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https://mathoverflow.net/questions/258480 | 19 | Is there a 4-manifold whose Stiefel-Whitney classes satisfy $w\_1\neq 0$, $w\_3 \neq 0$, and $w\_1^2=0$?
This question follows on from [this one](https://math.stackexchange.com/q/2071685/39599) where the condition $w\_3 \neq 0$ is replaced by $w\_2 \neq 0$. On a closed four-manifold $\operatorname{Sq}^1(w\_2) = w\_3$... | https://mathoverflow.net/users/17787 | A 4-manifold with Stiefel-Whitney classes $w_1\neq 0$, $w_3 \neq 0$, and $w_1^2=0$? | Let's begin by reformulating the question a bit. Note that any orientable 4-manifold is spin-c, so in particular has $w\_3 = 0$. The condition that $w\_1 \not= 0 $ is thus redundant.
Wu's theorem gives that on a closed 4-manifold
* $w\_1 = v\_1$, so $Sq^1(c) = w\_1 c$ for any $c \in H^3$ ($\mathbb{Z}/2$ coefficient... | 17 | https://mathoverflow.net/users/13061 | 259407 | 117,010 |
https://mathoverflow.net/questions/259379 | 5 | In my [earlier MO question](https://mathoverflow.net/questions/259104/a-neat-evaluation-of-an-infinite-matrix), I was seeking for a proof for $\det A\_{\infty}:=\det(I\_{\infty}-M\_{\infty}^2) =\sqrt[4]{1-x^2}$ where $M\_n$ is the $n\times n$ matrix:
$$M\_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j... | https://mathoverflow.net/users/66131 | a matrix of Onsager-Kaufman vs Schwarz-Wu | If the question is *is there a real (deeper) reason and connection between the Ising model (in 2D) and free fermion (in 1D)* the answer is yes. I don't know how deep. I believe this was shown for the first time neatly in H. Lieb's solution of the 2D classical Ising model.
The steps are the following
1) Write the p... | 1 | https://mathoverflow.net/users/74539 | 259414 | 117,012 |
https://mathoverflow.net/questions/259415 | 2 | $E$ is a real, positive-definitive 3x3 symmetric tensor (I am thinking about the strain tensor in solid mechanics). We perform eigendecomposition and get:
$$E\_p=\sum\_{i=1}^{3}λ\_iN\_i⊗N\_i$$
into its principal components, where $λ\_i$ are its eigenvalues and $N\_i$ are its eigenvectors.
My question is, is there a... | https://mathoverflow.net/users/103502 | Derivative of eigenvalues w.r.t. a tensor | Let $p(\lambda)$ be the characteristic polynomial $p(\lambda)=\det(E-\lambda I)$. Then $p(\lambda)=(\lambda\_1(E)-\lambda)(\lambda\_2(E)-\lambda)(\lambda\_3(E)-\lambda)$. Differentiate in $E$ and then set $\lambda=\lambda\_i(E)$:
$$
\frac{\partial \lambda\_i}{\partial E}
=
\frac{1}{(\lambda\_j-\lambda\_i)(\lambda\_k-\... | 5 | https://mathoverflow.net/users/13268 | 259417 | 117,013 |
https://mathoverflow.net/questions/259428 | 1 | Let $ M $ be a class of L-functions such that whenever $ F $ and $ G $ belong to $ M $, then so do their product $ F.G $ and their tensor product $ F\otimes G $ defined by $ F\otimes G : s\mapsto\sum\_{n>0}\frac{a\_{n}(F)a\_{n}(G)}{n^s} $ for $\Re(s)>1 $ if $ F : s\mapsto\sum\_{n>0}\frac{a\_{n}(F)}{n^s} $ and $ G : s\m... | https://mathoverflow.net/users/13625 | Galois group of an L-function | Let $M$ be the set of finite products of Dirichlet $L$-functions. These surely form a class of $L$-functions as in the question. Now take some prime $p$ congruent to 1 mod 4 and let $\chi$ be one of the two Dirichlet $L$-functions of conductor $p$ and order 4 (the other one will then be $\overline{\chi}$). Let $F$ be $... | 4 | https://mathoverflow.net/users/1384 | 259432 | 117,019 |
https://mathoverflow.net/questions/259436 | 3 | Can somebody explain why the cumulative binomial equals an integral expression?
Thanks!${}$
$$
\sum\_{j=0}^{k-1}\binom{n}{j}\theta^j(1-\theta)^{n-j} = 1- \binom{n}{k}k\int\_0^\theta t^{k-1}(1-t)^{n-k}dt\,.
$$
| https://mathoverflow.net/users/103514 | cumulative binomial equals integral | **Key players:** $m\binom{n}m=n\binom{n-1}{m-1}$, derivative $\frac{d}{d\theta}$ and re-indexing $j-1\rightarrow j$.
Denote the LHS by $f(\theta)$ and the RHS by $g(\theta)$. Take derivatives w.r.t. $\theta$ to get
$$g'(\theta)=-k\binom{n}k\theta^{k-1}(1-\theta)^{n-k}=-n\binom{n-1}{k-1}\theta^{k-1}(1-\theta)^{n-k},$$... | 5 | https://mathoverflow.net/users/66131 | 259439 | 117,022 |
https://mathoverflow.net/questions/259427 | 3 | A continuum $X$ is called *minimal* if it is not a single point and is homeomorphic to all its nontrivial subcontinua.
Here a trivial continuum is a single point.
>
> What is an example of a minimal continuum not homeomorphic to the interval?
>
>
>
This question is motivated by the following post and its relat... | https://mathoverflow.net/users/36688 | A minimal continuum | As Andreas points out in the comments, [the pseudo-arc](https://en.wikipedia.org/wiki/Pseudo-arc) provides an example of a continuum like this. This fact was first proved by Moise in 1948, in
>
> Edwin Moise, "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua," *Transa... | 6 | https://mathoverflow.net/users/70618 | 259441 | 117,023 |
https://mathoverflow.net/questions/259304 | 3 | Let $\Omega\subset\mathbb{R}^n$ be a bounded strictly convex domain and $\nu:\partial\Omega\rightarrow\mathbb{R}$ be a lower semi-continuous functions. It is well known that the function $\overline{\nu}:\overline{\Omega}\rightarrow\mathbb{R}$ defined by
$$
\overline{\nu}(x):=\sup\Big\{\alpha(x)\,\Big|\,\alpha:\mathbb{R... | https://mathoverflow.net/users/17294 | Finding a smooth convex function with prescribed boundary value and small Monge-Ampère measure | In the case $n \geq 3$ it is false. One way to see this is to use the Pogorelov example
$$w(x', \,x\_n) = |x'|^{2-2/n}f(x\_n),$$
which for the appropriate choice of $f$ solves $\det D^2w = 1$ in $\{|x\_n| < \rho\}$. This example is $C^{1,1-2/n}$ and is $0$ on the $x\_n$-axis. (This example arises from the affine invar... | 2 | https://mathoverflow.net/users/16659 | 259443 | 117,024 |
https://mathoverflow.net/questions/259445 | 6 | Suppose $M^4$ is a compact hyperbolic (i.e. curvature $-1$) $4$-manifold and $\Gamma\cong\pi\_1(M^4)$.
Is there any expectation whether $\Gamma$ acts properly and co-compactly on a $\rm CAT(0)$ cube complex?
Note that by work of Bergeron Wise based on work of Kahn-Markovic the answer to the question is always posit... | https://mathoverflow.net/users/13441 | Fundamental groups of hyperbolic $4$-manifolds and $\rm CAT(0)$ cube complexes | This question is certainly open in general. I don't know if anyone has formally expressed an 'expectation' in print, but you might be interested in the following pieces of positive evidence.
I believe that all known examples of such 4-manifolds essentially come either from arithmetic constructions or from Coxeter gro... | 8 | https://mathoverflow.net/users/1463 | 259455 | 117,027 |
https://mathoverflow.net/questions/259374 | 8 | In a recent [paper](https://arxiv.org/abs/1610.05803) studying some generalizations of Stirling numbers, my coauthors and I needed the following result:
If $f(x)=\sum\_{n \geq 1} a\_n x^n/n!$ is a power series with $a\_1 \neq 0$, and $g(x) = \sum\_{n \geq 1} b\_n x^n/n!$ is its series reversion (so $f(g(x))=g(f(x))=x... | https://mathoverflow.net/users/21690 | Combinatorial interpretation of series reversion coefficients | Yes, this is known. As Tom Copeland mentioned, this is in [Drake's thesis](http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf), Example 1.4.7.
Drake gives a broad generalization of your result. His main theorem (Theorem 1.3.3) is roughly as follows. Suppose you have an alphabet $A$ of certain trees,... | 6 | https://mathoverflow.net/users/103524 | 259463 | 117,030 |
https://mathoverflow.net/questions/259462 | 1 | Let $T$ be a scheme (probably integral noetherian) and $X$ a smooth projective variety. Let $K,K',A,A'$ be locally free coherent sheaves on $X\times T$. There are exact sequences:
$0\rightarrow K\rightarrow A\rightarrow E\rightarrow 0$
and
$0\rightarrow K'\rightarrow A'\rightarrow E'\rightarrow 0$
which are exa... | https://mathoverflow.net/users/83786 | Sheaf Hom is flat | No. For example, let $X = T = C$ be a smooth projective curve (but we keep the notation $X \times T$ to indicate the separate functions of the two). Let $E = \mathcal O\_{\Delta\_C}$, and let $E' = \mathcal O\_{x \times T}$ for a point $x \in X$.
Since both the maps $\Delta\_C \to X \times T \to T$ and $x \times T \t... | 5 | https://mathoverflow.net/users/82179 | 259470 | 117,033 |
https://mathoverflow.net/questions/259466 | 6 | I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and see the beauty of numbers. For instance, I have enough informations about the golden ratio.
I watched Professor Ken Ono... | https://mathoverflow.net/users/76102 | Beauty of some numbers discovered by Ramanujan | I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, *Ramanujan's Notebooks*, Part I, II, etc, Springer.
Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special ca... | 5 | https://mathoverflow.net/users/66131 | 259471 | 117,034 |
https://mathoverflow.net/questions/259464 | 1 | Let $f(x)$ be a nonconstant polynomial over $\mathbb Z$.
Must $f$ have a zero in $\mathbb F\_p$ for some prime number $p$?
More generally, let $f\_1,\dots,f\_k$ be such polynomials, must there exist a $p$ such that each $f\_i$ has a zero in $\mathbb F\_p$?
| https://mathoverflow.net/users/4600 | Integers mod $p$ "together" algebraically closed | Easy answer to the first question: for $n \gg 0$, we have $f(n) > 1$ or $f(n) < -1$. Hence $f(n)$ is divisible by some prime number $p$, so $n$ is a root mod $p$.
| 2 | https://mathoverflow.net/users/82179 | 259473 | 117,036 |
https://mathoverflow.net/questions/259433 | 5 | Let $K$ be a field, and let $L/K$ be an **algebraically closed** field extension (i.e. the only elements of $L$ that are algebraic over $K$ are already in $K$). Let $R$ be a $K$-algebra that is an integral domain. Does it follow that $R \otimes\_K L$ is an integral domain? I'm particularly interested in the case where ... | https://mathoverflow.net/users/19045 | Is a base-change of an integral domain by an extension of its base field without algebraic elements still a domain? | No. Let $K = \mathbf{F}\_p(s,t)$, $A = K[x,y]/(sx^p + t y^p - 1)$. One checks $A$ is a domain, even Dedekind, so we can define $L = {\rm{Frac}}(A)$. By exploring residue fields of $A$ at maximal ideals and using that $A$ is Dedekind one shows with some thought that $K$ is algebraically closed in $L$. But for the field ... | 8 | https://mathoverflow.net/users/81332 | 259474 | 117,037 |
https://mathoverflow.net/questions/259410 | 3 | I am thinking about some problems in noncommutative projective geometry, to do with graded module categories and their quotients by torsion modules. The following question arose in this more specific setting but I'm phrasing it in generality as I don't believe the answer is likely to be specific to graded modules.
>... | https://mathoverflow.net/users/13215 | Do equivalences descend to Serre quotients? | You get an isomorphism of functors $\bar{G}\bar{F}\simeq1\_\mathcal{A/C}$ by applying $\pi\_\mathcal{A}$ to an isomorphism $GF\simeq1\_\mathcal{A}$.
Just to check we're thinking of the quotient category in the same way: $\mathcal{A/C}$ has the same objects as $\mathcal{A}$, and the maps are constructed by adjoining i... | 2 | https://mathoverflow.net/users/22989 | 259493 | 117,040 |
https://mathoverflow.net/questions/259461 | 2 | **[From Wikipedia:](https://en.wikipedia.org/wiki/Min-max_theorem)**
Let $A$ be an $n \times n$ Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient $R\_A :
\mathbf C^n \setminus \{0\} \to \mathbb{R}$ defined by
$$R\_{A}(x)={\frac{(Ax,x)}{(x,x)}}$$ where $... | https://mathoverflow.net/users/40747 | Under what condition does Courant–Fischer–Weyl min-max principle hold in general? | To avoid the various ambiguities in your statement I'll assume that $A$ is *real* and symmetric. $\newcommand{\bR}{\mathbb{R}}$ Choose an *orthonormal* basis $\newcommand{\be}{\mathbf{e}}$ $\be\_1,\dotsc,\be\_n$ consisting of eigenvectors of $A$ and denote by $x\_1,\dotsc, x\_n$ the coordinates determined by this basis... | 1 | https://mathoverflow.net/users/20302 | 259494 | 117,041 |
https://mathoverflow.net/questions/259488 | 0 | Given two finite dimensional algebras $A$ and $B$ over a field. The Gorenstein dimension of an algebra A is defined as the injective dimension of the module A. The finitistic dimension of an algebra A is defined as the supremum of projective dimensions of modules having finite projective dimension.
Are there exact form... | https://mathoverflow.net/users/61949 | Homological dimensions of tensor products of algebras | For the finitistic dimension, this is Theorem 16 in
*Samuel Eilenberg, Alex Rosenberg, and Daniel Zelinsky*, MR 98774 [**On the dimension of modules and algebras. VIII. Dimension of tensor products**](http://www.ams.org/mathscinet-getitem?mr=98774), *Nagoya Math. J.* **12** (1957), 71--93.
| 6 | https://mathoverflow.net/users/18756 | 259495 | 117,042 |
https://mathoverflow.net/questions/259486 | 2 | Let $G=(V,E)$ be a simple, undirected graph. A *clique decomposition* is a set ${\cal C} \subseteq {\cal P}(V)$ such that
1. $\emptyset \notin {\cal C}$,
2. $C\in {\cal C}$ and $x\neq y \in C$ imply that $\{x,y\}\in E$ (that is every member of ${\cal C}$ is a clique),
3. $\bigcup {\cal C} = V$, and
4. $e=\{x,y\} \in ... | https://mathoverflow.net/users/8628 | Minimal clique decompositions | There can be multiple distinct decompositions of minimum size. Consider a triangular lattice tiling of a torus. The maximal cliques are triangles, so we certainly need $e(G) / 3$ cliques in any minimum-sized clique decomposition. But both the upwards pointing and downwards pointing triangles are a clique decomposition ... | 3 | https://mathoverflow.net/users/25485 | 259504 | 117,045 |
https://mathoverflow.net/questions/259503 | 6 | [Indifference graphs](https://en.wikipedia.org/wiki/Indifference_graph) are those graphs $G=(V,E)$ for which there exists a real-valued function $f$ defined on $V(G)$ such that, if $u$ and $v$ are distinct vertices, $|f(u)−f(v)| \lt 1$ if and only if $\{u,v\}\in E$.
A famous result of [Roberts (1969)](http://www.ams.o... | https://mathoverflow.net/users/13639 | Have this generalization of Indifference graphs been studied before? | Seeing that indifference graphs are the same as unit interval graphs is very easy: we simply identify each vertex $v$ with a unit interval centred at $f(v)$.
We can try to do something similar here. Identify each vertex $v$ with the point $(f(v), g(v))$ in $\mathbb R^2$. Then two points are adjacent if and only if th... | 7 | https://mathoverflow.net/users/25485 | 259506 | 117,047 |
https://mathoverflow.net/questions/259491 | 8 | Let $G$ be a simple complex algebraic group. What are its complex irreducible finite-dimensional representations?
Before you start voting to close the question, I never said "rational". I am asking about abstract representations of $G$ as a group.
I can a make an obvious conjecture but I have no idea how to prove i... | https://mathoverflow.net/users/5301 | Irreducible representations of simple complex groups | If $\rho:G(\mathbf{C}) \rightarrow {\rm{GL}}(V)(\mathbf{C})$ is such an abstract linear representation, by irreducibility (or mere semisimplicity) the Zariski closure $H \subset {\rm{GL}}(V)$ of the image has connected reductive identity component (due to Lie-Kolchin). But $G(\mathbf{C})$ is perfect with simple quotien... | 17 | https://mathoverflow.net/users/81332 | 259512 | 117,049 |
https://mathoverflow.net/questions/259487 | 13 | Below we assume any simple Lie group $G$ to be simply connected.
$\pi\_3(G)=\mathbb{Z}$ for any simple Lie group $G$ and there is a uniform proof for that.
Now the textbooks say $\pi\_4(G)$ is trivial except for $G=Sp(n)$, for which it is $\mathbb{Z}/2\mathbb{Z}$.
My question is the following: is there a uniform... | https://mathoverflow.net/users/5420 | Computation of $\pi_4$ of simple Lie groups | As a partial answer, here's at least a uniform statement, which can be found as Theorem 3.10 in Mimura's survey "Homotopy theory of Lie groups" in the Handbook of Algebraic Topology:
>
> Let $G$ be a compact, connected, simply connected, simple Lie group, $T$ a maximal torus of $G$, $\mathbb{R}^r$ its universal co... | 13 | https://mathoverflow.net/users/50846 | 259527 | 117,053 |
https://mathoverflow.net/questions/259467 | 3 | This is a follow-up to my previous question [*An explicit series representation for the analytic tetration with complex height*](https://mathoverflow.net/q/259278/9550).
Recall the definition $(11)$ from there:
$$t(z) = \sum\_{n=0}^\infty \sum\_{k=0}^n (-1)^{n-k} \, q^{\binom {n-k} 2} {z \brack n}\_q {n \brack k}\_q ... | https://mathoverflow.net/users/9550 | Reconstructing analytic tetration with a complex height from a thinner set of points | Well, this problem can be handled exactly as the last one was handled. Let me give you a rough reasoning as to why. I won't elaborate in detail as quite literally my last answer handles this case with little generalization.
Any bounded function $\phi(z)$ in the right half plane $\Re(z) > -\delta$ for some $\delta > 0... | 3 | https://mathoverflow.net/users/nan | 259529 | 117,055 |
https://mathoverflow.net/questions/259480 | 3 | Suppose that $A$, $B$ are deterministic $n\times n$ matrices and $G$ a Gaussian matrix of i.i.d. entries $N(0,1)$.
I'd like to establish an upper bound of the trace norm of $AGB$ as
$$
\mathbb{E}\|AGB\|\_\ast \leq \max\{\|A\|\_\ast \|B\|\_F, \|A\|\_F\|B\|\_\ast \},
$$
where $\|\cdot\|\_\ast$ denotes the trace norm an... | https://mathoverflow.net/users/48609 | trace norm of AGB, where G is Gaussian random matrix | [Edit: Now I answer all questions.]
The answer to the first question is yes, the answer to the second question is no, and the answer to the third question is *if and only if $p \geq 2$* (only a guess in the case $p<2$).
* First question
The inequality
$$ \mathbb E \|A G B\|\_\* \leq \min( \|A\|\_\* \|B\|\_F,\|A\|... | 2 | https://mathoverflow.net/users/10265 | 259532 | 117,058 |
https://mathoverflow.net/questions/259509 | 6 | I have a question regarding a confusion from reading the Princeton Companion to Mathematics on the topic of Ergodics Theorems. It is about proving a stronger version of Poincare Recurrence Theorem using Neumann's Mean Ergodic Theorem. I apologize if this question is too easy for this site.
Let $A$ be a subset of posi... | https://mathoverflow.net/users/80191 | Poincare Recurrence by Mean Ergodic Theorem | $A\_{N,M}(f)$ converges to some $U$-invariant function $g$ that satisfies $\langle g, f\rangle$ = $\langle g, g\rangle$.
We also have $\langle g, 1\rangle = \lim \langle A\_{N,M}(f), 1\rangle = \langle f, 1\rangle$ since $\langle A\_{N,M}(f), 1\rangle = \langle f, 1\rangle$ for all $N, M$.
So we have $\lim \ \lang... | 5 | https://mathoverflow.net/users/6129 | 259533 | 117,059 |
https://mathoverflow.net/questions/259535 | 5 | Let us fix a base ring $k$. The category of $\mathbb{Z}$-graded $k$-algebras is equivalent to the category of $\mathbb{G}\_m$ equivariant affine $k$-schemes. The following 2 properties often come up when constructing the $Proj$ of a graded ring: Let $A$ be a graded $k$-algebra.
1. The zeroth graded piece is the base ... | https://mathoverflow.net/users/22810 | Expressing properties of graded algebras in terms of the $\mathbb{G}_m$action | Let $V$ be an affine variety with a $\mathbb{G}\_m$-action. Then you should think of $A\_0$ as a good model for $V/\mathbb{G}\_m$. So being connected says something like $\mathbb{G}\_m$ acts transitively- at least from the point of view of functions. Being positively graded is saying something like the action of $\math... | 2 | https://mathoverflow.net/users/6936 | 259538 | 117,060 |
https://mathoverflow.net/questions/259539 | 7 | Given positive semidefinite matrices $A,B \succeq 0$, $A, B \in \mathbb{R}^{n \times n}$, if we have $$\langle A, B \rangle = 0,$$ where $\langle \cdot, \cdot \rangle$ denotes the Frobenius inner product, then
1. What are tight necessary conditions of ranks of $A, B$? For example, $\mbox{rank} (A) + \mbox{rank}(B) \... | https://mathoverflow.net/users/103308 | Orthogonality of positive (semi-)definite matrices | If $A$ and $B$ are positive semidefinite and $A^{1/2}$ and $B^{1/2}$ their positive semidefinite square roots,
$$\text{Tr}(AB) = \text{Tr}(A^{1/2} B A^{1/2}) = \text{Tr}((B^{1/2} A^{1/2})^T B^{1/2} A^{1/2})$$
and this is $0$ if and only if $B^{1/2} A^{1/2} = 0$.
That in turn is equivalent to $\text{Ran}(A) \subset... | 9 | https://mathoverflow.net/users/13650 | 259541 | 117,061 |
https://mathoverflow.net/questions/256158 | 5 | Let $W$ be a set of words of length $n$ on the three letters $a$, $b$ and $\ast$. Say that an element $w$ in $W$ ``matches" a word $w'$ of length $n$ on the letters $a$ and $b$ if each $\ast$ in $w$ can be replaced with an $a$ or a $b$ in such a way to give $w$'. Say that two words in $W$ are disjoint if, for all $i=1,... | https://mathoverflow.net/users/101909 | Probability of finding one sub-object vs a million disjoint ones | As pointed out to me by Noga Alon, this was known as the Van den Berg - Kesten conjecture, which was solved by David Reimer:
*If p is the probability of matching at least one word in W, then the probability of matching at least two disjoint words in W is at most p^2.*
<http://journals.cambridge.org/article_S0963548... | 7 | https://mathoverflow.net/users/103562 | 259542 | 117,062 |
https://mathoverflow.net/questions/259490 | 13 | On [groupprops](https://groupprops.subwiki.org/wiki/Jacobson_radical), the Jacobson or Baer radical of a group $G$ is defined to be the intersection of all maximal normal subgroups of $G$. This is similar to, but distinct from, the Frattini subgroup which is the intersection of all maximal subgroups of $G$.
Obviousl... | https://mathoverflow.net/users/59158 | Has the Jacobson/ Baer radical of a group been studied? | The Jacobson radical $\mathfrak{J}(G)$ of a group $G$ is defined by Reinhold Baer in [2] as the intersection of the maximal normal subgroups of $G$ (as noted by Baer the identity $G = \mathfrak{J}(G)$ holds if and only if $G$ has no maximal normal subgroup). His seminal article [2] was written in German and doesn't see... | 18 | https://mathoverflow.net/users/84349 | 259545 | 117,064 |
https://mathoverflow.net/questions/259553 | 3 | Let $X$ be a compact set. Given probability measures $\mu,\mu\_n: \mathcal{B}(X)\to R$. Is it possible that $$\langle f,\mu\_n\rangle \rightarrow \langle f,\mu\rangle$$
for all convex function $f:X\to R$ but $\mu\_n$ does not converge weakly to $\mu$?
| https://mathoverflow.net/users/94894 | Convex function and weak convergence of measures | Let $\mathscr{X}$ be a locally convex space and $X \subset \mathscr{X}$ compact.
Let $E = \{f \in C(X) : \langle f, \mu\_n \rangle \to \langle f, \mu \rangle\}$. It's easy to see that $E$ is a closed linear subspace of $C(X)$ (use the triangle inequality).
Note that for any continuous linear functional $\lambda \in... | 7 | https://mathoverflow.net/users/4832 | 259555 | 117,067 |
https://mathoverflow.net/questions/259548 | 2 | I want to prove the existence and uniquness of the following problem by the semigroup method
$$
\eqalign{
& {u\_{tt}} + {u\_{xxtt}} + {u\_t} - {u\_{xx}} = 0 \cr
& u(t,0) = u(t,l) = 0 \cr
& u(0) = {u\_{0{\rm{ }}}} ,u'(0) = {u\_1} \cr}
$$
i wrote it as a Cauchy problem :
$$U' = AU$$ with : $U = ({v^1},{v^2})$ an... | https://mathoverflow.net/users/106804 | Evolution equation with inverse operator | There are two cases.
>
> **Case 1. $l\notin \pi\mathbb N$.** $\,$ In this case, the operator $A$ is bounded on $L^2(0,l)\oplus L^2(0,l)$ and generates a uniformly
> continuous group $\{e^{At}\}\_{t\in\mathbb R}$ (i.e., one can solve
> forward and backward in time).
>
>
>
To see that the operator $(I+\partial... | 2 | https://mathoverflow.net/users/69194 | 259556 | 117,068 |
https://mathoverflow.net/questions/259549 | -1 | I'm writing my dissertation on symplectic structure-preserving algorithms for Hamiltonian systems simulation, and I'm trying to figure out how much exposition is necessary for it to be readable by scientists, engineers and related professionals.
I'm actually in love with the chapter-opening sentence "Manifolds are po... | https://mathoverflow.net/users/8948 | Are manifolds typically taught to undergraduates outside mathematics (and possibly theoretical physics) tracks? | I wouldn't even assume that mathematics undergrads understand manifolds. I think the majority our math majors never see the definition of one before graduating, and only a small handful could actually tell you the definition (manifold is actually a really tricky notion!). In fact, one can even finish our "graduate prep... | 5 | https://mathoverflow.net/users/66 | 259557 | 117,069 |
https://mathoverflow.net/questions/90009 | 26 | Is it possible to get widely available math software (Maple/Matlab/Mathematica, etc) to symbolically differentiate vector and scalar functions of matrices, returning the result in terms of the original matrices and vectors involved? I have in mind the simple sort of rules collected [here](http://www.colorado.edu/engine... | https://mathoverflow.net/users/8719 | Software for symbolic matrix calculus? | Indeed, I was having the same problem. Hence, I implemented a matrix calculus toolbox myself. You can find it at [www.matrixcalculus.org](http://www.matrixcalculus.org). It can compute vector and matrix derivatives and will return the result in terms of the original vectors and matrices involved.
| 24 | https://mathoverflow.net/users/103578 | 259567 | 117,072 |
https://mathoverflow.net/questions/259442 | 2 | Consider the setup of the [$k$-means problem](https://en.wikipedia.org/wiki/K-means_clustering) and assume that the data points are confined to $k$ balls of radius $\varepsilon$ while the pairwise distances between the centers of the balls are $> 2 \varepsilon$, and each ball contains the same number of points. It seem... | https://mathoverflow.net/users/36687 | Solution of the k-means problem in a simple case | There are counterexamples in $\mathbb{R}^2$ under these conditions.
For $(d,k,n) = (2,2,4)$, place the balls such that they fit (neighborhoods of) the corners of a square. Here's such a configuration, with the K-means solution $(\theta\_1,\theta\_2)$ outside of the balls:
[Image.](https://i.stack.imgur.com/ZQvvP.png)... | 1 | https://mathoverflow.net/users/69359 | 259571 | 117,074 |
https://mathoverflow.net/questions/259574 | 9 | Let $M$ be a finitely presented module over the ring $R$. Suppose that for all primes $P\subset R$ the $k(P)$ vector space $M\otimes \_Rk(P)$ has a dimension $d(P)$ independent of $P$.
Can I conclude that $M$ is flat over $R$?
I am asking because I want to better understand the criterion for a family of projecti... | https://mathoverflow.net/users/103579 | Flatness from constancy of dimension of fibers | Yes if $R$ is *reduced* (no nilpotent elements). This is (for instance) Lemma 1 p. 51 in Mumford's *Abelian Varieties* (2nd edition). No in general: just take $R=k[\varepsilon ]/(\varepsilon ^2)$, $M=k$.
| 14 | https://mathoverflow.net/users/40297 | 259576 | 117,075 |
https://mathoverflow.net/questions/259519 | 4 | Let $G$ be a countable group with trivial center. Define $k(G)$ to be the smallest value of $k$ such that there is a subgroup $H$ of $G$ generated by $k$ elements having trivial centralizer in $G$, and $k(G)=\infty$ if no such subgroup exists. Is there a name for this invariant of $G$? Have its properties been studied?... | https://mathoverflow.net/users/8112 | Smallest subgroups with trivial centralizer? | The number $k(G)$ is the domination number of the non-commuting graph of $G$.
See Proposition 2.14 of [J. Algebra, 298 (2006) 468–492].
By Corollary 2.17 of [J. Algebra, 298 (2006) 468–492], if $k(H)$ is finite for some finite index subgroup $H$ of $G$, then $k(G)$ is also finite.
Another meaning for $k(G)$ is th... | 4 | https://mathoverflow.net/users/19075 | 259577 | 117,076 |
https://mathoverflow.net/questions/259579 | 1 | The following is known:
>
> Let $s \in (0,1)$ and $p \in [1,\infty)$ be such that $sp < n$. Let $q \in [1, p^\*\_{n,s})$ with $p^\*\_{n,s} = np/(n-sp)$, $\Omega \subset \mathbb R^n$ be a bounded extension domain for $W^{s,p}$ and $\mathscr F$ be a bounded subset of $L^p(\Omega)$. Suppose that
> $$
> \sup\_{f \in \... | https://mathoverflow.net/users/11512 | Fractional-order Rellich–Kondrashov Theorem | The answer is **yes** if $\Omega$ admits a uniform extension operator for $W^{1,p}$ and $L^p$. I suspect that it is not written down in the Hitchhiker's guide because they seem to avoid interpolation.
Suppose $0 \leq s \leq 1$ and $1 < p < \infty$. We know that $$W^{s,p}(R^d) = \bigl(L^p(R^d),W^{1,p}(R^d)\bigr)\_{s,... | 5 | https://mathoverflow.net/users/85906 | 259591 | 117,079 |
https://mathoverflow.net/questions/259592 | -2 | Let $G$ be a connected Lie Group of dimesion $m<\infty$ and let $g\in G$. The Maurer-Cartan form allows us to define a map from $G$ to the space of $\mathfrak{g}$-valued forms, via
$$g\rightarrow g^{-1}dg$$
Is this map surjective, i.e. can every $\mathfrak{g}$-valued form be written as $g^{-1}dg$ for some $g\in G$?
... | https://mathoverflow.net/users/85641 | Does Maurer-Cartan form define surjection from Lie Group to Algebra-valued forms? | These forms are at different points $g$ of $G$ for different values of $g$, so these are not in the same cotangent space, and the question is not meaningul. If you pick only one point $g$ of $G$, you only get one $\mathfrak{g}$-valued 1-form.
| 3 | https://mathoverflow.net/users/13268 | 259593 | 117,080 |
https://mathoverflow.net/questions/259315 | -1 | I recently noticed that many even numbers formed as 2p (p prime number) have at least two pairs of prime numbers that they dercribe that even number via Goldbach's Conjecture.
Examples: 10=2\*5=5+5=7+3
34=2\*17=17+17=23+11=29+5=31+3
I wanted to know if there's any research on how many pairs exist that they descr... | https://mathoverflow.net/users/103447 | Is there any research on how many pairs of prime numbers exist that they describe an even number via Goldbach's Conjecture? | All approaches to Goldbach's problem give not only the existence of solutions, but lower bounds for the number of solutions.
Using the circle method one can show that the asymptotic formula
$$
\#\{(p,q):p+q=n, p, q\mbox{ prime}\} \sim \mathfrak{S}(n)\frac{n}{\log^2 n}
$$
holds for almost all integers $n$. The first r... | 5 | https://mathoverflow.net/users/37555 | 259595 | 117,081 |
https://mathoverflow.net/questions/259580 | 0 | According to formula 163 at page 47 in the paper [A theory for the zeros of Riemann Zeta and other L-functions](https://arxiv.org/abs/1407.4358) by Guilherme França and André LeClair, the Gram points can be approximated with the formula:
$$g\_n \approx \frac{2 \pi \left(n-\frac{7}{8}\right)}{W\left(\frac{n-\frac{7}{8... | https://mathoverflow.net/users/25104 | Can there be more than two zeta zeros in between a Gram point and a França-LeClair point? | It is generally believed that a positive proportion of zeros of $\zeta$ satisfy your condition. In fact, for each fixed $k$, random matrix theory predicts a distribution of the renormalized tuples $(\gamma\_{n+1}-\gamma\_n, \gamma\_{n+2}-\gamma\_n, \ldots, \gamma\_{n+k}-\gamma\_n)$, and the distribution is non-negative... | 5 | https://mathoverflow.net/users/37555 | 259604 | 117,084 |
https://mathoverflow.net/questions/259596 | 7 | Let $F$ be a totally real number field, $E$ a totally imaginary quadratic extension over $E$, and $V$ an hermitian $n$-dimensional vector space over $F$. I assume $n=2m$ is even. Let $U$ be a unitary group, *i.e.* the group au automorphism preserving a given hermitian form on $V$.
The structure of local components of... | https://mathoverflow.net/users/43737 | Type of place versus type of unitary group | Things are perhaps a bit messier than you hope. In particular it is not true that the unitary group is non-quasi-split if and only if $v$ ramifies. Disclaimer: I did not know the answer to this question off the top of my head but I did want to know, so I just figured it out below; hopefully there are no errors (hopeful... | 10 | https://mathoverflow.net/users/1384 | 259605 | 117,085 |
https://mathoverflow.net/questions/254316 | 5 | On reading about matroids representable over partial fields, one learns about the 6th root of unity partial field, but other even-th root of unity partial fields seem to be absent from the standard repertoire of examples. Why is this? Is the matroid theory boring or not useful for those partial fields? I'd be particula... | https://mathoverflow.net/users/100907 | matroids representable over root of unity partial fields that are not the 6th root of unity partial field | Consider the partial field $P:=\{p\in \mathbb{C}: |p|=1\}$. It is a theorem of Whittle that if a matroid $M$ is representable over $P$, then $M$ is sixth-root of unity.
$P$ contains each $P\_k:=\{k^\text{th}\text{ roots}\}$, so all the $k$-root of unity matroids are also sixth-root-of-unity.
To represent $U\_{2,4}$... | 2 | https://mathoverflow.net/users/27941 | 259610 | 117,089 |
https://mathoverflow.net/questions/259601 | 1 | Given $\textbf{P}$ independent and identically distributed random variables, $X\_1, X\_2, ..., X\_P \sim \Gamma(M,2c)$ how can we prove that:
$$U = X\_1 + X\_2 + ... + X\_P$$
and
$$V = \frac{X\_1}{X\_1 + X\_2 + ... + X\_P}$$
are independent?
Where $U \sim \Gamma(MP,2c)$ and $V \sim \beta(M,M(P-1))$.
| https://mathoverflow.net/users/103291 | Independence of Gamma and Beta random variables with common term | Just for the record: this is answered in item 25 of <http://www.randomservices.org/random/special/Beta.html>
| 1 | https://mathoverflow.net/users/11260 | 259617 | 117,092 |
https://mathoverflow.net/questions/259627 | 4 | As shown in title, I am studying the LDPC code recently. However, I still can not calculate the error correction capacity of it, maybe due to complex decoding algorithm. And there lacks easy understanding properties to infer the error correction.
An easy example is RS code (Reed solomon code). For the setting RS(N,k)... | https://mathoverflow.net/users/88968 | How to quantify the error correction capacity of LDPC code? | It is a difficult problem to find the minimum distance of an LDPC. Vardy shows in [The Intractability of Computing the Minimum Distance of a Code](https://ieeexplore.ieee.org/document/641542) that it is NP-hard to find the minimum distance. Looking around one can find papers with various algorithms for minimum distance... | 6 | https://mathoverflow.net/users/51668 | 259632 | 117,095 |
https://mathoverflow.net/questions/259575 | 1 | If $X$ is a non-empty set and ${\cal A}, {\cal B}$ are covers, then we say that ${\cal A} \leq\_{\text{fin}} {\cal B}$ if for all $A\in {\cal A}$ there is $B\in{\cal B}$ such that $A\subseteq B$ and we say that ${\cal A}$ *refines* ${\cal B}$.
Let $G=(V,E)$ be a simple, undirected graph. A *clique decomposition* is a... | https://mathoverflow.net/users/8628 | Clique decompositions that are maximal with respect to refinement | Sure. Given the way that you have asked it, simply take the set of all cliques: every singleton point, every edge, all finite and infinite cliques. That is surely maximal.
All you really need is the set of all maximal cliques, those not properly contained in any other clique. You must have those and you are free to ... | 2 | https://mathoverflow.net/users/8008 | 259638 | 117,098 |
https://mathoverflow.net/questions/258989 | -1 | Given
\begin{equation}\label{eq:definition\_of\_z}
\begin{split}
\textbf{Z} = \left[\begin{array}{cccc}
{z}\_{11} & {z}\_{12} & \cdots & {z}\_{1P} \\
{z}\_{21} & {z}\_{22} & \cdots & {z}\_{2P} \\
{z}\_{31} & {z}\_{32} & \cdots & {z}\_{3P} \\
\vdots & \vdots & \ddots & \vdots \\
{z}\_{M1} & {z}\_{M2} & \cdots & {z}\_{MP... | https://mathoverflow.net/users/103291 | Expectation of the ratio between Beta and Gamma random variables | For the following case I have found the following equation for its expectation
$$
\mathbb{E} \left\lbrace y\_1 \right\rbrace = \mathbb{E} \left\lbrace \frac{\textbf{z}\_{i}^{H}\textbf{z}\_{j}}{\left( \textbf{z}\_{1}^{H}\textbf{z}\_{1} + \textbf{z}\_{2}^{H}\textbf{z}\_{2} + \cdots + \textbf{z}\_{P}^{H}\textbf{z}\_{P}... | 0 | https://mathoverflow.net/users/103291 | 259641 | 117,100 |
https://mathoverflow.net/questions/259640 | 5 | Let $G$ be a compact Lie group having a left-invariant complex structure $J$.
Is there a hermitian metric $h$ in $G$, compatible with the complex structure $J$, such that $G$ is a Kähler manifold?
In the case it is not possible to turn $G$ into a Kähler manifold, does it hold that exist a hermitian metric such that... | https://mathoverflow.net/users/12233 | When is a compact Lie group endowed with a left-invariant complex structure a Kähler manifold of balanced manifold? | To answer the first question: Only when $G$ is abelian. In fact, if $G$ is not abelian and has dimension $2d$, then there is no element of $H^2(G,\mathbb{R})$ whose $d$-th power is nonzero in $H^{2d}(G,\mathbb{R})$ (just look at the representation by bi-invariant forms), and there would have to be such an element if th... | 7 | https://mathoverflow.net/users/13972 | 259645 | 117,103 |
https://mathoverflow.net/questions/259395 | 10 | The Hasse norm theorem says a nonzero element $x$ of a number field $K$ is a norm for a cyclic extension $L/K$ iff $x$ is a norm for all completions $L\_p/K\_p$ at primes $p$ of $L$ (writing $p$ also for the restriction to $K$).
The global squares theorem says $x$ is a square in number field $K$ iff it is a square in... | https://mathoverflow.net/users/38783 | Does the Hasse norm theorem easily imply the global squares theorem? | Well, I would say that this crucially depends on what you define to be "quick". If you admit global class field theory, at least in its idelic formulation, the fact that all primes in $K$ are split in the extensions $K(\sqrt{x})/K$ proves that $x$ is a square quite rapidly (as Timo Keller has observed in the comments, ... | 2 | https://mathoverflow.net/users/18238 | 259648 | 117,105 |
https://mathoverflow.net/questions/259628 | 4 | Suppose I have a transitive model $M$ of ZFC, and - in $M$ - $U$ is a measure on $\kappa$. Then the transitive collapse of the ultrapower of $M$ along $U$ is an inner model, $N\subset M$.
My question is:
>
> Can we ever have $M$ be a class-generic extension of $N$?
>
>
>
*Context*: The set-forcing version of... | https://mathoverflow.net/users/8133 | Can an ultrapower be undone by class forcing? | The answer is negative, if one considers parameter-free definable ZFC-preserving class forcing satisfying the forcing theorem, which means that the forcing
relation is definable and true statements in the extension are
forced.
**Theorem.** Suppose that $\newcommand\P{\mathbb{P}}\P$ is a
ZFC-preserving definable class... | 5 | https://mathoverflow.net/users/1946 | 259658 | 117,110 |
https://mathoverflow.net/questions/259659 | 3 | Assume $X$ be a normal projective variety with $\mathbb Q$-Cartier divisor $D$, then can we extend adjunction formula on pair $(X,D)$?
| https://mathoverflow.net/users/103611 | Adjunction formula on pair | If the pair $(X,D)$ is log canonical and $S$ be a component of $D$ with coefficient 1, then we have adjunction type formula as $$K\_S+D\_S=(K\_X+D)|\_S$$
[edited]: If $f:Z\to X$ be a finite Galois map, then there exists a branch divisor $B$ on $X$ s.t, $K\_X+B$ is $\mathbb Q$-Cartier and $K\_Z=f^\*(K\_X+B)$.
As an ... | 4 | https://mathoverflow.net/users/nan | 259661 | 117,111 |
https://mathoverflow.net/questions/259660 | 0 | If $f:\mathbb{R}\to\mathbb{Q}$ is continuous, then it is constant. Are there infinite connected $T\_2$-spaces $X,Y$ such that the only continuous maps $f:X\to Y$ are the constant maps?
| https://mathoverflow.net/users/8628 | Connected $T_2$-spaces with only constant maps between them | Any real-valued function on the positive integers with the prime integer topology (subbasis of sets of the form $U\_p(b)=\{b+np:n\in \mathbb{Z}, p\nmid b\}$) is constant. This is on [page 82](https://books.google.com/books?id=Gc3DAgAAQBAJ&pg=PA82) of [Counterexamples in Topology](https://en.wikipedia.org/wiki/Counterex... | 3 | https://mathoverflow.net/users/3075 | 259665 | 117,112 |
https://mathoverflow.net/questions/259365 | 0 | Is there an extension of Ito's Lemma where $X\_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any reference is appraciated.
| https://mathoverflow.net/users/36886 | Non-smooth Ito lemma for semi-martingales | There are versions available for convex $f$ and for $f\in H^1$. Some places to start are [On semimartingale decompositions of convex functions of semimartingales](http://projecteuclid.org/euclid.ijm/1255987418) (Carlen and Protter) and [On Itô s formula for multidimensional Brownian motion](http://link.springer.com/a... | 1 | https://mathoverflow.net/users/42851 | 259670 | 117,114 |
https://mathoverflow.net/questions/259671 | 5 | Let $X$ be a smooth cubic threefold over $\mathbb{C}$ and let $L \subset X$ be a line.
The projection from $L$ yields a rational map $X \dashrightarrow \mathbb{P}^2$. Resolving the indeterminacy by blowing up $L$, we obtain a conic bundle morphism $\pi: \tilde{X} \to \mathbb{P}^2$.
Consider the relative anticanonic... | https://mathoverflow.net/users/5101 | The conic bundle of a cubic threefold | Choose coordinates $(U,V,X,Y,Z)$ in $\mathbb{P}^4$ so that $L$ is the line $X=Y=Z=0$. The equation of $X$ is of the form
$$ AU^2+2BUV+CV^2+2D U + 2EV +F=0\ ,$$where $A,B,\ldots F$ are homogeneous forms in $X,Y,Z$ of degree $1,1,1,2,2,3$. You can view this equation as a section $s \in H^0(\mathbb{P}^2,\mathrm{Sym}^2\mat... | 10 | https://mathoverflow.net/users/40297 | 259672 | 117,115 |
https://mathoverflow.net/questions/259637 | 18 | Let $a\_i,b\_i\in\mathbb{R}$ and $n>1$, does the inequality
$$
\left(\sum\_{i=1}^n a\_i^2\right)\left(\sum\_{i=1}^n b\_i^2\right)+\left(\sum\_{i=1}^na\_i b\_i\right)^2\ge \sqrt{\left(\sum\_{i=1}^n a\_i^4\right)\left(\sum\_{i=1}^n b\_i^4\right)}+\sum\_{i=1}^na\_i^2b\_i^2
$$
hold true?
Observe that $a\_i$ and $b\_i$ a... | https://mathoverflow.net/users/62673 | A seemingly simple inequality | The inequality is equivalent to
$$
\left(\sum\_{i>j} (a\_ib\_j+a\_jb\_i)^2+\sum\_{i} a^2\_ib^2\_i \right)^2\geq \sum\_{i} a^4\_i \sum\_{i} b^4\_i.
$$
The left hand side is greater or equal to
$$
\sum\_i a\_i^4b\_i^4+\sum\_{i>j} (a\_ib\_j+a\_jb\_i)^4+2(a\_ib\_j+a\_jb\_i)^2(a\_i^2b\_i^2+a\_j^2b\_j^2)+2a\_i^2b\_i^2a\_j^2b... | 19 | https://mathoverflow.net/users/100908 | 259675 | 117,116 |
https://mathoverflow.net/questions/259623 | 7 | Given a morphism of algebraic stacks $f: X \to Y$ (possibly non-representable) what sufficient conditions will guarantee that the derived push-forward of the structure sheaf on $X$ is isomorphic to the structure sheaf on $Y$? I am looking for a stacky version of a result of Buch-Mihalcea (Thm 3.1 of Quantum K-theory of... | https://mathoverflow.net/users/6223 | Derived push-forwards of structure sheaves for morphisms of algebraic stacks | I am amplifying the above comments. The theorem of Buch-Mihalcea also holds for stacks.
Let $k$ be a field of characteristic $0$. Let $Y$ be a Deligne-Mumford stack that is finite type over $k$ and that is normal: there exists a surjective, étale morphism $h:\widetilde{Y}\to Y$ with $\widetilde{Y}$ a normal scheme. L... | 3 | https://mathoverflow.net/users/13265 | 259678 | 117,118 |
https://mathoverflow.net/questions/259664 | 54 | Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell\_{01}$, $\ell\_{02}$, $\ell\_{03}$, $\ell\_{12}$, $\ell\_{13}$, and $\ell\_{23}$. Now consider the tetrahedron $T'$ with edge lengths
$$\begin{aligned}
\ell'\_{02} &= \ell\_{02} &
\ell'\_{13} &= \ell\_{13}\\
\ell'\_{01} &= s-\ell\_{01} &
\e... | https://mathoverflow.net/users/5010 | Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra | These are the so-called Regge symmetries, described by T. Regge in a 1970-ish paper. For a bit on it, with references, see the paper
*Philip P. Boalch*, MR 2342290 [**Regge and Okamoto symmetries**](http://dx.doi.org/10.1007/s00220-007-0328-x), *Comm. Math. Phys.* **276** (2007), no. 1, 117--130.\
| 39 | https://mathoverflow.net/users/11142 | 259680 | 117,119 |
https://mathoverflow.net/questions/259692 | 2 | I need to find a $3$-approximation algorithm for finding a $3$-hitting set.
The set-up is that I have a set $S$ and a family $\mathcal{F}$ of subsets of $S$, where each member of $\mathcal{F}$ **contains exactly $3$ elements**. I need to find a hitting set (a set which intersects all members of $\mathcal{F}$) with th... | https://mathoverflow.net/users/103623 | 3-Approximation Algorithm for 3-Hitting Set | Let $\mathcal{H}$ be a hypergraph where each hyperedge has size $3$. A *vertex cover* is a set of vertices $X$ such that every hyperedge is incident to a vertex in $X$. Rephrased, our goal is to find a small vertex
cover. One way to do this is to solve a dual problem. A *matching* in $\mathcal{H}$ is a set $M$ of hyper... | 2 | https://mathoverflow.net/users/2233 | 259695 | 117,123 |
https://mathoverflow.net/questions/259698 | 31 | I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949).
One key step is to prove that, with $\vartheta(x) = \sum\_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \log x+\sum\_{p\leq x}(\log p) \vartheta(x/p) = 2x\log x+O(x)$$
1. **What's the idea behind this identity?**
Here... | https://mathoverflow.net/users/85239 | Ideas in the elementary proof of the prime number theorem (Selberg / Erdős) | The complex-analytic proof of the prime number theorem can help inform the elementary one.
The von Mangoldt function $\Lambda$ is related to the Riemann zeta function $\zeta$ by the formula
$$ \sum\_n \frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)},$$
at least in the region $\mathrm{Re}(s) > 1$. The right-hand s... | 52 | https://mathoverflow.net/users/766 | 259719 | 117,131 |
https://mathoverflow.net/questions/259714 | 3 | Let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ such that $a\subseteq b$. This relation defines a lattice structure on $\text{Part}(X)$.
Is there a distributive lattice $L$ such that for n... | https://mathoverflow.net/users/8628 | Quotients of $\text{Part}(X)$ | In the paper
Ore, Oystein,
Theory of equivalence relations,
Duke Math. J. 9, (1942), 573–627
it is proved that $\textrm{Part}(X)$ is simple. (This is very easy to prove.)
So the only nontrivial quotient of $\textrm{Part}(X)$ is $\textrm{Part}(X)$ itself, which is not distributive when $|X|>2$. Therefore the ans... | 4 | https://mathoverflow.net/users/75735 | 259722 | 117,133 |
https://mathoverflow.net/questions/259721 | 4 | Let
$$ \forall\_{n=1\ 2\ \ldots}\quad I(n)\ :=\
\frac {(6\cdot n-3)!!}{(2\cdot n-1)!!\cdot(4\cdot n-3)!!} $$
Let $\ M(n)\ $ be the smallest natural number such that
$$ M(n)\cdot I(n)\ \in\ \mathbb Z $$
is an integer. What is the behavior of sequence $\ M(n)$? As a minimum, the Prime Number Distribution Theorem... | https://mathoverflow.net/users/8385 | Odd Chebyshev, part 2 | We have
$$
I(N)=\frac{(6N-2)!(N-1)!}{2(4N-2)!(3N-1)!}.
$$
We count the exponent of a prime $p$ in $k!$ using a standard formula $[k/p]+[k/p^2]+\dots$. This gives the sum $w(p):=\sum\_{s} h(n/p^s)$ for the exponent of $p$ in the number $I(n)\frac{6n-1}{4(4n-1)}$, where we denote $h(x)=[x]+[6x]-[3x]-[4x]$. $h$ takes the ... | 12 | https://mathoverflow.net/users/4312 | 259731 | 117,134 |
https://mathoverflow.net/questions/259616 | 4 | Here's [a very silly mistake I made recently](https://mathoverflow.net/questions/257613/getting-measures-especially-on-omega-2-from-potential-clubs): I claimed that if $\mathbb{P}\in L(\mathbb{R})$ is a forcing which adds a real, then $$(\*)\quad L(\mathbb{R})^{V^\mathbb{P}}=L(\mathbb{R})^\mathbb{P}.$$ While it is true... | https://mathoverflow.net/users/8133 | Forcing conflation for $L(\mathbb{R})$ | $(FC)\_{\text{proper}}$ is false. If there are infinitely many Woodin cardinals with a measurable above, then after adding a single Cohen real, we do not have $L(\mathbb R)[c] = L(\mathbb R)^{V[c]}$. The reason is that $L(\mathbb R)[c]$ does not satisfy $\text{AD}$: in $L(\mathbb R)[c]$, the set of ground model reals i... | 8 | https://mathoverflow.net/users/102684 | 259733 | 117,135 |
https://mathoverflow.net/questions/259734 | 8 | Basically, what the title says.
Presumably, one could use the fact that monoidal categories (resp. strict monoidal categories) are one-object bicategories (resp. 2-categories) and use the Lack model structure on those, but I am unsure if this would work or not.
| https://mathoverflow.net/users/68468 | Is there a model structure on (strict?) Monoidal Categories? | I'm not sure about the case of general monoidal categories. (Although I seem to recall a remark that there is no such structure since the category of monoidal categories is not cocomplete and a suitable replacement would be the category of multicategories. Perhaps somebody can confirm this.)
However, the case of stri... | 9 | https://mathoverflow.net/users/12547 | 259737 | 117,136 |
https://mathoverflow.net/questions/259726 | 6 | I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know that for a system of differential equations $P$, the holonomicity of the $D$-module $D\_X / D\_X P$ means that the system $... | https://mathoverflow.net/users/86286 | Example of non-holonomic D-module and explicit computation of characteristic variety | (These examples are shamelessly pilfered from *Gröbner Deformations of Hypergeometric Differential Equations* by Saito, Sturmfels and Takayama, which is perhaps *the* place to learn computational D-module stuff.)
Holonomic: $D\cdot\{z\_1\partial\_2,z\_2\partial\_1\}$. This left ideal has characteristic ideal of dimen... | 8 | https://mathoverflow.net/users/1481 | 259738 | 117,137 |
https://mathoverflow.net/questions/214985 | 9 | Let $(M, \omega)$ be a symplectic manifold and $L \subseteq M$ - a Lagrangian submanifold. I am trying to understand under what circumstances the Maslov homomorphism $I\_{\mu, L} \colon \pi\_2(M, L) \to \mathbb{Z}$ is in fact induced by an element $\mu\_L \in H^2(M, L;\mathbb{Z})$. I am encountering a few related issue... | https://mathoverflow.net/users/78275 | Maslov class as a relative cohomology class in $H^2(M, L)$ | I think it is always true, and one can explicitly define the class as follows, at least in the compact case (this construction is surely well-known to some people, but I'm not sure where to find it written down).
Let $M^{2n}$ be an almost complex manifold and $L^n \subseteq M$ a totally real submanifold (of course in... | 7 | https://mathoverflow.net/users/nan | 259746 | 117,140 |
https://mathoverflow.net/questions/259712 | 3 | **QUESTION** Find all triples of odd natural numbers $\ a < b\ $ and $\ c\ $ such that $\ a+b = c-1\ $ and
$$ \frac {c!!}{a!!\cdot b!!}\ =\ \frac {P(c)}{P(b)} $$
where $\ P(x) \ $ is the product of all primes $\le x$.
>
> The above fraction on the left looks somewhat similar to binomial coefficients but they ar... | https://mathoverflow.net/users/8385 | Odd Chebyshev, part 1 | Let me prove that there are only finitely many solutions for $a>1$. Assume the contrary, then for large $a$ we may suppose $b/a\to \lambda$, where $\lambda\geqslant 1$ is either finite constant or $+\infty$ (for fixed $a>1$ there are finitely many solutions of course). Using Stirling approximation (or its proof) and PN... | 4 | https://mathoverflow.net/users/4312 | 259748 | 117,142 |
https://mathoverflow.net/questions/247388 | 10 | Recall that a mouse is a structure of the form $(J\_\alpha[U],\in,U)$ with $U$ being an amenable ultrafilter with some iterability properties.
One of the interesting facts about mice is that given two mice, $m$ and $n$, there is an ordinal $\lambda$, such that if we iterate both mice $\lambda$ times, one is going to ... | https://mathoverflow.net/users/7206 | How verminous are mice? | Looking at your example of mouse, it seems you are going to use, and are asking about, the ‘old-fashioned’ fine-structural mouse, as used originally by Dodd and Jensen in their papers, and by Dodd in his book. It is crucial to say this since the ‘modern’ definition involved a reorganisation of these hierarchies followi... | 8 | https://mathoverflow.net/users/6942 | 259752 | 117,145 |
https://mathoverflow.net/questions/259728 | 4 | Let $p$ be an odd prime number,
let $\mathbb{Q}\_p$ be the field of $p$-adic numbers,
and let $\overline{\mathbb{Q}\_p}$ be an algebraic closure of it.
For a primitive $p$-th root of unity $\zeta\_p \in \overline{\mathbb{Q}\_p}$,
set $K = \mathbb{Q}\_p(\zeta\_p)$.
>
> Is it true that for every $a \in K^\*$ there e... | https://mathoverflow.net/users/38889 | Hilbert Symbols, Norms, and p-adic roots of unity | I think I can construct an explicit counterexample with $a\in\mathbb{Q}\_p$.
Choose a compatible sequence $\zeta\_{p^m}$ of $p^m$th roots of unity in $\overline{\mathbb{Q}}\_p$. Write $q=p^n$ with $n\geq1$. By local class field theory, $(a,\zeta\_p)\_q=1$ if and only if $a$, considered as an element of $\mathbb{Q}\_... | 6 | https://mathoverflow.net/users/1384 | 259756 | 117,146 |
https://mathoverflow.net/questions/259476 | 6 | *Let $X$ be the Hopf vector field on the three-sphere. Is there a smooth nowhere zero function $f$ so that the modified vector field $fX$ is not the Reeb vector field of *any* contact form on the three-sphere?*
This question is a follow-up on [aglearner's](https://mathoverflow.net/users/13441/aglearner) interesting [... | https://mathoverflow.net/users/21123 | Non-Reeb vector fields on the three-sphere | The answer is yes.
**Proposition.** *Let $\alpha$ be the standard contact form on the three-sphere (for which the Reeb vector field is the Hopf vector field $X$). If $f$ is a strictly positive function on $S^3$ such there exist two Hopf circles $\gamma\_1$ and $\gamma\_2$ for which
$$
\int\_{\gamma\_1} f^{-1} \alpha... | 2 | https://mathoverflow.net/users/21123 | 259761 | 117,148 |
https://mathoverflow.net/questions/253337 | 2 | Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied?
$$\text{(P) If }\,xy = x\,\text{ or }\, yx=x\,\text{ for some }\,x,y \in H,\,\text{ then }\,y \in H^\times.$$
The property implies that $xy \... | https://mathoverflow.net/users/16537 | Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit | Sorry for answering my own question, but it's definitely clear that there is no consolidated terminology for the kind of properties mentioned in the OP. One reason could be that they have never been considered before, which is the impression I've drawn from talking to various semigroup theorists. So we resolved to use ... | 1 | https://mathoverflow.net/users/16537 | 259790 | 117,163 |
https://mathoverflow.net/questions/259837 | 5 | Consider a graph $G$ with at least two unavoidable crossings, say, the disjoint union of two copies of $K\_5$. Can such a graph always be drawn so that there is only one singular point (where all crossings happen)? I guess there is an easy proof that this is not possible.
| https://mathoverflow.net/users/11504 | Can all crossings in a graph be moved to one point? | **No**, this is not always possible.
**Lemma.** Let $G$ be an $n$-vertex graph with at least $3n-2$ edges. Then $G$ cannot be drawn in the plane so that all crossings occur at the same point.
*Proof.* We make the standard assumption that every pair of edges which intersect in a drawing are not 'tangent' at the poin... | 17 | https://mathoverflow.net/users/2233 | 259839 | 117,176 |
https://mathoverflow.net/questions/259779 | 1 | Let $X = [x\_1 \cdots x\_N] \in \mathbb{R}^{d \times N}$ and $ T= [t\_1 \cdots t\_N] \in \mathbb{R}^{1 \times N}$. Define
$$
E(w) = \text{tr}(T - w^TX)(T - w^TX)^T
$$
as least square energy. When defining
$$
\epsilon = T- w\_{LS}^T X,
$$
with $w\_{LS}^T$ being the solution that minimizes $E$, can we find a simple bou... | https://mathoverflow.net/users/43967 | error bound for least square minimization | $\epsilon$ is the orthogonal projection of $T$ onto the orthogonal complement of the space spanned by the rows of $X$. Hence we have
$$
\|\epsilon\|\_\infty=\frac{\|\epsilon\|\_\infty}{\|T\|\_\infty}\|T\|\_\infty \leq \sup\_{(u,v)=0,u,v\neq 0} \frac{\|u\|\_\infty}{\|u+v\|\_\infty} \|T\|\_\infty\leq \frac{\sqrt{N}+1}{2}... | 2 | https://mathoverflow.net/users/100908 | 259856 | 117,184 |
https://mathoverflow.net/questions/259823 | 1 | Let $A$ be a ring (commutative and noetherian if it helps).
Suppose we are given an inverse system $M\_i$ of complexes of $A$-modules (where $i$ is a natural number),
and integers $a<b$
such that for each $i$, the complex $M\_i$ has non-zero cohomologies only in degrees $a<j<b$.
Consider the complex $\varprojlim M\_i... | https://mathoverflow.net/users/103691 | Does the inverse limit of complexes with bounded cohomology have a bounded cohomology? | No, even for $A=\mathbf{Z}$.
Take your favourite example of a complex of short exact sequences of abelian groups whose projective limit is not exact (see for example [this math.stackexchange question](https://math.stackexchange.com/questions/1153414/example-that-inverse-limit-is-not-exact)).
Now just make a complex... | 4 | https://mathoverflow.net/users/1384 | 259861 | 117,187 |
https://mathoverflow.net/questions/259863 | 8 | My teacher of Riemannian Geometry told the class about the problem of finding Einstein Manifolds in 5 dimension. My question is, what is the difficulty of this problem? Possessing a first course in Riemmanian geometry can I understand this?
Thanks
| https://mathoverflow.net/users/80399 | Einstein Manifolds in dimension five? | **Answer for the compact homogeneous case:** Due to Alekseevsky, Dotti and Ferraris (Homogeneous Ricci positive 5-manifolds, Pacific. J. Math, 175, 1-12, 1996) we know that in the non-symmetric case, the classification of 5-dimensional compact homogeneous Einstein spaces reduces to the classification of Einstein metric... | 17 | https://mathoverflow.net/users/20783 | 259870 | 117,193 |
https://mathoverflow.net/questions/258966 | 1 | Let $D$ be an unbounded pseudoconvex domain in $\mathbb{C}^2$. I would like to study the peak set of $D$.
1)Can the peak set of $D$ be empty? Or
2) Does $D$ always admit a nonconstant bounded holomorphic function?
Any reference or idea or example will be appreciated. Thanks.
Here is an example.
Let $f\in O(\m... | https://mathoverflow.net/users/59021 | Bounded holomorphic functions in unbounded domain | See MR0549981 (81b:32005)
Diederich, Klas; Sibony, Nessim
Strange complex structures on Euclidean space.
J. Reine Angew. Math. 311/312 (1979), 397–407.
In this paper, the authors construct various examples of Stein (hence pseudoconvex) open subsets of $\mathbb{C}^2$ diffeomorphic to $\mathbb{R}^4$ and not isomorphi... | 1 | https://mathoverflow.net/users/14493 | 259871 | 117,194 |
https://mathoverflow.net/questions/259875 | 4 | For given $n\times n$ real symmetric positive definite matrices $X\_{1}, X\_{2}$, let $Y := X\_{1}^{-1}X\_{2}$, and let $I$ be the identity matrix.
I would like to solve the following equation for the unknown $p>0$, in terms of functions of the matrices $X\_{1}$ and $X\_{2}$:
$$\text{trace}\left[\left(I + pY\right)... | https://mathoverflow.net/users/18526 | Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$ | Write $Y=SDS^{-1}$, where $D$ is diagonal (since $X\_1$ and $X\_2$ are psd, $Y$ is diagonalizable). Then, observe that
\begin{equation\*}
f(p) = \text{tr}(S^{-1}(I+pSDS^{-1})^{-1}SS^{-1}(I-p^2SDS^{-1})S).
\end{equation\*}
This simplifies into a simpler function of $p$ involving only the diagonal values in $D$. That ... | 10 | https://mathoverflow.net/users/8430 | 259879 | 117,198 |
https://mathoverflow.net/questions/259829 | 8 | $\require{AMScd}$Are there references for a construction of the enriched slice category of $\mathcal A \in \mathcal{V}\text{-Cat}$? A reasonable definition should be
1. Fix an object $a\in\mathcal A$ and let the objects of ${\cal A}/a$ be the set ${\cal V}(J, {\cal A}(x,a))$ for all $x\in\cal A$. $J$ is the monoidal... | https://mathoverflow.net/users/7952 | Enriched slice categories | There is at least one sense in which this "works". Namely, it is the comma object in the 2-category $\mathcal{V}$-Cat of $\mathrm{Id} : \mathcal{A} \to \mathcal{A}$ over $[a] : \mathcal{J} \to \mathcal{A}$, where $\mathcal{J}$ is the unit $\mathcal{V}$-category with one object and $J$ as its hom-object. It's true that ... | 7 | https://mathoverflow.net/users/49 | 259884 | 117,200 |
https://mathoverflow.net/questions/259745 | 2 | Adjoints between posets are (monotone or antitone) Galois connections; monads correspond to closure operators; what is a [two-variable adjunction](https://ncatlab.org/nlab/show/two-variable+adjunction) in this low-categorical setting?
I'm able to write the bare definition, of course. What I seek is intuition, and pos... | https://mathoverflow.net/users/7952 | Two variable adjunctions in poset-categories | One important special case of a 2-variable adjunction is a biclosed monoidal structure on a category, such as a cartesian closed category. A cartesian closed poset (with finite joins as well) is called a [Heyting algebra](https://ncatlab.org/nlab/show/Heyting%20algebra), and corresponds to intuitionistic logic in the s... | 4 | https://mathoverflow.net/users/49 | 259885 | 117,201 |
https://mathoverflow.net/questions/259307 | 11 | If $k$ is an algebraically closed field of characteristic zero and $H$ is a cocommutative Hopf algebra, then
$$
H \cong U(P(H)) \ltimes kG(H).
$$
What happens if the field is not algebraically closed? Is the theorem still true or is there any counterexample? What about characteristic different from zero?
| https://mathoverflow.net/users/103443 | Cartier-Kostant-Milnor-Moore theorem | When $k$ fails to be algebraically closed the theorem is false but the discrepancy can be understood in terms of Galois descent and so in principle understood in terms of Galois cohomology.
Suppose $H$ is a cocommutative Hopf algebra over $k$, not algebraically closed but characteristic $0$. Then the classification ... | 10 | https://mathoverflow.net/users/290 | 259886 | 117,202 |
https://mathoverflow.net/questions/259836 | 7 | In Sheaf theory one can obtain the Mayer Vietoris spectral sequence for cohomology. For $\mathcal{U}$ an open cover of $X$ we get the convergence
$E\_2^{pq} = \check H^p(\mathcal{U},H^q(-,F)) \Longrightarrow H^{p+q}(X,F)$.
In topological K theory it is a fact (see for exemple Karoubi's book II.4.18) that we have th... | https://mathoverflow.net/users/86526 | Mayer Vietoris Spectral sequence for topological K theory | As suggested by [Denis Nardin](https://mathoverflow.net/users/43054/denis-nardin) I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.
So, choose a cohomology theory $h^\*$ like e. g. $K$-theory, and, given a cover $(U\_i)\_{i\in I}$ of $X$, ... | 4 | https://mathoverflow.net/users/41291 | 259903 | 117,206 |
https://mathoverflow.net/questions/259905 | 2 | Suppose we have $\alpha \in \mathbb{R}$. Then we know that
$$\sum\_{1 \leq n \leq X} e(n \alpha) \ll \min \{ X, \|\alpha\|^{-1} \}$$
where $\| \cdot \|$ is the distance to the nearest integer.
I was wondering if we put von Mangoldt function as a weight and consider the exponential sum
$$S(\alpha) = \sum\_{1 \l... | https://mathoverflow.net/users/84272 | Exponential sum (linear in the argument) over primes | It's a classical sum going back to Vinogradov, but the estimate is more complicated. Usually, it is phrased in terms of an approximation $a/q$ of $\alpha$, with $a$ coprime to $q$, such that $|\alpha-a/q|\leq 1/q^2$ as
$$S(\alpha)\ll ((Xq)^{1/2}+Xq^{-1/2}+X^{4/5})$$
(up to powers of $\log X$). This is explained in many... | 4 | https://mathoverflow.net/users/103735 | 259914 | 117,212 |
https://mathoverflow.net/questions/259833 | 9 | I've heard that Reshetikhin-Turaev (RT) is stronger than homotopy, and it can distinguish certain homotopy-equivalent, but non-homeomorphic Lens spaces (I think $L(7,1)$ and $L(7,2)$). Now the Turaev-Viro-Barrett-Westbury (TVBW) invariant for a spherical fusion category $\mathcal{C}$ is the Reshetikhin-Turaev invariant... | https://mathoverflow.net/users/13767 | Is Turaev-Viro-Barrett-Westbury stronger than homotopy? | I think the following example seems to work: consider a special class of TVBW, the Dijkgraaf-Witten invariant whose input are a finite $G$ and a 3-cocycle from $H^3(G, C^\*)$ (together they define a spherical fusion category). More specifically, let $G$ be an Abelian group. The invariant evaluated on $L(p,q)$ is given ... | 8 | https://mathoverflow.net/users/39342 | 259923 | 117,216 |
https://mathoverflow.net/questions/259621 | 3 | **Context:**
I must find the Cartan 1-form for $Sp(2)$ before I start dealing with the natural connection of the Hopf fibration $S^3 \hookrightarrow S^7 \overset{\mathcal P}\to S^4$. To do so, the idea is to look at the restriction of the Cartan 1-form of $G = GL(2,\mathbb H)$ to $Sp(2)$. Now if $g = \begin{pmatrix} ... | https://mathoverflow.net/users/58158 | Computing the Cartan1-form for $Sp(2)$ | Here is the answer.
For each $g = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \in G = GL(2,\mathbb H)$, $\boldsymbol \Theta\_G (g)$ is given by $g^{-1}dq$. Now if $g \in Sp(2)$ then $g^{-1} = \bar g^T$ and $\color{red}{\bar g^T g = id}$ (this is what I was missing). In particular, $\bar \alpha\al... | 1 | https://mathoverflow.net/users/58158 | 259936 | 117,222 |
https://mathoverflow.net/questions/259920 | 7 | Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}\_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners:
**By its associated L-function**
Godement and Jacquet associate to it the automorphic L-function $L(s, \pi)$. This L-function... | https://mathoverflow.net/users/43737 | On the consistency of the definition of the conductor for automorphic forms | These definitions are consistent, though it's not immediate.
The conductor quantifies the extent to which $\pi$ is ramified. As an aside, I prefer to write $c(\pi)$ for the conductor exponent of $\pi$, which is a nonnegative integer, so that $\mathfrak{p}^{c(\pi)}$ is the conductor of $\pi$, and $q^{c(\pi)}$ is the a... | 12 | https://mathoverflow.net/users/3803 | 259940 | 117,225 |
https://mathoverflow.net/questions/259937 | 2 | Consider an almost periodic trigonometric polynomial $f(t)=e^{i2\pi t} + e^{i 2\pi \lambda t}$ for some irrational $\lambda$. I'm interested in distribution of such polynomial. In other words, is there a Borel measure $\mu$ defined on $X = Cl(f(\mathbb{R}))=2\mathbb{D}$ ($\mathbb{D}$ is the unit disk) such that for eve... | https://mathoverflow.net/users/85336 | Distribution of an almost periodic trigonometric polynomial | Map the torus $\mathbb T^2$ to $2 \mathbb D$ by the projection $\pi: (\theta, \phi) \mapsto e^{i\theta} + e^{i\phi}$. The trajectory is uniformly distributed on $\mathbb T^2$, and what you have is the image of that trajectory under the projection. Thus $\mu(E) = m(\pi^{-1}(E))$ where $m$ is normalized Lebesgue measure.... | 1 | https://mathoverflow.net/users/13650 | 259946 | 117,228 |
https://mathoverflow.net/questions/259942 | 1 | I have posted this question [in mathSE](https://math.stackexchange.com/questions/2082244/determining-the-rate-of-spread-of-geodesics-when-the-sectional-curvature-is-zero) a few weeks ago (and proposed a bounty) but so far got no response.
In the book Riemanian geometry (by do-Carmo), the following result is proved (C... | https://mathoverflow.net/users/46290 | Determining the rate of spread of geodesics when the sectional curvature is zero | I'm not 100% sure this works, since I haven't carried it out carefully. Here's what I would do:
1. Fix an orthonormal frame $e\_1, \dots, e\_n$ that is parallel along the geodesic and $e\_n = T$ is tangent to the geodesic.
2. Let $J\_1, \dots, J\_n$ be the Jacobi fields along the geodesic such that $J(0) = 0$ and $\n... | 2 | https://mathoverflow.net/users/613 | 259948 | 117,229 |
https://mathoverflow.net/questions/259951 | 11 | It's known that the Seifert–Weber space (obtained from a dodecahedron by gluing opposite faces with a 3/10 turn) is an example of a non-Haken 3-manifold. Since every closed 3-manifold is virtually Haken, I was wondering: is there was a known finite cover of the Seifert-Weber space that is Haken?
| https://mathoverflow.net/users/103758 | What is a finite Haken cover of the Seifert–Weber space? | This is constructed (reasonably explicitly) in John Hempel's 1982 paper.
*John Hempel*, MR 664329 [**Orientation reversing involutions and the first Betti number for finite coverings of $3$-manifolds**](http://dx.doi.org/10.1007/BF01393377), *Invent. Math.* **67** (1982), no. 1, 133--142.
| 8 | https://mathoverflow.net/users/11142 | 259958 | 117,232 |
https://mathoverflow.net/questions/259970 | 5 | This question is a sort of a follow-up to these two: [reference on classfication of multiply transitive permutation groups](https://mathoverflow.net/questions/222639/reference-on-classfication-of-multiply-transitive-permutation-groups) and [Multiply transitive groups, continued](https://mathoverflow.net/questions/16128... | https://mathoverflow.net/users/11142 | More on multiply transitive permutation groups | Yes. In the paper
P.J. Cameron, P.M. Neumann, and D.N. Teague, On the degrees of primitive permutation groups, Math. Z. 180 (1982), 141-149,
the stronger statement is proved that, for almost all $n$, the only primitive permutation groups of degree $n$ are $A\_n$ and $S\_n$. (By a fortunate coincidence I saw a refe... | 11 | https://mathoverflow.net/users/35840 | 259971 | 117,234 |
https://mathoverflow.net/questions/259995 | 3 | Let $G=(V,E)$ be a finite, simple, undirected graph. A *dominating set* is a set $D\subseteq V$ such that for all $v\in V\setminus D$ there is $d\in D$ such that $\{v,d\}\in E$. The *dominating number* $\gamma(G)$ is defined to be the minimum cardinality of a dominating set.
The complete graphs show that the dominati... | https://mathoverflow.net/users/8628 | Domination number and chromatic number | **No.** Suppose that such an $r$ exists. Choose $t \in \mathbb{N}$ such that $\frac{1}{t-1} < r$. Let $G$ be the disjoint union of $t-1$ copies of $K\_{t}$. Then $|V(G)|=(t-1)t$ and $\chi(G)=t$, so $\frac{\chi(G)}{|V(G)|}=\frac{1}{t-1} < r$. But $\gamma(G)=t-1 < \chi(G)$.
Note that this example can be made connected... | 3 | https://mathoverflow.net/users/2233 | 259998 | 117,244 |
https://mathoverflow.net/questions/259997 | 1 | Let $p\_n$ be the $n$-th prime, then from Wikipedia I got that
$p\_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.
What is a better approximation that includes
$O\big(\frac{(\ln \ln n)^3}{\ln^3 n}\big)$?
| https://mathoverflow.net/users/95470 | $n$th prime: a better approximation | You can find an in-depth answer to your question in this [paper of de Reyna and Jeremy](https://arxiv.org/abs/1203.5413). See in particular (65)-(66) along with (30) and Theorem 4.9. See also Theorem 6.2.
| 10 | https://mathoverflow.net/users/11919 | 260011 | 117,248 |
https://mathoverflow.net/questions/260010 | 3 | Let $x,y$ be two different points in a normed space. A bisector is defined as a set $B(x,y):=\{p: ||p-x||=||p-y||\}$, i.e. points of equal distance to both $x$ and $y$ in the space.
Theorem 25 in the second paper below basically says a finite-dimensional normed vector space is euclidean if and only if all bisectors a... | https://mathoverflow.net/users/103790 | Characterizing Euclidean metric with Bisectors (and others?) | These results can be pushed quite far for general metric spaces, and Herbert Busemann did so in The Geometry of Geodesics (1955), secs. 46 and 47. He worked in the context of [G-spaces](https://mathoverflow.net/questions/34394/g-spaces-and-manifolds), where the G is for geodesic, and proved
>
> (Theorem 47.4) If ea... | 5 | https://mathoverflow.net/users/nan | 260013 | 117,249 |
https://mathoverflow.net/questions/260009 | 4 | Let $f\_1,\ldots,f\_r\in\Bbb C[x\_1,\ldots,x\_n]$ define a map $f\colon\Bbb C^n\to\Bbb C^r$. Let $Z:=Z(f\_1,\ldots,f\_r)\subseteq \Bbb C^n$ be the variety cut out by the $f\_i$, and assume that $Z$ is nonempty. Define
\begin{align\*}
d:\Bbb C^n &\longrightarrow \Bbb R\_+ \\
x &\longmapsto \operatorname{dist}(x,Z)=\inf... | https://mathoverflow.net/users/9947 | Bounding the value of a polynomial map by the distance from the variety | Here is a counter-example. Take
$$f(x,y) = (x(1-x),xy-1)$$
So $Z = \{ (1,1) \}$. Consider now consider the family $(1/R,R)$ as $R \to \infty$. The distance from $(1/R,R)$ to $(1,1)$ goes to $\infty$, but $|f(1/R,R)| = |(1/R-1/R^2,0)|$ goes to $0$. So, if we were to have a bound of the form $|f(x,y)| > g(d(x,y))$, we'd ... | 3 | https://mathoverflow.net/users/297 | 260024 | 117,254 |
https://mathoverflow.net/questions/260005 | 13 | Let $M$ be a smooth $n$-manifold. Here are three constructions which produce manifolds which are homeomorphic to $M\times S^1$, but might not be diffeomorphic to it:
1. Take $M'\times S^1$, where $M'$ is homeomorphic to, but not diffeomorphic to $M$.
2. Take a connected sum of $M\times S^1$ with an exotic $n+1$-spher... | https://mathoverflow.net/users/85994 | Smooth structures on $M\times S^1$ | I assume that you are interested in this question in high dimensions, which should be 6 (or possibly 5) for $M \times S^1$. Construction 2 is actually a special case of construction 3, for if you take $(M \times S^1) \# \Sigma$ (the last being a homotopy sphere) then you can split along a copy of $M$ to get an s-cobord... | 7 | https://mathoverflow.net/users/3460 | 260046 | 117,261 |
https://mathoverflow.net/questions/260042 | 6 | Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$.
>
> **Question.** Is it true that $D^nf(z)=0$ has only real roots that are simple?
> If so, any slick proof?
>
>
>
| https://mathoverflow.net/users/66131 | roots of higher derivatives of exponential | The (physicists') [Hermite polynomials](https://en.wikipedia.org/wiki/Hermite_polynomials) are
$$ H\_n(x) = (-1)^n e^{x^2} D^n e^{-x^2}$$
And their roots are real. For that you don't need to know they are Hermite polynomials: just Rolle's theorem. See [this](https://math.stackexchange.com/questions/104845/the-roots... | 19 | https://mathoverflow.net/users/13650 | 260047 | 117,262 |
https://mathoverflow.net/questions/260049 | 3 | suppose I have a matrix $A\_{1}\in C^{m\*n}, m\geqslant n$ so the QR decomposition of $A\_{1}$ is $A\_{1}=Q\_{1}\*R\_{1}$. Now, define the augmented matrix $A\_{2}=\begin{bmatrix}
A\_{1}\\
I\_{n\*n}
\end{bmatrix}$ where $I\_{n\*n}$ is $n\*n$ identity matrix, so the QR decomposition of $A\_{2}$ is $A\_{2}=Q\_{2}\*R\_{2... | https://mathoverflow.net/users/77969 | Relation between the QR decomposition of matrix A and the QR of the augmented version of A | First consider the simple case, if $Q\_1 = I$, then you want to find $Q\_2$ and $R\_2$ such that $Q\_2 R\_2 = \begin{bmatrix} R\_1 \\ I \end{bmatrix}$. Split the columns of $Q\_2^T$ into left and right halves such taht $Q\_2^T = [U V]$. Note that $U^TU = V^TV = I$ and $U^TV = 0$. Then basically you want to find $R\_2 =... | 3 | https://mathoverflow.net/users/29887 | 260054 | 117,266 |
https://mathoverflow.net/questions/260060 | 24 | I am interested in what are the possible directions for new research in persistent homology (more of the mathematical theoretical aspects rather than the computer algorithm aspects).
So far from googling, proving stability (not affected by small changes) seems to be one direction.
Another direction I have seen is t... | https://mathoverflow.net/users/83274 | Research directions in persistent homology | Persistent homology is related to spectral sequences. There is already a discussion [Persistence barcodes and spectral sequences](https://mathoverflow.net/questions/208406/persistence-barcodes-and-spectral-sequences) about if this leads to new theoretical insides and there are stated further references (for example, th... | 14 | https://mathoverflow.net/users/75338 | 260078 | 117,275 |
https://mathoverflow.net/questions/260076 | 0 | Let $I$ be an ideal in a Noetherian quasi-unmixed local ring $(R,\mathfrak m)$ of dimension $d.$
Let $q(I)=\overline{I}\cap I^{sat}$ where $I^{sat}=\cup\_{n\geq 0}I:\mathfrak m^n.$ Then $q(I)$ is called *relative integral closure* of $I.$
**Question** Is there any example of a principal ideal $I$ (i.e. $I=(a)$) su... | https://mathoverflow.net/users/9485 | Example of a principal ideal which is properly contained in its relative integral closure | There are examples in arbitrary dimension. Begin with the polynomial ring $S=k[N]=k[s\_1,\dots,s\_d]$, i.e., the semigroup $k$-algebra on the semigroup $N=(\mathbb{Z}\_{\geq 0})^d$ of exponent vectors for monomials in $S$. Now for any subsemigroup $M\subset N$, let $R=k[M]\subset k[N]$ be the corresponding $k$-subalgeb... | 2 | https://mathoverflow.net/users/13265 | 260083 | 117,278 |
https://mathoverflow.net/questions/260097 | 9 | Let $X$ be an affine variety of dimension $n$ over $\mathbb{C}$.
>
> Does the analytic space associated with $X$ have the homotopy type of a $n$-dimensional CW complex?
>
>
>
| https://mathoverflow.net/users/nan | Homotopy type of a complex affine variety | If $X$ is smooth, and if you ask to have the same homotopy type of a CW complex of real dimension *at most* $n$, this is precisely the statement of the Andreotti-Frankel theorem.
It is true, more generally, for a Stein manifold of complex dimension $n$.
If $X$ is arbitrarily singular, the same theorem holds, provid... | 12 | https://mathoverflow.net/users/9871 | 260098 | 117,289 |
https://mathoverflow.net/questions/260106 | 6 | The Joyal model structure on the category of simplicial sets, has monomorphisms as cofibrations and quasi-categories as fibrant objects (these model $(\infty,1)$-categories). In HTT (section 2.3.4) Lurie defines the notion of an $n$-category (these model $(n,1)$-categories) and proves some theorems about them. For exam... | https://mathoverflow.net/users/50409 | "Joyal type" model structure for (n,1)-categories? | I don't know if this will work with the definition of $(n, 1)$-categories exactly as stated by Lurie, but it will work with a reasonable modification.
Definition 2.3.4.1 from HTT basically says that a quasicategory is an $(n, 1)$-category if it has no non-trivial morphisms above dimension $n$ and it insists that this... | 7 | https://mathoverflow.net/users/12547 | 260110 | 117,292 |
https://mathoverflow.net/questions/260107 | 15 | Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
In the general case, each successive projection can increase the number of facets from $f$ to $\left\lfloor \frac{f^... | https://mathoverflow.net/users/103831 | Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$? | Consider the polytope in $\mathbb{R}^3$ with $8$ vertices at coordinates $(\pm 1, \pm 2, 1), (\pm 2, \pm 1, -1)$. Geometrically this looks like a cube where the top face is stretched in the direction of the $y$-axis and the bottom face is stretched in the direction of the $x$-axis, but it still has the face structure o... | 25 | https://mathoverflow.net/users/39120 | 260114 | 117,295 |
https://mathoverflow.net/questions/260126 | 4 | It seems to me that the answer should be yes, but my naive attempts to come up with an example have failed.
Just to clarify, by finite hyperbolic geometry I mean a finite set of points and lines such that
1. every pair of points determines determines a unique line
2. every line contains at least two points
3. there... | https://mathoverflow.net/users/17798 | Does there exist a finite hyperbolic geometry in which every line contains at least 3 points, but not every line contains the same number of points? | Take a 2-$(v,4,1)$ design on $v$ points and delete one block, along with the four points on it. In the original system each point is on exactly $(v-1)/3$ blocks, so if we assume $v\ge25$ the geometry we get by deleting one block is hyperbolic, all blocks have at least three points, and there are blocks of size three an... | 4 | https://mathoverflow.net/users/1266 | 260135 | 117,303 |
https://mathoverflow.net/questions/260125 | 5 | It is common knowledge that every Hamiltonian system is locally integrable (away from singular points of the Hamiltonian), meaning that, in a neighborhood of each point of the $2n$-dimensional symplectic manifold on which the Hamiltonian vector field is defined, it is possible to find $n$ integrals of motion in involut... | https://mathoverflow.net/users/27069 | Why is every Hamiltonian system locally integrable? | For non-singular Hamiltonian systems, you can see it as a consequence of a slight generalization of the Darboux Theorem which is known as the Carathéodory--Jacobi--Lie Theorem (see, e.g. Libermann, Marle, Symplectic Geometry and Analytical Mechanics).
>
> **Carathéodory--Jacobi--Lie Theorem** Let $(M,\omega)$ be a ... | 10 | https://mathoverflow.net/users/12617 | 260145 | 117,308 |
https://mathoverflow.net/questions/260165 | 3 | It is known that $M\_n(R/J(R))\simeq M\_n(R)/M\_n(J(R))=M\_n(R)/J(M\_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R\_R)$, where $J(R)$ and $Soc(R\_R)$ stand for the Jacobson radical and the right socle of the ring $R$ (as a right $R$-module), but to no avail. Any help or suggestion would be ... | https://mathoverflow.net/users/48889 | Isomorphism concerning $Soc(M_n(R))$ | You can use the Morita equivalence between the categories $Mod(R)$ and $Mod(M\_n(R))$. The equiavlence is given by tensoring a right $R$ module with the bimodule $R^n$ over $R$ (in the other direction we also tensor with $R^n$, but over $M\_n(R)$). Under this equivalence, the $M\_n(R)$-module $M\_n(R)$ corresponds to t... | 3 | https://mathoverflow.net/users/41644 | 260180 | 117,315 |
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