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https://mathoverflow.net/questions/234845 | 1 | Let $A\longrightarrow M$ and $B\longrightarrow N$ be [Lie algebroids](https://en.wikipedia.org/wiki/Lie_algebroid) with anchors $\#\_A$ and $\#\_B$, respectively.
A morphism of Lie algebroids is a morphism of vector bundles $\Phi:A\longrightarrow B$ covering let's say $\phi:M\longrightarrow N$ such that the induced ... | https://mathoverflow.net/users/80464 | Property of Lie algebroid morphism: $\#_B\circ \Phi=d\phi\circ \#_A$? | A vector is determined by its action on functions as a directional derivative. This is just the Lie derivative in degree $0$ mentioned at the end of $(iii)$. Together with the naturality $(ii)$ of $d$ and the definition of $d\phi$, you get $$(\#\_b\circ\Phi\_p)(\alpha)(f)=d\_B(f)\_{\phi(p)}(\Phi\alpha)=(\Phi^\*d\_Bf)\_... | 2 | https://mathoverflow.net/users/70808 | 234866 | 108,824 |
https://mathoverflow.net/questions/234833 | 7 | Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$.
Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in the power series expansion is less or equal than $n$. I am looking at the convex hull of class $S$. Any function there ... | https://mathoverflow.net/users/44316 | Convex Hull of univalent functions and Bieberbach Conjecture | By extremality, a function of the form $f(z)= z+\sum\_{n\geq 2} n a\_n z^n$ with $|a\_n|=1$ belongs to the convex hull of $S$ if and only if it belongs to $S$.
According to [wikipedia](https://en.wikipedia.org/wiki/De_Branges's_theorem), the only univalent such functions are those for which $a\_n= \omega^{n-1}$ for ... | 4 | https://mathoverflow.net/users/10265 | 234872 | 108,825 |
https://mathoverflow.net/questions/234869 | 11 | Let $A,B$ be two rational rotations:
$$ A = \left[\begin{array}{rcc} \frac{3}{5} & \frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{3}{5} & 0 \\
0 & 0 & 1 \end{array}\right]
\quad\text{ and }\quad
B = \left[\begin{array}{crc}
1 & 0 & 0 \\
0 & \frac{3}{5} & \frac{4}{5} \\
0 & -\frac{4}{5} & \frac{3}{5} \end{array}\right]
$$... | https://mathoverflow.net/users/1358 | Can two rational rotations $F_2 = \langle A, B \rangle \to SO(3)$ efficiently approximate the $3 \times 3$ identity matrix? | Setting $a\_0=A^7, b\_0=B^7$ and then $a\_{n+1}=[b\_n^{-1},a\_n], b\_{n+1}=[a\_n,b\_n]$ seems very efficient. The length grows like $C \alpha^n$ with $\alpha= \frac{3 + \sqrt{17}}2$ and the operator norm distance to the identity matrix gets squared in each step, since
$$\|1 - [a,b]\| \leq 2 \|1-a\| \cdot \|1-b\|.$$ I u... | 19 | https://mathoverflow.net/users/8176 | 234873 | 108,826 |
https://mathoverflow.net/questions/234878 | 1 | Let $(M,\omega)$ be a sympletic manifold and $\{ \cdot, \cdot \}$ the corresponding Poisson-bracket. Assuming $M$ is completely integrable w.r.t $f=f\_1$, so we find $n = \frac{1}{2}\dim M$ functions $f\_1, \dots , f\_n \colon M \to \mathbb{R}$ such that they are functionally independent and mutually Poisson-commute on... | https://mathoverflow.net/users/75382 | Lagrangian foliation | Yes since functions which Poisson commute are constant on one another's Hamiltonian flows.
| 3 | https://mathoverflow.net/users/13268 | 234880 | 108,830 |
https://mathoverflow.net/questions/234868 | -1 | **Inequality**
What values of $n$ satisfy the following inequality?
>
> $$2(n-2) < Ap\_n\prod\_{i=3}^n \left(\frac{p\_i-2}{p\_i}\right)$$
>
>
>
$p$ are prime numbers and the notation $p\_i$ indicates the $i$th prime number.
A comes from the relationship $p\_n = 6A + r : 0 \leq r < 6$
(This is a slight va... | https://mathoverflow.net/users/41928 | Find the values of $n$ that satisfy this inequality involving a product over prime numbers | We already have $p\_n \gt 2(n-2)$ for $n \gt 0$ by elementary (non-analytic) methods. Similar methods can establish the inequality for sufficiently large n.
Suppose the inequality $1 \lt A \prod\_{i=3}^n \frac{p\_i - 2}{p\_i}$ holds for positive integers $n=j$ and $j+1$. I will show the inequality holds for $j+2$, l... | 3 | https://mathoverflow.net/users/3402 | 234892 | 108,835 |
https://mathoverflow.net/questions/234784 | 1 | I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan.
I have a problem understanding one step of the proof needed to get a $ L^\infty$ estimate for the solution $u$. The step goes like this:
$$ \int f(x)u^{a+kp+1}... | https://mathoverflow.net/users/89594 | Nonlinear elliptic problem involving the p-laplacian, Hölder inequality | Some typing mistakes are detected, especially in these two lines. First, it should be mentioned that at this step, we have to consider the case where $p-1< \alpha.$ Take $t=\frac{p^\*qp}{\alpha(q-p^\*)+q+(p-1)p^\*}.$ One can easily check that $\frac{t}{p}>1$ and $t<p^\*.$ Then write in the integral $u^{\alpha+kp+1}$ as... | 2 | https://mathoverflow.net/users/75814 | 234896 | 108,837 |
https://mathoverflow.net/questions/234925 | 4 | I am considering the Cauchy IVP for the evolution equation
$$u\_t + \Psi u =0$$
where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$.
The corresponding solution semigroup is
$$u = \mathcal{F}^{-1}\left( e^{-\widehat{\Psi} t} \mathcal{F}(f) \right)$$
What ar... | https://mathoverflow.net/users/64620 | Strongly continuous semigroups and symbols of pseudo differential operators | The answer may depend upon which functional space you are dealing with. But since you insist upon the symbol and the Fourier transform, let me assume that you have $L^2({\mathbb R}^d)$ in mind. Because $\cal F$ is an isometry, it is equivalent to ask whether $t\mapsto[\xi\mapsto\exp(-t\hat\Psi(\xi))]$ is a strongly con... | 5 | https://mathoverflow.net/users/8799 | 234933 | 108,846 |
https://mathoverflow.net/questions/234931 | 4 | Given a topological space $C$ and points $c\_0, c1\in C$ we say that a topological space $(X,\tau)$ is $(C,c\_0,c\_1)$-connected if and only if for all $x,y\in X$ there is $f:C\to X$ continuous with $f(c\_0) = x$ and $f(c\_1)=y$.
(In this language, path-connectedness equals $([0,1],0,1)$-connectedness.)
What is an ... | https://mathoverflow.net/users/8628 | Stronger form of connectedness than path-connectedness | Let $C$ be the space nº74 ("double origin topology") in Steen & Seebach's *Counterexamples of Topology*, chosen because it is the only one listed there that is T2 and path-connected but not T3:
>
> $C$ consists of the set of points of the plane $\mathbb{R}^2$ together with an additional point $0^\*$. Neighborhoods ... | 5 | https://mathoverflow.net/users/17064 | 234934 | 108,847 |
https://mathoverflow.net/questions/234772 | 2 | This question is related to this question: "[Solutions of equations characterizing a complex structure.](https://mathoverflow.net/questions/230574/solutions-of-equations-characterizing-a-complex-structure)" Where, here we suppose the Euclidean space instead of Sphere and the following equations happen if and only if th... | https://mathoverflow.net/users/86401 | Existence of non-constant solutions for this equations | The answer is "Yes, there are many solutions with $u$ and $v$ non-constant."
Here is one way to understand this problem: First, set $z = u+iv$ and note that the above equations become the complex system
$$
\frac{\partial z}{\partial x^\ell} + z\,\frac{\partial z}{\partial y^\ell} = 0,
\qquad \ell = 1,\ldots,n.
\tag1... | 2 | https://mathoverflow.net/users/13972 | 234940 | 108,849 |
https://mathoverflow.net/questions/231390 | 7 | **My Motivation:** I'm having a hard time following the description of the topology in the Satake Compactification of locally symmetric spaces. The group theory is something I'm finding a bit tricky to follow, so I'd prefer answers in terms of explicit matrix representations, though a concrete descriptions of cylindric... | https://mathoverflow.net/users/4181 | An explicit description of neighborhoods of the rank 2 boundary in the Satake Compactification of $\mathbf{A}_2$ | This answer is incomplete but I'll try to finish it later.
First note that you can't express the answer only using $Y$, even in the $g=1$ case. If the imaginary part of $\tau$ goes to $0$, you do not know anytihng about what $j(\tau)$ is doing.
I think you want to work with the induced quadratic form on pairs $(v,w... | 3 | https://mathoverflow.net/users/18060 | 234948 | 108,854 |
https://mathoverflow.net/questions/234946 | 11 | Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties:
>
> Is every infinite dimensional Banach space stable?
>
>
>
Is this question still open?
| https://mathoverflow.net/users/12156 | Do non-stable Banach spaces exist? | No, see the following paper of Gowers
<https://blms.oxfordjournals.org/content/26/6/523.full.pdf>
| 13 | https://mathoverflow.net/users/3675 | 234953 | 108,856 |
https://mathoverflow.net/questions/234950 | -1 | Let $\mathbb{K}$ be a quadratic extension of $\mathbb{Q}$ and $\mathbb{L}$ be a cyclic extension of $\mathbb{Q}$ of odd degree. Given a rational $r\neq 0$, does there always exist $k\in \mathbb{K}^\*$ and $l \in \mathbb{L}^\*$ such that
$$N\_{\mathbb{K}}(k)= rN\_{\mathbb{L}}(l) \,\,?$$
If not, would this kind of equa... | https://mathoverflow.net/users/89704 | Equation with norms of cyclic extensions of coprime degrees | Yes, this follows from the fact that the degrees of $\mathbb{K}$ and $\mathbb{L}$ are coprime.
The result follows from a simple application of Bezout's identity and the fact that, for an extension $k$ of $\mathbb{Q}$ of degree $n$ and $x \in \mathbb{Q}$, we have $N\_k(x) = x^n$.
| 2 | https://mathoverflow.net/users/5101 | 234956 | 108,858 |
https://mathoverflow.net/questions/234902 | 7 | In "Reminiscences of Grothendieck and his school" Luc Illusie says:
"*I heard from Deligne that there were problems in some parts. (of Monique Hakim thesis).*
Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).
My question is for a refer... | https://mathoverflow.net/users/83957 | Problems in some parts of Monique Hakim thesis? | The [published version of the paper](http://www.ams.org/notices/201009/rtx100901106p.pdf) (it is free), yet apparently not all versions in circulation, has a footnote on that sentence:
>
> (13) Added in April 2010: Deligne doesn’t think there was
> anything wrong but remembers that the objects she
> defined over... | 13 | https://mathoverflow.net/users/nan | 234963 | 108,864 |
https://mathoverflow.net/questions/234955 | 4 | I am reading a paper on coding theory, and it uses a statement, which was claimed to be a reformulation of Maschke's Theorem. But I felt that was false...
Let's say $\mathcal(V):=\mathcal{F}\_2^n$ is the big ambient space. $\sigma \in S\_n$ is a permutation of odd prime order $p$. $\sigma$ acts on our ambient space b... | https://mathoverflow.net/users/89027 | Is the direct sum in Maschke's Theorem an orthogonal decomposition? | The decompositions are the same because actually there is only one $H$-invariant complement to $\mathcal{V}(\sigma)$ in $V$. Consider $e = \sum\_{g\in H} g$ as operator on $V$. Then $v\in \mathcal{V}(\sigma) = \operatorname{Fix}\_V(H)$ iff $ev=v$, as is easy to show. So if $W$ is any complement to $\mathcal{V}(\sigma)$... | 4 | https://mathoverflow.net/users/10266 | 234964 | 108,865 |
https://mathoverflow.net/questions/234965 | 7 | My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any surjective endomorphism of a finitely generated module for a Noetherian ring is injective.
Let $R$ be a unital ring, not necess... | https://mathoverflow.net/users/7709 | Hopfian modules | Let $R$ be Shepherdson's ring, that is, a domain for which the two by two matrix ring, $M\_2 R$, contains
elements $x$ and $y$ such that $xy = 1$, but $yx \neq 1$. With right module $M =
R$, every homomorphism $M \to M$ is given by left multiplication by an
element of $R$, so every nonzero endomorphism of $M$ is one to... | 7 | https://mathoverflow.net/users/42278 | 234975 | 108,869 |
https://mathoverflow.net/questions/234979 | 1 | I need to find the final state (i.e. the state at $t\to+\infty$) of the following ODE system:
$$\begin{eqnarray\*}\frac{dA}{dt}&=&-aAX\\
\frac{dB}{dt}&=&-bBX\\
\frac{dX}{dt}&=&-(aA+bB)X\end{eqnarray\*}$$
where $A$, $B$, and $X$ are non-negative variables (with known initial values at $t=0$: $A(0)\ge 0$, $B(0)\ge 0$, $X... | https://mathoverflow.net/users/89716 | Is it possible to find the final state of the bilinear ODE system? | EDIT: It's obvious there can be no equilibria with all $A,B,X > 0$, and $A,B,X$ are decreasing. The only possibilities are a limit where $X=0$ or a limit where $A=B=0$. As noted, $A+B-X$ is invariant, so the first occurs, with $A(\infty) + B(\infty) = A(0) + B(0) - X(0)$, if $A(0) + B(0) - X(0) \ge 0$ while the latter ... | 1 | https://mathoverflow.net/users/13650 | 234981 | 108,871 |
https://mathoverflow.net/questions/234985 | 8 | **Quillen's Theorem A** is formulated as follows:
Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category
$F/y$ is contractible. Then $F$ induces a weak equivalence between the nerves $N(X)\to N(Y)$.
I am not a topologist, but it seems can prove the following statement, which I ... | https://mathoverflow.net/users/89514 | Relative version of Quillen's theorem A | The condition that appears in the assumption of what you call "Relative Theorem A" was introduced by Grothendieck in *Pursuing Stacks*. It is a part of the definition of a [basic localizer](https://ncatlab.org/nlab/show/basic+localizer), i.e. a class of functors between small categories that behaves like the class of w... | 10 | https://mathoverflow.net/users/12547 | 234987 | 108,873 |
https://mathoverflow.net/questions/234922 | 4 | What is known about existence of lattices in reductive Lie groups? The best results I know about existence of lattices in connected Lie groups are either about semisimple groups or nilpotent groups but never about "mixed" cases like reductive groups.
Let me be a bit more precise. Let G be a reductive Lie group. For m... | https://mathoverflow.net/users/89687 | Existence of lattices in reductive Lie groups | My recollection is that not every reductive group has a lattice, and an example is provided by a slight modification of your construction. (I think this is essentially in an old paper of Starkov.) Let $H$ be a simple Lie group of large real rank (maybe greater than 1 is enough) whose center is $\mathbb{Z}$, choose an i... | 2 | https://mathoverflow.net/users/68305 | 234996 | 108,876 |
https://mathoverflow.net/questions/234994 | -1 | This problem has probably been solved somewhere but I could not find it. We have $n$ Bernoulli random trials $X\_i$ with different occurrence probabilities, $\mathrm{Pr}[X\_i=1]=p\_i>p\_{\min}>0$ for $i=1,2,\dots,n$ and some constant $p\_{\min}$ what is the probability of having even number of ones when $n$ is very lar... | https://mathoverflow.net/users/82990 | Equal probability of having even/odd number of ones in many Bernoulli trials with different probabilities? | The probability that an even number of trials succeed is exactly
$$ \frac12\biggl( 1 + \prod\_{i=1}^n (1-2p\_i)\Biggr).$$
This is a standard elementary application of probability generating functions.
| 2 | https://mathoverflow.net/users/9025 | 234999 | 108,877 |
https://mathoverflow.net/questions/234881 | 8 | *This is a more carefully worded version of [this](https://math.stackexchange.com/questions/1719656/factor-a-square-matrix-bf-a-into-product-bf-b-bf-c-where-bf-c) question, here tailored to professional mathematicians.*
Consider a matrix ${\bf A}\in{\bf M}\_{n\times n}({\mathbb R})$ with possibly positive, negative a... | https://mathoverflow.net/users/89654 | Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative entries and ${\bf B}$ has few non-zero entries | Let $v\_1, \dots, v\_n$ denote the rows of $A$.
Let $u$ be a vector with all positive entries that is not a linear combination of $v\_2, \dots, v\_n$. Then we may write $v\_1 = c u + a\_2 v\_2 + \dots a\_n v\_n$ for some scalars $c, a\_2, \dots, a\_n$.
Now choose $\epsilon$ small enough that for all $j$ from $2$ t... | 8 | https://mathoverflow.net/users/18060 | 235001 | 108,878 |
https://mathoverflow.net/questions/234930 | 7 | A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and similarly for the higher Cantor space $2^\kappa$, with the initial segment topology, as defined [here](https://mathoverflow... | https://mathoverflow.net/users/2415 | References for higher descriptive set theory surveys | I would like suggest the following three course notes given by Sy Friedman:
>
> 1) Invariant descriptive set theory (on classical and generalised Baire space),
>
>
> 2) Cardinal characteristics of the uncountable,
>
>
> 3) The higher descriptive set theory of isomorphism.
>
>
>
Also the monograph
>
> G... | 6 | https://mathoverflow.net/users/11115 | 235024 | 108,887 |
https://mathoverflow.net/questions/235025 | -2 | When proof reading and correcting a mathematical text, I sometimes see people use special notation symbols in the margin to indicate correction, deletion, replacement and so on. Is there any standard for the correction symbols used?
| https://mathoverflow.net/users/82568 | Correction symbols used for mathematical texts | **Chicago Manual of Style**, 12 edition, p. 71, Proofreader's Marks.
| 4 | https://mathoverflow.net/users/13268 | 235027 | 108,889 |
https://mathoverflow.net/questions/235015 | 7 | Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the corresponding map in the rational cohomology rings.
Andrew Ranicki claims in his paper "On the construction and topologi... | https://mathoverflow.net/users/89741 | Source request for $H^*(B\mathrm{TOP},\mathbb{Q}) \cong H^*(BO,\mathbb{Q})$ | 1. The canonical map $BPL\rightarrow BTOP$ is a rational homotopy equivalence by works of Thom, Novikov, Kirby-Siebenmann and others. In fact the homotopy fiber $TOP/PL$ is a $K(\mathbb{Z}/2,3)$. A nice reference is the survey "Piecewise Linear Structures on Topological Manifolds" by Rudyak.
2. The canonical map $BO\ri... | 9 | https://mathoverflow.net/users/27816 | 235030 | 108,892 |
https://mathoverflow.net/questions/234943 | 2 | Suppose $p\geq 5$ is a prime, and $C$ a genus-one curve, defined over $\mathbf{Q}$. Is there always an extension $K/\mathbf{Q}\_{p}$ whose degree divides a power of $6$, so that
$C(K)$ is not empty?
(I posted that question before, but it was badly formulated. I hope there is no mistake this time.)
| https://mathoverflow.net/users/70751 | Local nontriviality of genus-one curves over extensions of degree dividing $6^n$ | I am just posting my comment above as an answer. As JSE points out, surely there are more methodical approaches to the cohomology groups $H^1\_{ \acute{e}t }(\text{Spec}(\mathbb{Q}\_p),E)$. However, already the Brauer group of a field gives many nontrivial elements of these cohomology groups. Let $r$ be an integer. Let... | 5 | https://mathoverflow.net/users/13265 | 235037 | 108,896 |
https://mathoverflow.net/questions/235034 | 6 | *Notation:* A word $w$ on the alphabet $A=\{a,b\}$ having $2p$ letters can be viewed as a word $w'$ having $p$ letters on the alphabet $A'=A^2$. I denote by $\beta(w)$ the number of occurences of the letter "$bb$" in the word $w'$. For example $\beta(aaab)=0$, $\beta(abbb)=1$, $\beta(abba)=0$.
Let $T$ be the rotation... | https://mathoverflow.net/users/21339 | Question about a certain coding of rotations | This is true for every irrational $\theta$; the question can be rephrased in terms of Sturmian sequences. Given a sequence $z \in \{a,b\}^\mathbb{Z}$ and indices $i<j$, let $z\_{[i,j]} \in \{a,b\}^{j-i+1}$ be the subword of $z$ given by $z\_i \cdots z\_j$. Let $c(z\_{[i,j]})$ be the number of times the symbol $b$ appea... | 6 | https://mathoverflow.net/users/5701 | 235044 | 108,898 |
https://mathoverflow.net/questions/235032 | 0 | During my research I came across the following problem:
I need to find a root of the following function:
$$\Gamma\_{N}(x) = \sum\limits\_{i=0}^{M}\left(\frac{\sum\limits\_{n=0}^{n\_F}n\ \alpha\_{i,n} x^n}{\sum\limits\_{n=0}^{n\_F}\alpha\_{i,n} x^n}\right)- N,$$
where $\alpha\_{i,n}$ are positive coefficients such tha... | https://mathoverflow.net/users/57699 | Root of a special rational function with positive coefficients | More elementary argument than in the answer by Alexandre Eremenko:
it suffices to prove that if $a\_k\geqslant 0$ for all $k$ and $x>y>0$, then $$\frac{\sum na\_nx^n}{\sum a\_n x^n}\geqslant \frac{\sum na\_ny^n}{\sum a\_n y^n}.$$
Multiplying by common denominator this reduces to $$\sum\_{n,k}a\_na\_k(n-k)(x^ny^k-x^ky... | 4 | https://mathoverflow.net/users/4312 | 235055 | 108,905 |
https://mathoverflow.net/questions/235057 | 2 | Is there a necessary and sufficient condition for a linearly ordered topological space to be pseudo-metrizable? (by a pseudo-metric, I mean a map $X\times X\rightarrow\mathbb{R}$ in which all the metric axioms are satisfied except that the distance between two distinct points may be zero)
A negative answer is, of co... | https://mathoverflow.net/users/80352 | The pseudo-metric and linear orders | As Joel Hamkins observed above, a pseudometric may generate the topology of a linearly ordered topological space only if it is a metric.
A necessary and sufficient condition for a linearly ordered topological space $X$ to be metrizable is that is has a $G\_\delta$-diagonal (that is, the set $\Delta=\{(x,x):x\in X\}$... | 2 | https://mathoverflow.net/users/48481 | 235086 | 108,917 |
https://mathoverflow.net/questions/235084 | 5 | Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another example (with parameters) is "$S$ is a nonstationary subset of $\kappa$."
Question: Is there an analogue of the second exampl... | https://mathoverflow.net/users/11145 | A button for individual reals | The way you've stated it, you haven't said that $\varphi$ isn't already true, and so technically any tautological statement would work. But I assume that you want $\varphi(r,a)$ to start out false in the original model $V$. This would be an *unpushed* button, which hasn't yet been pushed.
It is consistent with ZFC t... | 7 | https://mathoverflow.net/users/1946 | 235089 | 108,918 |
https://mathoverflow.net/questions/235080 | 5 | For $n \ge 2$, there is at least one binary DeBruijn sequence beginning with $n$ zeros followed by $n$ ones. Is there a straightforward way to construct such a sequence for each $n \ge 2$? Examples:
$n=2: 0011$
$n=3: 00011101$
$n=4: 0000111101100101$
$n=5: 00000111110111001101011000101001$
| https://mathoverflow.net/users/61426 | How to construct particular De Bruijn sequences | The answer is the construction of the De Bruijn sequence by concatenating certain Lyndon words as indicated [here](https://en.wikipedia.org/wiki/Lyndon_word) or, in this [article](http://emoreno.uai.cl/publicaciones/PREPRINTS/preprint-WORDS03-Lyndon_words_and_de_Bruijn_sequences_in_a_subshift_of_finite_type.pdf); that ... | 4 | https://mathoverflow.net/users/31310 | 235095 | 108,921 |
https://mathoverflow.net/questions/235094 | 4 | I asked this [on Math.StackExchange](https://math.stackexchange.com/questions/1693831/polar-forms-of-algebraic-curves-surfaces), but received no response, so trying here ...
A paper I'm reading says the following ...
>
> With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equatio... | https://mathoverflow.net/users/50265 | Polars of algebraic curves and surfaces | Geometrically, one first meets polar hypersurfaces when studying tangent lines from a point to a hypersurface. In fact, this concept generalizes the classical polarity of a point with respect to a conic.
More precisely, let $X \subset \mathbb{P}^k$ be a hypersurface of degree $n$ given by the zero locus of a homogene... | 8 | https://mathoverflow.net/users/7460 | 235096 | 108,922 |
https://mathoverflow.net/questions/235105 | 0 | Is every (left) finitely generated projective modules over the matrix ring $M\_n(\mathbb{C})$ isomorphic to a trivial module? Is there a good reference to look at this problem?
Apologies for asking what is likely a very simple question - note it is really about the isomorphism classes, not K-theory.
| https://mathoverflow.net/users/29625 | Finitely generated projective modules over matrix rings | The ring $\mathbb{M}\_n(\mathbb{C})$ is a semisimple ring, and so *every* module is a sum of simple modules, and is projective. For this ring, there is only one simple module $S$, up to isomorphism. Thus, every finitely generated module is $S^{(k)}$ for some $k\in \mathbb{N}$. (The module $S$ is isomorphic to an $n\tim... | 5 | https://mathoverflow.net/users/3199 | 235106 | 108,925 |
https://mathoverflow.net/questions/235104 | 6 | Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta\_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to Laplace and vice versa.
For example, what I know, for $G$ being the Heisenberg group $H^3= \mathbb C \times \mathbb R$, the... | https://mathoverflow.net/users/84558 | Difference between the Laplacian and the sub-Laplacian of a Lie group | As Sebastian Goette explained in his comment, the sub-Laplacian $\Delta\_{sub}$ depends in general from an additional structure. And so does the Laplace-Beltrami $\Delta$ that you use to compute the difference. Let me elaborate.
**SUB-LAPLACIANS**
On a given smooth manifold $M$, we consider a sub-Riemannian structu... | 8 | https://mathoverflow.net/users/13915 | 235110 | 108,926 |
https://mathoverflow.net/questions/220201 | 7 | [Here](http://www.ams.org/journals/proc/1996-124-04/S0002-9939-96-03199-1/S0002-9939-96-03199-1.pdf) Y. Sekine introduces a one-parameter family of finite quantum groups of dimension $2n^2$. Let $n\geq 3$ be fixed and $\zeta=e^{2\pi i/n}$. Set
$$\mathcal{B}\_n=\mathbb{Z}\_n\times\mathbb{Z}\_n=\{(i,j):i,j=0,1,\dots,n-1\... | https://mathoverflow.net/users/35482 | The Irreducible Representations of the Sekine Quantum Groups | Thank you to [Zahlendreher](https://mathoverflow.net/users/33854/zahlendreher) and [Sébastien Palcoux](https://mathoverflow.net/users/34538/s%C3%A9bastien-palcoux) for the help.
Zahlendereher led me to the states
With Sébastien's help I was happy that brute force would reveal the matrix elements of the corepresent... | 1 | https://mathoverflow.net/users/35482 | 235111 | 108,927 |
https://mathoverflow.net/questions/234332 | 6 | Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int\_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\mathbb{G})$ given by
$$a\star\_A b=b\_{(2)}\int\_{\mathbb{G}}S\left(b\_{(1)}\right)a,$$
where
$$\Delta(b)=\sum b\_{... | https://mathoverflow.net/users/35482 | Quantum group representations from (convolution) matrix units? | As explained [here](https://mathoverflow.net/a/235111/35482), with help from Sébastien I was able to show that, in the [context of what I was working on](https://mathoverflow.net/questions/220201/the-irreducible-representations-of-the-sekine-quantum-groups), the one-dimensional minimal central projections satisfied
$... | 1 | https://mathoverflow.net/users/35482 | 235112 | 108,928 |
https://mathoverflow.net/questions/182672 | 4 | Let $G$ be a reductive group over a $p$-adic field. My understanding is that the Plancherel measure on $G$ is a measure on the unitary dual $\hat{G}$.
But at the same time, for example, in his famous 1990 Annals paper, Shahidi defines a Plancherel measure using induced representations and intertwining operators.
Ar... | https://mathoverflow.net/users/32746 | A question on Plancherel measure for $p$-adic group | They are related but not the same. The Pleancherel measure is strictly speaking a measure on $\hat{G}$. If one were to parametrize $\hat{G}$, say by an integral on the real line, then one could get the Plancherel measure to be given by density functions. In other words, there exists some function $\mu$ such that the Pl... | 2 | https://mathoverflow.net/users/40832 | 235125 | 108,934 |
https://mathoverflow.net/questions/235033 | 7 | Let $\mathrm{MGL}$ be the $\mathbb{P}^1$-ring spectrum over a field $k$ representing algebraic cobordism. Suppose, for simplicity, that $k$ is of characteristic 0. Let $H\mathbb{Z}$ be the motivic Eilenberg-Maclane spectrum. My question is:
>
> Can $\mathrm{MGL}$ be given the structure of an $H\mathbb{Z}$-module?
>... | https://mathoverflow.net/users/39193 | Is $MGL$ an $H\mathbb{Z}$-algebra? | $MGL$ does not admit a structure of $H\mathbb Z$-module. There are many ways to prove this. As Sean said in the comments, if it were true over $\mathbb C$, topological realization would imply that $MU$ is an $H\mathbb Z^{top}$-module, which is false. By [rigidity](http://folk.uio.no/paularne/rigidityma.pdf) this takes ... | 10 | https://mathoverflow.net/users/20233 | 235127 | 108,935 |
https://mathoverflow.net/questions/235133 | 2 | Let $G=(V,E)$ be a directed graph. For $v\in V$ set $\text{In}(v)=\{x\in V: (x,v)\in E\}$.
Is it possible to find a partition $P\_1,P\_2,P\_3$ of $V$ such that for every $P\_i$ and every vertex $v\in P\_i$ we have $$|\text{In}(v)\cap P\_i| \leq |\text{In}(v)\cap(V\setminus P\_i)|$$?
| https://mathoverflow.net/users/8628 | Partitioning finite directed graphs into 3 "incoming-sparse" sets | Suppose you have a edge (A,B), and B has at most two incoming edges. Then A and B must get different colors in order for the inequality to be preserved. Now have C get an edge from each of A and B. With no other incoming edges to C or B, this "mini-tournament" must get three colors.
Now add D, E, F so that B and C le... | 1 | https://mathoverflow.net/users/3402 | 235142 | 108,941 |
https://mathoverflow.net/questions/235138 | 16 | It is well-known, that there are a lot of applications of classical Hopf algebras in QFT, e.g. Connes-Kreimer renormalization, Birkhoff decomposition, Zimmermann formula, properties of Rota-Baxter algebras, Hochschild cohomology, Cartier-Quillen cohomology, motivic Galois theory etc.
But all these structures are bas... | https://mathoverflow.net/users/75934 | Braided Hopf algebras and Quantum Field Theories | Some particular braided Hopf algebras known as [Nichols algebras](https://en.wikipedia.org/wiki/Nichols_algebra) are useful in conformal field theories. Here you have some references:
* Semikhatov, A. M.; Tipunin, I. Yu. Logarithmic $\widehat{s\ell}(2)$ CFT models from Nichols algebras: I. J. Phys. A 46 (2013), no. 4... | 7 | https://mathoverflow.net/users/17845 | 235147 | 108,944 |
https://mathoverflow.net/questions/235146 | 11 | The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains
$$F(k)=\frac {4\pi}{k} \int\_0^\infty \sin(kr) dr.$$
The standard way to obtain a sensible value is to multiple the integrand by $f(\alpha,r)=e^{-\alpha r}$ and after doing the integral,... | https://mathoverflow.net/users/76815 | Is the regularization of a Fourier transform unique? | Yes, the answer is unique. What this "regularization" is doing is computing the Fourier Transform in the sense of distributions.
Editing to add (see comment of Christian Remling below): For your precise question, the hypotheses you propose for the regularization are too weak: it's not enough that $f(\alpha,r)$ conver... | 15 | https://mathoverflow.net/users/327 | 235148 | 108,945 |
https://mathoverflow.net/questions/235162 | -1 | Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module
and $E$ be an injective $R$-module. When $Hom(M,E)$ is injective?
Thanks.
| https://mathoverflow.net/users/49325 | When Hom(M,E) is injective? | 1. $Hom(K,Hom(M,E))\cong Hom(K\otimes M,E)$
2. $F$ is flat iff $F\otimes -$ is exact.
3. Let $E$ be [injective cogenerator](https://math.stackexchange.com/questions/229168/injective-cogenerators-in-the-category-of-modules-over-a-noetherian-ring). Then $0 \longrightarrow X \longrightarrow Y \longrightarrow Z \longrighta... | 0 | https://mathoverflow.net/users/47763 | 235172 | 108,955 |
https://mathoverflow.net/questions/235152 | 3 | Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that for every $f\in\mathbb{C}[X]$ there exists $k\in\mathbb{N}$ such that $V^k(f) = 0$, does it necessarily follow that $V$... | https://mathoverflow.net/users/nan | Locally nilpotent algebraic section of tangent bundle is complete? | Yes, it is complete. The flow is given by the operator $e^{tV}$ acting on the coordinate functions $x^i$. This is defined globally because on Zariski open sets the operator is a unipotent linear transformation, so given by polynomials in $t$.
| 1 | https://mathoverflow.net/users/13268 | 235174 | 108,956 |
https://mathoverflow.net/questions/235114 | 3 | Let $G$ be a group acting on a set $\Omega$ faithfully. Then 2-closure of $G$ denoted by
$G^{(2)}$ is the largest subgroup of the symmetric group of $\Omega$ with
the same orbits as $G$ on $\Omega\times\Omega$. Clearly $G\leq G^{(2)}$. If $G=G^{(2)}$ then $G$ is called 2-closed. I am collecting some group-theoretic pro... | https://mathoverflow.net/users/27831 | 2-closure of a permutation group | Let's call an abstract group $2$-closed if all of its faithful permutation representations are $2$-closed.
You observed that cyclic groups are $2$-closed.
Also a group $G$ for which the only faithful permutation representation is the regular representation must be $2$-closed, because in any faithful action there mu... | 3 | https://mathoverflow.net/users/35840 | 235176 | 108,957 |
https://mathoverflow.net/questions/235170 | 1 | Given a Segre product $\mathbb P^m \times \mathbb P^n$, or more generally $\mathbb P^{m\_1}\times\cdots\times\mathbb P^{m\_n}$, is there a characterization in terms of $m$ and $n$, or the $m\_i$, for the Segre product to be Gorenstein?
Similar question for $\nu\_d(\mathbb P^n)$
Added: I am interested in the case wh... | https://mathoverflow.net/users/82465 | When are Segre- and Veronese embeddings Gorenstein? | I assume that you mean arithmetically Gorenstein, i.e. the cone over the variety is Gorenstein. Then:
$\bullet$ The Veronese variety $\nu\_d(\mathbb{P}^n)\subset \mathbb{P}^N$ is always arithmetically Cohen-Macaulay; the extra condition you need is $\omega \_{\mathbb{P}^n}\cong \nu\_d^\*\mathcal{O}\_{\mathbb{P}^N}(\e... | 9 | https://mathoverflow.net/users/40297 | 235179 | 108,959 |
https://mathoverflow.net/questions/235178 | 0 | Also asked here: <https://math.stackexchange.com/questions/1725787/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz>
A reference on totally geodesic submanifold (TGS):
<http://www.map.mpim-bonn.mpg.de/Totally_geodesic_submanifold>
Let "|" stand for restricted metric. Let $(S,g|)$ b... | https://mathoverflow.net/users/35936 | When are the minimizing geodesics of a totally geodesic submanifold also minimizing in the underlying manifold? | The totally geodesic submanifold might not be complete. Take the usual metric on the upper half of Euclidean 3-space, and some generic metric on the lower half, so that they agree where they meet to infinite order. There won't be any totally geodesic surfaces in a generic metric. So the flat planes in the upper half wi... | 4 | https://mathoverflow.net/users/13268 | 235183 | 108,961 |
https://mathoverflow.net/questions/235180 | 1 | By a *nilmanifold* I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group $G$ by the left action of a maximal lattice , i.e. a discrete cocompact subgroup.
Suppose we have two maximal lattices $\Gamma\_1$ and $\Gamma\_2$ of $G$. What can be said about the nilmanifolds $\G... | https://mathoverflow.net/users/51380 | Two nilmanifolds of the same Lie group | The manifold $\Gamma\_1/G$ is homeomorphic to $\Gamma\_2/G$ implies that $\Gamma\_1$ is isomorphic to $\Gamma\_2$ since $G$ is $1$-connected and $\Gamma\_1,\Gamma\_2$ are discrete, thus $\pi\_1(\Gamma\_1/G)=\Gamma\_1$ and two homeomorphic manifolds have isomorphic fundamental groups. This implies that $\Gamma\_1$ is is... | 7 | https://mathoverflow.net/users/80891 | 235185 | 108,962 |
https://mathoverflow.net/questions/235169 | 2 | Let $G$ be a compact Lie group and let $\mathcal{P}\_G$ denote the family of proper subgroups of $G$. The universal space for the family $\mathcal{P}\_G$ is a cofibrant $G$-space which does not have $G$-fixed points and such that for every proper subgroup $H<G$, the fixed point space $(E\mathcal{P}\_G)^H$ is contractib... | https://mathoverflow.net/users/88384 | Universal space for the family of subgroups of a finite cyclic group | Let $X$ be a space with an action of $G$ such that $X^G=\emptyset$, and for every proper subgroup $H$, $X^H\ne \emptyset$. Then the infinite join of $X$ with itself is a universal space for the family of proper subgroups. This follows from the following facts: fixed points commute with join, infinite join of non-empty ... | 6 | https://mathoverflow.net/users/6668 | 235187 | 108,963 |
https://mathoverflow.net/questions/235194 | 1 | Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$.
Let $E$ be a norm closed convex subset of positive operators in $K(H)$ and let $a$ be a non-zero positive compact operator where $a\notin E$.
Q: Is there any vector $\zeta\in H$ which separa... | https://mathoverflow.net/users/84390 | A point-wise separation Hahn-Banach theorem in C*-algebras | $2 \times 2$ counterexample, $E = \left\{\left[\matrix{\lambda& 0\cr 0&\lambda}\right]: \lambda \geq 0\right\}$ and $A = \left[\matrix{1&1\cr 1&1}\right]$. Then for any nonzero $\zeta$ we have $\{\langle B\zeta,\zeta\rangle: B \in E\} = [0,\infty)$, so no $\zeta$ can separate.
The general idea is that you can separat... | 5 | https://mathoverflow.net/users/23141 | 235196 | 108,965 |
https://mathoverflow.net/questions/122199 | 16 | Is it true that every finitely generated infinite simple group has
exponential (word-)growth?
Remark: As Mark Sapir has pointed out, the question whether
every finitely generated group of subexponential growth is even residually finite
has been answered in the negative in
Anna Erschler. Not residually finite groups... | https://mathoverflow.net/users/28104 | Is it true that every f.g. infinite simple group has exponential growth? | No, there exists a finitely generated infinite simple group of intermediate
growth. This has meanwhile been found out by Volodymyr Nekrashevych, cf.
[*Palindromic subshifts and simple periodic groups of intermediate growth*](http://arxiv.org/pdf/1601.01033.pdf),
arXiv, 2016.
| 10 | https://mathoverflow.net/users/28104 | 235198 | 108,966 |
https://mathoverflow.net/questions/235199 | 1 | I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers):
$$\int\_{0}^{\infty} x \;K\_{a}(nx) \; K\_{b}(kx) \; dx$$
I could not find it in litterature with a and b complex (We have $Re(a)<1$ and $Re(b)<1$ for convergence in zero!).
Any reference or help on th... | https://mathoverflow.net/users/38290 | Definite intergal with two K-Bessel functions and x | Mathematica:
$$\int\_{0}^{\infty} x \;K\_{a}(nx) \; K\_{b}(kx) \; dx=\frac{1}{2n^2}(k n)^{-b} $$
$$\qquad\times \left[n^{2 b} \Gamma (b) \Gamma \left(-\frac{a}{2}-\frac{b}{2}+1\right) \Gamma \left(\tfrac{1}{2} (a-b+2)\right) \, \_2F\_1\left(\tfrac{1}{2} (-a-b+2),\tfrac{1}{2} (a-b+2);1-b;\frac{k^2}{n^2}\right)\right.$$
... | 1 | https://mathoverflow.net/users/11260 | 235200 | 108,967 |
https://mathoverflow.net/questions/160399 | 10 | Let $F(n;i)$ be the number of labeled $i$-edge forests on $n$ vertices ([A138464](http://oeis.org/A138464) on the OEIS). The first few values of $F(n;i) \pmod n$ are listed below:
$$\begin{array}{r|rrrrrrrrrrr}
& i=0 & 1 & 2 & 3 & 4 & 6 & 7 & 8 & 9 & 10 & 11 \\
\hline
n=2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ... | https://mathoverflow.net/users/48278 | A published proof for: the number of labeled $i$-edge ($i \geq 1$) forests on $p^k$ vertices is divisible by $p^k$ | The proof is now Lemma 2 here:
>
> A. P. Mani, R. J. Stones, Congruences for the weighted number of labeled forests. Integers, 16 (2016): A17.
>
>
>
which is freely available from: <http://www.integers-ejcnt.org/vol16.html>
| 4 | https://mathoverflow.net/users/48278 | 235229 | 108,975 |
https://mathoverflow.net/questions/234681 | 3 | Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$
be positive integers $\geq 2$. I want to prove that there exists a primitive
polynomial $$F(x) = x^{mn}-\sum\limits\_{j=0}^{mn-1}f\_jx^j\in R[x]$$
with the property that there is a $k\in\{1,2,\ldots,n-1\}$ such that $f\_{km}\neq 0$.
I think tha... | https://mathoverflow.net/users/85489 | Non-zero coefficients of primitive polynomials | I post here a Hansen-Mullen's conjecture, which was proved in 2007. The validity of my assumption in the question is a straightforward consequence of this conjecture.
**Conjecture**.(Hansen and Mullen, 1992)
*Let $a\in \mathbb{F}\_q$ and let $n\geq2$ be a positive integer. Fix and integer $m$ with $0<m<n$. Then there... | 1 | https://mathoverflow.net/users/85489 | 235235 | 108,977 |
https://mathoverflow.net/questions/235237 | -2 | (A version of [this question](https://mathoverflow.net/questions/235133/partitioning-finite-directed-graphs-into-3-incoming-sparse-sets) for undirected graphs.)
Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ set
$$
N(v) := \{x\in V: \{x,v\}\in E\}.
$$
Is it possible to find a partition $P\_1,P\_... | https://mathoverflow.net/users/8628 | Splitting the vertices of undirected graphs into 2 sparse sets | Yes, it is a variant of Lovasz partition lemma. Choose partition $V=P\_1\sqcup P\_2$ for which the number $E(G|\_{P\_1})+E(G|\_{P\_2})$of edges which join vertices of the same part is minimal possible. It works for each vertex $v$: else moving $v$ to other part decreases this value.
| 3 | https://mathoverflow.net/users/4312 | 235238 | 108,978 |
https://mathoverflow.net/questions/234160 | 16 | Let $S$ be a compact connected orientable surface, and let $G$ be a nontrivial finite group acting freely on $S$ and preserving orientation (note the the action being free is a strong condition, since automorphisms usually have fixed points). Then $H^1(S)$ also has an action of $G$. I know how to prove, using Riemann-H... | https://mathoverflow.net/users/40821 | $G$-action on the integral homology of a compact surface | I don't see how to answer this is general, but the following partial result might be of interest to you.
Theorem: If $H^1(S;\mathbb{Z}) \cong \mathbb{Z}^2 \oplus \mathbb{Z}[G]^{2k}$ then $G$ does not contain $(\mathbb{Z}/p)^3$ for any prime number $p$.
Proof: Suppose $P = (\mathbb{Z}/p)^3 \leq G$. Then considered a... | 3 | https://mathoverflow.net/users/318 | 235242 | 108,980 |
https://mathoverflow.net/questions/235188 | 5 | **In short:** For a given smooth or continuous function, how can we obtain the best $L^{\infty }$ approximating polynomial?
Jackson (1911) proved that there is a best approximating polynomial in the $L^{\infty}$ sense. The proof Can be found in the references below. The theorem is
Let $I=\lbrack -1,1]$, then there ... | https://mathoverflow.net/users/42864 | $L^{\infty}$ polynomial approximation | You are looking for the well-known [Remez algorithm](https://en.wikipedia.org/wiki/Remez_algorithm), which dates to 1934. For a practical implementation (with an excellent description), I highly recommend the [Chebfun function `remez()`](http://www.chebfun.org/examples/approx/BestApprox.html) and [this paper that descr... | 5 | https://mathoverflow.net/users/20507 | 235256 | 108,986 |
https://mathoverflow.net/questions/235255 | 2 | The standard type III discrete cosine transformation (DCT) is defined as [follows](https://en.wikipedia.org/wiki/Discrete_cosine_transform):
$${X\_k} = \frac{1}{2}{x\_0} + \sum\limits\_{n = 1}^{N - 1} {{x\_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} \right)} \right]\quad \quad k = 0, \ldots ,N - 1.$$
I... | https://mathoverflow.net/users/52878 | How to relate this summation to standard discrete cosine transformation? | in the second transformation the integer $k$ runs from $0$ to $\tilde{N}-1$, not to $N-1$; so you have twice as many data points $X\_k$ than you have moments $x\_n$, which is perfectly OK (oversampling); to use the conventional formulas for the DCT, just extend the list of moments $x\_n$ by padding it with zeros ($x\_n... | 1 | https://mathoverflow.net/users/11260 | 235261 | 108,987 |
https://mathoverflow.net/questions/235216 | 4 | I would like to consider three sheaves $\mathcal{C}^0$, $\mathcal{H}$ and $\mathcal{S}$ on $\mathbb{C}$ (endowed with the euclidean topology): the first is the sheaf of continuous $\mathbb{C}$-valued functions, the second is the sheaf of holomorphic functions and the third is the sheaf of "overconvergent analytic funct... | https://mathoverflow.net/users/18238 | Grothendieck topologies on $\mathbb{C}$ | There is (as always) a finer topology $T$ making this into a sheaf, the question is whether this topology is different from the usual one or not.
The cover for $T$ of an open subset $V \subset \mathbb{C}$ are the family $V\_i \subset V$ such that for any open subset $U \subset \mathbb{C}$, $\mathcal{S}$ satisfies the... | 3 | https://mathoverflow.net/users/22131 | 235264 | 108,989 |
https://mathoverflow.net/questions/235252 | 4 | **In Short:** I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate.
**Longer:** If $p\_n$ is the $n$-th Legendre polynomial, and the Legendre expansion of a real function $f$ is $f(x) = \sum\limits\_{n=0}^{\infty} \hat{f}(n) p\_n (x)$, ... | https://mathoverflow.net/users/42864 | Reference for the exponential decay of Legendre coefficients | This is theorem 2.1 in [On the convergence rates of Legendre approximation](https://www.researchgate.net/profile/Shuhuang_Xiang2/publication/220577137_On_the_convergence_rates_of_Legendre_approximation/links/5461ee8c0cf27487b453afa5.pdf) (2012) [yes, with a proof in English]
| 4 | https://mathoverflow.net/users/11260 | 235265 | 108,990 |
https://mathoverflow.net/questions/235233 | 6 | (Everything is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E\_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L\_\kappa$ isomorphic to $L\_\mu$}\},$$ and let $$E\_\kappa^+=\{\mu<\kappa: \mbox{$\exists$ an elementary substructure of $(L\_{\kappa^+}, L\_\kappa)$ which is ... | https://mathoverflow.net/users/8133 | Fine structure question: when do levels of $L$ look "a lot" like each other? | Under your assumptions, the set $E\_\kappa^+$ is empty. Indeed, we don't even need the predicates that you mention. (In any case, since $\kappa$ is definable in $L\_{\kappa^+}$, the predicate for $L\_\kappa$ in $\langle L\_{\kappa^+},\in,L\_\kappa\rangle$ would be definable and therefore offer no additional expressive ... | 6 | https://mathoverflow.net/users/1946 | 235275 | 108,994 |
https://mathoverflow.net/questions/235257 | 0 | **Short and informal version**: Does the [stable marriage problem](https://en.wikipedia.org/wiki/Stable_marriage_problem) have a solution if there are $\kappa$ men and $\kappa$ women for any cardinal $\kappa \geq \aleph\_0$?
**Long and formal version**: Let $\kappa$ be an infinite cardinal. For any set $X$ we set $\t... | https://mathoverflow.net/users/8628 | Stable marriages for infinite bipartite graphs | Your formal version does not look correct. For each boy $b$, there should be a total order $\leq\_b$ on the set of girls $G$ (this is the preference order for $b$) and for each girl $g$, there should be a total order $\leq^g$ on the set of boys (this is the preference order for $g$). In the special case that $G$ and $B... | 4 | https://mathoverflow.net/users/2233 | 235281 | 108,998 |
https://mathoverflow.net/questions/235279 | 16 | Let $m$ and $n$ be distinct odd positive integers. The equality
$$
\prod\_{k=0}^{mn-1} \left( e^{\frac{2\pi i k}{m}}
+ e^{\frac{2\pi i k}{n}} \right)
\ = \ 2^{\gcd(m,n)}
$$
holds for all pairs of such $m$ and $n$ less than $50$ except for
$(3,21)$, $(3,39)$, $(15,33)$ and $(21,39)$, where the left-hand side
takes th... | https://mathoverflow.net/users/28104 | An equality involving roots of unity which holds most of the times, but not always | It always equals $2^{\gcd(m-n,mn)}$. Proof:
denote $x=e^{2\pi i/mn}$. Denote by $A$ the set of residues $k$ modulo $mn$ for which $mn$ divides $k(m-n)$, $|A|=\gcd(m-n,mn)$, $B$ is the set of other residues. We have
$$
\prod\_k (x^{km}+x^{kn})=\prod\_{k\in A} 2x^{km}\cdot \prod\_{k\in B} \frac{x^{2km}-x^{2kn}}{x^{km}-x^... | 30 | https://mathoverflow.net/users/4312 | 235283 | 108,999 |
https://mathoverflow.net/questions/235286 | 3 | (Everything below is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E\_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L\_\kappa$ isomorphic to $L\_\mu$}\},$$ and let $$E\_\kappa^+=\{\mu<\kappa: \mbox{$\exists M\prec (L\_{\kappa^+}, L\_\kappa)$ with $M\cong (L\_\alpha, L\_... | https://mathoverflow.net/users/8133 | Levels of L resembling each other, take 2 | If $\mu$ is the minimum of $E\_\kappa$, then $L\_\mu$ is the pointwise definable. However, if $\mu \in E\_\kappa^+$, then $L\_\mu$ is not pointwise definable. (See Joel's comment above. I am too lazy to type.)
| 5 | https://mathoverflow.net/users/64308 | 235289 | 109,001 |
https://mathoverflow.net/questions/235274 | 3 | Consider the Cantor space $\mathcal C={}^\omega2$ with the usual product measure, and let $r$ be a random real (over a transitive model $V$ of ZFC). Let $B\subset \mathcal C^V\times\mathcal C^V$ a Borel set of positive measure in $V$ and let $B^{V[r]}$ the Borel set in $\mathcal C^{V[r]}\times \mathcal C^{V[r]}$ with t... | https://mathoverflow.net/users/41274 | Generic sections of non-null sets are non-null | No, let $(r,y)\in B $ iff $r (0)=0$.
Then if $r $ is a random real with $r (0)=1$ you have a counterexample.
| 3 | https://mathoverflow.net/users/4600 | 235291 | 109,002 |
https://mathoverflow.net/questions/161919 | 6 | Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r\_N$. The resulting graph can be thought of as a discrete approximation to the 2-sphere.
>
> Is there a way to recover the eigenvalues an... | https://mathoverflow.net/users/396 | Recovering Spherical Harmonics from Discrete Samples | Tl;dr: The answer to your question is, **Yes,** but you picked the wrong weights on the graph, which is why the eigenvalues were off.
---
Before I get into some generalities about how to think about this kind of approximation problem, in addition to the paper Steve Huntsman posted above, you may find [this](http:... | 8 | https://mathoverflow.net/users/20796 | 235296 | 109,004 |
https://mathoverflow.net/questions/235298 | 7 | I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^\*$-algebras than $\*$-isomorphism.
Can someone explain this sentence or know some text that could be helpful?
Does anybody know some comparisons of different $C^\*$-algebras categories? ... | https://mathoverflow.net/users/49443 | Most natural equivalence between $C^*$-algebras in NCG | Here I list some facts that may be useful for building your intuition:
**1.** Two *commutative* Morita equivalent $C^\*$-algebra are in fact $\*$-isomorphic.
**2** If $A$ is $C^\*$-algebra and you take $B=M\_n(A)$ then $A$ and $B$ are Morita equivalent.
**3** Many invariants for $C^\*$-algebras such as $K$-the... | 11 | https://mathoverflow.net/users/24078 | 235300 | 109,005 |
https://mathoverflow.net/questions/235128 | 4 | Zheng and Weihrauch (<http://www-sst.informatik.tu-cottbus.de/~wwwti/zheng/publications/1999/mfcs99.pdf>) define a real number $x$ to be $\Sigma\_n$ if and only if there is a computable function $f:\mathbb{N}^n\rightarrow\mathbb{Q}$ such that $x=\sup\_{i\_1}\inf\_{i\_2}...f(i\_1,...,i\_n)$. Another possible definition ... | https://mathoverflow.net/users/83073 | Are these two definitions of arithmetical hierarchy of real numbers equivalent? | They are the same as the following induction proof shows.
*Base case:* For $\Sigma\_1$, if $x = \sup\_i f(i)$ for $f$ computable, then $q < x$ is equivalent to the $\Sigma\_1$ statement $\exists i [q<f(i)]$. Conversely, if $\{q\mid q < x\}$ is computably enumerable, then $x = \sup\_i f(i)$ where $f(i)$ is a computabl... | 1 | https://mathoverflow.net/users/12978 | 235302 | 109,006 |
https://mathoverflow.net/questions/235287 | 3 | Is there a relationship between $ \mathcal{D} $ - modules, Tannakian formalism and Galois theory of monodromy representations ?
Thanks in advance for your help.
| https://mathoverflow.net/users/89900 | Relationship between $ \mathcal{D} $ - modules, Tannakian formalism and Galois theory of monodromy representations | I don't know what level of depth you want, so here is the short version. Given a smooth complex variety $X$, as Daniel Barter points out, the categories of $\mathcal{O}\_X$-coherent $D$-modules, vector bundles with integrable connections, and complex representations of $\pi\_1(X)$ are equivalent via monodromy. Moreover... | 4 | https://mathoverflow.net/users/4144 | 235303 | 109,007 |
https://mathoverflow.net/questions/235301 | 7 | Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\mathcal C^V\stackrel\cong\longrightarrow \mathcal C^V\times \mathcal C^V\longrightarrow \mathcal C^V$ we obtain two conti... | https://mathoverflow.net/users/41274 | Iteration of random reals | Since $x$ is random over $V$, the fact that $C\_x^{V[x]}$ is null is (like any fact about $x$ in $V[x]$) forced by some condition in $V$. This condition is the equivalence class, modulo the null ideal, of some positive-measure Borel set $P$ in $V$. Let $P'$ be the Borel set in $V[x]$ with the same code. Then $x\in P'$,... | 8 | https://mathoverflow.net/users/6794 | 235305 | 109,009 |
https://mathoverflow.net/questions/235307 | 2 | While playing with some PDE I came across a singular integral that looks something like
$$T(f\_1,f\_2,\ldots,f\_n)(x)=p.v.\int\_{-\infty}^\infty\frac{(f\_1(x)-f\_1(y))(f\_2(x)-f\_2(y))\cdots(f\_n(x)-f\_n(y))}{(x-y)^n}dy$$
where the functions are really nice, say Schwartz. Clearly if $n=1$ then $T$ is just a multiple of... | https://mathoverflow.net/users/70155 | A singular integral of several functions | Without loss of generality, $x=0$.
Since $T(f\_1,\dots,f\_n)$ is $n$-linear in $(f\_1,\dots,f\_n)$, you can express each $f\_i$ as a mixture of harmonics $\exp(it\cdot)$, $t\in\mathbb R$ (with, in general, complex coefficients), so that $T(f\_1,\dots,f\_n)$ is expressed as a mixture of the values $T(\exp(it\_1\cdot),\... | 1 | https://mathoverflow.net/users/36721 | 235313 | 109,011 |
https://mathoverflow.net/questions/235267 | 8 | I have the following question and since I am not an expert on C\*-algebras, I thought I ask it here:
I know that in general the sum and product of normal elements need not be normal. It is even true that every element in a C\*-algebra is the sum of two normal elements. What I do not know is if that is true for multip... | https://mathoverflow.net/users/58628 | is every element in a C* algebra a product of normal elements? | Since the question was asked wrt C$\*$-algebras, I guess there is room for a general remark. Suppose that $xy = 1$ where $x$ is a product of normal elements, say
$v\_1v\_2\cdots v\_n$. Then $v\_1$ is normal and has a right inverse;
therefore it is (two-sided) invertible. Thus $v\_2\dots v\_n y $ is
invertible, so $v\_2... | 10 | https://mathoverflow.net/users/42278 | 235318 | 109,012 |
https://mathoverflow.net/questions/228557 | 3 | I've noticed that the name 'Szemeredi's regularity lemma' is used for several closely related yet different statements about graphs.
Specifically, I'm interested in the distinction between two of them:
**Variant 1** Given $\varepsilon>0$ and $l>0$, there exists an $L=L(\varepsilon,l)$ such that any graph $G$ can b... | https://mathoverflow.net/users/85349 | Variants of Szemeredi's regularity lemma | I think this recent arXiv paper <http://arxiv.org/abs/1604.00733v1> explains precisely the relationship between these two versions of the regularity lemma.
| 3 | https://mathoverflow.net/users/25028 | 235323 | 109,015 |
https://mathoverflow.net/questions/235330 | 4 | *Question*: Assume that $a,b\in Z$, and $4a^3+27b^2\neq 0$. Prove that there exist infinitely many positive integers $n$ such $n^3+an+b$ is square-free.
I have following
>
> **There exist infinitely many postive integers $n$ such $n^2+1$ is squarefree.**
>
>
>
**Idea of proof**
Let $S(n)$ be the number of ... | https://mathoverflow.net/users/38620 | Do there exist infinitely many $n$ such that $n^3+an+b$ is squarefree? | [Erdos](https://www.renyi.hu/~p_erdos/1953-02.pdf) was the first to show that cubic polynomials (meeting obvious necessary conditions) do take on square-free values. More generally he considered polynomials of degree $\ell$ taking on $\ell-1$-th power-free values infinitely often. [Granville](http://www.dms.umontreal.c... | 14 | https://mathoverflow.net/users/38624 | 235332 | 109,018 |
https://mathoverflow.net/questions/235340 | 3 | Let $H$ be a separable Hilbert space and consider the Calkin algebra $C(H)=\frac{B(H)}{K(H)}$.
Q) True or false: Any representation of $C(H)$ is a direct sum of irreducible representations.
| https://mathoverflow.net/users/84390 | Representations of Calkin algebra | If you google "representations of the Calkin algebra", you find the 1967 paper of Sakai which answers your question in the negative: It says that the Calkin algebra has a type III factor representation. By the Lemma of Schur, however, every irrducible representation is a type I representation and a direct sum of more t... | 6 | https://mathoverflow.net/users/nan | 235342 | 109,020 |
https://mathoverflow.net/questions/235322 | 8 | The Triangle Removal Lemma states that any graph with $o(n^3)$ triangles can be made triangle-free by removing only $o(n^2)$ edges. More generally, the Graph Removal Lemma states that for any graph $H$ on a constant $|V(H)| = k$ number of nodes, any graph with $o(n^k)$ copies of $H$ can be made $H$-free by removing at ... | https://mathoverflow.net/users/25121 | Can the graph removal lemma be proved directly from the triangle removal lemma? | A proof of the Graph Removal Lemma that avoids using the regularity lemma can be found in [A new proof of the graph removal lemma](http://arxiv.org/abs/1006.1300) (2010). For an explanation why a direct proof of the Graph Removal Lemma from the Triangle Removal Lemma is not viable, see this [discussion:](https://matheu... | 8 | https://mathoverflow.net/users/11260 | 235352 | 109,024 |
https://mathoverflow.net/questions/235353 | 2 | I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
>
> Suppose that $X$ is a compact Hausdorff topological space and $\mu$ is a regular Borel measure on $\mathcal B(X)$. Let $\mu\_{0}$ be ... | https://mathoverflow.net/users/50259 | Integration against Borel measures on compact Hausdorff spaces | Your highlighted proposition is true as a particular case of the following more general proposition.
>
> Let $(S,\Sigma,\mu)$ be a measure space. Let a function
> $f\colon S\to\mathbb R$ be $\Sigma\_0$-measurable and $\mu$-integrable, where
> $\Sigma\_0$ is a sub-sigma-algebra of $\Sigma$. Let $\mu\_0$ be the
>... | 2 | https://mathoverflow.net/users/36721 | 235355 | 109,026 |
https://mathoverflow.net/questions/235347 | 33 | In a [recent publication](http://arxiv.org/abs/1603.04246) by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved.
Admittedly it is very difficult to grasp the work for someone not involved in the field, but it would be incredib... | https://mathoverflow.net/users/nan | Understanding sphere packing in higher dimensions | There are two things you need to understand. The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds"). My [lecture notes from PCMI 2014](http://arxiv.org/abs/1603.05202) give an exposition of this theory, which covers the period up to, but not including, Viazovska's paper on e... | 42 | https://mathoverflow.net/users/4720 | 235360 | 109,027 |
https://mathoverflow.net/questions/235357 | 3 | Let $G$ be any finite solvable group with Fitting subgroup
$F(G)$. Which conditions on $F(G)$ makes $G$ to be supersolvable?
(It is well-known that any finite solvable group with cyclic Fitting subgroup
is supersolvable). In particular, if $F(G)$ is abelian which conditions on
$F(G)$ (and $G$) are needed for $G$ being ... | https://mathoverflow.net/users/27831 | Finite solvable groups with abelian Fitting subgroup | The relation between supersolvability of a finite group and its Fitting
subgroup is that for a finite group $G$ the following are equivalent:
1. $G$ is supersolvable.
2. $G' \leq {\rm Fit}(G)$ and ${\rm Fit}(G)$ is the product of cyclic
and weak $S$-quasinormal subgroups of $G$ of prime power orders.
Here a subgro... | 6 | https://mathoverflow.net/users/28104 | 235361 | 109,028 |
https://mathoverflow.net/questions/235371 | -1 | I am trying to understand how the Heisenberg group is defined because I would like to understand the (irreducible) representations.
Following [this article](http://www.pvamu.edu/PDFFiles/edu/Include/Math/MDTRS/DMTRS11_b.pdf) given a symplectic bilinear form $\langle, \rangle$ on a finite-dimensional vector space $W$ ... | https://mathoverflow.net/users/14574 | Representations of the $3\times 3$ Heisenberg group | Assuming your symplectic form is $\langle (w\_1,w\_2), (w'\_1, w'\_2) \rangle = w\_1 w'\_2 - w\_2 w'\_1$, the isomorphism is
$$((w\_1,w\_2),a) \mapsto \begin{pmatrix} 1 & w\_1 & \tfrac{1}{2}(a+w\_1 w\_2)\\ 0 & 1 & w\_2 \\ 0 & 0 & 1 \end{pmatrix}.$$
| 4 | https://mathoverflow.net/users/297 | 235372 | 109,031 |
https://mathoverflow.net/questions/235299 | 21 | Given the [march 2016 breakthrough concerning sphere packings](https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/) by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that the known cases $\Delta\_1=1$, $... | https://mathoverflow.net/users/50912 | Sphere packings : what next after the recent breakthrough of Viazovska (et al.)? | Here's an attempt at answering your Question A:
Currently, one of the most powerful methods for proving upper bounds on sphere packing densities is the linear programming bound of [Cohn and Elkies](http://arxiv.org/abs/math/0110009) (which is what Viazovska used).
### Exact answers
According to the numerical comp... | 18 | https://mathoverflow.net/users/8297 | 235382 | 109,035 |
https://mathoverflow.net/questions/235380 | 0 | Let $(X,\mathcal{O}\_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact.
Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. $E$ is a finitely generated locally free sheaf of $\mathcal{O}\_X$-modules on $U$.
$\textbf{My question}$ is: can we ... | https://mathoverflow.net/users/24965 | Can we always extend a vector bundle on an open subset of a ringed space with soft structure sheaf? | No, this is false. Take $X=\mathbb{R}^3$ and $U=\mathbb{R}^3\smallsetminus\{0\} $,
with $\mathcal{O}\_X$ the sheaf of complex $C^{\infty}$ functions.
Line bundles on $U$ are parametrized by $H^1(U, \mathcal{O}\_U^\*)$, which is isomorphic to $H^2(U,\mathbb{Z})=\mathbb{Z}$ by the exponential exact sequence. Similarly $... | 6 | https://mathoverflow.net/users/40297 | 235384 | 109,036 |
https://mathoverflow.net/questions/235385 | 11 | Let we have a sequence $\{a\_{n}\}$, such that $\forall n \,\, a\_{n}>0$ and $a\_{n} \rightarrow\infty, n\rightarrow\infty$. Also let's suppose that we have a subsequence $\{a\_{n\_{k}}\}$ such that $\exists C>0$ $\forall k:$ $a\_{n\_{k}}<Clog\, n\_{k} $. How can we prove that the generating function of $\{a\_{n}\}$ ca... | https://mathoverflow.net/users/89943 | Generating function of a sequence is not algebraic | Yes, we can. Asymtotics of the coefficients of an algebraic function is determined
by the finitely many singularities on the circle of convergence. At every singularity the algebraic function has a Puiseux exansion, from which follows
that the function has a power asymptotics (no logs). This paper is a convenient refer... | 13 | https://mathoverflow.net/users/25510 | 235392 | 109,040 |
https://mathoverflow.net/questions/235383 | 1 | What does it mean if the two smallest eigenvalues of the Laplacian matrix of a graph are equal to zero?
| https://mathoverflow.net/users/89944 | If the two smallest eigenvalues of the Laplacian matrix of a network are equal to zero, then does it mean that the network is not connected? | If by "network" you mean "graph", then yes, the eigenvalue 0 having multiplicity at least two means that the graph has at least two connected components. This depends on the fact that the characteristic vector of each connected component is always an eigenvector for the eigenvalue 0.
If by network you mean something ... | 2 | https://mathoverflow.net/users/26039 | 235398 | 109,045 |
https://mathoverflow.net/questions/235412 | 4 | Let $B(p)$ denote the Bernoulli distribution over $\{0,1\}$ and $B(p)^n$ the corresponding product distribution over $\{0,1\}^n$. For $n>1$ and $0<x<1$, define
$$P\_n(x):=B(\frac12+\frac x2)^n$$
and
$$Q\_n(x):=B(\frac12-\frac x2)^n.$$
I can show $ ||P\_n(x)-Q\_n(x)||\_1 \le 2.1\sqrt{n}x$ using Pinsker's inequality.
*... | https://mathoverflow.net/users/12518 | Sharpened Pinsker inequality for special case | Looking at **[[Pinsker's inequality](https://en.wikipedia.org/wiki/Pinsker's_inequality)]**, I am assuming that the left-hand side of the conjectured inequality is understood as the the total variation norm of the signed measure $P\_n(x) - Q\_n(x)$, which equals
\begin{equation}
s(y):=s\_n(y):=2\sum\_{j=0}^m\binom nj... | 6 | https://mathoverflow.net/users/36721 | 235424 | 109,051 |
https://mathoverflow.net/questions/235415 | -1 | $\mathbf{Question}$. Let us assume that $M^n$ is a topological manifold of dimension n, with a group action $\Gamma$, which acts discontinuously and freely on $M^n$. Is the orbit space $M^n / \Gamma$ still a manifold of dimension n?
"Discontinuously" here means the orbit of a certain point does not have a limit point... | https://mathoverflow.net/users/87755 | When is the orbit space of a manifold still a manifold of the same dimension? | No, here is the standard counterexample. Let $M$ be the complement of the origin in $\mathbb{R}^2$, let $T$ be the linear transformation $T(x,y) = (2x,y/2)$, and let $\Gamma$ be the cyclic group generated by $T$. Then $\Gamma$ acts discontinuously and freely on $M$, but the orbit space $M/\Gamma$ is not Hausdorff, so i... | 3 | https://mathoverflow.net/users/68305 | 235429 | 109,052 |
https://mathoverflow.net/questions/235452 | 7 | For a group $G$ generated by a finite set $S$ we denote by $B\_{G,S}(n)$ the ball of radius $n$, that is the set of all elements in $G$ which are expressible as products $x\_1x\_2\ldots x\_n$ where $x\_i\in S\cup S^{-1}\cup\{1\}$. One calls the set $Q$ generic in $G$ with respect to $S$ if $$\lim\_{n\to\infty}\sup \fra... | https://mathoverflow.net/users/10443 | Generic set that is a proper subgroup | No, it's not possible. For notational convenience assume $S$ is symmetric and $1\in S$, so that $B\_{G,S}(n) = S^n$. Suppose
$$
|H\cap S^{n\_i}|/|S^{n\_i}|\to 1
$$
for some subsequence $(n\_i)$. Let $x$ be an element of $S$ not in $H$. Since $H$ and $Hx$ are disjoint we have
$$
|H\cap S^{n\_i}x^{-1}|/|S^{n\_i}| = |Hx... | 9 | https://mathoverflow.net/users/20598 | 235460 | 109,065 |
https://mathoverflow.net/questions/235461 | 6 | In the paper "Arakelov's theorem for abelian varieties" Faltings proves the Shafarevich conjecture for abelian varieties.
The statement is the following:
>
> Let B be smooth projective a curve, S a finite set in B. There exist
> only finitely many families of principally polarized abelian varieties
> of dimens... | https://mathoverflow.net/users/59377 | Shafarevich conjecture for abelian varieties | Let $B$ be a smooth projective curve over an algebraically closed field of characteristic zero. Let $K$ be the function field of $B$. Let $S$ be a finite set of closed points of $B$.
You might find the following reformulation of Faltings's theorem less confusing.
**Theorem 1.** (Faltings, geometric Shafarevich con... | 11 | https://mathoverflow.net/users/4333 | 235467 | 109,067 |
https://mathoverflow.net/questions/235458 | 4 | I know about equational logic, cf. <https://en.wikipedia.org/wiki/Lattice_(order)#As_algebraic_structure>, and understood that lattices are expressed equationally, i.e., in terms of equational logic (with function symbols $\wedge, \vee$ and by introducing the order $p \le q \; :\Leftrightarrow \; p = p \wedge q$).
Th... | https://mathoverflow.net/users/89916 | Posets (partially ordered sets) in equational logic | No. The category of models of an equational theory (i.e. a variety in the sense of universal algebra) is always a [regular category](https://ncatlab.org/nlab/show/regular+category), but the category of posets is not regular.
| 8 | https://mathoverflow.net/users/11640 | 235473 | 109,070 |
https://mathoverflow.net/questions/235475 | 3 | As far as I remembered there is an inverse Hadamard inequality for the determinant of the form
$$
|D|>\prod\_j \sqrt{(a\_{jj}^2-\sum\_{i\neq j}a\_{ij}^2)}
$$
providing all values in $(\cdot)>0$.
Please help me with exact references to this inequality, its possible generalizations and modifications and comments.
| https://mathoverflow.net/users/49208 | Inverse Hadamard determinant inequality | Did you mean Ostrowski's theorem $$|D|>\prod\_j \left(|a\_{jj}|-\sum\_{i\neq j}|a\_{ij}|\right)$$
for diagonally dominant matrices (see e.g. <http://planetmath.org/propertiesofdiagonallydominantmatrix>)?
| 5 | https://mathoverflow.net/users/35593 | 235482 | 109,073 |
https://mathoverflow.net/questions/235477 | 8 | Let $k$ be an algebraically closed field and $C$ be a smooth projective curve over $k$. Let $p$ be a prime number. Does there exists a finite morphism $f : C \to \mathbb{P}^1$ such that the degree of the Galois closure of the field extension $k(\mathbb{P}^1) \to k(C)$ is coprime to $p$ ?
| https://mathoverflow.net/users/85020 | Finite morphism from a smooth projective curve. | Following Jason Starr's suggestion, let $C$ be a very general genus $2$ curve, forming a covering of degree $n$ of $\mathbb P^1$. Without loss of generality there is no intermediate curve between $C$ or $\mathbb P^1$ (it would have to have genus $0$ or $1$. In the first case, we may simplify by replacing $\mathbb P^1$ ... | 8 | https://mathoverflow.net/users/18060 | 235485 | 109,074 |
https://mathoverflow.net/questions/235483 | 4 | Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}\_E$ be the constant sheaf on $E$. Let $f\_\*(\underline{\mathbb{C}}\_E)$ denote its direct image in the derived category $D(Sh\_B)$ of sheaves of $\mathbb{C}$-vector spaces. Since $F$ is connected, it is clear ... | https://mathoverflow.net/users/16183 | On push-forward of the constant sheaf for fibrations | No, it is not true. For example, let $E=\mathbb C^2\backslash \{ 0\}$, $B=\mathbb C\mathbb P^1$ (with the obvious map $f$). Then the pushforward as a complex of sheaves on $\mathbb C\mathbb P^1$ has the following cohomology: constant sheaf in degree 0 and constant sheaf in degree +1. If the triangle you are asking abou... | 3 | https://mathoverflow.net/users/3891 | 235492 | 109,076 |
https://mathoverflow.net/questions/235463 | 19 | Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
\limsup\_{m \rightarrow \infty} \left( \limsup\_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \right)?
$$
If needed, answers may be conditional under the assumption that a suitable generalization
of the [Bunyakovsk... | https://mathoverflow.net/users/28104 | How many primes can there be in a short interval? | As observed by Hensley and Richards in
*Douglas Hensley and Ian Richards*, [**Primes in intervals**](http://www.ams.org/mathscinet-getitem?mr=396440), *Acta Arith.* **25** (1973-74), 375--391,
if the prime tuples conjecture is true, then $\limsup\_{n \to \infty} \pi(n+m) - \pi(n) = \rho^\*(m)$, where $\rho^\*(m)$ ... | 36 | https://mathoverflow.net/users/766 | 235497 | 109,078 |
https://mathoverflow.net/questions/235472 | 7 | Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over a field $k$. An elementary finite correspondence from $X$ to $Y$ is defined to be an irreducible closed subset $W$ of $X... | https://mathoverflow.net/users/39193 | Intuition behind the definition of finite correspondences | Traditionally correspondences were defined simply as cycles on the product, but then you need a moving lemma just to define composition. This limits you to working on smooth varieties. The beauty of finite correspondences is that composition can be defined much more directly, essentially as a composition of multivalued... | 13 | https://mathoverflow.net/users/4144 | 235500 | 109,080 |
https://mathoverflow.net/questions/235503 | 16 | I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong):
Let $G$ and $H$ be topological groups and $f : G \to H$ be a continuous homomorphism. Applying the classifying space functor gives a map $Bf: BG \to BH$.
1. The homotopy fiber ... | https://mathoverflow.net/users/644 | Homotopy fiber of a map between classifying spaces | One source for some of this is section 8 of [May's "Classifying spaces and fibrations"](http://www.math.uchicago.edu/~may/BOOKS/Classifying.pdf)
He has G and H interchanged, unfortunately uses a coset notation for what turns out to be the homotopy fibre, and doesn't talk about the special cases you are after (his int... | 4 | https://mathoverflow.net/users/4648 | 235519 | 109,087 |
https://mathoverflow.net/questions/235538 | 9 | Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally isomorphic to an analytic subset of $\mathbb{C}^{n}$ (with the sheaf of holomorphic functions). Denote the category of these ... | https://mathoverflow.net/users/22810 | Is the analytification functor part of a geometric morphism of topoi? | I am not very familiar with the analytic side of the pictures or with the analytification functor but here is what I can claim, it seems from your comment that this answer your question:
If you have two subcanonical site $C$ and $D$ (it means that representable presheaves are sheaves, so it is the case with your exam... | 8 | https://mathoverflow.net/users/22131 | 235545 | 109,094 |
https://mathoverflow.net/questions/234040 | 5 | This question is motivated purely by curiosity. In algebraic geometry there is a major distinction between the world of characteristic $0$ and that of characteristic $p > 0$ with different methods, different results available etc.
From reading a number of books and papers I got the idea that in the case of rigid ana... | https://mathoverflow.net/users/1220 | Rigid analytic geometry in characterstic 0 vs positive characteristic | Resolution of singularities for rigid analytic varieties of equal characteristic zero follows from resolution of singularities for schemes of characteristic zero (Nicaise, A trace formula for rigid analytic varieties etc., 2009, Proposition 2.43).
There are more examples where the characteristic plays a role, e.g. in... | 5 | https://mathoverflow.net/users/62434 | 235552 | 109,097 |
https://mathoverflow.net/questions/235526 | 18 | I'm wondering about when the colimit and the homotopy colimit agree with diagrams of simplicial sets. I know that hocolim$(F)=$colim$(F\_c)$ where $F\_c$ is the cofibrant replacement of $F$. However, it is not always necessary for $F$ to be cofibrant for the colimit and homotopy colimit to be the same. For example, let... | https://mathoverflow.net/users/89997 | When do colimits agree with homotopy colimits? | I don't think we can expect to have one general answer to this question, only a collection of unrelated specialized results. Here are two more:
* In the category of simplicial sets all filtered colimits are homotopy colimits. **Added:** The gist of the argument can be found in Proposition 1.3 in Quillen's *Higher Alg... | 12 | https://mathoverflow.net/users/12547 | 235555 | 109,099 |
https://mathoverflow.net/questions/235505 | 8 | Chebyshev got famous showing that if the limit $l:=\lim\_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime number theorem conjectured by Gauss and Legendre. What I would like to know is whether other famous similar results are know... | https://mathoverflow.net/users/13625 | Famous results about the value of a given limit assuming it exists | The story with sharp thresholds for random constraint satisfaction problems somewhat fits into this picture. In the random [k-SAT](https://en.wikipedia.org/wiki/Boolean_satisfiability_problem#3-satisfiability) problem with $n$ Boolean variables, one includes each of the $2^k \binom{n}{k}$ potential clauses independentl... | 4 | https://mathoverflow.net/users/658 | 235561 | 109,102 |
https://mathoverflow.net/questions/54661 | 18 | The classical Borel Lemma states that for an arbitrary sequence $(v\_n)\_{n \in \mathbb{N}\_0}$ of complex numbers there is a smooth function $f\colon \mathbb{R} \longrightarrow \mathbb{C}$ with Taylor coefficients at $0$ given by the $v\_n$. Some generalizations work for functions of $d$ variables and also for values ... | https://mathoverflow.net/users/12482 | Borel Lemma for vector-valued functions | Yes, there are other spaces in which Borel's Theorem holds (property (BT)).
Example 1. Note that every cartesian product of locally convex spaces with (BT) has (BT). In particular, any uncountable power ${\mathbb R}^I$ has (BT). Yet, ${\mathbb R}^I$ cannot be given a topology ${\mathcal O}$ which makes it a Fréchet s... | 6 | https://mathoverflow.net/users/90040 | 235564 | 109,104 |
https://mathoverflow.net/questions/235430 | 10 | Suppose that a bounded sequence of real numbers $s\_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu\_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu\_2\in \beta{\Bbb N}\setminus{\Bbb N}$, surely there exists some rearrangement $s\_{r(i)}$ of $s\_i$ that has the same limit $\... | https://mathoverflow.net/users/10909 | Limits of rearranged sequences along ultrafilters | If $\mathfrak{p} > \kappa$, then you can get such a $\pi$ for every family of $\kappa$-many bounded sequences.
To see this, suppose $\mathfrak{p} > \kappa$, $\mu\_1$ and $\mu\_2$ are nonprincipal ultrafilters, and $s^\xi\in\mathbb{R}^\mathbb{N}$ ($\xi < \kappa$) is a family of bounded sequences. Let $\lambda^\xi$ be ... | 7 | https://mathoverflow.net/users/11233 | 235565 | 109,105 |
https://mathoverflow.net/questions/235560 | 3 | Let $BG$ is classifying space of $G$ topological group.
If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the
inclusion map $i:H\rightarrow G$ induces
\begin{equation\*}
G/H\rightarrow BH\rightarrow BG
\end{equation\*}
a fiber bundle?
If $G$ is any compact group and $H$ is a closed subgroup of ... | https://mathoverflow.net/users/86099 | Fiber bundle and fibration of classifying space | No, this is not true for Q1 and Q2.
To be specific we probably need to pick a particular model for $BG$, and there are several. For the argument I will give, the following property is sufficient: there exist isomorphisms $\pi\_{i+1}(BG) \cong \pi\_i(G)$ of homotopy groups.
Let $G = S^1$ and let $H$ be the subgroup ... | 5 | https://mathoverflow.net/users/360 | 235571 | 109,108 |
https://mathoverflow.net/questions/235553 | 1 | Suppose, $ A $ is a unitary matrix in $ M\_n(\mathbb{C}) $ given by $ (a\_{i,j})\_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e\_i $, $ Ae\_i\ne |\lambda| e\_j $ for all $i,j $ and $ |\lambda|=1 $. Then consider the matrix $ B=(a\_{i,j}^4)\_{1\le i,j\le n} $. Then is it true that $ ||B||... | https://mathoverflow.net/users/87890 | Norm of an operator formed using a unitary operator | I will show $||B||<1$ (in the finite-dimensional case). Suppose $|Bv|=|v|$ for some nonzero $v$. Then as $|Bv|$ is the projection of $A^{\otimes 4} ( \sum\_i v\_i e\_i^{\otimes 4})$ onto the subspace generated by $e\_i^{\otimes 4}$, and $|A^{\otimes 4} ( \sum\_i v\_i e\_i^{\otimes 4})|=| \sum\_i v\_i e\_i^{\otimes 4}|=... | 4 | https://mathoverflow.net/users/18060 | 235572 | 109,109 |
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