(z\_0,...,z\_{n-1})$$ for a homogeneous p... | 3 | https://mathoverflow.net/users/nan | 235972 | 109,241 |
https://mathoverflow.net/questions/235502 | 2 | Consider the group $GL\_n(\mathbb{R})$ with its standard topology.
It is not hard to show that there exists Riemannian metrics on it which are left-$GL\_n$ and right-$O\_n$ invariant. (In fact it's possible to describe all of them explicitly). An immediate corollary is that there exists left-$GL\_n$ and right-$O\_n$... | https://mathoverflow.net/users/46290 | Characterizing left invariant and right-$O_n$ invariant distances on $GL_n$ | $\newcommand{\al}{\alpha}$
The answer is no, there are many more such metric which are not induced by a Riemannian metric.
(This answer is based on the comments above, made by user89334).
Examples:
1) Take any smooth norm on the space of $n \times n$ real matrices which is invariant under conjugation by the ortho... | 0 | https://mathoverflow.net/users/46290 | 235989 | 109,244 |
https://mathoverflow.net/questions/235987 | 2 | Let $S^{2n+1}$ be the $m$-dimensional sphere in $\mathbb{C}^{n+1}$. Endow $S^{2n+1}$ with the standard metric. Let $S^1$ act by multiplication on $S^{2n+1}$. Then $S^1$ and the canonical action of $SU(n+1)$ are by isometries. Since the actions of $S^1$ and $S^{2n+1}$ commute, $G = SU(n+1) \times S^1$ acts by isometries... | https://mathoverflow.net/users/75382 | coisotropic action on $TS^{2n+1}$ | 1) Since $S^{2n+1}$ is a Riemannian manifold one can identify $TS^{2n+1}$ with the cotangent bundle $T^\*S^{2n+1}$. The latter is well-known to carry a canonical symplectic structure and a momentum map.
2) This condition can be rephrased as $\Phi/G:T^\*S^{2n+1}/G\to\mathfrak{g}^\*/G$ being finite to one. Let $G\_0\co... | 4 | https://mathoverflow.net/users/89948 | 235991 | 109,245 |
https://mathoverflow.net/questions/235977 | 3 | I am looking for the English translation of the paper by V. M. Borok (originally in Russian)
[The Cauchy problem for finite-infinite systems of linear differential equations](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=6842&option_lang=eng). This work is about the Cauchy problem $x'=Ax$ where ... | https://mathoverflow.net/users/90243 | An English version Borok's work on finite-infinite systems of ordinary differential equations | "Soviet mathematics" is a publication which translated Russian papers from various journals which were not translated cover-to cover. This publication was quite expensive and few libraries subsctibed it. Selected papers from Izvestiia Vysshikh Uchebnykh Zavedenii, where this paper is published were translated. Most uni... | 2 | https://mathoverflow.net/users/25510 | 235992 | 109,246 |
https://mathoverflow.net/questions/235915 | 9 | It is known that adjunctive set theory interprets Robinson arithmetic, and that extensionality is not needed for that. (Montagna and Mancini, "A minimal predicative set theory", Notre Dame Journal of Formal Logic, 1994).
I wonder whether it is known if the following weak set theory (without extensionality) also inte... | https://mathoverflow.net/users/90219 | Interpreting Robinson arithmetic in a very weak set theory | This theory was introduced by Vaught, and it does *not* interpret Robinson’s arithmetic. See Visser [1] for a thorough discussion of related theories; Vaught’s theory is denoted VS in the paper. (Note that the axioms are stated more concisely there: axiom 1 is a special case of axiom 2 for $n=0$.)
That VS does not in... | 9 | https://mathoverflow.net/users/12705 | 235998 | 109,251 |
https://mathoverflow.net/questions/235985 | 1 | I have problems understanding two (for my research important) details in the Proof of Theorem 4 (page 14) in this paper: <http://arxiv.org/abs/1209.1518>. Notation is very straightforward and is found on page 4.
I expect that both issues are in reality the same problem:
1. In the first line of page 14 Schottdorf clai... | https://mathoverflow.net/users/88808 | Estimating sums with restrictions to different Frequencies | To answer your question 1 (and I expect the answer to question 2 is similar):
Note that $I^+(f,g)$ first take the product $fg$ and act on it as a Fourier multiplier. So the frequency support of $I^+(f,g)$ is the same as that of $fg$.
Now, when $L \ll H$ you have that the frequency support of $u\_L v\_H$ is $\appro... | 1 | https://mathoverflow.net/users/3948 | 236000 | 109,253 |
https://mathoverflow.net/questions/235986 | 1 | I am looking for formulas for writing a basis element of $ Sym^k(H) $ as sum of elements of the form $ v^{\otimes k} $ where $ v\in H $. Here $ H $ is a hilbert space and by basis element I mean the image of elements of the form $ e\_{i\_1}\otimes e\_{i\_2}\otimes....\otimes e\_{i\_k} $ under the map $ H^{\otimes k}\to... | https://mathoverflow.net/users/87890 | Symmetric tensors as sum of powers | There are of course many ways to do that; I like the following formula, for $e\_1,\ldots ,e\_k\in H$ (I write the product in $Sym^k(H)$ as a usual product):
$$e\_1\ldots e\_k=\frac{(-1)^k}{k!} \sum\_{I\subset [1,k]} (-1)^{\# I}(\sum\_{i\in I}e\_i)^k\ .$$
To prove it, take the term of degree $k$ in the equality $$\pro... | 6 | https://mathoverflow.net/users/40297 | 236004 | 109,254 |
https://mathoverflow.net/questions/235973 | 9 | Is anything known about the behavior of the function $$f(x)=\prod\_{i=1}^n x\_i^{x\_i}$$ on the standard simplex, i.e. the set $\{x\in\mathbb{R}^n:\sum\_{i=1}^n x\_i=1, x\_i\geq0\}$? I ask because I have done a lot of numerical experiments that suggest that $$1\leq\left(\prod\_{i=1}^{n}x\_{i}^{x\_{i}}\right)\left(\sum\... | https://mathoverflow.net/users/90239 | An inequality on the simplex involving $x^x$ | The upper bound is not correct. E.g., let $n=101$, $x\_1=\dots=x\_{100}=1/1000$, $x\_{101}=9/10$. Then
$$\left(\prod\_{i=1}^{n}x\_{i}^{x\_{i}}\right)\left(\sum\_{i=1}^{n}x\_{i}^{1/2}\right)^{2}>7>2.$$
More generally, take any $a\in(0,1)$ and any natural $k$, and then let $n=k+1$, $x\_1=\dots=x\_k=a/k$, $x\_{k+1}=1-a... | 12 | https://mathoverflow.net/users/36721 | 236007 | 109,256 |
https://mathoverflow.net/questions/117513 | 26 | I asked this [question](https://math.stackexchange.com/questions/258736/limit-of-sequence-of-growing-matrices) on math.stackexchange and haven't received an answer in two weeks, so I'm repeating it here.
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \cr
1/2 & 0 & 1/2 & 0 \cr
1/2 & 0 & 0 & 1/2\cr
0 & 1/2 & 1... | https://mathoverflow.net/users/30264 | Singular values of sequence of growing matrices | First let's change the matrix $H$ to $H=\frac{1}{2}\left(\begin{array}{cc}I\_2 & I\_2 \\ I\_2& P\_2\end{array}\right)$, where $P\_2$ is a 2x2 permutation matrix. This swapping of the rows of $H$ won't affect the singular values of the matrices. Now lets consider what affect the iteration has on a matrix of the form $\l... | 10 | https://mathoverflow.net/users/90258 | 236011 | 109,259 |
https://mathoverflow.net/questions/235979 | 6 | Voevodsky defined the slice filtration on the motivic stable homotopy category $SH(S)$ over a Noetherian scheme $S$. In the article [Open Problems in the Motivic Stable Homotopy Theory, I](http://www.math.illinois.edu/K-theory/0392/newopen.pdf), Section 2, he defined $SH^{eff}(S)$ to be the smallest triangulated subcat... | https://mathoverflow.net/users/39193 | Doubt regarding the definition of slice filtration | The key here is that $SH^{eff}(S)$ is closed under suspensions, so there's an inclusion $j\_{n+1}:\Sigma^{n+1}\_T SH^{eff}(S)\subseteq \Sigma^n\_T SH^{eff}(S)$. Hence you can write $i\_{n+1}=i\_n \circ j\_{n+1}$ and $r\_{n+1} = l\_{n+1}\circ r\_n$ where $l\_{n+1}$ is the right adjoint of $j\_{n+1}$ (which exists becaus... | 5 | https://mathoverflow.net/users/43054 | 236016 | 109,260 |
https://mathoverflow.net/questions/236013 | 4 | So, last year I got obsessed with the idea of finding a way to calculate π that wasn't already done. After reading some history, the Greek idea of measuring polygons inscribed within circles and cutting their vertices to increase their side counts and get closer to calculating π intrigued me. From this I decided to use... | https://mathoverflow.net/users/89987 | π based on the perimeter of inscribed polygons | The formulas that you give (and many others of a similar flavor) appear in the article [Nested Square Roots 2](https://www.ams.org/mathscinet-getitem?mr=1984573) by L. D. Servi. The full citation is
Servi, L. D.. “Nested Square Roots of 2”. The American Mathematical Monthly 110.4 (2003): 326–330.
According to this ... | 8 | https://mathoverflow.net/users/nan | 236018 | 109,261 |
https://mathoverflow.net/questions/236012 | 3 | Let $1\leq p<\infty$. We say that an operator $T:X\rightarrow Y$ is unconditionally $p$-converging if $T$ takes a weakly $p$-summable sequence to a norm null sequence.
Question: Is every unconditionally $p$-converging operator $T$ from $L\_{1}[0,1]$ to every Banach space $Y$ weakly compact?
It is known that this q... | https://mathoverflow.net/users/41619 | Unconditionally $p$-converging operators on $L_{1}[0,1]$ | No. Take a projection $P$ from $L\_1$ onto a subspace isometrically isomorphic to $\ell\_1$ and use the fact that $\ell\_1$ has the Schur property.
| 1 | https://mathoverflow.net/users/2554 | 236021 | 109,263 |
https://mathoverflow.net/questions/236023 | 2 | tanh is not absolutely integrable, so a direct fourier transform does not exist. But even for the signum function, which is not absolutely intrgrable, we can get the fourier transform by applying some tricks.
| https://mathoverflow.net/users/85600 | Fourier transform of tanh | Mathematica says: $$i \sqrt{\frac{\pi }{2}} \text{csch}\left(\frac{\pi t}{2}\right).$$
| 4 | https://mathoverflow.net/users/11142 | 236024 | 109,265 |
https://mathoverflow.net/questions/236003 | 10 | When reading about homotopy algebras (e.g. $L\_\infty$-algebras, $A\_\infty$-algebras), an $\infty$-morphism $f$ is called an $\infty$-quasi-isomorphism if $f\_1$ is a quasi-isomorphism.
**Recall/Example ($A\_\infty$-algebras):**
An **$A\_\infty$-morphism** between two $A\_\infty$-algebras $(A,\mathfrak{m})$ and $(... | https://mathoverflow.net/users/47810 | Why are quasi-isomorphisms of homotopy algebras only defined for arity 1? | Q1: The conventional theory of homotopy algebras is built on the premise that the lower-degree operations dominate over the higher-degree ones, in some sense. A discussion of this can be found in the introduction to my preprint "Weakly curved $\mathrm A\_\infty$-algebras over a topological local ring", <http://arxiv.or... | 12 | https://mathoverflow.net/users/2106 | 236026 | 109,266 |
https://mathoverflow.net/questions/236032 | 4 | Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H\_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent the quotients $G/H\_i$ in some computer algebra system like GAP or Magma.
I already tried to do this by defining $G$ as a ... | https://mathoverflow.net/users/42374 | Torsion-free, normal subgroups of certain Coxeter groups | There has been quite a bit of activity in "[abstract regular polytopes](https://en.wikipedia.org/wiki/Abstract_polytope)", i.e. certain kinds of quotients of Coxeter groups with string diagram. See e.g. the book by P.McMullen and E.Schulte "Abstract regular polytopes", CUP, 2002.
One way to construct such examples is ... | 3 | https://mathoverflow.net/users/11100 | 236037 | 109,270 |
https://mathoverflow.net/questions/236029 | 15 | A *homotopy sphere* is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$.
A *homology sphere* is a topological $n$-manifold $M$ such that $H\_i(M) \cong H\_i(S^n)$ for all $i$.
Note, by (one version of) Whitehead's Theorem, every simply connected homology sphere is a homotopy sphere. However, the... | https://mathoverflow.net/users/21564 | Are there non-smoothable homotopy/homology spheres? | Since the Poincaré conjecture is known in all dimensions any homotopy $n$-sphere is homeomorphic to $S^n$ and hence admits a smooth structure.
Any manifold of dimension $\le 3$ admits a smooth structure.
Kirby-Siebenmann famously showed that a closed manifold $M$ of dimension $>4$ admits a PL structure if and only... | 24 | https://mathoverflow.net/users/1573 | 236050 | 109,273 |
https://mathoverflow.net/questions/235516 | 6 | Consider a sequence of real numbers $s=(s\_0,s\_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that
$$
s\_k = \int\_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?
$$
The well known [Hausdorff moment problem](https://en.wikipedia.org/wiki/Hausdorff_moment_problem) asks the same qu... | https://mathoverflow.net/users/58456 | Moment problem on [-1,1]: necessary and sufficient conditions | If $\mu$ is a measure supported on $[-1,1]$, the change of variables $y = (1+x)/2$ gives you a measure $\rho$ supported on $[0,1]$, and the corresponding moments are related by
$$\int\_0^1 y^k\; d\rho(y) = \int\_{-1}^1 ((1+x)/2)^k \; d\mu(x)
= 2^{-k} \sum\_{j=0}^k {k \choose j} \int\_{-1}^1 x^j \; d\mu(x)$$
Moreov... | 6 | https://mathoverflow.net/users/13650 | 236051 | 109,274 |
https://mathoverflow.net/questions/236041 | 10 | If $R \in L\_\alpha$ is a binary relation so that $L\_\alpha$ thinks $R$ is well-founded, must $R$ truly be well-founded? (Edit) That is, if $L\_\alpha$ thinks that every nonempty subset of the domain of $R$ has a least element, is the same true in $V$?
(Edit) If $L\_\alpha$ satisfies a sufficiently strong fragment o... | https://mathoverflow.net/users/64676 | Must $L_\alpha$ be correct about well-foundedness? | *Coming back to this years later I've just noticed that my original answer was massively flawed; I've corrected it now. The issue is essentially that I took it for granted that all admissible sets look like $L\_{\omega\_1^{CK}}$ more than they actually do - see e.g. [here](https://mathoverflow.net/questions/277343/orde... | 7 | https://mathoverflow.net/users/8133 | 236056 | 109,275 |
https://mathoverflow.net/questions/236052 | 4 | Everything is in the title. I wonder if there is known good asymptotic (when $n\to+\infty$) for the quantity $LCM(\binom{2k}k)\_{1\le k\le n}$
Thanks in advance
| https://mathoverflow.net/users/33128 | Asymptotic of $Lcm(\binom{2k}k)_{1\le k\le n}$ | Put $A\_n$ to be the lcm of $\binom{2k}{k}$ for $1\le k\le n$, and let $B\_n$ denote the lcm of the numbers at most $2n$ (thus $B\_n = \exp(\psi(2n))$). We claim that $A\_n = B\_n/2$ unless $n=2^r-1$ in which case $A\_n =B\_n$. Since asymptotics for the $\psi(2n) = \sum\_{r\le 2n} \Lambda(r)$ are well known (the prime ... | 5 | https://mathoverflow.net/users/38624 | 236064 | 109,278 |
https://mathoverflow.net/questions/236014 | 5 | Let $(M,\omega)$ be a symplectic manifold and let $\operatorname{Ham}^c(M,\omega)$ denote the group of compactly supported Hamiltonian diffeomorphisms of $(M,\omega)$. Is the Hofer topology on $\operatorname{Ham}^c(M,\omega)$ second countable?
By the Hofer topology I mean the topology induced by the Hofer metric.
| https://mathoverflow.net/users/23500 | Is the Hofer topology second countable? | Since it is a metric space, your question is equivalent to asking whether it is separable (i.e. has a dense countable subset). Now, under some assumption on the manifold (compact is ok for sure, but second-countable is probably enough too), $\mathrm{Ham}\_c(M,\omega)$ is second-countable (hence separable) for the $C^1$... | 3 | https://mathoverflow.net/users/58620 | 236080 | 109,281 |
https://mathoverflow.net/questions/236046 | 2 | Let $T(t)$ be a $C\_0$-semigroup on a Hilbert space $H$ with a generator $A$.
It is well known that for all $x\in H,$ we have: $ \int\_0^t T(s)x ds \in D(A) $ and $ A\int\_0^t T(s)x ds = T(t)x-x$.
How is this formula changed when $x$ is replaced by a continuous function $t \mapsto x(t)$ such that: $x(t)\in D(A),\, ... | https://mathoverflow.net/users/90277 | On extending a semigroup property | For a function $f\in W^{1,1}(\mathbb R\_+,X)$ (let me stress: only $X$, not $D(A)$) or else for a function $f\in C\_0(\mathbb R\_+,D(A))$ it is well-known that the convolution
$$
\int\_0^t T(t-s)f(s)ds
$$
is in $D(A)$ for all $t$, see e.g. Corollaries VI.7.6 and VI.7.8 in the book by Engel and Nagel. This holds in gen... | 3 | https://mathoverflow.net/users/26039 | 236082 | 109,282 |
https://mathoverflow.net/questions/236017 | 23 | Near the top of the second page of [this paper](http://arxiv.org/abs/math/0012096), it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? It is trivially correct for 3-manifolds and lower dimensions. What about higher di... | https://mathoverflow.net/users/51450 | Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic? | It follows by smoothing theory. If $h : X \to Y$ is a homeomorphism between smooth 4-manifolds, one obtains two maps $X \to BO$ which become homotopic in $BTOP$. The difference between them is therefore a map $d: X \to TOP/O$.
Now $TOP/O$ is [No it isn't, see comments] 6-connected, so $d$ is nullhomotopic. Therefore ... | 11 | https://mathoverflow.net/users/318 | 236101 | 109,286 |
https://mathoverflow.net/questions/236087 | 6 | While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the question is - Why does it work?
**The Method:**
1. We construct the **Matrix** $A \in M\_{N\times N} (\mathbb{R} )$ b... | https://mathoverflow.net/users/42864 | Symmetric matrix formula for Gauss-Legendre quadrature | This is a particular implementation of a more general method, described in John Boyd's *Why Eigenvalues Are Roots: A Derivation of the One-Dimensional Companion Matrix for General Orthogonal Polynomials* ([restricted access](http://epubs.siam.org/doi/pdf/10.1137/1.9781611973525.appa)). Gauss-Legendre quadrature of orde... | 3 | https://mathoverflow.net/users/11260 | 236102 | 109,287 |
https://mathoverflow.net/questions/236063 | 1 | I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in \mathbb{C}^{d\times d}$ be an DCT matrix whose elements are given by
$$ W\_{jk} = 1/\sqrt{d} \cdot \exp (2\pi i\cdot jk/ d) .$$
Let $A... | https://mathoverflow.net/users/81633 | Restricted Isometry Property for Discrete Fourier Transform Matrix | This is a solved problem, see for instance
<http://www.mathc.rwth-aachen.de/~rauhut/files/LinzRauhut.pdf> .
See for instance theorem 4.3 for non uniform recovery results.
For more general estimations of RIP, see Theorem 8.1:
Let $A \in \mathbb{C}^{n \times d}$ be the sampling matrix from a Bounded Orthonormal System... | 2 | https://mathoverflow.net/users/40231 | 236103 | 109,288 |
https://mathoverflow.net/questions/236097 | 1 | A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the sum, or is this a consequence?
Also, assuming this is true, then the decomposition must be unique by a cosimplicity argu... | https://mathoverflow.net/users/89074 | Definition of a cosemisimple Hopf algebra | For the finite dimensional case, cosemisimple is the same as the dual being semisimple. So you can apply your knowledge of semisimple algebras. So the answer to your first question is no, you're allowed to have repeats just like semisimple algebras can have the same matrix factor repeated.
For uniqueness of decomposi... | 3 | https://mathoverflow.net/users/22 | 236107 | 109,290 |
https://mathoverflow.net/questions/236104 | 4 | Let $G$ be a compact Lie group with Haar measure $dg$ and (finite-dimensional real) Lie algebra $\frak g$. Endow $\frak g$ with an $\hbox{Ad}$-invariant norm $\|\cdot\|\_{\frak g}$ so that $\frak g$ becomes a normed space (one can use the norm induced by the negative of the Killing form).
Define the function
$$
f:{\f... | https://mathoverflow.net/users/90303 | When this Ad-invariant function on a Lie algebra is zero? | (this is an edit made after user65432 caught me assuming implicitly that $G$ is connected.)
In case $G$ is connected,
$f=0$ iff $\mathfrak{g}$ has no center.
In the general case, $G/G^0$ acts on the center $\mathfrak{z}<\mathfrak{g}$ and $f=0$ iff there are no invariant vectors for this action ($G^0$ denotes here th... | 6 | https://mathoverflow.net/users/89334 | 236109 | 109,291 |
https://mathoverflow.net/questions/236001 | 22 | Let $\{M\_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov compactness theorem, one can choose a subsequence converging in the Gromov-Hausdorff sense to a compact Alexandrov space wit... | https://mathoverflow.net/users/16183 | Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces | As mentioned in the comments, if the limit is a circle, then by the Yamaguchi fibration theorem, $M\_i$ fibers over the circle, and hence it is a torus (or Klein bottle in the non-orientable case).
If the limit is a segment, then $M\_i$ is $S^2$ for all large $i$. One (somewhat heavy handed) way to see this is to app... | 10 | https://mathoverflow.net/users/1573 | 236110 | 109,292 |
https://mathoverflow.net/questions/236058 | 6 | Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note that I assume $n$ is the dimension of $M$).
Let $\omega:(M,g) \to (\mathbb{R}^n,e)$ be the harmonic function corresp... | https://mathoverflow.net/users/76815 | Harmonic function with injective boundary conditions is an immersion? | Let $M$ be the closed unit disk in the plane. Let $f$ be a diffeomorphism of $M$ onto a smooth Jordan region $D$ in the plane. If $D$ is not convex, we can easily arrange that the average
$$\int\_{\partial{M}}f(z)ds$$
does not belong to $\overline{D}$. This implies that the harmonic extension $F$ of $f$ maps
some inter... | 10 | https://mathoverflow.net/users/25510 | 236115 | 109,294 |
https://mathoverflow.net/questions/236108 | 4 | For $A\neq B\in {\cal P}(\omega)$ we set $$\mu(A,B) = \min\big((A\setminus B)\cup (B\setminus A)\big).$$ We define $A < B$ if and only if $A \neq B$, and
* $A = B\cap \mu(A,B)$ (that is $A$ is an initial segment of $B$), **or**
* $\mu(A, B)\in A$ and there is $b\in B$ with $b > \mu(A,B)$.
For example we have $\{0,1... | https://mathoverflow.net/users/8628 | "Lexicographic" ordering on ${\cal P}(\omega)$ | **Update.** My original post had some wrong statements, which I
have now corrected.
As Emil had noted in the comments, the order is dense on the
infinite subsets. Suppose $A<B$ and both are infinite. In this
case, the least difference element is in $A$. Let $A^+$ agree with
$A$ up to and including that least differen... | 6 | https://mathoverflow.net/users/1946 | 236118 | 109,296 |
https://mathoverflow.net/questions/236122 | 8 | It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$:
(1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in V$ over $M$ such that $V = M[G]$.
(2) There is a cardinal $\kappa$ such that for every ordinal $\alpha$ and every func... | https://mathoverflow.net/users/11145 | Theorem of Bukovsky characterizing ground models | The original paper is ``[Characterization of generic extensions of models of set theory](https://eudml.org/doc/214703)''
A proof can be found in the master thesis of Giorgio Audrito, "Characterizations of set generic extensions", which is available at Viale's home page [here](http://www.logicatorino.altervista.org/ma... | 4 | https://mathoverflow.net/users/11115 | 236126 | 109,298 |
https://mathoverflow.net/questions/236074 | 3 | This was inspired by the following paper:
* J. Arias de Reyna, J. van de Lune, *"How many $1$s are needed?" revisited*, [arXiv link](http://arxiv.org/abs/1404.1850).
It might help explain my question better, because my question is actually similar to those in the problem.
So, I was curious, what if we were only a... | https://mathoverflow.net/users/nan | Time-efficient way of calculating the least number of 1s in a representation of $n$ using only the operations $+,!$ | Arjafi's comment shows how to compute $\|a\|$, so this variant is much simpler than using $\{+,\times\}$.
Suppose $n = \sum\_i n\_i i!$ with $n\_i \le i$. This is the "base factorial" expression for $n$. Then $\| n\|= \sum\_i n\_i \| i!\| = \sum\_i n\_i \|i\|$.
Proof: Suppose $n$ is the smallest counterexample. For... | 6 | https://mathoverflow.net/users/2954 | 236130 | 109,300 |
https://mathoverflow.net/questions/236131 | 7 | **Question.** Let $S$ be a closed surface of genus $> 1$. Can $\pi\_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree.
Things I know related to this: a minimal action on an $\mathbb{R}$-tree $T$ (like a simplicial tree) gives a canonica... | https://mathoverflow.net/users/5010 | Can a surface group act on a finite-valence simplicial tree? | Theorem 1.2 of Breuillard-Gelander-Souto-Storm in "Dense embeddings of surface groups" (<http://arxiv.org/abs/math/0602635>) says the following:
Let $G$ be a locally compact group. Suppose that $G$ contains a nondiscrete
free subgroup $F$ of finite rank $r > 1$. Then $G$ has a subgroup $\Gamma$ containing $F$ such th... | 6 | https://mathoverflow.net/users/89334 | 236135 | 109,302 |
https://mathoverflow.net/questions/235916 | 18 | Consider the fundamental unit $\varepsilon$ of a real quadratic number field $k = {\mathbb Q}(\sqrt{p})$ for primes $p \equiv 1 \bmod 4$, and let $h$ denote its class number. By Dirichlet's work on class number formulas, $\varepsilon^h$ is a norm of a cyclotomic unit in the maximal real subfield $K^+$ of the field $K =... | https://mathoverflow.net/users/3503 | Are quadratic units cyclotomic norms? | The answer is no, and it fails in the very first example. Let $L/\mathbf{Q}$ be the degree six field inside $M = \mathbf{Q}(\zeta\_{229})^{+}$.
The unit group has rank five, and can be computed explicitly via pari. Both you and I will have no difficulty computing that the norm of each unit in $L$ to $K = \mathbf{Q}(\sq... | 16 | https://mathoverflow.net/users/90316 | 236139 | 109,304 |
https://mathoverflow.net/questions/236134 | 3 | Story
-----
I want to prove [Euler's reflection formula](https://en.wikipedia.org/wiki/Reflection_formula) by showing that
\begin{equation\*}
f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s)
\end{equation\*}
is constant, where $s = \sigma + it$. It's easy to see that $f$ is entire and $f(s + 1) = f(s)$, so for fixed $... | https://mathoverflow.net/users/37398 | Proof of Euler's reflection formula via rapidly decreasing Fourier series | The function $\sigma\mapsto f(\sigma+it)$ is periodic with period $1$, for any value
of $t$. Therefore
$$f(\sigma+it)=\sum\_{n\in Z} c\_n(t)e^{2\pi i n\sigma}.$$
It follows that
$$c\_n(t)=\int\_0^1 f(x+it) e^{-2\pi i n x}\,dx.$$
The $n$-th term in the Fourier expansion is, with $s=\sigma+it$
\begin{multline\*}
f\_m(s)... | 2 | https://mathoverflow.net/users/7402 | 236140 | 109,305 |
https://mathoverflow.net/questions/236143 | 3 | Consider the following optimization problem. I have $n$ advisors and $dn$ students. I want to assign each student an advisor so that each advisor has exactly $d$ students. Each advisor/student pair has a weight, and I want to maximize the total weight. Equivalently, I have a weight function on $K\_{n,cn}$, and I want t... | https://mathoverflow.net/users/8604 | A modified bipartite assignment problem | Edit: Creating $d$ copies of each advisor and running the standard weighted matching algorithm solves this, if I'm not missing something.
If I understand your question correctly, this is an instance of [Minimum-cost flow problem](https://en.wikipedia.org/wiki/Minimum-cost_flow_problem), which can be solved in polynom... | 3 | https://mathoverflow.net/users/58456 | 236145 | 109,308 |
https://mathoverflow.net/questions/226880 | 1 | I am reading the paper [A Gröbner fan method for biochemical network modeling](https://dl.dropboxusercontent.com/u/6980502/Uni/Paper-1-Gr%C3%B6bner-Fan-Method.pdf).
In Chapter 4.3 (i.stack.imgur.com/h2O8B.png) they calculate the vanishing ideal of some tuples (input points of Series 1-4) in the integer ring modulo $2... | https://mathoverflow.net/users/84486 | Vanishing ideal of a finite set of points does not have expected amount of cones in Gröbner fan | Thank you for your question. I am one of the authors of the paper.
To be clear, the points from which the ideal of points is created are the inputs (not the outputs) of the data. What you have done seems correct: you computed the ideal of the input points over ZZ/2 and then computed the Groebner fan of the ideal. I ... | 0 | https://mathoverflow.net/users/90317 | 236153 | 109,313 |
https://mathoverflow.net/questions/236151 | 34 | I was wondering if there is some description known for the conjugacy classes of $$\mathrm{SL}\_2(\mathbb{Z})=\{A\in \mathrm{GL}\_2(\mathbb{Z})|\;\;|\det(A)|=1\}.$$ I was not able to find anything about this. Most references only give solutions for $\mathrm{SL}\_2(\mathbb{R})$.
Thank you for your help.
| https://mathoverflow.net/users/74051 | Conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$ | One can proceed as follows for $SL\_2(\mathbb{Z})$.
1. First, the trace is a conjugacy invariant.
2. For trace $0$ there are two conjugacy classes represented by $\pmatrix{0 & 1 \\ -1 & 0}$ and $\pmatrix{0 & -1 \\ 1 & 0}$. These representatives can be thought of as $90^\circ$ and $270^{\circ}$ degree rotations of a l... | 43 | https://mathoverflow.net/users/20787 | 236162 | 109,317 |
https://mathoverflow.net/questions/236138 | 5 | Let $G$ be a connected finite simple graph with vertex set $V$, $F$ a finite set and let $\Delta(G)$ denote the degree of $G$, i.e. $\Delta(G)= \max\_{v\in V} \deg(v)$. We say that a coloring $\phi\colon V \to F$ is *asymmetrical* if $\text{Iso}(G,\phi) = \{Id\_V\}$ , where $\text{Iso}(G,\phi)$ denotes the set of bijec... | https://mathoverflow.net/users/44172 | Finite graph colorings without symmetries | I think $k+1$ always suffice. Here is an algorithm that produces such a coloring.
Suppose the set of colors is $\{0,\ldots,k\}$. First, pick a vertex, say $v$, and color it $0$. We will never use that color again. Now, repeat the following procedure, iteratively: choose a vertex $u$ which is colored but has at least... | 2 | https://mathoverflow.net/users/22377 | 236163 | 109,318 |
https://mathoverflow.net/questions/236164 | 1 | In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least
$$\big(\frac{1}{\pi}+o(1)\big)R\log R$$
zeroes in a disk $|s|<R$. Is there a reference and/or a proof of this statement that I can cite?
| https://mathoverflow.net/users/48554 | Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$ | I doubt that this is what he's claiming exactly, since if you position your disk in the strip $\Re s > 1$ then the count is zero. I realize now that most likely the disk is expanding (that is $R$ goes to infinity) in that case this follows just from the usual count of zeros which is known to be $(1/2\pi) T \log T + (\t... | 1 | https://mathoverflow.net/users/90326 | 236166 | 109,319 |
https://mathoverflow.net/questions/236156 | 4 | I asked this question on MSE [here](https://math.stackexchange.com/q/1720572/254733) some time ago, but I couldn't get an answer. There was a suggestion in the comments for a counterexample using a fat Cantor set, but I couldn't show a contradiction with the statement in my question.
Let $\Omega\subset \mathbb R^d$ (... | https://mathoverflow.net/users/75968 | $\forall g\in L^2(\Omega)$ exists $g_n\in H_0^1(\Omega)$ and $\epsilon>0$ s.t $g_n(x)\to g(x),\,a.e$ and $|g_n(x)|\leq |g(x)|+\epsilon$ | I just stumbled on the following fact in Conway's complex analysis book, vol. 2, Lemma 19.11.6: If $f\in H\_0^1$, then, after modification on a null set, $f(x,y)$ will be (in fact: absolutely) continuous as a function of $x$ for fixed $y$. (Since it's a complex analysis book, it's done for $d=2$ there, but the proof wo... | 2 | https://mathoverflow.net/users/48839 | 236167 | 109,320 |
https://mathoverflow.net/questions/236154 | 3 | In his book *Young Tableaux*, Fulton asserts, in Exercise 9.4.18 on p. 152, that the Schubert variety $\Omega\_{\lambda}$ is defined by the conditions $\text{dim}(V \cap F\_{n+i- \lambda\_{i}}) \geq i$ for those $i$ such that the coordinate $(i,\lambda\_{i})$ is an outside corner of the Young diagram $\lambda$. I take ... | https://mathoverflow.net/users/86315 | Schubert varieties and Young diagrams | All values of $(i,\lambda\_i)$ appear. The ones that aren't outside corners are redundant.
| 3 | https://mathoverflow.net/users/88133 | 236169 | 109,321 |
https://mathoverflow.net/questions/236175 | -2 | Due to the fact that $$\Gamma(x)\Gamma(1-x)\sin \pi x=\pi$$
is this necessarily true:
$$\Gamma(x)\Gamma(1-x)\sin(\Gamma(1+x)\Gamma(1-x)\sin(\cdots))=\pi$$
Is this at all used as a tactic to create different identities in general?
(The Reflection Property example was just to illustrate my point)
| https://mathoverflow.net/users/85704 | When are/ if recursive identities used? | As follows is what I assume you are asking. Taking
$$A = \Gamma(z)\Gamma(1-z)\sin(z\pi) = \pi$$
then rather trivially substituting $A$ for $\pi$
$$\Gamma(z)\Gamma(1-z)\sin(z\Gamma(z)\Gamma(1-z)\sin(\pi z)) = \pi$$
Continually making the substitution for $\pi$ on the inside of the $\sin$ with $A$ produces the ex... | 0 | https://mathoverflow.net/users/nan | 236178 | 109,325 |
https://mathoverflow.net/questions/236186 | 12 | For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum number that such any two points with distance less than that number have a unique geodesic length minimizer between them.
... | https://mathoverflow.net/users/58103 | how to define the injectivity radius of manifolds with boundary? | <http://arxiv.org/abs/math/0001108>
This paper by Schick is nicely written and contains coordinate-wise and coordinate-free definitions of bounded geometry for manifolds with boundary.
In particular, it says that in this case the condition of having injectivity radius bounded from below translates to the following ... | 8 | https://mathoverflow.net/users/44172 | 236194 | 109,328 |
https://mathoverflow.net/questions/236204 | 2 | This is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, of course), we can take the classical phase space with polarization as a high-dimensional Kahler manifold $K$ (A priori, o... | https://mathoverflow.net/users/69531 | Hessians on Kahler Manifolds | Williamson's theorem explains the normal forms of matrices under symplectic linear transformation. <http://www.ime.usp.br/~piccione/Downloads/LecturesIME.pdf> p. 23
| 4 | https://mathoverflow.net/users/13268 | 236205 | 109,332 |
https://mathoverflow.net/questions/236208 | 3 | Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation)
$$
p:S^2S^3\mathbb{C}^2\longrightarrow S^6\mathbb{C}^2
$$
is the 3-dimensional subspace generated by
$$
XZ-Y^2\, ,\quad XW-YZ\, ,\quad YW-Z^... | https://mathoverflow.net/users/22606 | How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic? | In the 3-dimensional space of quadrics through the twisted cubic curve consider the subset, parameterizing degenerate quadrics. An easy computation shows, these form a double smooth conic. So, the plane $P^2$ of quadrics comes with a distinguished conic, that identifies the corresponding 3-space with a symmetric square... | 5 | https://mathoverflow.net/users/4428 | 236209 | 109,333 |
https://mathoverflow.net/questions/236207 | 8 | Welcome octonions friends !
Long time ago when I travelled through octonion land, I conjectured that every $SO\_8$ element can be expressed as product $L\_a L\_b R\_c R\_d$ for unit octonions $a$, $b$, $c$, $d$. Is this true ?
It can be seen as generalization of the fact that $SO\_4$=$S^3 \otimes S^3$ i.e. every el... | https://mathoverflow.net/users/nan | Expressing $SO_8$ element as product of $L_u$ and $R_u$ for unit octonions $u$ | Your conjecture on octonions is false. In Conway and Smith's book *On Quaternions and Octonions* (A.K. Peters 2003), §8.5 theorem 9 on page 94, it is shown that the set of $L\_a L\_b R\_c$ for $a,b,c$ unit octonions is $20$-dimensional. So the set of $L\_a L\_b R\_c R\_d$ for $a,b,c,d$ unit octonions is at most $27$-di... | 9 | https://mathoverflow.net/users/17064 | 236213 | 109,335 |
https://mathoverflow.net/questions/236198 | 3 | Let $\pi$ denote a saturated set of weights. Let $S\_q(\pi)$ denote the associated generalised $q$-Schur algebra. I was wondering if the following claim is true:
Claim: The algebra $S\_q(\pi)$ is Koszul if and only if the decomposition numbers of $S\_q(\pi)$ are given by the associated Kazhdan-Lusztig polynomials.
... | https://mathoverflow.net/users/19113 | Is Koszulity equivalent to the Lusztig character formula holding? | Since Geordie already started the shameless self-promotion: I believe you can apply the results of section 1 of [Canonical bases and higher representation theory](https://arxiv.org/abs/1209.0051) to show that this will happen whenever the graded version of this $q$-Schur algebra (which you may need to believe [a conjec... | 2 | https://mathoverflow.net/users/66 | 236216 | 109,336 |
https://mathoverflow.net/questions/236215 | -7 | Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|\_{2} \leq 1/(1-\|A\|\_{2})$ hold true?
$\|A\|\_{2}$ denotes maximum eigenvalue of A.
If it is correct can somebody give me link to the proof for this inequality?
| https://mathoverflow.net/users/89565 | Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1? | Since $A$ is positive semidefinite, norm of $A$ is the same as spectral radius of $A$. Thus
$$\|(I-A)^{-1}\|=\|I+A+A^2+\dots\|\leqslant 1+\|A\|+\|A\|^2+\dots=\frac1{1-\|A\|}.$$
| 3 | https://mathoverflow.net/users/4312 | 236217 | 109,337 |
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