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https://mathoverflow.net/questions/207642 | 3 | In the web page <http://www.encyclopediaofmath.org/index.php/Moore_space> it can be found the following statement:
>
> If $K(\mathbb Z,n)$ is the Eilenberg–MacLane space of the group of integers $\mathbb Z$ and $M\_k(G)$ is the Moore space with $\tilde{H}\_k(M\_k(G))=G$, then
>
>
> $$lim\_{N\rightarrow\infty}[\Si... | https://mathoverflow.net/users/74168 | Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$? | Let $H\mathbb{Z}$ denote the spectrum for cohomology with coefficients in $\mathbb{Z}$, so your spectrum is $H\mathbb{Z}\wedge M\_k(G)$. Let $$0\to\bigoplus\_I\mathbb{Z}\stackrel{f}{\to}\bigoplus\_J\mathbb{Z}\to G\to 0$$ be a presentation of $G$. Then we can explicitly construct a Moore space $M\_k(G)$ as the cofiber o... | 16 | https://mathoverflow.net/users/75 | 207647 | 99,211 |
https://mathoverflow.net/questions/207640 | -1 | Let $(X,d)$ be a non-compact, complete metric space and $K\subseteq X$ compact. Pick $x^\* \in X\setminus K$.
Let $E$ be the connected component of $X\setminus K$ that contains $x^\*$. Let ${\cal C}$ be the collection of connected components of $K$. For each $C\in {\cal C}$ let $E\_C$ be the connected component of $X... | https://mathoverflow.net/users/8628 | Intersection of complements of connected components (2) | No.
[I'm assuming you means to write "let $E\_C$ be the connected component of $X \setminus C$ that contains $x^\*$." Otherwise I don't see how your question makes sense. Let me know if I'm guessing wrong and you meant to ask something else.]
Let $X\_0 = (\omega+1) \times \mathbb{R}$ (where $\omega+1$ has its usual... | 2 | https://mathoverflow.net/users/70618 | 207648 | 99,212 |
https://mathoverflow.net/questions/207664 | 9 | Let $X$ and $Z$ be two (possibly dependent) random variables. Is it necessarily the case that there exists a Borel function f and a random variable $Y$ that is independent of $X$ such that $Z = f(X, Y)$? If this is true, this would help with a result I am trying to prove. By random variable, the more general the better... | https://mathoverflow.net/users/74186 | Decompose dependent random variables into function of dependent and independent parts | I interpret the question as suggested in my comment above: we would like to find random variables $X,Z$ on *some* probability space with the given joint distribution, and then a third random variable $Y$ with the requested properties.
Given $X:\Omega\to\mathbb R$ with the right distribution, we can take $Y$ as unifor... | 6 | https://mathoverflow.net/users/48839 | 207671 | 99,218 |
https://mathoverflow.net/questions/207237 | 12 | I am trying to search on MathSciNet for articles which contains $C^\*$ in their title (as in $C^\*$-algebras) however I can't figure out how to get MathSciNet not to interpret the '\*' as a stand in for an arbitrary sequence of characters.
(I am moving this question from [MathStackExchange](https://math.stackexchange... | https://mathoverflow.net/users/8106 | Searching for $C^*$ | I have emailed technical support at MathSciNet. Unfortunately they do not know how to search for a string containing a \*, as \* is interpreted as a wild card. However, if you know the word before and after the \* then you can use the proximity search tool adjN which is a stand in for at most N-1 words.
For example,... | 6 | https://mathoverflow.net/users/8106 | 207674 | 99,219 |
https://mathoverflow.net/questions/207681 | 9 | This question arose from an [answer](https://mathoverflow.net/a/207571/41291) to my recent question [How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras?](https://mathoverflow.net/q/207540/41291)
What I need from that answer here is combination of the... | https://mathoverflow.net/users/41291 | Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation? | Yes. $C\_n$ counts the number of paths from $(0, 0)$ to $(2n, 0)$ which involve moving by either the vector $(1, 1)$ or $(1, -1)$. Each such path passes through $(n, k)$ for a unique $0 \le k \le n$, and the number of paths passing through $(n, k)$ for a fixed $k$ is the square of the number of paths from $(0, 0)$ to $... | 10 | https://mathoverflow.net/users/290 | 207682 | 99,221 |
https://mathoverflow.net/questions/207668 | 8 | I have two questions regarding to $p$-groups.
1. A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of course there are many other $p$-groups with cyclic center that are not extraspecial. I would to know if there is any clas... | https://mathoverflow.net/users/8419 | classification of $p$-groups | Every $p$-group is a homomorphic image of a $p$-group with cyclic center of order $p$, so a classification (whatever that means) of $p$-groups with cyclic center would (more-or-less) yield a construction for all $p$-groups, and I would not hold my breath waiting for that.
To see why a $p$-group $P$ is a homomorphic ... | 23 | https://mathoverflow.net/users/9694 | 207686 | 99,222 |
https://mathoverflow.net/questions/207660 | 28 | If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph\_1$, and then collapse $\aleph\_1$ to be countable.
There are many ways of doing that, but all of them (that I could think of, w... | https://mathoverflow.net/users/7206 | How hard is it to destroy a diamond? (with a real) | The answer is yes, assuming the existence of $\aleph\_2$-many measurable cardinals. To see this, assume $GCH+\Diamond$ holds and $S$ is a discrete set of measurable cardinals of size $\aleph\_2.$
**Step 1.** Force with Prikry product forcing $P\_S$ to change the cofinality of each element of $S$ into $\omega.$
Note... | 10 | https://mathoverflow.net/users/11115 | 207700 | 99,229 |
https://mathoverflow.net/questions/207621 | 4 | There is a following theorem:
$H$ is a commutative Hopf algebra over a field $k$. Then there exists a bijective correspondence between
$$\{ \textrm{Hopf subalgebras }K\subset H \} \quad \leftrightarrow\quad\{\textrm{ normal Hopf ideals } I\subset H\}$$
Where Hopf subalgebra is a subalgebra $K$ satisfying $\Delta K... | https://mathoverflow.net/users/45974 | Normal Hopf ideals and Hopf subalgebras, quotient group schemes | I am not sure I have an answer to your question but it seems to me that the statement you refer to needs a property which is really of somewhat geometric nature: i.e. that every normal algebraic subgroup of an affine algebraic group is observable. Observability is precisely the property that selects subgroups behaving ... | 1 | https://mathoverflow.net/users/6032 | 207707 | 99,233 |
https://mathoverflow.net/questions/207704 | 6 | Let $G$ be a finite group with the discrete topology. To set terminology:
* a **$G$-sphere** is a sphere equipped with a continuous $G$-action
* a **$G$-representation sphere** is a $G$-sphere obtained from an orthogonal $G$-representation $V$ by taking its unit sphere $S(V)\subset V$
From what I've heard, there ex... | https://mathoverflow.net/users/1916 | Example of a $G$-sphere that is not a $G$-representation sphere | There are involutions on some Brieskorn spheres which are true spheres whose fixed point sets are only Z/2-homology spheres. (But fixed sets of linear actions on spheres are spheres.)
Here it is easy to write down the Brieskorn sphere and the involution, it is more work to see that the Brieskorn sphere is topological... | 5 | https://mathoverflow.net/users/69091 | 207712 | 99,236 |
https://mathoverflow.net/questions/207701 | 8 | A "geometric" rough path is a rough path such that $Sym(\mathbb{X}\_{s,t})=\frac{1}{2}X\_{s,t}\otimes X\_{s,t}$. For example the Ito rough path is not geometric because $Sym(\mathbb{X}\_{s,t})=\frac{1}{2}X\_{s,t}\otimes X\_{s,t}-\frac{1}{2}I(t-s)$ but the Stratonovich rough path is geometric.
Why the term "geometric"... | https://mathoverflow.net/users/nan | Why the term "geometric" rough path? | Geometric rough paths have the property that if you want to solve an equation with values in a manifold, choose a coordinate chart, and write in local coordinates
$$
dY^i = V\_0^i(Y)\,dt + \sum\_j V\_j^i(Y)\,dX\_j
$$
for some vector fields $V\_i$ (with the obvious abuse of notation that the solution actually depends on... | 17 | https://mathoverflow.net/users/38566 | 207717 | 99,237 |
https://mathoverflow.net/questions/207699 | 4 | Consider the differential operator $P$ on $\mathbb{R}^2$, given by $P = \frac{\partial^2}{\partial x^2} + x^2\frac{\partial^2}{\partial y^2}$. Clearly it is elliptic everywhere except on the $y$-axis. I am looking to make a statement like $P\varphi \in H^s\_{\text{loc}} \Rightarrow \varphi \in H^{s + r}\_{\text{loc}}$.... | https://mathoverflow.net/users/74210 | Loss of derivative of subelliptic operator | You have $P=X\_1^2+X\_2^2$ with the real vector fields $X\_1=\partial\_x$ and $X\_2=x\partial\_y$. Since $X\_1$ and the commutator $[X\_1,X\_2]$ generate all vector fields, Hörmander's theorem applies and gives hypoellipticity with loss of $2-\varepsilon$ derivatives. Chapter 9 of A. Unterberger, Pseudo-differential op... | 2 | https://mathoverflow.net/users/nan | 207722 | 99,239 |
https://mathoverflow.net/questions/207726 | 11 | Suppose $\kappa$ is a regular cardinal. Does there necessarily exist a poset $\mathbb P$ that collapses $\kappa^+$ while preserving all other cardinals?
| https://mathoverflow.net/users/11145 | minimal collapsing without GCH | For $\kappa=\aleph\_1,$ the answer is yes, and is due to Abraham: "[On Forcing Without the Continuum Hypothesis](http://www.jstor.org/stable/2273457?seq=1#page_scan_tab_contents)". See also Todorcevic's paper "[A note on the Proper Forcing Axiom](http://www.math.toronto.edu/~stevo/Todorcevic_Note_On_PFA.pdf)", for a di... | 12 | https://mathoverflow.net/users/11115 | 207727 | 99,241 |
https://mathoverflow.net/questions/207716 | 10 | I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form
$$A \wedge dA + \frac{2}{3}A \wedge A \wedge A,$$
where $A$ is a **Lie algebra valued** 1-form. My question is "how is the last term $A \wedge A \wedge A$ d... | https://mathoverflow.net/users/74161 | $A \wedge A \wedge A$ in Chern-Simons | **Option (1)** Use the definition $(\omega \otimes S) \wedge (\eta \otimes S) = (\omega \wedge \eta) \otimes (S\otimes T)$ of the wedge product for Lie algebra valued forms. Define Lie bracket and Killing form as bilinear maps $[S\otimes T] = [S,T]$ and $\langle S \otimes T \rangle = \langle S, T\rangle$. Then the form... | 13 | https://mathoverflow.net/users/2622 | 207729 | 99,242 |
https://mathoverflow.net/questions/207738 | 7 | Throughout the question ITTM refers to Hamkins' infinite Turing machines, though I will be interested in results related to stronger models.
Recently I was wondering, is it consistent that there is an ITTM which recognizes a set of reals which is not Lebesgue measurable? Here with "recognizable" I mean that the set h... | https://mathoverflow.net/users/30186 | Can ITTM recognize a non-measurable set? | Nice question!
The answer is no, because if a set of reals $A$ is infinite time semi-decidable, then it is $\Delta^1\_2$, and not only $\Delta^1\_2$, but *absolutely* $\Delta^1\_2$, in the sense that the equivalent $\Sigma^1\_2$ and $\Pi^1\_2$ characterizations continue to be equivalent in all forcing extensions. Thi... | 12 | https://mathoverflow.net/users/1946 | 207746 | 99,249 |
https://mathoverflow.net/questions/207721 | 5 | Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in V\big\}$.
We endow $[X]^2$ with the topology $[\tau]^2$, which is generated by $\{[U,V]: U,V\in \tau\land U\cap V =\emp... | https://mathoverflow.net/users/8628 | Hausdorff space $X$ with $X\cong [X]^2$ | It is easy to see that the operation $X \mapsto [X]^2$ preserves the properties: countable, second-countable, regular, no isolated points. Hence $\mathbb{Q} \cong [\mathbb{Q}]^2 $.
| 12 | https://mathoverflow.net/users/17836 | 207752 | 99,251 |
https://mathoverflow.net/questions/207763 | 8 | Are there some outstanding results using some version of Helly's theorem in a totally different area (whatever that means) than convex geometry?
| https://mathoverflow.net/users/34575 | Helly's theorem in other areas of mathematics | * Applications of Helly's theorem to linear programming are discussed
in this [thesis](http://web.cs.ucdavis.edu/~amenta/pubs/ninathesis.pdf).
* Then there are applications to the theory of approximation of
continuous functions by polynomials ([Chebyshev
approximation](http://www.maths.manchester.ac.uk/~khudian/Etudes/... | 8 | https://mathoverflow.net/users/11260 | 207770 | 99,257 |
https://mathoverflow.net/questions/207764 | 10 | The following problem is somehow hidden in [this recently asked question](https://mathoverflow.net/questions/204020/is-the-set-aaa-always-at-least-as-large-as-aa), but I believe that it deserves to be asked explicitly.
>
> Is it true that for any finite set $A$ of real numbers, and any real $\lambda\notin\{0,-1\}$,... | https://mathoverflow.net/users/9924 | Sumsets and dilates: does $|A+\lambda A|<|A+A|$ ever hold? | It seems the energy version is true if make the additional assumption that $\lambda=c/d$ is rational, meaning that $T\_A(\lambda)$ counts the number of solutions in $A$ to
$$d(a\_1-a\_2)-c(a\_3-a\_4)=0 \, \, \, \, \, \, (\*).$$ Using a Freiman isomorphism together with translation invariance, we can assume that $A$ is ... | 4 | https://mathoverflow.net/users/405 | 207773 | 99,260 |
https://mathoverflow.net/questions/207414 | 2 | Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma\_1(N) \cap \Gamma\_0(p)$. I seem to recall the existence of a theorem to the following effect:
>
> Let $\rho\_{f,p}: G\_{\mathbf{Q}\_p} \to \mathrm{GL}\_2(\bar{\mathbf{F}... | https://mathoverflow.net/users/10547 | Level-Lowering in Weight 2 | The answer is yes. This result (and more) is contained in Theorem 6.4 of Diamond's "The refined conjecture of Serre".
| 2 | https://mathoverflow.net/users/10547 | 207784 | 99,267 |
https://mathoverflow.net/questions/207743 | 7 | Let $T\_G(x,y)$ denote the Tutte polynomial of a graph. Of course we may have $T\_G(x,y) = T\_H(x,y)$ for $G$ and $H$ non-isomorphic graphs.
Now let $c(G)$ denote the *cone graph* of $G$, i.e., the graph obtained from $G$ by adding a new vertex connected by an edge to every vertex of the original graph $G$. Let $c^{n... | https://mathoverflow.net/users/25028 | Does the Tutte polynomial of iterated cone graphs detect isomorphism? | I don't think that's true - I think the idea from [this](http://arxiv.org/abs/1306.0864) paper works out.
Let $G$ be a simple graph and let $H$ be a spanning subgraph of $G$ having connected components of order $h\_1 \geq h\_2 \geq \cdots \geq h\_k.$ We say that $(|E(H)|,h\_1,h\_2, \ldots, h\_k)$ is a s*ubgraph descr... | 5 | https://mathoverflow.net/users/1737 | 207786 | 99,268 |
https://mathoverflow.net/questions/207766 | 2 | It is a classical theorem. For given integer $n \ge 1$, among ${n\choose{n/2}} = 2^{(1-o(1)n)}$ strings in the cube $\{0, 1\}^n$ with weights $n/2$, i.e., $n/2$ indices are 1, there are at least $2^{cn}$ of these strings such that each pair has Hamming distance at least $n/4$, where $c$ is a constant between $0$ and 1.... | https://mathoverflow.net/users/8369 | Ask the name of a combinatorial theorem | I don't have a name for the theorem, but I can give a quick proof in case you don't get hold of a name: Let $S$ be a maximal set of $n/4$-separated strings in the weight $n/2$ slice. Then the union of $n/4$-Hamming balls centred at elements of $S$ covers the entire slice. But each Hamming ball has $\binom{n}{0}+\ldots+... | 3 | https://mathoverflow.net/users/11054 | 207789 | 99,269 |
https://mathoverflow.net/questions/207788 | 1 | Some papers I am reading talk about an "adelic" object $PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A})$ . This has sparked a lot of confusion since I don't know what such a quotient could mean.
A crude way of looking at the [adéles](https://mathoverflow.net/questions/106709/why-are-s-arithmetic-groups-interesting)... | https://mathoverflow.net/users/1358 | How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})$? | Yes, the quotient $\mathrm{PGL}\_2(\mathbb{Q})\backslash\mathrm{PGL}\_2(\mathbb{A})$ and its generalizations for other (reductive) algebraic groups is a complicated object, and this is to a large extent the reason why the theory of automorphic forms is a deep subject. The diagonal embedding of $\mathrm{PGL}\_2(\mathbb{... | 3 | https://mathoverflow.net/users/11919 | 207794 | 99,271 |
https://mathoverflow.net/questions/207793 | 3 | Let $X$ be a CW complex such that for all extraordinary homology theories, if you plug $X$ into them you get the same value as plugging in a point. Must $X$ be contractible?
| https://mathoverflow.net/users/74249 | Acyclic complexes for extraordinary cohomology theories | First of all I claim that asking that $X$ is acyclic for ordinary homology with integral coefficients is the same as asking that it is acyclic with respect to every homology theory. This is because of the Atiyah-Hirzebruch spectral sequence
$H\_p(X;E\_q) \Rightarrow E\_{p+q}X$
So if $X$ is acylic with respect to or... | 10 | https://mathoverflow.net/users/43054 | 207802 | 99,273 |
https://mathoverflow.net/questions/207675 | 1 | I have the two spaces $W\_0^{1,p}$ with the norme $$||u||^p=||u||^p\_{L^p}+||\nabla u||^p\_{L^p}$$ and $$L^{p^\*}\_{\alpha}=\{ u~\text{measurable}, \int\_{\Omega} (|x|^{\alpha} u(x)|)^{p^\*} dx<\infty\}$$ equiped with the norm $$||u||\_{L^{p^\*}\_{\alpha}}^{p^\*}=\int\_{\Omega} (|x|^{\alpha}|u(x)|)^{p^\*} dx$$
How to... | https://mathoverflow.net/users/49045 | Condition to obtain a not compact embedding | The failure of the compactness of the embedding $W^{1, p} (\Omega) \subset L^{p^\*} (\Omega)$ is local, that is, it can be exhibited in any ball.
An abstract way of seeing the failure of the embedding is to consider a ball $B$ such that $\overline{B} \subset \Omega \setminus \{0\}$. The compactness of the embedding ... | 1 | https://mathoverflow.net/users/42047 | 207813 | 99,277 |
https://mathoverflow.net/questions/207814 | 1 | Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in V\big\}$.
We endow $[X]^2$ with the topology $[\tau]^2$, which is generated by $\{[U,V]: U,V\in \tau\land U\cap V =\emp... | https://mathoverflow.net/users/nan | If $X$ is compact, is $[X]^2$ compact, too? | Not necessarily, because in some cases of $X$ Hausdorff, compact, infinite, $[X]^2$ is isomorphic to an open subset of $X\times X$ that is not closed (therefore not compact).
Here's an example. Let $I$ be the real interval $[0,1]$. Then it is not hard to prove that $[I]^2 \cong \{(x,y) \in I^2: x < y\}$. The latter s... | 4 | https://mathoverflow.net/users/8628 | 207815 | 99,278 |
https://mathoverflow.net/questions/207804 | 1 | Let $k$ be a field, and let $R$ be a finitely generated $k$-algebra. (If it helps, you may assume $R$ is an integral domain.) Let $I$ be an ideal of finite colength. Note that $A:=k+I$ is a subring of $R$. Indeed, it is the pullback of the diagram $k \hookrightarrow R/I \twoheadleftarrow R$. Here's my question: $$\text... | https://mathoverflow.net/users/19045 | finite generation of a certain type of subring | Yes. Both $R$ and $A=k+I$ are filtered by powers of $I$ and we may look at the associated graded rings.
Now $I/I^2$ is a finitely generated $R/I$ module and $R/I$ is a finitely generated vector space, by Zariski's lemma
on finitely generated $k$ algebras that are fields. So $I/I^2$ is a finitely generated vector space ... | 3 | https://mathoverflow.net/users/4794 | 207819 | 99,281 |
https://mathoverflow.net/questions/205930 | 13 | Two (not necessarily convex) poygons of equal area are scissor-congruent, i.e. both can be cut along a finite number of straight lines or segments into isometric pieces.
What can be said about the complexity of scissor congruence?
Remarks: The number of cuts cannot be bounded in terms of the number of sides:
A very... | https://mathoverflow.net/users/4556 | Complexity of planar scissor congruence | Let's assume you have a good linear bound $O(A)$ when both polygons are triangles of the same area. I am pretty sure you are right and the standard proof gives this. More precisely, a sequence triangle $\to$ parallelogram $\to$ square depends only on the smallest angle in a triangle, and thus on $A$.
Now use the fol... | 2 | https://mathoverflow.net/users/4040 | 207829 | 99,286 |
https://mathoverflow.net/questions/207452 | 6 | The general question has been treated [here](https://mathoverflow.net/questions/116123/how-to-find-define-eigenvectors-as-a-continuous-function-of-matrix), and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not only on th... | https://mathoverflow.net/users/13093 | Eigenvectors as continuous functions of matrix - diagonal perturbations | As @Joe Silverman said in his comment, the key here is that $\lambda$ is simple.
I copy here a few Theorems on the subject that might be helpful.
>
> From the book [Linear Algebra and Its Applications](http://rads.stackoverflow.com/amzn/click/0471751561) (by Peter D. Lax):
>
>
>
**Theorem 7, p.130:**
Let $A... | 3 | https://mathoverflow.net/users/54403 | 207833 | 99,288 |
https://mathoverflow.net/questions/207836 | 1 | I just found this related question in here [Q1](https://math.stackexchange.com/questions/1237061/change-of-eigenvectors-by-hadamard-product).
Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary matrix $\mathbf{B}$,
>... | https://mathoverflow.net/users/11825 | Hadamard Product and Eigendecomposition | There won't be a nontrivial closed form in general. A trivial "closed-form" is mentioned below.
The Hadamard product $A \circ B$ is a principal submatrix of the Kronecker product $A\otimes B$. Thus, let $A=U\Lambda U^{-1}$ and $B=VDV^{-1}$ (I assumed $B$ is diagonalizable and not fully arbitrary as required in the qu... | 7 | https://mathoverflow.net/users/8430 | 207840 | 99,290 |
https://mathoverflow.net/questions/207838 | 6 | Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps (between topological spaces). Assume these hypotheses:
* $f : X \to Y$ is a split surjection, i.e. has a section.
* $g \circ f : X \to Z$ is a local homeomorphism, i.e. there is an open cover $\{ U\_i : i \in I \}$ of $X$ such that, for each $i \in I$, the compos... | https://mathoverflow.net/users/11640 | Does the property of being a local homeomorphism descend through split surjections? | First, note that $f:X\to Y$ must be locally injective. Now choose a splitting of $f$ and consider $Y$ as a subspace of $X$ via this splitting. For any $y\in Y$, there then some neighborhood $U\subseteq X$ of $y$ on which $f$ is injective. But $f$ is the identity on $U\cap Y$, and so $f$ must map $U\cap (X\setminus Y)$ ... | 5 | https://mathoverflow.net/users/75 | 207843 | 99,291 |
https://mathoverflow.net/questions/207362 | 1 | Let $X$ be a complex Banach space. Does validity of
$$
\mbox{ker}\left(e^{2\pi \imath \, T} - 1\right) = \overline{\sum\nolimits\_{k\in \mathbb{Z}} \mbox{ker} (T-k) }\quad \forall \, T \in B(X,X)
$$
imply that $\mbox{dim}X<\infty$?
| https://mathoverflow.net/users/66292 | A nullspace identity for operator exponentials | No: every complex Banach space satisfies this condition, and you don't need the closure.
For any bounded linear operator $T$, $K = \sigma(T) \cap \mathbb Z$ is finite.
Write $$e^{2\pi i z} - 1 = g(z) \prod\_{k \in K} (z - k)$$
where $g(z)$ is analytic and nonzero on a neighbourhood of $\sigma(T)$. Therefore
by the h... | 1 | https://mathoverflow.net/users/13650 | 207847 | 99,292 |
https://mathoverflow.net/questions/207755 | 6 | Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free (i.e., all stabilizers are trivial) and minimal (i.e., every orbit is dense). I would like to show (or disprove) that if th... | https://mathoverflow.net/users/23661 | Free actions of non-amenable groups | Let $G$ be a non-compact locally compact amenable group which contains a dense countable group $\Gamma$ which is non-amenable as a discrete group (e.g., $G = E(3)$ by Banach-Tarski). Then the action of $\Gamma$ on $G$ is free and minimal since $\Gamma$ is dense, has an invariant mean since $G$ is amenable, and does not... | 7 | https://mathoverflow.net/users/6460 | 207851 | 99,293 |
https://mathoverflow.net/questions/207848 | 11 | It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda\_1(M)$ of the (positive) Laplace-Beltrami operator $\Delta$ satisfies
\begin{equation}
\lambda\_1(M) \geq \frac{\pi^2}{d^2}, \tag{1}
\en... | https://mathoverflow.net/users/13915 | Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold | The answer is no.
To see this consider two (2D) sphere's of constant curvature 1 and a (flat) cylinder of unit length and radius $\epsilon$. Cut out a disk from each sphere and glue in the cylinder (and smooth everything out in a small neighborhood of the gluing). By Gauss Bonnet this must introduce negative curvatur... | 10 | https://mathoverflow.net/users/66607 | 207852 | 99,294 |
https://mathoverflow.net/questions/207844 | 19 | This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that.
Let $X$ be a (nice) connected topological space. Let $LX=Map(S^1,X)$ be the free loop space and $\Omega X = Map\_\*(S^1,X)$ the su... | https://mathoverflow.net/users/50409 | When does the free loop space fibration split? | Let $X$ be a H-space with multiplication $\mu$ and unit $e$, let us consider the map
$$\Phi:\Omega\_e X\times X\rightarrow \mathcal{L}X$$
given by $\Phi(\omega,x)=\mu(\omega(t),x)$. This continuous map satisfies:
$$ev\_0(\Phi(\omega(t),x))=\mu(\omega(0),x)=0=pr\_2((\omega,x)).$$
Then it induces a morphism of fibrations... | 19 | https://mathoverflow.net/users/27816 | 207856 | 99,295 |
https://mathoverflow.net/questions/207405 | 6 | Is the braid group with $n$ strings $\mathcal{B}\_n$ known to be a lattice in a connected semi-simple Lie group ? (for $n$, say, bigger than $3$)
Or is it known that it cannot be such a lattice ?
| https://mathoverflow.net/users/25511 | Is the braid group with $n$ strings $\mathcal{B}_n$ a lattice in a connected semi-simple Lie group? | **Proposition:** *no finite index subgroup of $B\_n$, for any $n\ge 4$, is isomorphic to a lattice in any virtually connected Lie group.*
Assume that some finite index subgroup $H$ of $B\_n$, $n\ge 4$, is a lattice in a virtually connected Lie group $G$. Modding out if necessary, we can assume that the compact radic... | 5 | https://mathoverflow.net/users/14094 | 207866 | 99,298 |
https://mathoverflow.net/questions/207751 | 8 | Let $L$ be a (non-restricted) Lie algebra over a field of prime characteristic $p,$ $UL$ be its universal enveloping algebra and $a\_1,\dots, a\_p \in L$ (the number of elements is equal to the characteristic). I can prove that the element $\sum\_{\sigma \in S\_p} a\_{\sigma(1)}\cdot{}\dots{}\cdot a\_{\sigma(p)}$ of $U... | https://mathoverflow.net/users/23310 | One identity in Lie algebras | If you expand the commutator by the possible $2^{n-1}$ sign choices you get
$$[a\_1,a\_2,\dots,a\_n]=\sum\_{r=1}^{n-1}(-1)^r\sum\_{k\_1> k\_2>\cdots> k\_r>1}a\_{k\_1}a\_{k\_2}\cdots a\_{k\_r}W(k\_1,k\_2,\dots,k\_r)$$
where the expression $W(k\_1,k\_2,\dots,k\_r)$ stands for the word $a\_1a\_2\cdots a\_n$ with the lett... | 5 | https://mathoverflow.net/users/2384 | 207876 | 99,300 |
https://mathoverflow.net/questions/207801 | 39 | Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E\_\infty$ ring spectra over $\bar{\mathcal M}\_\text{ell}$, the (derived) compactified moduli stack of elliptic curves. Further, Behrens and Law... | https://mathoverflow.net/users/48554 | What can topological modular forms do for number theory? | Here are two applications that I know of which involve direct crossover.
---
In section 5 of the [Hopkins ICM address](http://arxiv.org/abs/math/0212397v1) referenced in Drew's answer to the other question, he gives a topological proof of the following congruence originally [due to Borcherds](https://math.berkele... | 16 | https://mathoverflow.net/users/360 | 207880 | 99,303 |
https://mathoverflow.net/questions/207807 | 5 | Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi\_{8s+1}(SO) = \mathbb Z/2$ in $\pi\_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres.
Now regard $\pi\_{8s+1}^s = \pi\_{8s+1} ((B\Sigma\_{\infty})^+)$ , where $\Sigma\_{\infty}$ is the group of compactly supported permutations of $\mathbb N$.
... | https://mathoverflow.net/users/14233 | Map between homotopy groups of O, related to J-homomorphism and K-theory of Z | The composition is trivial. Composition with $\eta$ acts trivially on the source and nontrivially on the target, for positive $s$, which implies the claim. This argument does not apply for $s=0$, because $J : SO \to QS^0$ does not deloop.
| 6 | https://mathoverflow.net/users/9684 | 207882 | 99,304 |
https://mathoverflow.net/questions/207812 | 7 | Let $X\_k$ be $\mathbb{P}^2$ blown up at $k$ points (where $k$ is $0$ to
$8$). Let $\beta \in H\_2(X\_k, \mathbb{Z})$ be the homology class given by
$$ \beta := n L + m\_1 E\_1 + \ldots + m\_k E\_k $$
where $L$ is the homology class of a line and $E\_i$ are the exceptional divisors.
Also, define
$$ \delta\_{\beta}... | https://mathoverflow.net/users/4463 | Are genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative? | I believe that the paper <http://arxiv.org/abs/alg-geom/9611012> by Pandharipande and Gottsche addresses exactly this question. The short answer is yes, the genus 0 invariants are enumerative up to k=8.
For k>8, one can still conclude that the genus 0 GW invariants are *weakly* enumerative, meaning that for $\delta\... | 6 | https://mathoverflow.net/users/9617 | 207893 | 99,308 |
https://mathoverflow.net/questions/207885 | 4 | I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}\_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch theorem. Does anyone know where I could find such a reference? Much thanks in advance.
| https://mathoverflow.net/users/74301 | Reference request, zeta function is rational function via Riemann-Roch? | See Theorem 4.1.11 and its proof in Niederreiter-Xing: Algebraic geometry in coding theory and cryptography.
Alternately, one can observe that the Riemann-Roch theorem in this setting is equivalent to the Poisson summation formula for the adeles of the function field (cf. Theorems 7-10 and 7-12 in Ramakrishnan-Valenz... | 12 | https://mathoverflow.net/users/11919 | 207894 | 99,309 |
https://mathoverflow.net/questions/207895 | 7 | Assuming that $p$ is an odd prime. How many primes have the form $(2^p+1)/3$? Is the number finite? Mathematica calculation shows that there are 23 such primes when $p$ ranges over the first 500 primes. Here are those primes $p$,
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199,
313, 347, 701, 17... | https://mathoverflow.net/users/56827 | How many primes have the form $(2^p+1)/3$? | I don't think anyone is going to answer this question, any more than anyone is going to decide the number of Mersenne primes any time soon, so for the sake of having an answer, I'll note that these numbers are tabulated at the [Online Encyclopedia of Integer Sequences](http://oeis.org/A000978) where they are called "Wa... | 10 | https://mathoverflow.net/users/3684 | 207896 | 99,310 |
https://mathoverflow.net/questions/207879 | 7 | A poset $\mathbb{P}$ is called *well-met* iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is $\lambda$-directed closed (for some regular infinite cardinal $\lambda$). Can we always view $\mathbb{P}$ as a dense subo... | https://mathoverflow.net/users/26319 | Introducing meets while preserving directed closure | $\newcommand\P{\mathbb{P}}$The answer is no. For a counterexample, consider the following
partial order $\P$. On the bottom layer, we have countably many
incompatible atoms $a\_n$ for $n<\omega$. On a second layer, we have a
collection of pairwise-incomparable elements $b\_k$, with $a\_n<b\_k$
just in case $k\neq n$. S... | 7 | https://mathoverflow.net/users/1946 | 207901 | 99,314 |
https://mathoverflow.net/questions/207900 | 3 | From my previous [question](https://mathoverflow.net/questions/207885/reference-request-zeta-function-is-rational-function-via-riemann-roch), I know that$$\zeta\_X(s) = {{P(u)}\over{(1-u)(1-qu)}}$$for some polynomial $P(u)$ of degree $2g$, where
* $X$ is the set of all places of $F$, a function field in one variable ... | https://mathoverflow.net/users/74301 | Question about zeta function of function field in 1 variable over $\mathbb{F}_q$ | Yes. From the functional equation (Theorem 4 in Chapter VII-6 of Weil: Basic number theory) we know that $P(u)=q^gu^{2g}P((qu)^{-1})$. We also know that (see same theorem) that $P(1)=h$. Hence
the limit in question equals
$$ \frac{P(q^{-1})}{1-q^{-1}}=\frac{q^{-g}P(1)}{1-q^{-1}}=\frac{P(1)}{q^{g-1}(q-1)}=\frac{h}{q^{g-... | 7 | https://mathoverflow.net/users/11919 | 207902 | 99,315 |
https://mathoverflow.net/questions/207765 | 1 | The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific representatives in terms of the so called Schubert polynomials. Does anyone know of a presentation of this material for the s... | https://mathoverflow.net/users/41562 | Schubert Polynomials for Complex Projective Space | In general for a Grassmannian $Gr\_k(\mathbb C^n)$, the Schubert polynomials are Schur polynomials in $k$ variables, one for each partition $\lambda$ in a $k\times (n-k)$ rectangle. In this case $S\_\lambda = \sum\_{SSYT(\lambda)} \prod\_{content} x\_{entry}$, where the sum is over semistandard Young tableaux with entr... | 2 | https://mathoverflow.net/users/391 | 207912 | 99,318 |
https://mathoverflow.net/questions/207870 | 0 | We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber at $0$?
In particular, how should the moment map images of the one-dimensional $T-$orbits change?
Any suitable re... | https://mathoverflow.net/users/7780 | Moment maps and flat degenerations of toric varieties | I assume you mean that $T$ acts preserving each fiber. Then the flatness says that the multigraded Hilbert polynomial is constant. As the Duistermaat-Heckman measure is the leading-order behavior of the multigraded Hilbert polynomial, it too is constant. Then the moment polytope is the support of the DH measure, so it,... | 4 | https://mathoverflow.net/users/391 | 207913 | 99,319 |
https://mathoverflow.net/questions/207830 | 11 | Clearly the title needs clarifying. Allow me to let "how many" to mean a set larger than a skeleton of the category of Fréchet manifolds and smooth maps, if this category is indeed essentially small.
Since I suspect this category is *not* essentially small, I would be satisfied with knowing the answer to the question... | https://mathoverflow.net/users/4177 | How many Fréchet manifolds are there? | Andrew Stacey points out elsewhere that arbitrary Hilbert spaces have smooth partitions of unity, so that criterion is not enough to get an essentially small category of Fréchet manifolds. For the purposes of considering mapping spaces between finite-dimensional manifolds, however, arbitrary Hilbert spaces are a bit of... | 1 | https://mathoverflow.net/users/4177 | 207916 | 99,320 |
https://mathoverflow.net/questions/207915 | 0 | In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom\_T(X,\Omega Y)\to Hom\_T(SX,Y)
$$
is defined. And a map
$$
\eta\_n=\phi^{-n}(1\_{S^nX}):X\to \Omega^n S^nX.
$$
Question: I do not quite understand what does $\eta\_n$ mean? What means $\ph... | https://mathoverflow.net/users/41075 | iterated loop spaces and configuration spaces | $\Phi^n:Hom(X,\Omega^nS^nX)\to Hom(SX,\Omega^{n-1}S^nX)\to \dots\to Hom(S^{n-1}X,\Omega S^nX) \to Hom(S^nX,S^nX)$.
| 1 | https://mathoverflow.net/users/26935 | 207920 | 99,322 |
https://mathoverflow.net/questions/207821 | 1 | Let $(0\in X)$ be the germ of a normal surface singularity and let $f: Y \to X$ be the minimal resolution.
**Questions>**
(1) How can I define a map $f\_\*\mathcal{O}\_Y(K\_Y)\hookrightarrow \mathcal{O}\_X(K\_X)$?
(2) If $f\_\*\mathcal{O}\_Y(K\_Y)=\mathcal{O}\_X(K\_X)$ and $K\_X$ is Catier, then $f^\*\mathcal{O}\... | https://mathoverflow.net/users/64057 | canonical divisors of a resolution of a normal surface singularity | Concerning (1), the map in question can be described in more down-to-earth terms as follows. Let $E = \sum E\_i \subset Y$ be the exceptional divisor of $f$. A section of $f\_\* \mathcal O\_Y(K\_Y)$ is by definition nothing but a section of $\mathcal O\_Y(K\_Y)$, which you can restrict to $Y \setminus E$. As $f$ restri... | 0 | https://mathoverflow.net/users/44860 | 207928 | 99,326 |
https://mathoverflow.net/questions/207937 | 8 | An immediate consequence of Bott periodicity is that the infinite unitary space is an infinite loop space and so an $E\_\infty$-space. I wonder if there is a direct proof (not using $U = \Omega^2 U$) giving an explicit construction of an $E\_\infty$-operad actiong on $U$? I'm also interested in the case of the infinite... | https://mathoverflow.net/users/51663 | Direct proof that $U$ is an $E_\infty$-space | The linear isometries operad is the $E\_\infty$ operad that most naturally acts on $O(\infty):=colim\_n O(n)$.
The $n$-ary operations in that operad is the space of linear isometric maps (not necessarily surjective) from $\underbrace{\mathbb R^\infty\oplus\ldots \oplus\mathbb R^\infty}\_{n}$ to $\mathbb R^\infty$.
... | 16 | https://mathoverflow.net/users/5690 | 207938 | 99,328 |
https://mathoverflow.net/questions/207862 | 0 | Is this group known outside of the stable range? If so, what is it? If not, what is known about it?
| https://mathoverflow.net/users/74288 | The kernel of the natural map $\pi_k(BU(r)) \to \pi_k (BU)$ | For the homotopy groups of $U(r)$, hence also of $BU(r)$, you can look at the survey by M. Mimura, "Homotopy theory of Lie groups", which is chapter 19 in the "Handbook of Algebraic Topology", edited by I. M. James. It gives $\pi\_k(SU(n))$ for $k\le15$ and for $k\le 2n+6$, in section 3. The survey also lists many furt... | 5 | https://mathoverflow.net/users/9684 | 207944 | 99,330 |
https://mathoverflow.net/questions/207845 | 40 | I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x\_1,x\_2, \ldots , x\_n, y\_1,y\_2,\ldots, y\_n>0$ such that for all
$k\in\{1,\ldots, n-1\}$
it holds
$\sum\_{i\_1<\ldots<i\_k} x\_{... | https://mathoverflow.net/users/74281 | The sum of squared logarithms conjecture | Is there anything wrong with the following argument?
First of all, by scaling all $x\_i$ and all $y\_i$ by a positive constant, we may safely assume that $\prod\_i x\_i = \prod\_i y\_i =1$.
The result would now follow from the following more general conjecture.
$\bf Conjecture:$ For ${\bf a}=(a\_1,\ldots,a\_{n-1}... | 33 | https://mathoverflow.net/users/38468 | 207950 | 99,332 |
https://mathoverflow.net/questions/207931 | 3 | Let $G,H$ be infinite simple undirected graphs with the property that for any graph $X$ we have $|\text{Hom}(X,G)| = |\text{Hom}(X,H)|$. Does this imply that $G$ is isomorphic to a subgraph of $H$, and vice versa?
(Note that in the finite case the condition above implies $G\cong H$, see [here](http://www.maths.qmul.a... | https://mathoverflow.net/users/8628 | Infinite graphs with "similar" Hom-sets | Did you want both G to be a subgraph of H and H to be a subgraph of G? Then I think the answer is negative. For example if G is infinite and have no isolated vertices and H is G together with an isolated vertex. If X have no isolated vertices the Hom-sets are the same. The set of maps from the set of isolated vertices ... | 6 | https://mathoverflow.net/users/6066 | 207951 | 99,333 |
https://mathoverflow.net/questions/207954 | 10 | I have the book "Handbook of Computational Group Theory", by Derek Holt, and in it is a section on finding the transversal of a subgroup. Recall a transversal of a subgroup $H$ of $G$ is a single representative from each of the cosets of $H$ in $G$ (either left or right cosets). Two methods are given to find a transver... | https://mathoverflow.net/users/73518 | Computing a transversal of a subgroup $H$ of $G$ in expected $O(|G : H|^2 \log |G : H| + |H|)$ time | I guess I should try and answer that!
I don't really have a good answer to the question of why the random method is not mentioned in the book, and I would agree that it might to be the fastest method if you simply want to compute and store a complete transversal.
I guess the point is that in many applications you w... | 10 | https://mathoverflow.net/users/35840 | 207960 | 99,336 |
https://mathoverflow.net/questions/207962 | 3 | A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal precategory satisfying the Segal condition.
Namely, we want the natural maps $$\phi\_k:X\_k \to X\_1 \times\_{X\_0} \cdot... | https://mathoverflow.net/users/57280 | Segal maps for Segal precategories | For any $a,b \in X\_0$, let $X\_1(a,b)$ denote the subspace of $X\_1$ lying over $(a,b) \in X\_0 \times X\_0$. Since $X\_0$ is discrete, this iterated fiber product breaks up as a disjoint union of product spaces:
$$
\bigcup\_{(a\_0, \ldots, a\_k)} X\_1(a\_{k-1}, a\_k) \times \cdots \times X\_1(a\_0,a\_1)
$$
Generally,... | 4 | https://mathoverflow.net/users/360 | 207968 | 99,340 |
https://mathoverflow.net/questions/207875 | 7 | Is the following fact true?
>
> Let $N$ be a finitely generated nilpotent group, and denote its lower central series by $(N\_r)\_{r\ge 1}$, that is, $N\_1=N$ and $N\_{k+1}=[N,N\_k]$ is the commutator group of $N$ and $N\_k$. Then there is a finite index subgroup $H$ of $N$ which has the following property - if $H\... | https://mathoverflow.net/users/56465 | Quotients of finitely generated nilpotent groups | I believe that for class $2$ and the original question (finding a subgroup $H$ of finite index such that $H\_i/H\_{i+1}$ is torsionfree for all $i$) the following works.
Suppose that $N$ is of class $2$. Write $N^{\rm ab} = C\_{k\_1}\oplus\cdots \oplus C\_{k\_r}\oplus C\_0^{s}$, where $C\_k$ is the cyclic group of or... | 4 | https://mathoverflow.net/users/3959 | 207972 | 99,343 |
https://mathoverflow.net/questions/207973 | 8 | Let $X$ be an open subset of $\mathbb{R}^n$. Following the notation of Schwartz, we denote $\mathcal{D}$ the space of compactly supported complex-valued smooth functions on $X$ equipped with the topology defined by the following convergence condition: a sequence of functions $\{f\_i\}$ is said to converge to $f$ in $\m... | https://mathoverflow.net/users/74343 | Topologies on spaces of distributions and test functions | They are all continuous with dense images. See e.g. Trèves, *Topological Vector Spaces, Distributions and Kernels*, [p. 272](https://books.google.com/books?id=kClvQ1qk9r8C&pg=PA272), Theorem 28.2, [p. 301](https://books.google.com/books?id=kClvQ1qk9r8C&pg=PA301), and Remark 28.3, [p. 303](https://books.google.com/books... | 6 | https://mathoverflow.net/users/19276 | 207977 | 99,344 |
https://mathoverflow.net/questions/207985 | 9 | For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega\_1$-tree with all levels finite has a cofinal branch. A similar fact holds also for much taller trees with all levels finite. I used... | https://mathoverflow.net/users/1946 | Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch? | The following theorem of Kurepa answers the question.
**Theorem (Kurepa)** Suppose that $\kappa$ is regular, $\lambda< \kappa$, and $T$
is a $\kappa$-tree each of whose levels has cardinality less than $\lambda$. Then $T$ has a
cofinal branch.
The theorem is stated in Kanamori's book ``The higher infinite'', as [P... | 12 | https://mathoverflow.net/users/11115 | 207986 | 99,346 |
https://mathoverflow.net/questions/207842 | 3 | Suppose we have a large collection of standard normal random variables $a\_i\in\mathbb{R}^n$. We know by standard concentration results that if we take $m \geq C\left(t/\epsilon\right)^2n$ samples, then with probability greater than $1-\exp\left(-ct^2n\right)$ we have
$$
\|\frac{1}{m}\sum\_{i=1}^ma\_ia\_i^T-Id\|\_{op} ... | https://mathoverflow.net/users/57784 | Concentration and Correlation for Magnitudes of Gaussian Vectors | In case anyone is interested in situations like this, the answer is **yes**. The proof relies on an $\epsilon$-perturbation of the following lemma:
Suppose $w,x,y\in\mathbb{R}^m\_+$ are three vectors on the probability simplex which satisfy
$$
\langle w, x \rangle < \alpha\langle w, y\rangle
$$
for some $\alpha < 1$... | 0 | https://mathoverflow.net/users/57784 | 207993 | 99,347 |
https://mathoverflow.net/questions/207996 | 4 |
>
> (1) I am looking for an example of a u.p (unique product) group which is not right orderable (RO).
>
>
>
Almost any group I pick up (obviously torsion-free, as u.p. group cannot have nontrivial torsion elements) turns out to be RO, whether it be easy groups like integers, modulo $n$ etc, or groups on matric... | https://mathoverflow.net/users/59205 | Unique product group which is not right orderable | Such a group has been found by N. Dunfield, see the appendix to
the paper
* Steffen Kionke, Jean Raimbault, Nathan Dunfield, *On geometric aspects of diffuse groups*, Documenta Mathematica, Vol. 21 (2016), 873-915, [journal](https://www.math.uni-bielefeld.de/documenta/vol-21/24.html), arXiv:[1411.6449](https://arxiv.... | 7 | https://mathoverflow.net/users/32210 | 208001 | 99,350 |
https://mathoverflow.net/questions/208007 | 1 | I apologize if this is not a research level question (already tried asking <https://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion> stack exchange with no response), but it's not a homework question either: an applied mathematics/medical imaging paper I was reading tr... | https://mathoverflow.net/users/35936 | Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true? | The answer is yes. Since the Jacobi field satisfies the second order equation $$ J(s)'' + R(s)J(s)=0,$$ where $R(s)$ is the curvature tensor (details are skipped here); then for $J\_{t,W(t)}(s)$ it holds $J(s)=(s-t)W(s) + (s-t)^3 R(s)W(s)/6$ up to higher orders of $(s-t)$ as $s-t$ goes to 0 ($J(t)=0$ is used here). So,... | 1 | https://mathoverflow.net/users/1988 | 208010 | 99,353 |
https://mathoverflow.net/questions/208011 | 0 | How do I Calculate, if possible, in terms of well-known constants the integral :
$\int\_{0}^{1}x^{k}\psi(x)dx$ , where $k\geq 3$ is an integer ?
note: $\psi(x)$ is digamma function.
Any help would be greatly appreciated.
| https://mathoverflow.net/users/51143 | How do I Calculate :$\int_{0}^{1}x^{k}\psi(x)dx$ where $k\geq 3$ is an integer? | This integral has been observed by Donal F. Connor in 2010 (you can find the link [here](http://arxiv.org/abs/1005.3469), pg. 94). As far as I know, he found a closed form for the odd case, but I believe the even case is somewhere in that document (don't quote me on that yet). To solve it, as Feldmann Denis notes, use ... | 5 | https://mathoverflow.net/users/70508 | 208018 | 99,356 |
https://mathoverflow.net/questions/207998 | 13 | Let $B$ be a symmetric bilinear form over a Euclidean space $E$. Say that $|B(v,v)|\le c\|v\|^2$ for every $v\in E$, for some $c\ge0$. Then
$$4B(v,w)=B(v+w)+B(v-w)$$
yields $2|B(v,w)|\le c(\|v\|^2+\|w\|^2)$. Replacing $v$ by $tv$, $w$ by $t^{-1}w$ and minimizing over $t$, we get $|B(v,w)|\le c\|v\|\cdot\|w\|$. Notice t... | https://mathoverflow.net/users/8799 | Norm of $n$-linear symmetric forms | After a bit of searching, I found that $\gamma\_n=1$ for all $n$.
This is known as "van der Corput-Schaake inequality" (1935), discovered before by Szegö (1928), and mentioned to have been known to Banach in the Scottish Book.
The proof proceeds by showing that if $P$ is homogeneous of degree $n$ on a euclidean s... | 13 | https://mathoverflow.net/users/6451 | 208025 | 99,358 |
https://mathoverflow.net/questions/153227 | 1 | Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br\_{\mathfrak{g}}$ is generated by the $T\_i$ , $i \in I$ such that for all $i, j \in I$,
\begin{align}
\underbrace{T\_i T\_j \cdots}\_{m\_{ij}} = \underbrace{T\_j T\_i \cdots}\_{m\_{ij}},
\end{align}
where $m\_{ij}$ is the $(i,j)$-ent... | https://mathoverflow.net/users/11877 | Reference request: Lusztig's symmetries | The result $T\_{w\_0 s\_i} F\_i = F\_{i^\*} $ is a special case of Chari-Pressley, *A Guide to Quantum Groups*, Proposition 8.1.6. Presumably, if you look in the book you will find a reference to one of Lusztig's papers. In my paper with Peter Tingley, "The crystal commutor and Drinfeld's unitarized R-matrix", we used ... | 3 | https://mathoverflow.net/users/438 | 208026 | 99,359 |
https://mathoverflow.net/questions/208027 | 2 |
>
> Let $P(n)$ be a quadratic trinomial with integer coefficients.
> For each positive integer $n$, the number $P(n)$ has a proper divisor $d\_{n}$
> (i.e. $1 < d\_{n} < P(n)$), such that the sequence $d\_{1},d\_{2}, d\_{3},\ldots$ is increasing. Prove that either:
>
> i) $P(n)$ is the product of two linear po... | https://mathoverflow.net/users/70464 | Divisors of a quadratic trinomial | Assume the contrary.
At first, $P(n+2)-3P(n+1)+3P(n)-P(n-1)=0$, thus if the same number $m>1$ divides three consecutive values of $P$, it divides all values of $P$ and case ii) takes place.
Denote $P(n)=d\_na\_n$, $\ell(n)=P(n+1)-P(n)$ is a linear function. If $\ell(x)$ divides $P(x)$ as a polynomial, case i) take... | 4 | https://mathoverflow.net/users/4312 | 208032 | 99,361 |
https://mathoverflow.net/questions/204357 | 4 | Let $\mu, \mu\_1, \mu\_2, \dots$ be random measures on
a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$.
Suppose I know that
$$\int f d \mu\_n \to \int f d\mu$$
*in probability* for each bounded continuous real-valued function. This would be the definition of weak convergence ... | https://mathoverflow.net/users/64663 | convergence of integral for each bounded function in probability | It is interesting if you let the random index set depend on the realizations. For simplicity, restrict attention to random sequences $\{X\_1, X\_2, X\_3, \ldots\}$ that converge to 0 in probability, but not with probability 1. We say that a set $B$ contains almost all positive integers if:
$$ \lim\_{n\rightarrow\infty... | 0 | https://mathoverflow.net/users/73850 | 208048 | 99,365 |
https://mathoverflow.net/questions/208046 | 2 | Is there any lower bound for the order of a group with an irreducible character of degree $p$, where $p$ is a prime.
Is there any similar result for $p^2$ or $p^3$ instead of $p$?
Thanks for your helps
| https://mathoverflow.net/users/31045 | The lower bound of a group with characters of special degrees | Let me try to answer in some detail the case $p$. So $G$ is a finite group of minimal possible order subject to having a complex irreducible character $\chi$ of degree $p$, where $p$ is a chosen prime. Note that $\chi$ is necessarily faithful by minimality of $|G|$. As noted in my comment, by elementary character theor... | 8 | https://mathoverflow.net/users/14450 | 208057 | 99,369 |
https://mathoverflow.net/questions/208071 | 19 | Is $$\sum\_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$
In a slightly different vein, given a sequence of real numbers $\{a\_n\}\_{n=0}^\infty$, what are some necessary and sufficient conditions for $\sum a\_nz^n$ to be in $\mathbb{C}(z)$ with all poles simple?
| https://mathoverflow.net/users/38889 | Is this a rational function? | The function
$$ f(z)=\sum\_{n=1}^{\infty} \frac{z^n}{2^n-1} $$
defines a holomorphic function for $|z|<2$, and it satisfies
$$ f(2z) = f(z)+\frac{z}{1-z} $$
for $|z|<1$. Based on this identity, it is easy to prove that $f(z)$ extends to a meromorphic function on $\mathbb{C}$, and the set of poles is $\{2^n:\ n=1,2,\dot... | 44 | https://mathoverflow.net/users/11919 | 208075 | 99,375 |
https://mathoverflow.net/questions/208073 | 0 | Let $G$ be a finitely generated profinite group, and $p$ a prime number. Suppose that there exists some open pro-$p$ subgroup $H \leq\_o G$. Must $G$ have only finitely many maximal open subgroups?
Note that the case $H = G$ is trivial: A pro-$p$ group of rank $d$ has $\leq \frac{p^d-1}{p-1} < \infty$ maximal open su... | https://mathoverflow.net/users/38889 | Is the Frattini subgroup of a f.g virtually pro-p group open? | Yes, because $G$ is top. f.g. and its maximal subgroups have bounded index.
Let me prove the latter fact. Let $K$ be the core (=intersection of conjugates) of $H$, so $K$ is an open normal pro-$p$-subgroup. Let $M$ be a maximal open subgroup in $G$. Then either $MK=M$ or $MK=G$. In the first case, $K\subset M$, which... | 3 | https://mathoverflow.net/users/14094 | 208084 | 99,378 |
https://mathoverflow.net/questions/208092 | 1 | Fix a topological space $X$. Now consider a functor from the fundamental groupoid of $X$ to the category $Vect$. In other words, we assign a vector space to each point of $X$, we allow ourselves to flow our vectors around our space, and this flow is consistent with respect to homotopy.
This seems to give us a vector... | https://mathoverflow.net/users/57044 | Trying to relate the fundamental groupoid to vector bundles | This does give you a vector bundle, and it comes with a flat (i. e. path homotopy invariant) parallel transport. You cannot get all vector bundles, but you can get exactly the ones with such a parallel transport. (On a manifold, those are exactly the ones that admit a flat connection)
An easy way to justify this is t... | 6 | https://mathoverflow.net/users/39747 | 208094 | 99,382 |
https://mathoverflow.net/questions/207979 | 1 | Suppose we have a map of $E\_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there should be a lax $E\_{n-1}$-monoidal functor of quasicategories (where the domain is clearly a Kan complex): $$BX\to Top$$... | https://mathoverflow.net/users/11546 | Construction of Highly Structured Quotient Groups in Quasicategories | Your functor $BX \to \mathrm{Top}$ factors through the inclusion of the full subcategory of $\mathrm{Top}$ generated by the single object $Y$, which is just the data of $\mathrm{End}(Y)$ (an $E\_1$-algebra) -- you might call this subcategory $B(\mathrm{End}(Y))$. By adjunction, the multiplication on $Y$ gives a map $Y ... | 3 | https://mathoverflow.net/users/303 | 208104 | 99,387 |
https://mathoverflow.net/questions/208091 | 12 | I am currently going through Philip Walder's ["Proposition as Types"](http://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-types/propositions-as-types.pdf) and a passage of the introduction has struck me:
>
> for each way to simplify a proof
> there is a corresponding way to evaluate a program
>
>
>
Wh... | https://mathoverflow.net/users/74413 | What does "simplification of proofs as evaluation of programs" mean? | There are two inference rule of propositional logic involving implication (which I write using $\to$ instead of $\Rightarrow$):
1. The *introduction* rule says that we can prove $A \to B$ if we derive $B$ from the assumption $A$ (there are brackets around $[A]$ because $A$ here is a "temporary" assumption which is "d... | 17 | https://mathoverflow.net/users/1176 | 208113 | 99,390 |
https://mathoverflow.net/questions/208033 | 2 | Suppose that $(A,\mathfrak{m})$ is a complete intersection and $\mathbf{x}$ is a minimal basis for $\mathfrak{m}$. Consider the Koszul homologies $H\_\bullet(\mathbf{x},A)$. It is well-known that $\text{Vdim}\_{K}(H\_1(\mathbf{x},A))=\text{embdim}(A)-\text{dim}(A)$. My question is that do we have some information about... | https://mathoverflow.net/users/23240 | A question about Complete Intersections | If $n=\text{dim}\_{K}(H\_1(\mathbf{x},A))$ then $\text{dim}\_{K}(H\_2(\mathbf{x},A))=\frac{n(n-1)}{2}$. This was proved by [Assmus](https://projecteuclid.org/download/pdf_1/euclid.ijm/1255455121) in 1958. In general, $H\_\*(\mathbf{x},A)$ is the exterior algebra over $H\_1(\mathbf{x},A)$, even under weaker hypothesis (... | 2 | https://mathoverflow.net/users/36672 | 208120 | 99,395 |
https://mathoverflow.net/questions/208054 | 4 | The basic set up is the following:
Let $K$ be a number field. let $p$ be an odd prime. Let $\Sigma\_p$ be the set of primes of $K$ lying above $p$. Let $M$ be the composite of all finite $p$-extensions of $K$ which are unramified outside $\Sigma\_p$. $M^{ab}$ is the maximal abelian extension of $K$ contained in $M$. Le... | https://mathoverflow.net/users/69196 | A galois group is topologically finitely generated or finitely generated as a $\mathbb{Z}_p$-module | See also [Neukirch-Schmidt-Wingberg], Cohomology of Number Fields, Theorem (11.1.2) and Proposition (10.3.20)(ii). You can find an online version at <https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/index-de.html>
| 4 | https://mathoverflow.net/users/nan | 208122 | 99,397 |
https://mathoverflow.net/questions/208124 | 0 | In any lattice $L$, an *ideal* is a subset $I\subseteq L$ that is downward closed, and moreover $a,b\in I$ implies $a\vee b\in I$. We denote by ${\cal I}(L)$ the set of ideals of $L$, ordered by set inclusion.
Let $F(\mathbb{N})$ denote the lattice of finite subsets of $\mathbb{N}$, ordered by set inclusion. Is ${\ca... | https://mathoverflow.net/users/nan | Set of ideals of the set of finite subsets of $\mathbb{N}$ | There is quite an easy isomorphism that you may just have overlooked: ${\cal I}(F(\mathbb{N})) \cong {\cal P}(\mathbb{N})$, where ${\cal P}(\mathbb{N})$ denotes the powerset of $\mathbb{N}$. The isomorphism is very natural (not in a strict mathematical sense):
$$ I\in {\cal I}(F(\mathbb{N})) \mapsto \bigcup I.$$
Le... | 2 | https://mathoverflow.net/users/8628 | 208127 | 99,399 |
https://mathoverflow.net/questions/208123 | 1 | Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F\_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$.
Suppose that $F\_{\mathbb Q}$ is representable by a scheme (over $\mathbb Q$). Does it follow that $F\_{\mathbb Z[1/n]}$ is representable for some lar... | https://mathoverflow.net/users/74425 | On functors which are generically representable | Being a sheaf in the étale topology is insufficient. Begin with the locally ringed space $(\text{Spec}\mathbb{Z},\mathcal{O}\_{\text{Spec}\mathbb{Z}})$. Let $$i:X\hookrightarrow \text{Spec}\mathbb{Z}$$
be the complement of the generic point $\langle 0 \rangle$. Give $X$ the sheaf of rings $\mathcal{O}\_X := i^{-1}\mat... | 2 | https://mathoverflow.net/users/13265 | 208136 | 99,404 |
https://mathoverflow.net/questions/207861 | 3 | Suppose $A$ and $P$ are symmetric, positive definite matrices and that we factor $P^{-1}=EE^\top.$ Is it true that the condition number of $PA$ is upper-bounded by the condition number of $E^{-1}AE^{-\top}$?
The two matrices clearly have the same eigenvalues, but $PA$ may not be symmetric. So, I wasn't sure if this s... | https://mathoverflow.net/users/25311 | Condition number after preconditioning | Yes, $\kappa(E^{-1}AE^{-T}) \le \kappa(PA)$. For any matrix $X$, $|\lambda| \le \Vert X \Vert$ holds for any eigenvalue and any induced norm (since $\Vert \lambda u \Vert$ = $\Vert X u \Vert \le \Vert X \Vert \Vert u \Vert$ for the eigenvector $u$). Similarly, if $X$ is invertible, then $|\lambda^{-1}| \le \Vert X^{-1}... | 2 | https://mathoverflow.net/users/70005 | 208145 | 99,409 |
https://mathoverflow.net/questions/208100 | 6 | If $M$ is a convex-cocompact hyperbolic 3-manifold, and $S$ is a closed surface with genus $\geq$ 2. Suppose $f:S\to M$ is a minimal immersion, and $f(S)$ is negatively curved. I know that all the closed geodesics in $f(S)$ are closed geodesics in $M$. Can I conclude that $f(S)$ is totally geodesic in $M$?
| https://mathoverflow.net/users/51454 | Totally geodesic submanifold of a hyperbolic 3-manifold | Reading the first paragraph of the introduction of the following paper:
<http://homeweb.unifr.ch/parlierh/pub/BuserParlierOsaka.pdf>
it seems to me that the set of unit tangent vectors $v\_p$ to $S$ such that the $\gamma(t):=exp(tv\_p)$ is a closed geodesic is dense. If I am not misunderstanding such a density then $\a... | 4 | https://mathoverflow.net/users/43122 | 208161 | 99,413 |
https://mathoverflow.net/questions/207577 | 7 | Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum.
Is there a description of this spectrum as some sort of Thom spectrum?
I did find a construction in Costenoble's "An... | https://mathoverflow.net/users/35183 | Naive G-spectrum representing geometric equivariant cobordism | Since my comment answered Emanuele Dotto's answer, I post it as an answer:
Stefan Schwede discusses equivariant bordism in [his book project about global homotopy theory](http://www.math.uni-bonn.de/people/schwede/global.pdf) in detail.
| 6 | https://mathoverflow.net/users/32022 | 208166 | 99,415 |
https://mathoverflow.net/questions/200405 | 10 | The "signature" of rough path theory is defined by iterated integral as
$s(k)=\int\_{0 \le u\_1 \le \cdots \le u\_k \le t} \mathrm{d}X\_{u\_1} \otimes \cdots \otimes \mathrm{d}X\_{u\_k}$
in witch $X(t)$ is a $d$ dimensional rough path.
I'm new to tensor calculus, and still in struggle to figure out what the abov... | https://mathoverflow.net/users/64314 | Understand rough path iterated integral and how to compute it numerically? | $s(k)$ is made of $d^k$ numbers. They are labelled by the k-tuple $(j\_1,j\_2,\dots j\_k)$ - and there are $d^k$ possibilities.
For example, if $d$ is 3 and $k$ is 2, there are 9 values: $s^{(1,1)}$, $s^{(3,2)}$ etc.
For a concrete example, the value of $s^{(2,3)}$ is the integral $\int\_{0<u\_1<u\_2<t}\,dx\_2(u\_2)... | 8 | https://mathoverflow.net/users/74448 | 208173 | 99,416 |
https://mathoverflow.net/questions/208176 | 1 | We define $\mathcal A$ is a differential operator of order $n$ with variable coefficients if
$$ \mathcal A:=\sum\_{|\alpha|\leq n}A\_\alpha (x) D^\alpha $$
where $\alpha$ is an muti-index and $A\_\alpha(x)$ is a matrix in suitable dimension and $D^\alpha$ is the $\alpha$-th differential operator.
Next, take $\mathbb... | https://mathoverflow.net/users/62560 | The existence of differential operator of the form $AB=0$ | I think that the answer to (2), generally speaking is 'no'. For example, consider the case of $\mathcal{A}$ being the Cauchy-Riemann operator:
$$
\mathcal{A}(u,v) = (u\_x - v\_y,\ u\_y+v\_x).
$$
The kernel of $\mathcal{A}$ on $\mathbb{R}^2$ is the set of $(u,v)$ such that $u+iv$ is a holomorphic function of $z = x + i ... | 6 | https://mathoverflow.net/users/13972 | 208191 | 99,422 |
https://mathoverflow.net/questions/208217 | 3 | By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}\_2(V) := \{\{a,b\}: a\neq b \in V\}$. A *graph homomorphism* between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\}\in E(G)$ implies $\{f(v), f(w)\} \in E(H)$.
If $G,H$ are graphs and there is a graph homomorphism $f:G\... | https://mathoverflow.net/users/8628 | Uncountably many countable graphs with no homomorphism between them | Yes. Such cliques are called rigid families of graphs. For every infinite cardinal $\kappa$ there exists a rigid family of graphs of cardinality $2^\kappa$, such that each graph in the family has $\kappa$ vertices.
This is classical so I was surprised by the trouble of finding a neat reference tailored to your questi... | 10 | https://mathoverflow.net/users/16678 | 208220 | 99,432 |
https://mathoverflow.net/questions/208222 | 3 | Let $[\omega]^\omega$ denote the collection of all infinite subsets of $\omega$. Let us call $S\subseteq [\omega]^\omega$ *lower-bounding* if for all $a\in [\omega]^\omega$ there is $s\in S$ such that $s\subseteq a$.
What is the minimum cardinality that a lower-bounding subset of $[\omega]^\omega$ can have?
| https://mathoverflow.net/users/nan | Minimum cardinality of lower-bounding subset of $[\omega]^\omega$ | They have size continuum, since there is an almost disjoint family of infinite sets of integers of size continuum.
| 11 | https://mathoverflow.net/users/2689 | 208234 | 99,438 |
https://mathoverflow.net/questions/208227 | 9 | What is the asymptotic order of
$$
\int\_0^1 \left| \sum\_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.
| https://mathoverflow.net/users/46852 | $L^1$ norm of exponential sum of $n^2 x$ | [Jurkat and van Horne](https://projecteuclid.org/download/pdf_1/euclid.dmj/1077314936) showed that the $L^1$ norm is asymptotic to a constant times $\sqrt{N}$ (see Theorems 4 and 5 in their paper which compute all moments). For other related work see [Jurkat and van Horne](http://projecteuclid.org/download/pdf_1/euclid... | 17 | https://mathoverflow.net/users/38624 | 208247 | 99,441 |
https://mathoverflow.net/questions/208194 | 16 | Suppose that $X$ and $Y$ are independent identically distributed random vectors in a separable Banach space $B$. Does it always follow that $E\|X-Y\|\le E\|X+Y\|$?
Some background information on this question can be found at the end of the note posted on arXiv at [additive decomposition of norms](http://arxiv.org/ab... | https://mathoverflow.net/users/36721 | An inequality for two independent identically distributed random vectors in a normed space | To get dimension 4, I think the norm
\begin{equation} \|a\| = \max\_{\{i,j\}} |a\_i-a\_j| \vee \|a\|\_\infty,
\end{equation}
where $a=(a\_1,a\_2,a\_3,a\_4)$,
works. Again $X$ samples the unit vector basis uniformly.
EDIT: It looks like a variation takes care of dimension 3. Use again
\begin{equation} \|a\| = \max\_... | 6 | https://mathoverflow.net/users/2554 | 208250 | 99,442 |
https://mathoverflow.net/questions/208249 | 8 | Is $\lfloor(x+1/2)e\rfloor = \lfloor(x+1)(1+1/x)^x\rfloor$ for all $x > 0$?
The question occurred in connection with (nonhomogeneous) Beatty sequences, $\lfloor nr+h\rfloor$, where irrational $r>0$ and real $h$ are fixed, and $n = 1,2,\dots$.
Let
$$s\_n = (n+1/2)e - (n+1)(1+1/n)^n$$ I checked that $(s\_n)$ is str... | https://mathoverflow.net/users/61426 | A question involving e, floor, and all x > 0 | The question is whether there is an integer $m$ with $(n+1/2)e < m \le (n+1)(1+1/n)^n$. Note that $s\_n \sim e/n$, so this would mean (approximately)
$$ 0 > e - \dfrac{2m}{2n+1} > \dfrac{2e}{n(2n+1)}$$
Now the continued fraction for $e$ is well-known:
$$ e = [2;1,2,1,1,4,1,1,6,1,1,8,\ldots]$$
The corresponding ev... | 6 | https://mathoverflow.net/users/13650 | 208253 | 99,443 |
https://mathoverflow.net/questions/207922 | 15 |
>
> Let $L$ be the Hilbert class field of a number field $K$, and let $\mathfrak{p}$ be a prime ideal of $K$. Then $\mathfrak{p}$ splits completely in $L$ if and only if $\mathfrak{p}$ is a principal ideal.
>
>
>
What is the quickest and/or most elementary proof of this fact, and if so, where can I find it? I un... | https://mathoverflow.net/users/nan | Quickest and/or most elementary proof of "principal iff splits completely"? | Let $K/F$ be an extension of number fields. We say that $K/F$ is
* a Weber-Hilbert class field if the prime ideals of $F$ that split
completely in $K$ are exactly the principal prime ideals.
* a Takagi-Hilbert class field if the norms of all ideals from $K$
are principal ideals in $F$: $N\_{K/F} D\_K \subseteq P\_K... | 7 | https://mathoverflow.net/users/3503 | 208255 | 99,444 |
https://mathoverflow.net/questions/208266 | 5 | Is either of these inequalities true?
$$\lambda(tA + (1-t)B)\geq t\lambda(A) + (1-t)\lambda(B)$$
or
$$\lambda(tA + (1-t)B)\leq t\lambda(A) + (1-t)\lambda(B),$$
where $0\leq t \leq 1$, $A,B$ are bounded domains in $\mathbb{R}^n$ and $\lambda(\Omega)$ is an eigenvalue of the problem
$$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,... | https://mathoverflow.net/users/74492 | An inequality for eigenvalues of the Dirichlet problem | **Update:** I intended this to be a complete answer originally, but my "counterexample" was based on a miscalculation. So this is now more a collection of remarks on what I think the question is about.
We need to be more specific about which eigenvalue we want to take. I will discuss the ground state energy, which se... | 4 | https://mathoverflow.net/users/48839 | 208268 | 99,448 |
https://mathoverflow.net/questions/208256 | 12 | Quick Review:
The Baas-Sullivan construction cones off generators $\alpha\_1, ..., \alpha\_n \in \pi\_\*(MU)$ from $MU$ to get a new spectrum $MU/(\alpha\_1, ..., \alpha\_n)$, which is isomorphic to some spectrum $E$ we wish to present as a bordism theory with singularities. This is a "geometric" alternative to tens... | https://mathoverflow.net/users/56462 | Can we construct a Baas-Sullivan presentation of TMF? | There are a couple of possible variant questions of this, and I'm not quite sure which is appropriate. The first question question is whether, without knowledge of the functor $T$, we could construct $TMF$ by Baas-Sullivan theory.
If you do not invert 6, then this is not possible. Any object constructed by Baas-Sulli... | 18 | https://mathoverflow.net/users/360 | 208270 | 99,449 |
https://mathoverflow.net/questions/207693 | 0 | Find all possible rational solutions pairs $(z,a)$ of the equation
$a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 (44281626624 - 44281626624 z + 13332971520 z^2 - 1131282432 z^3 + 28901376 z^4) = 0$.
I have the following solutions for $a = (... | https://mathoverflow.net/users/62471 | Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter | It seems hopeless to try to provably find all the rational points on the genus $7$ curve $C$ mentioned in the question using either Chabauty's method, or etale descent. I will simply explain how to find a map from the genus $7$ curve to the rank three elliptic curve mentioned in the comment.
First, I noticed that all... | 2 | https://mathoverflow.net/users/48142 | 208275 | 99,452 |
https://mathoverflow.net/questions/208276 | 0 | Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R\_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C\_A(z) = Tr[R\_A(z) ]$.
* I want to compare between the numbers $[R\_A(z)]\_{ii}$ and $\frac{1}{n} C\_A(z)$.
Do we know a... | https://mathoverflow.net/users/38852 | Is there any way to compare between diagonals of a resolvent and a Cauchy transform? | The trace is the sum of the diagonal elements, so (when $z$ is real) it's always true for at least one $i$, and the only way it can be true for all of them is that all diagonal elements are equal. For example, this is the case for the matrix
$$ A = \pmatrix{3 & 0 & 1 & -1\cr
0 & 3 & 1 & -1\cr
1 & 1 & 3 & 0\cr
-1 & ... | 2 | https://mathoverflow.net/users/13650 | 208278 | 99,453 |
https://mathoverflow.net/questions/208175 | 8 | I Ask [this question](https://math.stackexchange.com/questions/1292642/symplectic-reversing-diffeomorphisms) in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there always a diffeomorphism $f$ on M with $f^{\*}... | https://mathoverflow.net/users/36688 | Symplectic reversing diffeomorphisms on a compact symplectic manifold | I'd like to mention the work of [Castaño-Bernard-Matessi-Solomon](http://arxiv.org/abs/0908.0966), who proved the existence of an anti-symplectic involution for symplectic manifolds carrying a Lagrangian torus fibration of a certain class. Such a class of Lagrangian fibration is constructed by gluing local models of La... | 6 | https://mathoverflow.net/users/43423 | 208286 | 99,457 |
https://mathoverflow.net/questions/208301 | 6 | We say that a set $A\subseteq \mathbb{N}$ has *lower density 0* if $$\text{lim inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$
Given $A,B\subseteq \mathbb{N}$ we set $A\cdot B = \{a\cdot b: a\in A, b\in B\}$. The set $A^n$ for $n\in\mathbb{N}$ is defined inductively in the obvious manner.
Let $P$ be the se... | https://mathoverflow.net/users/8628 | Lower density of {primes} times themselves | There is no such $m\_0$, due to the [Erdős–Kac theorem](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem).
| 15 | https://mathoverflow.net/users/28104 | 208304 | 99,459 |
https://mathoverflow.net/questions/207230 | 6 | A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor\_{C}(d,e)=\{0\}\ \forall d\in D$; which "operations" respect $E$? Note that $E$ is "extension-stable" (i.e., for any $C$-triangle $e\_1\to e\_2\to e\_3$ the ... | https://mathoverflow.net/users/2191 | On various "extension closures" and "orthogonals" in triangulated categories | The argument described in the proof of Proposition 1.3 of
Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317,
yields that any pseudo-extension (of $N$ by $A$, in the sense described in my question) is a retract of an "honest extension" (of $N$ by $A$). So, pseudo... | 2 | https://mathoverflow.net/users/2191 | 208306 | 99,460 |
https://mathoverflow.net/questions/208310 | 1 | In C.D.Sogge's *Fourier Integrals in Classical Analysis* pp.128-129, he proved Lemma4.2.3(Tauberian Lemma):
**Lemma.** Let$g(\lambda)$ be a piece-wise continuous tempered function of $\mathbb{R}$. Assume that for $\lambda>0$
$$|g(\lambda+s)-g(\lambda)|\leq C(1+\lambda)^a,0<a\leq1$$
Then if $\hat{g}(t)=0$ when $|t|\le... | https://mathoverflow.net/users/25437 | Ask the validity of Tauberian lemma in Sogge's book | The identity $\hat{\psi}(t)=(it)^{-1}\eta(t)$ you quoted is indeed hold except for the typo that $\eta\in C^{\infty}$ instead of Schwartz functions. Although the function on the right hand side is not integrable as you have noticed, itself can be the Fourier transform of a $L^1$ function. The more proper setting here i... | 2 | https://mathoverflow.net/users/35702 | 208317 | 99,465 |
https://mathoverflow.net/questions/208319 | 4 | Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that $C/I \cong \mathcal P(\omega)/ \mathrm{fin}$?
| https://mathoverflow.net/users/11145 | Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$ | Let $p=\{a\_{n}|n\in\omega\}$ be a countable partition of $C$. Let $p^{\*}=\{\bigvee R|R\subseteq p\}$. Then $p^{\*}$ is a Boolean algebra and there is an isomorphism $i:p^{\*}\rightarrow P(\omega)$. By the Sikorski extension theorem (one does not need the full Sikorski extension theorem here, but just the strictly wea... | 7 | https://mathoverflow.net/users/22277 | 208321 | 99,466 |
https://mathoverflow.net/questions/208327 | 4 | Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime.
>
> Is the natural morphism $\mathrm{Pic}(X) \otimes \mathbb{Z}\_\ell \to \mathrm{H}^2(X, \mathbb{Z}\_\ell(1))$ an isomorphism?
>
>
>
When $p = 0$, this is an easy ... | https://mathoverflow.net/users/5101 | Picard groups of Fano varieties in positive characteristic | First of all, that cycle class homomorphism is certainly surjective for all Fano manifolds that lift to characteristic $0$. Secondly, using the usual Kummer sequence, to prove that the cycle class map is surjective, it suffices to prove that the Brauer group is finite. I think this should not be too hard to prove via t... | 4 | https://mathoverflow.net/users/13265 | 208330 | 99,469 |
https://mathoverflow.net/questions/208328 | 1 | Let $X$ be a smooth projective rational variety over a field $k$. Let $CH^i(X)$ denote the Chow group of codimension $i$ algebraic cycles on $X$ modulo rational equivalence. What can one say about Chow groups $X$? Are they torsion-free?
| https://mathoverflow.net/users/74530 | Chow groups of rational varieties | You cannot say much. Suppose $X$ is obtained by blowing up $\mathbb{P}^n$ along a smooth subvariety $V$, say of codimension $c$; then $CH^p(X)=CH^p(\mathbb{P}^n)\oplus CH^{p-1}(V)\oplus\ldots \oplus CH^{p+1-c}(V)$. Thus the Chow groups of $X$ look like those of $V$, which are arbitrary. All you can say for a rational v... | 8 | https://mathoverflow.net/users/40297 | 208331 | 99,470 |
https://mathoverflow.net/questions/208324 | 8 | Let $P(x)=1+a\_1x+a\_2x^2+a\_3x^3+...$ be a series such that every $a\_i$ is an integer, $a\_1<0$, and $a\_i\ge 0$ for every $i\ge 2$. Are the following statements equivalent ?
* $P(y)=0$ for some $y>0$.
* Every coefficient of the series expansion of $\frac 1{P(x)}$ is positive.
| https://mathoverflow.net/users/71090 | Conjectured equivalent conditions on certain power-series | Yes. One direction is quite clear, the other one is carefully written in
D. I. Piotkovskii, "On the growth of graded algebras with a small number of defining relations", Uspekhi Mat. Nauk, 48:3(291) (1993), 199–200.
(It is also mentioned without proof in Lemma 5.3 of
D. Anick, "Generic algebras and CW complexes", ... | 8 | https://mathoverflow.net/users/1306 | 208332 | 99,471 |
https://mathoverflow.net/questions/208262 | 11 | $$y(t)=y\_0+\int\_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$
The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the paper of Walter "There is an Elementary Proof of Peano's Existence Theorem" the author showed there is an constructive ... | https://mathoverflow.net/users/69466 | Does Peano's existence theorem admits a constructive proof? | I think that the heart of the question is "Can one give a rigorous meaning to 'there is no constructive proof to the Peano's theorem'?" The answer to this is yes, but the answer is not as simple as one might naively hope.
The way that one would give an affirmative answer is to first choose a constructive framework. ... | 11 | https://mathoverflow.net/users/5442 | 208337 | 99,473 |
https://mathoverflow.net/questions/208302 | 4 | We know that we can put two different structures on $\mathbb R^5$ in the topological category. First is $C(S^{4})$ and second is $C(\Sigma X^3)$, where $X^3$ is a homology sphere and $C(\cdot)$ stands for the open cone construction and $\Sigma(\cdot)$ stands for the suspension construction.
The question is: do other ... | https://mathoverflow.net/users/1190 | Cone structures on $\mathbb R^n$ | You should look at this paper:
<http://www.ams.org/journals/bull/1978-84-05/S0002-9904-1978-14527-3/S0002-9904-1978-14527-3.pdf>
and in particular theorem 3.5.
And also
<http://www.maths.ed.ac.uk/~aar/books/haupt.pdf>
p. 23.
It follows from the double suspension theorem and its corollaries that when $n\geq 5$ ... | 4 | https://mathoverflow.net/users/27816 | 208344 | 99,477 |
https://mathoverflow.net/questions/208335 | 6 | Consider an extension of groups $$(E)\ :\ 1\to N\to G\stackrel{\pi}{\longrightarrow} Q\to 1$$ and assume
1. $N$ is abelian,
2. $Q$ is finite,
3. for any Sylow subgroup $S\_p$ of $Q$, the pullback $(E\_p)\ :\ 1\to N\to \pi^{-1}(S\_p)\to S\_p\to 1$ is split.
Then $(E)$ is split.
[A proof can be written using only... | https://mathoverflow.net/users/39552 | Extension splitting over Sylow subgroups | This didn't seem likely, so I looked for a counterexample and found one after a short search. Clearly $Q$ must not have any normal Sylow subgroups, so the first example to try is $Q=S\_4$. Then $N$ must not be abelian, so I tried $N=Q\_8$ and found an example.
The example is $\mathtt{SmallGroup}(192,988)$ in GAP or M... | 10 | https://mathoverflow.net/users/35840 | 208346 | 99,478 |
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