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https://mathoverflow.net/questions/156019 | 1 | Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, \,\,\,\delta\in (0,1), \,\,\epsilon\in (0, 1].
$$
My question is that for any $I=[a, b],$ which is $|I|<1$ the following inequali... | https://mathoverflow.net/users/46144 | A question which belongs to a class of Zygmund functions | The standard Zygmund class (with $\epsilon=0$) is the Besov space
$
B^1\_{\infty,\infty},
$
that is using a Littlewood-Paley decomposition
$$
1=\sum\_{\nu\ge 0}\varphi\_\nu(\xi),\quad \varphi\_\nu(\xi) =\phi(\vert\xi\vert 2^{-\nu}) \text{ for $\nu\ge 1$, supp$\phi=[r,R], r>0,\ \varphi\_0\in C^\infty\_c$,}
$$
$$
f\in B^... | 1 | https://mathoverflow.net/users/21907 | 156119 | 82,750 |
https://mathoverflow.net/questions/155986 | 5 | Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subset $A$ of $G$ with $d$ elements and a subset $B$ such that $G=AB$ and $|AB|=|A||B|$ (equivalently, the product $AB$ is direct)?
Note: If $G$ has the property that there exists a subgroup of $G$ with order $d$ or $n/d... | https://mathoverflow.net/users/40520 | Factor subset of finite group | You can handle several of the groups on the list from your edit with a modification of the argument of Marty Isaacs. For the rest, you can find guidance as to what a counterexample would look like.
Say that $G$ satisfies $P(a,b)$ if $ab = |G|$ and there are subsets $A$ and $B$ of cardinalities $a$ and $b$ such that ... | 6 | https://mathoverflow.net/users/19729 | 156134 | 82,753 |
https://mathoverflow.net/questions/156149 | 11 | Let $X$ be a complex Banach space. Let $(\sigma\_t)\_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma\_t(x)$ is continuous for each $x \in X$. An element $x \in X$ is said to be **entire** (for the given flow $\sigma$) if $t \mapsto \sigma\_t(x)$... | https://mathoverflow.net/users/12281 | Does the generator of a 1-parameter group of Banach space isometries know which elements are entire? | If I am not mistaken, then this is connected to the notion of analytic vectors and related object.
If I understand your question correctly, it is answered in this [paper](http://www.ams.org/journals/tran/1972-167-00/S0002-9947-1972-0295125-5/) of Chernoff.
Further, for group generators on Banach spaces the analytic... | 6 | https://mathoverflow.net/users/12898 | 156155 | 82,758 |
https://mathoverflow.net/questions/156154 | 8 | I've been trying to read [this paper](https://www.uni-due.de/~mat903/preprints/ec/quot_sing.pdf) to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other cyclic quotient singularities.
I've heard that in higher di... | https://mathoverflow.net/users/12402 | Why can you deform singularities in two dimensions but not in higher dimensions? | The rigidity of quotient singularities in dimension greater or equal than $3$ was established by Schlessinger in his paper *[Rigidity of quotient singularities](http://link.springer.com/article/10.1007/BF01418741)*, Invent. Math. **14** (1971). Roughly speaking, he proved that if $(X, \,x)$ is a local scheme with an is... | 11 | https://mathoverflow.net/users/7460 | 156159 | 82,760 |
https://mathoverflow.net/questions/156076 | 15 | My question is a simple one: is there a group with the properties in the title?
In the absence of the 'finitely presentable' hypothesis, an example is provided by [Juschenko--Monod](http://arxiv.org/abs/1204.2132)'s construction of a finitely generated, infinite, simple, amenable group. On the other hand, all the sta... | https://mathoverflow.net/users/1463 | An infinite, amenable, finitely presentable group with no non-trivial finite quotients | It's a classical open question. The finitely generated case was also open until the 2012 examples of Juschenko-Monod (for which finite generation and non-existence of finite quotients was established by Matui 2006). Matui also checked that these examples are not finitely presented.
Such groups can certainly not be e... | 4 | https://mathoverflow.net/users/14094 | 156174 | 82,764 |
https://mathoverflow.net/questions/156179 | 3 | Let $A$ be a closed subset of $\mathbb{R}^{n}$. Can the quotient space $\mathbb{R}^{n}/A$ be embedded in some Euclidean space $\mathbb R^{m}$? In particular, assume that $A$ is an algebraic variety of degree $k$, can we control $m$ in term of $n$ and $k$?
| https://mathoverflow.net/users/36688 | The quotient of $\mathbb{R}^{n}$ by a closed subset | I shall prove the case for $S^{n}$ rather than $\mathbb{R}^{n}$ since $S^{n}$ works better being compact (for closed non-compact sets of $\mathbb{R}^{n}$ it does not work; see Daniele Zuddas's answer). Suppose that $A$ is a closed subset of $S^{n}\subseteq\mathbb{R}^{n+1}$. Then define a mapping $f:S^{n}\rightarrow\mat... | 11 | https://mathoverflow.net/users/22277 | 156187 | 82,770 |
https://mathoverflow.net/questions/156123 | 8 | Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a\_1T+a\_2T^2+\cdots$, where $a\_1\in \mathcal{O}\_K^\times$, $a\_n\in\mathfrak{m}\_K$ for $n\geq 2$, and $a\_n\to 0$ as $n\to \infty$.
This question is about perfec... | https://mathoverflow.net/users/271 | What are the automorphisms of a perfectoid Tate algebra? | Recall the following elementary fact in commutative algebra:
**Fact**: Say $f:R \to S$ is a map of commutative rings. Assume there exists a $p \in R$ that is a non-zero divisor in both $R$ and $S$, and that $R$ and $S$ are $p$-adically complete. Then $f$ is an isomorphism if and only if $f\_1$ is so; here $f\_n$ is r... | 9 | https://mathoverflow.net/users/46217 | 156189 | 82,772 |
https://mathoverflow.net/questions/156192 | 2 | Let $X$ be a quasi-projective irreducible scheme, $\mathcal{F}\_1$ a globally generated $\mathcal{O}\_X$-module and $\mathcal{F}\_2$ a coherent sheaf over $X$. Suppose that $\mathcal{F}\_1$ is globally generated by the sections $s\_1,...,s\_n \in \Gamma(X,\mathcal{F}\_1)$. I would imagine that it follows from the defin... | https://mathoverflow.net/users/45397 | Morphisms between a globally generated sheaf and a coherent sheaf(Edited) | It is not true. Take for example $F\_1$ and $F\_2$ to be the structure sheaves of two distinct points on $X$. Then both are globally generated, the spaces of global sections are 1-dimensional, and there is a nontrivial morphism between these vector spaces. But there is no morphism between the sheaves.
On the other ha... | 6 | https://mathoverflow.net/users/4428 | 156197 | 82,774 |
https://mathoverflow.net/questions/156153 | 4 | Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L\_1 \ldots L\_N u \in {}^{\infty}H^{\mathrm{loc}}\_{(-m-n/4)}(X), \quad n = \dim X,
$$
for any smooth vector fields $L\_1$, $\ldots$... | https://mathoverflow.net/users/17896 | Practical way to check whether a distribution is conormal | Let us take a look at the case where $Y\equiv x\_n=0$ in $\mathbb R^n$.
Let $a(\underbrace{\overbrace{x\_1,\dots,x\_{n-1}}^{x'},x\_n}\_x; \xi\_n)$ be a symbol in $S^m(\mathbb R^n\times \mathbb R)$ and let us consider the following "oscillatory integral":
$$
u(x)=\int\_{\mathbb R} e^{ix\_n\cdot \xi\_n} a(x,\xi\_n) d\... | 4 | https://mathoverflow.net/users/21907 | 156198 | 82,775 |
https://mathoverflow.net/questions/156194 | 7 | Is every noncountable field of characteristic zero the ultraproduct (using a non principal ultrafilter over the set of prime numbers) of fields of positive characteristic?
| https://mathoverflow.net/users/46219 | Fields of characteristic zero via ultraproducts | No. Ultraproducts over nonprincipal ultrafilters on a countable index set are always $\aleph\_1$-saturated. This rules out many fields just on cardinality basis (the cardinality of the ultraproduct must satisfy $\kappa=\kappa^\omega$), but even if the field has cardinality $2^\omega$, it does not have to be $\aleph\_1$... | 12 | https://mathoverflow.net/users/12705 | 156199 | 82,776 |
https://mathoverflow.net/questions/156090 | 14 | I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows:
\begin{equation}\mathbf{P}=\begin{pmatrix}
0 & 0& \dots&0 & 0 &1\\
0 & 0& \dots&0 & \frac{1}{2}&\frac{1}{2}\\
\vdots& & & & & \vdots \\
0 &\frac{1}{n-1}& \dots&\frac{1}{n-1}&\frac{1}{n-1}&\frac{1}{n-... | https://mathoverflow.net/users/44464 | Eigenvectors of a particular transition matrix | *EDIT2*. **(5th Nov, 2014).** Based on Darij's comments, am editing the answer to improve its clarity. The answer below shows how to get both **eigenvalues** and **eigenvectors** (my original answer was just for eigenvectors).
Eigenvalues
-----------
The key idea is to consider $P^{-1}$. Some (Markovian) guessing l... | 12 | https://mathoverflow.net/users/8430 | 156202 | 82,777 |
https://mathoverflow.net/questions/156170 | 4 | Does one have a base change theorem in crystalline cohomology like in étale cohomology?
Suppose one has the following cartesian diagram
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int\_0^1}\right.}
%
\beg... | https://mathoverflow.net/users/4504 | Base change in crystalline cohomology? | You finish your question by insisting that the isomorphism be such but "not in the sense of derived categories", which I do not understand completely, since both objects you have at hands naturally live in a derived category. If you are happy with that, Corollary 7.12 of Berthelot&Ogus' *"Notes on Crystalline Cohomolog... | 4 | https://mathoverflow.net/users/18238 | 156204 | 82,779 |
https://mathoverflow.net/questions/156183 | 1 | Goal
----
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
-----------------
I know that a projective transformation in $\mathbb{CP}^1$ can be seen as a Möbius transformation, and in that setup I'm well used ... | https://mathoverflow.net/users/25563 | Möbius transformation by 3 points in the Minkowski model | In fact, you are looking for an imbedding
$\text{PGL}\_2{\mathbb C}=\text{PSL}\_2{\mathbb C}\hookrightarrow\text{PO}({\mathbb R};3,1)=\text{Isom}\,{\mathbb H}\_{\mathbb R}^3$
that provides the subgroup of all orientation-preserving isometries of
${\mathbb H}\_{\mathbb R}^3$.
Consider the $4$-dimensional ${\mathbb R}$... | 2 | https://mathoverflow.net/users/40352 | 156208 | 82,780 |
https://mathoverflow.net/questions/155703 | 9 | Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's *Martingale limit theory and its applications*, Theorem 2.10) that there exists a constant $C\_p$ such that if $(X\_i)\_{i=1}^n$ is a real valued martingale difference with respect to the filtration $(\mathcal F\_i)\_{i=1}^n$ (that is, $(... | https://mathoverflow.net/users/17118 | Rosenthal like inequality for weak $\mathbb L^p$-norms | The result you want is mentioned in Remark 6 in
Johnson, W. B.(1-TXAM); Schechtman, G.(IL-WEIZ)
Martingale inequalities in rearrangement invariant function spaces.
Israel J. Math. 64 (1988), no. 3, 267–275 (1989).
| 3 | https://mathoverflow.net/users/2554 | 156213 | 82,782 |
https://mathoverflow.net/questions/102615 | 3 | First I want to review some concept from quadratic form.
Let $V$ be quadratic space over finite field $F$ and $char(F)\neq 2$ with quadratic form $q$. For exmaple
$q:V\rightarrow V$ and $|F|=q$ and $\dim V=n$.
We may assume the vectors of $V$ are represented as column vectors $v=(a\_1,b\_1,…)\in V$, and that $q$ ta... | https://mathoverflow.net/users/16607 | Number of Totally Isotropic Subspaces | Let $q$ denote the order of the finite field, which assume has odd characteristic. If $Q$ is a non-degenerate quadratic form in $n$ variables, for $n$ odd (this is your case 3), we have that the number of totally isotropic subspaces of dimension $k$ (for $k \leq \frac{n-1}{2}$, otherwise the number is $0$) is
$$ \frac... | 2 | https://mathoverflow.net/users/630 | 156215 | 82,784 |
https://mathoverflow.net/questions/156209 | 11 | Let $G$ be a finite group. Suppose that we can write $G= A \rtimes B$ and also $A = C \rtimes D$. Further suppose that C is normal in $G$ (not just in $A$). Then can we write $G = C \rtimes E$ where $E=G/C$? Of course, if $|C|$ and $[G:C]$ are relatively prime, then this follows from the Schur-Zassenhaus theorem.
We ... | https://mathoverflow.net/users/7443 | Iterated semi-direct products | I believe that it must split if $C$ is abelian.
More precisely, I will prove that if $C$ is an abelian $p$-group and $D$ is a $p'$-group, then $C$ has a complement in $G$ and so $G \cong C \rtimes (D \rtimes B)$. We are assuming that $G$ is finite. Note that under these assumptions $C$ is a characteristic subgroup of... | 7 | https://mathoverflow.net/users/35840 | 156221 | 82,786 |
https://mathoverflow.net/questions/156228 | -1 | Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{\*}$ between $\mathbb{Z}$-cohomologies?
In this question $A$ is merely a proper subset.(not necessarily compact, not necessarily submanifold)
| https://mathoverflow.net/users/36688 | A closed manifold with a subset with the same ring cohomology | Suppose for simplicity that $M$ is connected and orientable. Choose a point $x\in M\setminus A$, and a closed disc $U$ centred at $x$. We can use the chart to deform $i$ into a homotopic map $j$ such that $j(A)\subseteq M\setminus\text{int}(U)$. Collapsing the complement of $U$ gives a map $p$ from $M$ to the one-point... | 7 | https://mathoverflow.net/users/10366 | 156229 | 82,789 |
https://mathoverflow.net/questions/156231 | 8 | I've been playing around with Riemann surfaces of cubics, and it seems to me that all coverings of the Riemann sphere from equations of the form
$w^3 = q(z)$, where $q(z)$ is a cubic with three distinct roots, must be isomorphic. Is this correct? (Argument given below.)
Mainly I want to know if there are any good re... | https://mathoverflow.net/users/44693 | Riemann surfaces of $w^3 = (z-a)(z-b)(z-c)$ | Yes, you can view this equation as a family of curves (cyclic covers of $\mathbb{P}^{1}$) , as you said we can move each three points to $0,1,\infty$ and hence your family in fact reduces to a point in the moduli space of curves (or $A\_{g}$). Your curve will have genus $1$ and is smooth so is an elliptic curve. You ca... | 11 | https://mathoverflow.net/users/37808 | 156233 | 82,790 |
https://mathoverflow.net/questions/155943 | 19 | It is known that [there are only 14 reasonable tensor norms](http://arxiv.org/pdf/1101.4195.pdf) in $Ban$. On the other hand it is well known fact for topologists that [one can obtain only 14 different sets](http://en.wikipedia.org/wiki/Kuratowski%27s_closure-complement_problem) from a given set applying closure and co... | https://mathoverflow.net/users/19593 | Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related? | I asked David Sherman this question and this was his response:
In the Kuratowski setup, if you omit complementation, you get 7 elements. Adding complementation gives you a disjoint upside-down copy.
In the Grothendieck setup, if you omit duality, you get 8 elements (two five-element chains with common inf and sup, ... | 9 | https://mathoverflow.net/users/15388 | 156239 | 82,792 |
https://mathoverflow.net/questions/156235 | 4 | Let $X$ and $Y$ be **affine** varieties over $\mathbb C$, and consider a
morphism $f:X\to Y$ and the induced homomorhism
$$ \varphi=f^\*:B=\mathbb C[Y]\to A=\mathbb C[X]. $$
It is very easy to see that if $\varphi$ is surjective then $f$ is injective.
The converse is not true, as the inclusion $\mathbb C\setminu... | https://mathoverflow.net/users/15155 | Is the induced ring homomorphism surjective for a finite injective morphism between affine varieties? | No - consider the normalization of a cuspidal rational curve.
I don't understand your condition that $A$ and $B$ are free polynomial rings: do you mean that $X$ and $Y$ are isomorphic to $\mathbb{C}^n$ for some $n$? In that case the answer is positive. More precisely, the answer is positive if $B$ is integrally clos... | 11 | https://mathoverflow.net/users/1508 | 156242 | 82,793 |
https://mathoverflow.net/questions/156240 | 6 | The rank is the number of linearly independent rows/cols of a matrix. Generally, we think of linear independence as a binary property. But we could imagine an alternative definition that allows for numbers in the range [0,1]. Then, we could have fractional ranks.
I'm curious if there's any use to such generalizations... | https://mathoverflow.net/users/40246 | Is there a generalization of linear algebra that allows fractional ranks? | von Neumann thought about this; the keyword is [continuous geometry](http://en.wikipedia.org/wiki/Continuous_geometry).
| 11 | https://mathoverflow.net/users/290 | 156243 | 82,794 |
https://mathoverflow.net/questions/156152 | 9 | A countably complete ideal $I$ on a set $Z$ ideal is c.c.c. when there is no uncountable family of pairwise disjoint $I$-positive subsets of $Z$. If such an ideal exists, then there exists a weakly Mahlo cardinal $\kappa$ and a $\kappa$-complete, c.c.c. ideal $J$ on $\kappa$. An example of such an ideal is the collecti... | https://mathoverflow.net/users/11145 | Suslin trees in ccc ideals | Such trees must have countable height because every $x \in \kappa$ has to leave the tree at some level below $\omega\_1$ so that by ccc-ness of the ideal the tree dies at a countable level.
| 9 | https://mathoverflow.net/users/2689 | 156244 | 82,795 |
https://mathoverflow.net/questions/156227 | 6 | Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask this. I want to undersatnd Grothendieck's section conjecture and its recent results and also I want to study anabelian g... | https://mathoverflow.net/users/44006 | Basics on anabelian geometry and Grothendieck's section conjecture | if you just read the Hartshorne, your background is not enough for anabelian geometry. You need
A) To learn about etale morphismes: Hartshorne just mentions that notion in exercises.
B) to learn about the étale fundamental group Grothendieck,
before C) learning anabelian geometry proper.
Let me begin by B), bec... | 15 | https://mathoverflow.net/users/9317 | 156250 | 82,797 |
https://mathoverflow.net/questions/156245 | 2 | (Some of the notational choices I"m about to make might be iffy; I'm happy to take suggestions for improvements.)
Let $G$ be a (discrete) group. Think of it as an object in the $2$-category of small categories; then it has an automorphism $2$-group $\text{Aut}(G)$, and this automorphism $2$-group has a classifying s... | https://mathoverflow.net/users/290 | What's the relationship between B Aut(G) and B Aut(BG) for a (discrete) group G? | The 2-nerve of the automorphism 2-group of the groupoid $(G\Rightarrow \*)$ is indeed homotopy equivalent to $BAut(BG)$.
Indeed, the
$$
\text{1-nerve of the 2-group of automorphisms of $(G\Rightarrow \*)$}
$$
is isomorphic, not just homotopy equivalent, as a group object in simplical sets to the
$$
\text{simpli... | 7 | https://mathoverflow.net/users/5690 | 156251 | 82,798 |
https://mathoverflow.net/questions/156210 | 5 | Let $(T\_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T\_t1 =1 $ for every $t\geq 0$.
(This is also called a Feller semigroup.)
Since $K$ is compact we know that there exists a probability measure $\mu$ on $K$
satisfying $\mu T^\*\_t = \mu $ f... | https://mathoverflow.net/users/21061 | Is irreducibility sufficient for uniqueness of invariant distribution for a Feller semigroup? | Take $K = \{0,1\}^{\mathbf{Z}^2}$ and take for $T\_t$ the Glauber dynamic for the Ising model below the critical temperature. Then $T\_t$ is Feller and irreducible, but it has two distinct ergodic invariant measures.
If however you know that $T\_t$ is strong Feller (or asymptotically strong Feller) then irreducibilit... | 5 | https://mathoverflow.net/users/38566 | 156255 | 82,799 |
https://mathoverflow.net/questions/155932 | 4 | Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P\_\theta$ is a probability measure on $(\cal X, \cal A)$. Suppose we have measurable function $J:\cal X\times \Theta\rightarrow\mathbb{R}^d$such that
\begin{equation}
\int{J(x,\theta... | https://mathoverflow.net/users/45305 | Integral wrt probability measure | Here is (I believe) a counterexample:
For $\theta\in\Theta=(-1,1)$ let $$P\_\theta= \frac{\theta^2}{2} \delta\_{-1/\theta} +\frac{\theta^2}{2} \delta\_{1/\theta}+(1-\theta^2)\delta\_0$$
(where $\delta\_x$ is the Dirac measure in $x$) if $\theta\neq 0$ and $P\_0=(\delta\_{-1}+\delta\_1)/2$.
For $J(\theta,x)= x$ your ... | 4 | https://mathoverflow.net/users/21051 | 156261 | 82,801 |
https://mathoverflow.net/questions/156266 | 14 | Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW-complex, i.e. is there a CW-complex $X$ with a subcomplex $Y$ and homotopy equivalences $g\colon X\rightarrow M$ and $h... | https://mathoverflow.net/users/32022 | Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex | I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absolute neighbourhood retracts, and thus have the homotopy type of CW-complexes. This is briefly stated in corollary 1 of Mi... | 10 | https://mathoverflow.net/users/21095 | 156272 | 82,805 |
https://mathoverflow.net/questions/156264 | 3 | Let $k$ be a field and $A$ a $k$-algebra with unit. The trace module is
$$
T(A)=A/[A,A],
$$
where $[A,A]$ is the left $A$-module generated by all elements of the form $ab-ba$ for $a,b\in A$. The natural trace map is the projection $T:A\to T(A)$.
For an $A$-module $P$ one wants to construct a trace map
$$
Tr\_P: End\_A... | https://mathoverflow.net/users/nan | Trace of finitely generated projective module | There's a standard way to define the trace (look up "Hattori-Stallings trace") that agrees with yours, but is clearly independent of choices.
For any (left) $A$-module $P$, there's a natural map
$$\operatorname{Hom}\_A(P,A)\otimes\_AP\to\operatorname{End}\_A(P),$$
sending $\varphi\otimes y$ to the endomorphism $x\ma... | 9 | https://mathoverflow.net/users/22989 | 156274 | 82,806 |
https://mathoverflow.net/questions/156279 | 4 | Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$.
Let $p\_1,\dots,p\_m$ be *all* lattice points in $P$.
**Question:** What is the condition on $P$ that guarantees that every lattice point in the dilation $nP$ can be expressed as $k\_1p\_1 + k\_2p\_2 + \cdots + k\_n p\_n$, ... | https://mathoverflow.net/users/1056 | Lattice points in dilated polytopes and sumsets | Your second question is known to be true by a theorem of Khovanskii: that is, $g(n)$ is
eventually a polynomial for large $n$. Khovanskii also compares this number with the Erhart polynomial -- that is he shows that the $n$ fold sums are approximately contained in the $n$-th dilate of the polytope (this is not so hard... | 5 | https://mathoverflow.net/users/38624 | 156284 | 82,810 |
https://mathoverflow.net/questions/156265 | 2 | Ref to : Sara Negri & Jan von Plato, *Structural Proof Theory* (2001).
In *Ch.6 : Structural Proof Analysis of Axiomatic Theories* [page 126-on], they
>
> give a method of adding axioms to sequent calculus, in the form of nonlogical rules of inference.
>
>
> **Theorem 6.4.1** [page 136] : If $\Gamma \implies \D... | https://mathoverflow.net/users/42676 | A question about consistent fragments of formalized mathematical theories with Natural Deduction | In negation-free arithmetic, replace the axiom $\neg 0=S(x)$ with $0=S(x) \rightarrow 0=1$.
This has the same negation-free theorems as ordinary arithmetic.
The amusing reason for this is that any arithmetical statement follows from $0=1$, even without negation: $0=1$, so $0=a$ and $0=b$ and therefore $a=b$, and so o... | 3 | https://mathoverflow.net/users/nan | 156287 | 82,813 |
https://mathoverflow.net/questions/155590 | 9 | **Edit:** The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a [new question](https://mathoverflow.net/questions/210996/an-algebraics-hamiltonian-vector-field-with-a-finite-number-of-periodic-orbits2).
Is There a polynomial Hamiltonian $H(x,y,z,w)=zP(x... | https://mathoverflow.net/users/36688 | An algebraic Hamiltonian vector field with a finite number of periodic orbits(1) | Answer to part two: No - this cannot be - even if you relax the conditions and only assume the vector field to be smooth.
Indeed, the question can be generalized somewhat: let $X : N \to TN$ be any smooth vector field on a manifold $N$. Then we may define a Hamiltonian
$H : T^\*N \to \mathbb{R} \qquad$ by $\qquad H... | 5 | https://mathoverflow.net/users/4500 | 156288 | 82,814 |
https://mathoverflow.net/questions/155596 | 30 | The objects of interest in motivic homotopy theory are "spaces"-which are simplicial sheaves of sets on the big Nisnevich site $Sm/k$ of smooth schemes of finite type over a field $k$. I understand that we need to enlarge the category of smooth schemes over $k$ to spaces, since $Sm/k$ is not closed under limits and col... | https://mathoverflow.net/users/42656 | Reasons for the use of Nisnevich topology in motivic homotopy theory | Here are some comments about the use of topologies in motivic homotopy theory. This is based on the discussion in Morel-Voevodsky's "A^1-homotopy theory of schemes" p.94-95 (MV below), I only add some background and references. I also comment on the differences between the development of the unstable and stable theorie... | 26 | https://mathoverflow.net/users/7878 | 156292 | 82,816 |
https://mathoverflow.net/questions/156238 | 51 | Yesterday I was shocked to discover that [function extensionality](http://coq.inria.fr/library/Coq.Logic.FunctionalExtensionality.html) (the statement that if two functions $f$ and $g$ on the same domain satisfy $f\left(x\right) = g\left(x\right)$ for all $x$ in the domain, then $f = g$) is not an axiom in the standard... | https://mathoverflow.net/users/2530 | Function extensionality: does it make a difference? why would one keep it out of the axioms? | I am going to draw heavily from Github discussion on [HoTT book pull request 617](https://github.com/HoTT/book/pull/617).
There are different kinds of equality. Let us say that equality is "intensional" if it distinguishes objects based on how they are defined, and "extensional" if it distinguishes objects based on t... | 59 | https://mathoverflow.net/users/1176 | 156295 | 82,818 |
https://mathoverflow.net/questions/155099 | 12 | Suppose the space $X$ has a countable basis and $X$ is $T\_{0}$. Must there exist a separable metrizable space $Y$ and a quotient map q:$Y \rightarrow X$?
(Some surrounding facts:
Every metrizable space is 2nd countable iff it's separable.
Every 2nd countable space is 1st countable and hence Frechet and hence seq... | https://mathoverflow.net/users/17029 | Is every T0 2nd countable space the quotient of a separable metric space? | Every second-countable $T\_0$ space is a quotient of a separable metric space and this essentially follows from the proof of the $T\_1$ case given by Paul Strong in [*Quotient and pseudo-open images of separable metric spaces*](http://www.ams.org/journals/proc/1972-033-02/S0002-9939-1972-0300253-7/home.html) [Proc. Ame... | 7 | https://mathoverflow.net/users/2000 | 156297 | 82,820 |
https://mathoverflow.net/questions/156282 | 3 | It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ decomposes into an element of $SU(N)$ in the adjoint representation, $[Ad\_U]$, times something else,
$$
O=M\cdot Ad\... | https://mathoverflow.net/users/21919 | $SO(N^2-1)$ and the adjoint representation of $SU(N)$ | Actually, it does not look like that. Take the case $N=3$. The representation of $\mathrm{SU}(3)$ on ${\frak{so}}(8)$ breaks up into the $8$-dimensional subspace ${\frak{su}}(3)$ and an irreducible $20$-dimensional space that is isomorphic to the complex homogeneous cubic polynomials in $3$ variables as an $\mathrm{SU}... | 10 | https://mathoverflow.net/users/13972 | 156300 | 82,822 |
https://mathoverflow.net/questions/156312 | 6 | Solve $f(x)=\int\_{x-1}^{x+1} f(t) \mathrm{d}t$.
When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?
| https://mathoverflow.net/users/46267 | Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$ | By differentiating we obtain
$$f'(x)=f(x+1)-f(x-1)$$
This type of equations was addressed in the MO question
[On equation f(z+1)-f(z)=f'(z)](https://mathoverflow.net/questions/114875/on-equation-fz1-fz-fz/114878#114878)
Let $\lambda$ be any (complex) root of the equation
$$\lambda=e^\lambda-e^{-\lambda},$$
which is eq... | 26 | https://mathoverflow.net/users/25510 | 156315 | 82,825 |
https://mathoverflow.net/questions/150199 | 4 | A relation T with domain and range in a Hilbert space is said to be accretive if the
transformation $ (T − \lambda)/(T + \bar \lambda\ ) $
with domain and range in the Hilbert space is contractive for some, and hence every,
complex number $\lambda$ in the right half–plane.
How does some $\lambda$ imply for all $\lamb... | https://mathoverflow.net/users/30081 | definition of accretive operator | Check Kato: [Perturbation theory...](http://poncelet.sciences.univ-metz.fr/~gnc/bibliographie/Functional%20Analysis/Perturbation_Theory_for_Linear_Operators__Second_Edition__Classics_in_Mathematics_%283%29.pdf), Section V.3.10.
| 2 | https://mathoverflow.net/users/12898 | 156331 | 82,828 |
https://mathoverflow.net/questions/156298 | 1 | On a domain $\Omega$ with $f \in L^2(0,T;H^{-1})$ such that $f(0) = f(T)$, consider
$$u\_t - \Delta u = f\quad\text{on $\Omega$}$$
$$\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$
$$u(0) = u(T)$$
Is there any well-posedness result in Sobolev/Bochner spaces for this problem?
I am wondering whe... | https://mathoverflow.net/users/43946 | well-posedness of heat equation with Neumann BC and periodic data | take a look [here](http://netlizama.usach.cl/Keyantuo-Lizama%28JEE-Final%29%282006%29.pdf). their setting is more abstract, but your specific equation should be just a special case of it.
| 0 | https://mathoverflow.net/users/26039 | 156335 | 82,830 |
https://mathoverflow.net/questions/156334 | 2 | The geodesic distance $d(p\_1, p\_2)$ on a geodesically complete, Riemanian manifold is a metric in the sense of a metric space metric. I'd like to know when other infinitesimal metric structures (e.g. finlser metrics) induce such a global metric and what the exact relation of the two things is.
What is the largest c... | https://mathoverflow.net/users/41654 | Local vs distance function metric structures | I answer only your last question.
Take a finite-dimensional $\mathbb C$-linear space $V$ equipped with a positive-definite hermitian form $\langle-,-\rangle$. The tangent space $\text{T}\_p{\mathbb P}\_{\mathbb C}V$ at a $1$-dimensional linear subspace $p\subset V$ is naturally identified with $\text{Lin}\_{\mathbb C... | 3 | https://mathoverflow.net/users/40352 | 156339 | 82,832 |
https://mathoverflow.net/questions/150575 | 1 | Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell\_i(x) \geq 0, i=1, \ldots, d$.
Of course, from the symplectic point of view, the inequalities $a\_i \ell\_i \geq 0$ still define the same polytope for all positive real $a\_i$.
What I wonder is the following. Does the function $g: P \ri... | https://mathoverflow.net/users/19545 | Legal potentials on delzant polytopes | Yes, but it might be singular. For instance, if $a\_i = 1/\beta\_i$, then the potential $u$ corresponds to a conical K\"ahler metric with cone angles $2\pi\beta\_i$ along the divisor corresponding to the face $[l\_i=0]$. In particular if you take $a\_i$ to be an integer it will correspond to an orbifold structure.
| 1 | https://mathoverflow.net/users/11945 | 156345 | 82,837 |
https://mathoverflow.net/questions/156338 | 24 | Who is responsible for the generalization of Hamilton's quaternions to other types of quaternion algebras, and when did this occur? In particular, Hamilton's quaternions are the 4-dimensional algebra generated over $\mathbb{R}$ by the elements $1, i, j, ij$, where $i^2=j^2=-1$, and $ij=-ji$, invented in the 1840s. In a... | https://mathoverflow.net/users/14835 | history of quaternion algebras | In the early 1900s, Dickson introduced what he called generalized quaternion algebras over any field $K$ of characteristic not 2. These are exactly what we'd call quaternion algebras over $K$. His definition was in terms of a basis with rules for their products, and he gave a criterion for these to be division rings. I... | 33 | https://mathoverflow.net/users/3272 | 156346 | 82,838 |
https://mathoverflow.net/questions/156353 | 3 | Let $X$ be a compact connected Hausdorff topological space.
We say $X$ is a cohomologicaly minimal space, briefly CM space, if $X$ satisfies the following property:
"For every proper subset $A\subset X$ with the inclusion map $i:A \to X$, $i^\*$ is NOT a ring isomorphism between $H^\*(X,\mathbb{Z})$ and $H^\*(A,\ma... | https://mathoverflow.net/users/36688 | Cohomologically minimal spaces | Let me say that $X$ has property $CM'$ if for every field $F$ and every point $a\in X$, the map $H\_\*(X\setminus a;F)\to H\_\*(X;F)$ is injective but not surjective. Then orientable closed manifolds have $CM'$, and $CM'$ implies $CM$. Now suppose that $X$ and $Y$ both have $CM'$. After noting that
$$ (X\times Y)\setm... | 5 | https://mathoverflow.net/users/10366 | 156356 | 82,842 |
https://mathoverflow.net/questions/156293 | 3 | As we all know, for a complex manifold $M$, its de Rham complex admits a decomposition into a double complex called the Dolbeault complex. If $M$ also admits a Kahler metric, then we get the wonderful [Kahler identities](http://mathworld.wolfram.com/HodgeIdentities.html).
Can I ask what extra structure we get on $\O... | https://mathoverflow.net/users/41562 | The de Rham complex of a quaternion-Kahler manifold | The fundamental 4-form on a quaternionic-Kahler manifold is closed and gives the same kind of decomposition as the Kahler form on a Kahler manifold. Reference: Bonan, Edmond, Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique. [Exterior algebra of a quaternionic almost Hermitian manifold] C. R. A... | 6 | https://mathoverflow.net/users/3377 | 156377 | 82,847 |
https://mathoverflow.net/questions/156373 | 2 | Let G be a subgroup of $E(n) = \mathbb{R}^n \rtimes O(n)$(the rigid motions of $\mathbb{R}^n$ ) with orbit space as a point.
Example: the group of all translations of $\mathbb{R}^n$ and of course any group containing it.
Q:
1.Are there other examples of G not containing all the translations? If such G exist , How m... | https://mathoverflow.net/users/18984 | Subgroups of $E(n) = \mathbb{R}^n \rtimes O(n)$ with trivial orbit space | In the simply-transitive case, there is the following result.
**Theorem**. Let $G$ be a simply-transitive group of affine rigid motions acting on a finite dimensional Euclidean space $V$. Then $G$ is a connected solvable group, and there exists an orthogonal decomposition $V=U\oplus W$, where $U\neq0$, and a homomor... | 3 | https://mathoverflow.net/users/15155 | 156382 | 82,849 |
https://mathoverflow.net/questions/156371 | 1 | Because I'm doing research in the area of harmonic function theory I would like to know are there any conjectures in the theory of harmonic functions in $\mathbb{R}^{n}$ still open. I know that there are many conjectures related to manifolds, but I don't want them. I want conjectures that are just related to harmonic f... | https://mathoverflow.net/users/44967 | Conjectures in classical harmonic function theory | There are many conjectures about harmonic functions in $R^n$ which are open. Not surprisingly, most of them are very hard. Even about harmonic polynomials in $R^3$ there
are many unsolved problems.
For example:
1. A question of of Nadirashvili.
Let $u$ be a harmonic function in $R^3$, and consider the set $\{ x:u(x... | 9 | https://mathoverflow.net/users/25510 | 156386 | 82,852 |
https://mathoverflow.net/questions/156394 | 8 | Let $G$ be a semisimple Lie group and $T$ be its maximal torus. Can we say that $G/T$ is a projective variety?. Is there any proof or counterexample for it?
| https://mathoverflow.net/users/nan | Is $G/T$ a projective variety? | Suppose that $G$ is compact, connected, and semisimple. Let $T\subseteq G$ be a maximal torus. Take the complexification $G\_{\mathbb{C}}$ of $G$, and choose a Borel subgroup $B\subseteq G\_{\mathbb{C}}$ containing $T$. Using the Iwasawa decomposition $G\_{\mathbb{C}}=GB$, we see that $G$ acts transitively on $G\_{\mat... | 7 | https://mathoverflow.net/users/25358 | 156398 | 82,854 |
https://mathoverflow.net/questions/156399 | 8 | Given a finite set $S$ of $n\times n$ integer matrices, it is known
that for $k\geq 3$ it is undecidable whether some product of them
(allowing repetitions) is the zero matrix (called the mortality
problem). It is also known
(<http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.102.448&rep=rep1&type=pdf>)
that if ... | https://mathoverflow.net/users/2807 | Decidability of a matrix product being the identity | This is undecidable in dimension 4 or up see <http://cgi.csc.liv.ac.uk/~igor/papers/matrixcomp.pdf>
The result is proved in Bell, P. C., Potapov, I.: On the undecidability of the identity correspondence problem and its applications for word and matrix semigroups, International Journal of Foundations of Computer Scien... | 7 | https://mathoverflow.net/users/15934 | 156401 | 82,855 |
https://mathoverflow.net/questions/156380 | 0 | Are there any known examples of analytic, globally continuous functions $f(x,y): (x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ such that
$$f(x,y)\lt 0\Leftrightarrow (x,y)\not\in\mathcal{P}$$
$$f(x,y)=0\Leftrightarrow (x,y)\in\partial\mathcal{P}$$
$$f(x,y)\gt 0\Leftrightarrow (x,y)\in\mathcal{P} \setminus... | https://mathoverflow.net/users/31310 | Examples of Bivariate Analytic, Globally Continuous Functions, whose Set of Zeros are Boundaries of Polygons | OK, here is a proof (mimic one-variable complex calculus). Assume that a power series $f(x,y)=\sum a\_{ij}x^iy^j$ vanishes on $y=0$, $-\epsilon<x\le0$. Then, as usual, $i!a\_{i0}=\partial^if/\partial x^i(0,0)=0$. (As we assume that the partial derivatives exist, we can use negative values to compute them.) But then $f(... | 3 | https://mathoverflow.net/users/44953 | 156404 | 82,857 |
https://mathoverflow.net/questions/156402 | 5 | Is there a standard principle in reverse math that is known to be equivalent (over $RCA\_0$) to the existence of a set of high (Turing) degree? I'm interested in the general case, but would be happy to learn of such a principle for $\omega$-models.
I haven't been able to find much discussion on this topic... but then... | https://mathoverflow.net/users/38989 | Reverse Math of High Sets? | In the paper "On a conjecture of Dobrinen and Simpson regarding almost everywhere domination", Binns, Lerman, Solomon and I constructed $\omega$-models of this "high" principle which demonstrate it does not imply WKL, WWKL, or $G\_\delta$-regularity.
| 8 | https://mathoverflow.net/users/4600 | 156409 | 82,861 |
https://mathoverflow.net/questions/155626 | 8 | I am interested in models of intuitionistic linear logic, that is, the logic that you get if you take classical linear logic and restrict the set of operators to $\otimes$, $1$, $\multimap$, $\times$, $\top$, $+$, $0$, and $!$. I know what categorical models of this logic are. However I am looking for something more co... | https://mathoverflow.net/users/25527 | Models of intuitionistic linear logic that reflect the resource interpretation | I think the Petri net semantics for linear logic probably best captures this intuition that it is a logic about resource manipulation. The idea is that Petri nets model the movement of tokens (i.e., resources) through a network. Here's the money quote from Lokhorst 1997.
>
> Petri nets are models of dynamic proces... | 5 | https://mathoverflow.net/users/7066 | 156415 | 82,864 |
https://mathoverflow.net/questions/156412 | 11 | I'm imagining a simple random walk on $\mathbb{Z}$ with three independent particles (maybe add laziness so they don't jump over each other). Suppose the particles are initially placed at, say, $-10$, $0$ and $10$. The distance between any two particles evolves like a random walk, so the expected hitting time for those ... | https://mathoverflow.net/users/20381 | Hitting time for two out of three random walk particles | The answer is the same for random walks and for Brownian motion.
If you project a $3$-dimensional Brownian motion perpendicular to $x=y=z$, you get a $2$-dimensional Brownian motion. The projection of the set where $x\lt y\lt z$ is a wedge of angle $\frac{\pi}{3}$. Your question is whether the first exit time from a ... | 10 | https://mathoverflow.net/users/2954 | 156417 | 82,865 |
https://mathoverflow.net/questions/111100 | 7 | Before asking my question, a caveat: The category theorist in me would like me to ask this question in more generality, but I will restrict my scope since what I'm really after is some geometric intuition.
Let $R$ be a commutative ring. A *$\mathbb Z$-filtered $R$-module* is a diagram $\cdots \hookrightarrow X\_{\leq... | https://mathoverflow.net/users/78 | A geometric characterization of Rees algebras in categories without Choice | Here is a formulation: if $\mathcal{F}$ is quasi-coherent $\mathbf{G}\_{m, R}$-equivariant $\mathcal{O}\_{\mathbf{A}^1\_R}$-module such that $\epsilon$ is a nonzero divisor on $\mathcal{F}$, then one gets a filtered module as in your question.
On the one hand, it is clear that the modules you've constructed have this... | 1 | https://mathoverflow.net/users/44817 | 156421 | 82,868 |
https://mathoverflow.net/questions/156422 | 3 | Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B\_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in $\operatorname{Spec}B$. (See, e.g., page 187 of Matsumara's *Commutative Ring Theory*.)
Is there a known example of a non finitely ge... | https://mathoverflow.net/users/16046 | Counterexample to Openness of Flat Locus | Here is one example. Let $S$ denote the set of positive prime integers.
Let $A$ be $\mathbb{Z}$. Let $R$ be the countably generated polynomial ring over $\mathbb{Z}$,
$$R = \mathbb{Z}[\{x\_p:p\in S\}].$$ Let $I\subset R$ be the ideal generated by $\{px\_p : p\in S\}$. Let $B$ be $R/I$. Then the ideal $\mathfrak{p}$ of... | 4 | https://mathoverflow.net/users/13265 | 156428 | 82,869 |
https://mathoverflow.net/questions/150197 | 7 | Background
==========
What I think of *Ehrhart theory* (<http://en.wikipedia.org/wiki/Ehrhart_polynomial>) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th dilation of $P$, the result is polynomial in $t$.
If, however, $P$ is only a *rational* polytope, then in gen... | https://mathoverflow.net/users/1102 | How "accidental" are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin? | This is an interesting and wide open question. The easiest case when considering Ehrhart quasipolynomials, namely, when *all* constituent polynomials are equal, goes by the name of *period collapse*. The most famous instance comes from representation theory (see [this paper by J. De Loera & T. McAllister](http://arxiv.... | 4 | https://mathoverflow.net/users/3193 | 156430 | 82,871 |
https://mathoverflow.net/questions/155677 | 4 | This question is coming from the fact that all the counter examples for which second order stochastical domination holds but first oder stochastical domination fails do not accept increasing likelihood ratio condition.
From this, a natural question is: If second order stochastical domination together with increasing ... | https://mathoverflow.net/users/36356 | Does second order stochastical domination with increasing likelihood ratio imply first order domination? | Increasing likelihood ratio alone implies first-order domination due to the following (very) special case of the FKG inequality:
>
> For any random variable $X$ and any two positive increasing functions $\phi,\psi$ we have $\mathsf{E} \, \phi(X) \psi(X) \ge \mathsf{E} \, \phi(X) \, \mathsf{E} \, \psi(X)$.
>
>
>
... | 3 | https://mathoverflow.net/users/22758 | 156433 | 82,873 |
https://mathoverflow.net/questions/156441 | 3 | Assume that $\Omega$ is a domain in $\mathbf{R}^n$ with the same area as a ball $B(x,r)$ and let $\alpha\in[1,0)$. I need the reference for the following inequality $$\int\_{\Omega} |x-y|^{\alpha-n} dy\le \int\_{B(x,r)} |x-y|^{\alpha-n} dy.$$
| https://mathoverflow.net/users/36162 | Riesz potential inequality | $$\int\_{\Omega}|x-y|^{\alpha-n}dy$$
$$=\int\_{\Omega\cap B(x,r)}|x-y|^{\alpha-n}dy+\int\_{\Omega\cap B(x,r)^c}|x-y|^{\alpha-n}dy$$
$$\leq\int\_{\Omega\cap B(x,r)} |x-y|^{\alpha-n}dy+\int\_{\Omega\cap B(x,r)^c}|r|^{\alpha-n}dy$$
$$=\int\_{\Omega\cap B(x,r)} |x-y|^{\alpha-n}dy+\int\_{\Omega^c\cap B(x,r)}|r|^{\alpha-n}dy... | 4 | https://mathoverflow.net/users/36119 | 156445 | 82,876 |
https://mathoverflow.net/questions/156151 | 0 | Currently I'm studying "Introduction to dyamical systems" by Stuck and Brin. In chapter 6 they define absolutely continuous and transversely absolutely continuous foliations. By proposition 6.2.2 if $W$ is transversely absolutely continuous then it is absolutely continuous. But the converse is not true in general. I'm ... | https://mathoverflow.net/users/43741 | An absolutely continuous foliation, which is not transversely absolutely continuous | Here is an 'non-dynamical' example. Let $C\subset [0,1]$ be the standard middle-third Cantor set, $f$ be a homeomorphism on $[0,1]$ such that $|f(C)|>0$. Then consider the segments $L\_y$ connecting $(0,y)$ with $(1,fy)$ for all $0\le y \le 1$, which form a foliation, say $\mathcal{L}$, of the unit rectangle $R=[0,1]^2... | 3 | https://mathoverflow.net/users/11028 | 156462 | 82,881 |
https://mathoverflow.net/questions/156290 | 29 | I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange).
I have been "brought up" as an algebraic geometer. Spectral sequences are obviously ubiqutous and useful in this subject. The conclusion I have drawn from my exposu... | https://mathoverflow.net/users/5181 | origin of spectral sequences in algebraic topology | It seems to me that everything written so far is addressing the title, not the body of the question, in particular focusing on the unstable. Part of that is people asking their own questions. But I think it is best to start with the easiest questions. Indeed, Tom, you have singled out the right place to start: Yes, the... | 15 | https://mathoverflow.net/users/4639 | 156483 | 82,889 |
https://mathoverflow.net/questions/147073 | 9 | The following definition is well known ($\kappa$ is regular uncountable cardinal):
**Definition:** a sequence $\mathcal{C} = \langle \mathcal{C}\_\alpha | \alpha < \kappa,\,\alpha \text{ limit ordinal} \rangle$ is a $\square (\kappa, <\lambda )$-sequence when for every $\alpha$, $|\mathcal{C}\_\alpha| < \lambda$, eve... | https://mathoverflow.net/users/41953 | Weak threads in $\square (\kappa, <\kappa)$ sequences | The answer is negative (at least at double successors of regular cardinals): it's possible that every square sequence is weakly threaded and there are Aronszajn trees.
For simplicity let work with $\omega\_2$.
We start with supercompact cardinal. Using Unger construction in [Fragility and indestructibility of the t... | 4 | https://mathoverflow.net/users/41953 | 156489 | 82,892 |
https://mathoverflow.net/questions/156495 | 6 | I am aware of some of the history of the gamma function $\Gamma(z)$, partly through
a 2009(!) MO question "[Who invented the gamma function?](https://mathoverflow.net/a/9790/6094)"—Euler, Bernoulli, etc.
My question does not seem to be answered in that discussion, or in other historical accountings I can easily locate:... | https://mathoverflow.net/users/6094 | Why does the gamma function use the symbol $\Gamma(\,)$? | The first use of $\Gamma$ in this sense is due to Legendre (1811). It is unknown why he choose that notation but some speculations are recorded at <http://jeff560.tripod.com/functions.html>
They range from the Gamma being an inverted L (from Legendre) and variants, to the in my opinion more integresting that the log... | 7 | https://mathoverflow.net/users/nan | 156498 | 82,895 |
https://mathoverflow.net/questions/72397 | 8 | Let $M$ be a smooth oriented $n$-dimensional manifold. My favorite model of $\operatorname{Chains}\_\bullet(M) \otimes \mathbb R$ is the space of smooth compactly-supported de Rham forms on $M$, shifted in degree by $[n]$. I like it because the intersection pairing of chains is well-defined: it corresponds after the gr... | https://mathoverflow.net/users/78 | Does there exist a model of chains on oriented manifolds with both a strict intersection pairing and strict functoriality for closed embeddings? | After work on this and other problems, I have come to the conclusion that the correct answer to my question is "No, there does not exist a model of chains with simultaneously strict intersection and diagonal maps". Even in characteristic 0, the problem is obstructed in dimension 1, as I prove in <http://arxiv.org/abs/1... | 2 | https://mathoverflow.net/users/78 | 156503 | 82,897 |
https://mathoverflow.net/questions/156519 | 4 | I'm thinking about coisotropic/involutive submanifolds of the symplectic phase space $T^\*\mathbb{R}^n$ (with coordinates $x\_1,\ldots,x\_n,\xi\_1,\ldots,\xi\_n$). As I understand, the smallest coisotropic submanifold is this setting is the Lagrangian $\{\xi\_1=\ldots=\xi\_n=0\}$ (aka the zero section). Of course, the ... | https://mathoverflow.net/users/46424 | coisotropic submanifolds | Locally, any codimension-$k$ submanifold can be described as the zero locus of $k$ smooth functions $f\_1,\dots,f\_k$. (This is true globally if and only if the submanifold has trivial normal bundle.) A submanifold $M \subset T^\*\mathbb{R}^n$, which is cut out by $f\_1,\dots,f\_k$ is coisotropic if the functions are i... | 4 | https://mathoverflow.net/users/394 | 156533 | 82,905 |
https://mathoverflow.net/questions/156252 | 3 | Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O\_q(Q\_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do for reals to find a Cartan decomposition?
Any reference would be highly helpful.
| https://mathoverflow.net/users/36735 | Indefinite orthogonal groups over p-adics | You should have a look at the following article by Delorme and Secherre :
Delorme, Patrick; Sécherre, Vincent, An analogue of the Cartan decomposition for $p$-adic symmetric spaces of split $p$-adic reductive groups.
Pacific J. Math. 251 (2011), no. 1, 1–21.
Aso see :
Y. Benoist and H. Oh, "Polar decomposition ... | 4 | https://mathoverflow.net/users/4767 | 156539 | 82,909 |
https://mathoverflow.net/questions/137149 | 2 | Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise orthogonal atoms of $L$. Is the converse also true?
| https://mathoverflow.net/users/6985 | Are there atomistic ortholattices which are not modular? | Yes there are. Any non-modular finite orthomodular lattice is an example.
A concrete example: take the Boolean algebras $2^2$,$2^3$ and identify their top and bottom elements. This is clearly an atomistic ortholattice, which is not modular, because it contains a pentagon as a sublattice.
| 2 | https://mathoverflow.net/users/4814 | 156547 | 82,913 |
https://mathoverflow.net/questions/156546 | 2 | Let $H$ be an infinite subgroup of a discrete group $G$. $H$ is called nearly normal if it is commensurable with a normal subgroup $K$ of $G$, that is $H\cap K$ is a finite index subgroup of both $H$ and $K$. If $H$ contains a finite index subgroup which is also a normal subgroup of $G$, then $H$ is obviously nearly no... | https://mathoverflow.net/users/nan | Does every nearly normal subgroup contain a normal subgroup? | Let $K$ be an infinite product of cyclic groups of order 2, and $H$ an index two subgroup of $K$. Let $A$ be the full automorphism group of $K$, and $G=K\rtimes A$.
Then since $G$ acts transitively on the non-trivial elements of $K$, $H$ has no non-trivial subgroup normal in $G$.
| 3 | https://mathoverflow.net/users/22989 | 156550 | 82,914 |
https://mathoverflow.net/questions/156551 | 2 | The term "distribution" is commonly associated with statistics and, less commonly known, to generalized functions.
**Questions:**
* what is known about the origin of the term in the two fields?
* are the terms related in some sense i.e. were the distributions in the context of generalized functions inspired by the... | https://mathoverflow.net/users/31310 | What are the Reasons for the Ambiguous Meaning of "Distribution" in Mathematics | The name "distributions" for generalized functions is explained by Laurent Schwartz in his autobiography "A Mathematician Grappling with His Century", p. 238 : he chose the name "Because, if $\mu$ is a measure, i.e. a particular kind of distribution, it can be considered as a distribution of electric charges in the uni... | 9 | https://mathoverflow.net/users/40297 | 156552 | 82,915 |
https://mathoverflow.net/questions/156554 | 20 | [Wikipedia states that](http://en.wikipedia.org/wiki/Archimedes%27_cattle_problem)
>
> [Archimedes' cattle problem] was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of forty-four lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773.
>
>
>
I have accepted that s... | https://mathoverflow.net/users/46454 | Original manuscript of Archimedes' cattle problem | Here is a link to the full document at the Hochschule für angewandte Wissenschaften in Augsburg:
<http://www.hs-augsburg.de/~harsch/graeca/Chronologia/S_ante03/Archimedes/arc_bous.html>
| 13 | https://mathoverflow.net/users/36090 | 156561 | 82,918 |
https://mathoverflow.net/questions/156456 | 0 | In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The equations governing the two phases
$$\frac{du\_1}{dt} - \sum\_i \frac{d}{dx\_i}\left(k(u\_1)\frac{du\_1}{dx\_i}\right)... | https://mathoverflow.net/users/46326 | A basic question about JL Lions' transformation of a Stefan problem | Lions called it a "Kirchhoff transformation", hence presumably the K (p. 197, bottom line). In Google, this points to a lot of informative papers.
| 0 | https://mathoverflow.net/users/46461 | 156570 | 82,921 |
https://mathoverflow.net/questions/156256 | 1 | Sorry, I asked this two days ago, but this time I modified it to be easily read and added more specific explanation. I hope to get your illuminating comment on whether my approach is right.
I am computing the order of some $GL(2)$-local $L$-function at $s=0$.
Let $F$ be a local field and $E=F \times F$. Let $\chi$ ... | https://mathoverflow.net/users/29422 | Order of some $L$-function at $s=0$ | $L\_E(s,BC(\pi)\otimes \chi)$ is not the product of L-functions but a product of local L-factors. So I think it should be $L\_F(s,\pi \otimes \chi\_1) \cdot L\_F(s,\pi^{\lor} \otimes \chi\_1^{-1})$ if $\chi=(\chi\_1,\chi\_1^{-1})$.
Thus if $\pi=B(\lambda\_1,\lambda\_2)$, then I think $L\_E(s,BC(\pi)\otimes \chi)=\fra... | 0 | https://mathoverflow.net/users/29422 | 156578 | 82,923 |
https://mathoverflow.net/questions/156575 | 2 | Let $M\_{g,[n]}$ be the moduli space of curves over complex number with level-$n$ structure. It is known that when $n\geq 3$, there is a universal family of curves over it: $f:\,C\_{g,[n]} \to M\_{g,[n]}$. There is a natural involution $\tau$ on $M\_{g,[n]}$ defined by $\tau\big([C, \alpha]\big) = [C,-\alpha]$, where $... | https://mathoverflow.net/users/46464 | Involutions on Moduli space of curves $C_{g,[n]} \to M_{g,[n]}$ | Yes. $C\_{g,[n]}$ is a fine moduli space for curves with one marked point + level $n$ structure. Just define $\sigma $ by $\sigma ([C,p,\alpha ])=[C,p,-\alpha ]$. If $C$ is hyperelliptic, the hyperelliptic involution $\iota $ induces an isomorphism $[C,p,-\alpha ]\cong [C,\iota (p), \alpha ]$, so $\sigma \_{|C}=\iota $... | 4 | https://mathoverflow.net/users/40297 | 156584 | 82,928 |
https://mathoverflow.net/questions/156581 | 5 | **Definition:** Suppose $\mathcal A$ is
the Boolean algebra of all Jordan measurable sets in $I=[0,1]$ (i.e $\mathcal A=\{A\subseteq I: \mu(\partial(A))=0\}$, where $\mu$ is the Lebesgue measure and $\partial$ means the topological boundary).
**Definition:** "Interval algebra" suppose $\mathcal B$ on $I=[0,1]$ is the... | https://mathoverflow.net/users/41345 | Uncountable atomless subalgebras of the Boolean algebra of all Jordan measurable sets in [0,1] | It seems to me that the cardinalities are off, so (1) fails, since $\cal A$ contains an atomless Boolean algebra of cardinality $2^{\frak c}$, and so many distinct atomless Boolean subalgebra of that size, but $\cal B$ has size continuum. So in these instances, they cannot be isomorphic.
Similarly, a cardinality argu... | 6 | https://mathoverflow.net/users/1946 | 156588 | 82,929 |
https://mathoverflow.net/questions/116137 | 24 | Recently I did some explicit computations that involved the BCH series, $\log(e^x e^y)$. Here $x$ and $y$ are non-commuting variables, and the BCH series lives in the graded completion $FL(x,y)$ of the free Lie algebra generated by $x$ and $y$.
Mostly by chance I found that when BCH is written in the Lyndon basis of ... | https://mathoverflow.net/users/8899 | The BCH series in terms of Lyndon words | This pattern of vanishing coefficients in the expansion of the Baker-Campbell-Hausdorff formula in the basis of Lyndon words is explained in section IV.C of the article [An efficient algorithm for computing the Baker–Campbell–Hausdorff series
and some of its applications](http://www.ehu.es/ccwmuura/research/bch.pdf) of... | 5 | https://mathoverflow.net/users/10881 | 156589 | 82,930 |
https://mathoverflow.net/questions/156518 | 3 | Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup\_i H^1\_i) \cup \{\text{2-handles}\}$ such that:
1. $L \cap H^0$ is a set of trivial arcs;
2. $L \cap H^1\_i$ is either empty or the core of $H^1\_i$ for all $i$;
3. $... | https://mathoverflow.net/users/23193 | Handlebody decomposition of a 3-manifold adapted to a link | Take a Heegaard splitting of the link exterior and then glue in solid tori with your link components as cores. The Heegaard splitting survives as a Heegaard splitting of the filled manifold and since the link is disjoint from the Heegaard surface it is disjoint from the 2-handles. To see that the other properties hold,... | 4 | https://mathoverflow.net/users/30679 | 156592 | 82,933 |
https://mathoverflow.net/questions/156585 | 1 | Given a nondiscrete compact Hausdorff space $K$, does there always exist a real-valued function $f$ on $K$ that is not locally constant? Why/why not?
In <http://arxiv.org/abs/math/9505204> the authors show that there are compact Hausdorff spaces $K$ such that all $f \in C(K)$ are locally constant on a dense subset (e... | https://mathoverflow.net/users/46472 | Existence of non-locally constant functions | A $P$-space is a completely regular space where the countable intersection of open sets is open. It is well known and easy to prove that that a completely regular space is a $P$-space if and only if every continuous function is locally constant.
In fact, there are several other necessary and sufficient conditions for w... | 4 | https://mathoverflow.net/users/22277 | 156593 | 82,934 |
https://mathoverflow.net/questions/156601 | 3 | My friend asked me to ask his question here. Where he can find (a paper or a book) *containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary tree*?
Thanks.
| https://mathoverflow.net/users/43954 | A structure of the group of automorphisms of an infinite binary tree | I am assuming you mean a binary rooted tree. I don't know your definition of complete. The group is the infinite permutational wreath product of symmetric groups of degree 2. Good references are [Automata, dynamical systems and infinite groups](http://www.math.tamu.edu/~grigorch/publications/PSIM128.PS) by R. Grigorchu... | 10 | https://mathoverflow.net/users/15934 | 156603 | 82,938 |
https://mathoverflow.net/questions/156507 | 1 | I was reading about the finite propagation speed of the wave equation on a Riemannian manifold. I was wondering if instead of the Laplace-Beltrami operator $\Delta$ we consider the equation $$\partial\_t^2 u = Lu$$ where $L$ is a negative-definite self-adjoint elliptic operator of order 2, would we still have finite sp... | https://mathoverflow.net/users/46397 | Finite propagation speed of second order operators | Your example is a special case of an operator with a wave-like principal symbol on a Lorentzian manifold. Namely, if the coefficients of the second order derivatives in $L$ are $L = h^{ij} \partial\_i \partial\_j + \cdots$, then $h^{ij}$ is an inverse Riemannian metric and it is always possible to write $L = \Delta\_h ... | 1 | https://mathoverflow.net/users/2622 | 156608 | 82,942 |
https://mathoverflow.net/questions/156568 | 3 | Independent set is polynomial in claw-free graphs,
so I am wondering if this can approximate independent set.
By adding enough edges to $G$ and gets claw-free $G'$.
IS in $G'$ is IS in $G$, so this is a bound for the MIS.
If one adds very few edges, the bound is better.
>
> Is it possible to efficiently find ... | https://mathoverflow.net/users/12481 | Making a graph claw-free by adding as few edges as possible | It's NP-complete, by a reduction from finding the largest triangle-free subgraph of a given graph $G$, which was proved NP-complete by M. Yannakakis, "Edge-deletion problems", *SIAM J. Comput.* 10(2):297–309, 1981.
Given a graph $G$ for which we want to find the largest triangle-free subgraph, let $H$ be the graph fo... | 8 | https://mathoverflow.net/users/440 | 156613 | 82,944 |
https://mathoverflow.net/questions/156606 | 3 | Let $Y$ be an affine scheme over a field of characteristic zero. Suppose we have a group $G$ acting on $Y$ and that the subset of $Y$ of points with non-trivial stabilizer is in codimension greater or equal than $3$. Then, by a theorem due to Schlessinger $X:=Y/G$ is rigid that is $X$ does not have non-trivial first or... | https://mathoverflow.net/users/14514 | Deformations of quotient singularities | Regarding your last question, the answer is **yes**, since there are terminal singularities that are *not* rigid.
For instance, in the recent preprint by Taro Sano *[On deformations of Fano threefolds with terminal singularities](http://arxiv.org/abs/1203.6323)* it is shown that any $\mathbb{Q}$-Fano threefold with "... | 2 | https://mathoverflow.net/users/7460 | 156616 | 82,947 |
https://mathoverflow.net/questions/156234 | 11 | Following [Anton Deitmar](http://arxiv.org/abs/1105.5290), let $\mathcal B$ be an "**$\mathbb F\_1$-linear category**" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is [balanced](http://ncatlab.org/nlab/show/balanced+category), [pointed](http://ncatlab.org/nlab/show/pointed+category), contains finite products, k... | https://mathoverflow.net/users/29322 | Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category? | Here is a counterexample (modulo a small statement I'm not sure how to prove). Building on the comments, let $B$ be (a skeleton of) the opposite of the category of at most countable groups. This satisfies all of the desired conditions. There is a functor $\text{Grp} \to \text{Lex}(B, \text{Set}\_{\ast})$ sending a grou... | 6 | https://mathoverflow.net/users/290 | 156618 | 82,948 |
https://mathoverflow.net/questions/156586 | 1 | The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ($\mathbb{R}$-valued) entries.
$$
A=A\_0+p\_1A\_1+\ldots+p\_kA\_k,\qquad A\_i=\begin{pmatrix}a\_{i,11}\ldots a\_{i,1n}... | https://mathoverflow.net/users/16044 | Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial | First there is a little fact about resultants. Let $(R(u),I(u))$ be a pair of polynomials in the variable $u$ of degrees $(m,m)$, resp. $(m,m-1)$. Define $x(v) = R(v^2) + vI(v^2)$, a polynomial in the variable $v$ of degree $n=2m+1$, resp. $n=2m$. Write, $$x(v) = c\_0v^n + c\_1v^{n-1} + \dots + c\_{n-1}v + c\_n.$$ Then... | 6 | https://mathoverflow.net/users/13265 | 156623 | 82,951 |
https://mathoverflow.net/questions/156393 | 2 | Which are the most powerful topological invariants of toroidal orbifolds?
In particular I am looking for topological invariants of two-dimensional toroidal orbifolds such as $T^{2}/Z\_{k}\times Z\_{k}$ and $T^{2}/Z\_{k}$.
I'm wondering if the corresponding orbifold Euler characteristics are zero or not and how I c... | https://mathoverflow.net/users/19938 | Topological invariants of toroidal orbifolds | There are 10 orbifolds that can be covered by the torus i.e. 10 compact Euclidean 2 orbifolds. However, only 7 of them are quotients of the torus by a cyclic group or Abelian product of cyclic groups. The intuition that Euler characteristics are zero is correct. Formulas for the orbifold Euler characteristic appear thr... | 7 | https://mathoverflow.net/users/27453 | 156624 | 82,952 |
https://mathoverflow.net/questions/156625 | 19 | Let $\mathfrak g$ be a simple Lie algebra.
By taking the specialization at $q^\ell=1$ of a certain integral version¹ of the quantum group $U\_q(\mathfrak g)$,
and by considering a certain quotient category² of the category of tilting modules³ over that Hopf algebra, one obtains a fusion category.
Moreover, by using ... | https://mathoverflow.net/users/5690 | Is the representation category of quantum groups at root of unity visibly unitary? | The answer is "Sometimes."
Following the notation of [Rowell's "From Quantum Groups to Unitary Modular Tensor Categories"](http://arxiv.org/abs/math/0503226), let $\mathcal C(\mathfrak g, l, q)$ be the category corresponding to $U\_q(\mathfrak g)$ such that $q=e^{\pi \imath/l}$.Denote by $m$ is the ratio of the squa... | 13 | https://mathoverflow.net/users/25642 | 156637 | 82,956 |
https://mathoverflow.net/questions/156638 | 4 | Let $M$ be a von Neumann algebra and $\tau$ a faithful (semi-finite?) normal trace on $M$; as is standard, the $L^p$-norm is defined as $||u||\_p=\tau(|u|^p)^{1/p}$. Let $\{u\_i\}\_{i=1}^\infty$ be a sequence of hermitian elements that converges to $u$ in the $L^p$-norm; i.e. $||u\_i-u||\_p\to 0$ as $i\to\infty$. If $u... | https://mathoverflow.net/users/15488 | Does noncommutative Lp-convergence respect orderings? | Yes. Use two facts: the first is that $x \geq 0$ if and only if $\tau(xq) \geq 0$ for all finite projections $q$. The second fact is Hölder's inequality, which implies that $x \mapsto \tau(xq)$ is continuous on $L^p$ for all finite projections $q$.
| 7 | https://mathoverflow.net/users/10265 | 156647 | 82,958 |
https://mathoverflow.net/questions/156645 | 6 | Let $G$ is a simple undirected graph. Suppose $G$ has two subgraphs $G\_1$ and $G\_2$, such that $E(G\_1)\cap E(G\_2) =\emptyset$ ($E(G\_i)$, stand for the set of edges of $G\_i$). Then is it true that genus of $G$ is greater than or equal to the sum of genera of $G\_1$ and $G\_2$?
| https://mathoverflow.net/users/46514 | Genus of a graph | No. The two subgraphs can share the surface more efficiently than that. Take a graph $G$ with genus $g\ge 1$ and duplicate each edge. If you don't like double edges, subdivide them with new vertices. Then you can divide the new graph into two edge-disjoint subgraphs homeomorphic to $G$, therefore each having genus $g$,... | 14 | https://mathoverflow.net/users/9025 | 156648 | 82,959 |
https://mathoverflow.net/questions/156667 | 2 | The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the normalization morphism. Being finite (I like so much excellent schemes!) we can define a push-forward map on $0$-cycles... | https://mathoverflow.net/users/11475 | Does the normalization morphism induce isomorphism on residue fields? | The residue field extension is not always trivial in characteristic $0$. For instance, in $\mathbb{A}^2\_{\mathbb{R}}$, consider the plane curve $C$ of points $(x,y)$ satisfying the equation $x^2+y^2 = y^3$. The closed point $(0,0)$ of $C$ has residue field $\mathbb{R}$, yet the inverse image in the normalization is a ... | 3 | https://mathoverflow.net/users/13265 | 156670 | 82,964 |
https://mathoverflow.net/questions/156661 | 1 | Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?
| https://mathoverflow.net/users/46522 | An L1 function whose Fourier series converges but not to itself | The formulation seems specific enough to me. If the partial sums of the Fourier series converge to a function $g$ a.e., then so do the Cesaro means. But these converge in the $L^1$-sense to the original function. So the answer to your question is no.
| 4 | https://mathoverflow.net/users/45681 | 156672 | 82,965 |
https://mathoverflow.net/questions/156504 | 0 | Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ such that they form Markov chain $V-X-Z$ and $Z$ and $V$ are independent.
| https://mathoverflow.net/users/41666 | Generating independent random variable from two correlated random variables | The necessary and sufficient condition for existence of such random variable which forms the Markov chain $V\to X\to Z$ is "row linear dependence" of the transition kernel from $X$ to $V$. Let $V\in\mathcal{V}$ and $X\in\mathcal{X}$ and also $|\mathcal{V}|=r$ and $|\mathcal{X}|=k$. Suppose $\mathcal{A}=\{P(\textbf{V}|x... | 0 | https://mathoverflow.net/users/41666 | 156687 | 82,969 |
https://mathoverflow.net/questions/156698 | 0 | I confess to be not an expert of analytic geometry, but I have come across the following problem, for which I need an help from experts in this specific field.
I was wondering myself if it is possible to find a real analytic set $X$ in $\mathbb R^n$, of codimension $1$, which comes equipped with a foliation into anal... | https://mathoverflow.net/users/46538 | Can an analytic set admit such a foliation? | Let $X$ be the cone $x^2+y^2=(z-1)^3$, foliated by circles $z$ constant. Or just take any curve in the plane and foliate by points.
| 1 | https://mathoverflow.net/users/13268 | 156699 | 82,972 |
https://mathoverflow.net/questions/156526 | 5 | A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following:
>
> $G$ is a $Q$-group if and only if elements which generate the same cyclic subgroup of $G$ lie in the same conjugacy class.
>
>
>
See fo... | https://mathoverflow.net/users/9672 | To whom is the internal characterization of $Q$-groups due? | It's hard to pin down who first made this remark because it is a corollary of the very basic fact that if $\sigma$ is an automorphism of the cyclotomic field $Q\_{|G|}$, then there exists an integer $m$ coprime to $|G|$ such that for every character $\chi$ of $G$, we have $\chi(x)^\sigma = \chi(x^m)$. The earliest expl... | 9 | https://mathoverflow.net/users/9694 | 156706 | 82,973 |
https://mathoverflow.net/questions/156653 | -2 | If $p$ is both Giuga and Carmichael number
then its known that
$1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1} \equiv -1\pmod{p}$
is it true that
if $p$ is both Giuga and Carmichael number then
$1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(r-1)^{p-1} \equiv (r-1)\pmod{p}$
where $2\le r\le p-2$
Thanks in advance :)
| https://mathoverflow.net/users/40834 | Giuga and Carmichael numbers | If $n$ is Carmichael, then $a^{n-1}\equiv1\pmod n$ for all $a$ with $\gcd(a,n)=1$. If $\gcd(a,n)\ne1$, then it is clearly impossible to have $a^{n-1}\equiv1\pmod n$. So, let $n$ be Carmichael, let $r$ be the smallest divisor of $n$ (other than 1); then it is impossible to have $1^{n-1}+2^{n-1}+\cdots+r^{n-1}\equiv r\pm... | 3 | https://mathoverflow.net/users/3684 | 156716 | 82,977 |
https://mathoverflow.net/questions/145168 | 3 | While working on another problem ([Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$](https://mathoverflow.net/questions/143599/solving-the-quartic-equation-r4-4r3s-6r2s2-4rs3-s4-1)), I came across a question which seems to be of [semi-] independent interest.
**Conjecture.** If $r > s \ge 1$ are ... | https://mathoverflow.net/users/19844 | Proving conditions on $(r+s)^2 \mid (4r^4+1)$, related to Pell oblongs | [*corrected $-$ see edit history for previous attempt*]
The conjecture is false: there are infinitely many "Pell" parametrizations,
some with larger values of $s/r$. For example,
$$
(r,s) = (307470495089672071303, \, 295528756570432706202)
$$
has $s/r \sim .961$.
This was obtained as follows. Recall that $4r^4 + 1$... | 8 | https://mathoverflow.net/users/14830 | 156717 | 82,978 |
https://mathoverflow.net/questions/156644 | 5 | I am trying to find the original reference for a lemma attributed to Cohn (as in Schur-Cohn method):
>
> Let $A(z)$ be a palindromic or skew-palindromic polynomial, and denote its derivative by $A'(z)$. Then $A(z)$ and $A'(z)$ have the same number of zeros outside the unit circle.
>
>
>
The lemma can be found ... | https://mathoverflow.net/users/24122 | A palindromic polynomial and its derivative have the same number of zeros outside the unit circle. Reference? | MR0058748 (15,419a)
Ancochea, Germán, Zeros of self-inversive polynomials,
Proc. Amer. Math. Soc. 4, (1953) 900–902.
41.1X
This is a simple and elegant proof of a theorem of A. Cohn [Math. Z. 14, 110–148 (1922)]. A polynomial $g(z)$ with its zeros symmetric with respect to the unit circle $C$ has the same number... | 6 | https://mathoverflow.net/users/3684 | 156720 | 82,980 |
https://mathoverflow.net/questions/156714 | 1 | Let $f$ be a strictly positive function such that $\int\_{0}^{\infty}f(x)dx=\int\_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative nonconstant function which satisfies $\int\_{0}^{\infty}g(ax)f(x)dx=a$, $\forall a>0$. Does this imply that $g(x)=x$ a.e.?
| https://mathoverflow.net/users/46546 | Characterization of a particular integrable function | The answer is no. Let $h$ be a positive function on $R$ with the following properties:
a) Fourier transform has a pair of non-real zeros $\lambda, -\overline{\lambda}$,
symmetric with respect to the imaginary axis.
b) $\int\_{-\infty}^\infty h(t)dt=1,$
c) $\int\_{-\infty}^\infty e^t h(t)dt=1.$
Condition a) mean... | 1 | https://mathoverflow.net/users/25510 | 156723 | 82,982 |
https://mathoverflow.net/questions/156731 | 2 | Let $R$ be a left noetherian ring, then it is well known that Matlis proved that any injective $R$-module decomposes into a sum of indecomposable injectives, each of form $E(R/I)$ for some irreducible left ideal $I$. If moreover $R$ is commutative, we know much more: indecomposable injectives correspond to prime ideals... | https://mathoverflow.net/users/25602 | Injective modules over noncommutative noetherian rings | The noncommutative analogue of this result is explained in Bo Stenstrom's *Rings of Quotients*, particularly in Chapter 7, “Hereditary Torsion Theories for Noetherian Rings”.
It's been a long time since I looked at this, so I forget the details, but my impression was that the noncommutative theory was not nearly as ... | 2 | https://mathoverflow.net/users/3711 | 156732 | 82,985 |
https://mathoverflow.net/questions/156734 | 1 | **Question:**
does the following mean value have a name?
$$v^\*=\sqrt{\frac{\sum\_{i=1}^{n}\alpha\_iv\_i^2}{\sum\_{i=1}^{n}\alpha\_i}}, \alpha\_i\in\mathbb{R}^+$$
where the $v\_i$ are individual speed limits and the $\alpha\_i$ are the cumulated lengths along a route, for which the individual speed limits hold.
... | https://mathoverflow.net/users/31310 | Does this Mean Value have a Name? | It is a weighted quadratic mean (= weighted power mean with parameter $p=2$), see <http://en.wikipedia.org/wiki/Generalized_mean>.
| 4 | https://mathoverflow.net/users/36090 | 156738 | 82,986 |
https://mathoverflow.net/questions/156711 | 5 | This is motivated by [this question](https://mathoverflow.net/questions/85719).
Let $f$ be the hypergeometric series
$ f(x) = 2 x \, \_{4}F\_3([1, 1, 4/3, 5/3], [2, 2, 2], 27 x) $
which is explictly given by
$ f(x) = \sum\_{n \geq 1} \frac{(3n-1)!}{n!^3} x^n $
and appears in some studies of Mahler measures.
... | https://mathoverflow.net/users/10881 | Exponential of a specific hypergeometric series | The integrality is a re-incarnation of the problem discussed by Jan Stienstra in ["Mahler Measure Variations, Eisenstein Series and Instanton Expansions"](http://arxiv.org/abs/math/0502193) (particularly eq. (6) there). The point is that the normalised derivative of the function is an explicit modular form after a suit... | 7 | https://mathoverflow.net/users/4953 | 156745 | 82,989 |
https://mathoverflow.net/questions/156739 | 3 | Let $X$ be a *strictly ergodic* shift space, and $\omega\_1$, $\omega\_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega\_1)=\omega\_2$? By an automorphism I mean a shift-commuting homeomorphism of $X$. The answer for a general minimal shift space is, I guess, negative as ... | https://mathoverflow.net/users/24676 | Automorphisms of strictly ergodic shift spaces | Strict ergodicity does not seem to change much, i.e. the answer is still negative, see the following theorem (W. Bułatek, J. Kwiatkowski, [Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers](http://matwbn.icm.edu.pl/ksiazki/sm/sm103/sm10323.pdf))
The exist a strictly ergodic Toeplitz flo... | 4 | https://mathoverflow.net/users/7827 | 156746 | 82,990 |
https://mathoverflow.net/questions/156730 | 0 | $\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous
and
$$\sum \_{i, j=1}^n\xi\_i\xi\_jh(x\_i+x\_j)\geq 0,$$
for all $n\in\mathbb{N}$ and all choices of $\xi\_i,x\_i\in\mathbb{R}$, $i = 1,\ldots ,n$, such that $x\_i+x\_j\in(a,b)$, $1 \leq i, j \leq n$.
The... | https://mathoverflow.net/users/43418 | Exponential Convexity | You just multiply or divide by 2, no?
To show that (i) implies (ii), take any $\xi\_i,x\_i\in\mathbb{R}$ such that $x\_i\in(a,b)$ for $i=1,\ldots,n$. Since the interval $(a,b)$ is convex, the midpoints $\frac{x\_i+x\_j}{2}$ are also all in $(a,b)$. Now set $y\_i=\frac{x\_i}{2}$ for $i=1,\ldots,n$. Then we have $y\_i+... | 2 | https://mathoverflow.net/users/36090 | 156747 | 82,991 |
https://mathoverflow.net/questions/156663 | 3 | Let $\ell$ be a prime, and $k$ a field of characteristic $\ne \ell$. Let $f \colon X \to Y$ be a proper map of smooth projective $k$-varieties. The Leray spectral sequence says
$$
E\_{2}^{pq} = H^{p}(\bar{Y},R^{q}f\_{\*}\mathbb{Q}\_{\ell}) \ \Longrightarrow\ H^{p+q}(\bar{X}, \mathbb{Q}\_{\ell}) .
$$
In particular this ... | https://mathoverflow.net/users/21815 | When does the filtration in the limit of the Leray spectral sequence split? | Credits to Francesco Polizzi for putting me on the right track, by pointing to Deligne's work.
The important references for this are
* Deligne's thesis [“Théorème de Lefschetz et critères de dégénérescence de suites spectrales”](http://publications.ias.edu/sites/default/files/Number3.pdf) Publications Mathématiques... | 3 | https://mathoverflow.net/users/21815 | 156750 | 82,992 |
https://mathoverflow.net/questions/156749 | -1 | Let be $f \in Diff(M)$. When is finite the subgroup $span\{f\}< Diff(M)$? What are implications in the structure of $f$ and $M$?
| https://mathoverflow.net/users/45393 | Let be $f \in Diff(M)$. What we can say about the subgroup $span{f}< Diff(M)$? What are implications in the structure of $f$ and $M$? | If $f$ generates a finite group, say of order $p$, then the action of $f$ can be linearized in a neighborhood of any fixed point, turning $f$ into an orthogonal matrix with eigenvalues that are all $p$-th roots of unity. So the dynamics are ``boring''. The action of $f$ on real homology must also be by diagonalizable l... | 4 | https://mathoverflow.net/users/13268 | 156756 | 82,995 |
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