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https://mathoverflow.net/questions/230379 | 4 | Let $M$ be a W\*-algebra and consider the following map:
$$\gamma: M\times M^\*\to M^\*: (a,f)\to af$$
where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^\*)$ and $M^\*$ under the weak-star topology $\sigma(M^\*,M)$.
Q: Is $\gamma$ weakly-weak-star jointly continuous? I mean, assume that $... | https://mathoverflow.net/users/84390 | Is the module action $M\times M^*\to M^*$ jointly continuous? | Before I answer, let me comment that duals of von Neumann algebras are highly pathological objects, and when you get to that level I suspect most operator algebraists would see you as doing set theory, not operator algebras. Just FYI. (Would the predual $M\_\*$ suit your purposes? It's a lot nicer.)
The answer is no,... | 5 | https://mathoverflow.net/users/23141 | 230387 | 107,231 |
https://mathoverflow.net/questions/230400 | 1 | Let $k$ be a field, $\ell$ a prime different from the characteristic.
If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}\_{\ell}$-local system on the henselization of $Y$ along $S$ comes from a $\mathbb{Z}\_{\ell}$-local system on $S$?
| https://mathoverflow.net/users/27398 | l-adic local system. on hensel schemes | I think you mean to ask if a $\mathbf{Z}\_{\ell}$ local system on $S$ comes by reduction of a unique one on the henselization of $Y$ along $S$ (which isn't quite what is written, but is likely the intent in view of the special case when $S$ is a point).
This is affirmative for any affine scheme $Y$. The main task is ... | 2 | https://mathoverflow.net/users/81332 | 230404 | 107,235 |
https://mathoverflow.net/questions/230366 | 28 | I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph theory is not my specialty and I did not take a graph theory course until grad school, I am seeking advice on the content.
... | https://mathoverflow.net/users/62562 | What (fun) results in graph theory should undergraduates learn? | Planar graph duality and its consequences, e.g. that a connected planar graph is Eulerian iff its dual is bipartite, or Hamiltonian iff its dual can be partitioned into two induced trees.
The emergence of the giant component in random graphs. The 0-1 law for first-order properties of random graphs, and its connection... | 12 | https://mathoverflow.net/users/440 | 230427 | 107,242 |
https://mathoverflow.net/questions/229994 | 7 | Set $\mathbb{N} := \{0,1,2,\ldots\}$. A *parking function* of length $n$ is a sequence $(\alpha\_1,\ldots,\alpha\_n) \in \mathbb{N}^n$ whose weakly increasing rearrangement $\alpha\_{i\_1} \leq \alpha\_{i\_2} \leq \cdots \leq \alpha\_{i\_n}$ satisfies $\alpha\_{i\_j} \leq j-1$ for all $1 \leq j \leq n$. Let $\mathrm{PF... | https://mathoverflow.net/users/25028 | Even parking functions and spanning trees of complete bipartite graphs | Let $n=2m$ and let $(\alpha\_1,\dots,\alpha\_{2m-1})$ be an even parking function. Apply Pollak's map: $c=(\alpha\_2-\alpha\_1,\dots,\alpha\_{2m-1}-\alpha\_{2m-2})\mod 2m$.
Thus $c$ is an $(2m-2)$-tuple of even numbers between $0$ and $2m-1$.
Let $a=(c\_1/2,\dots,c\_{m-1}/2)$ and let $b=(c\_m/2+m,\dots,c\_{2m-2}/2+m)... | 1 | https://mathoverflow.net/users/3032 | 230429 | 107,243 |
https://mathoverflow.net/questions/230426 | 64 | The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential geometry course (I was the Reeb foliation) and I didin't pay many attention to it. In the meanwhile I get interested in the... | https://mathoverflow.net/users/24078 | What is a foliation and why should I care? | Without any disrespect, let me say that I find it incredible that someone naturally cares about non-commutative geometry but needs convincing about actual geometry (this just goes to highlight that there is a wide variety of ways of thinking in mathematics). I would need convincing the other way around (e.g. How are C\... | 90 | https://mathoverflow.net/users/7631 | 230441 | 107,249 |
https://mathoverflow.net/questions/230451 | 3 | Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H\_2(M) \to \mathbb{R}$, whose image $\Gamma\_{\omega} \subseteq \mathbb{R}$ is called the group of periods of $\omega$. In the classical theory of prequantization one assumes that $\omega$ is integral, i.e. $\... | https://mathoverflow.net/users/17047 | Symplectic manifolds with dense group of periods | The discrete subgroups of $\mathbb{R}$ are the groups $a\mathbb{Z}$ for $a\in\mathbb{R}$; the group of periods of $\omega$ is of this form if and only if some (real) multiple of $\omega $ is integral. Any projective manifold with $h^{1,1}>1$ gives a counter-example. Indeed the Kähler classes form an open convex cone in... | 5 | https://mathoverflow.net/users/40297 | 230454 | 107,254 |
https://mathoverflow.net/questions/230444 | 3 | Consider the simple symmetric random walk on the integers starting from
the origin of length $n$. More precisely, I will denote an $n$ step random walk $w$ as
$$ w:= \omega\_0 \omega\_1 \ldots \omega\_n, $$
where $\omega\_0 =0$ and $\omega\_i := \omega\_{i-1}+1$ or $\omega\_{i-1}-1$.
Let me denote the set of all ... | https://mathoverflow.net/users/4463 | How many times does a simple symmetric random walk of length n return to the origin? | All these questions are answered in paragraph 6 of Chapter III of Volume 1 of "An Introduction to Probability Theory and its Applications" by Feller.
In particular:
(1) $p=1/2$ is indeed the "right" scale, but the (normalized) number of returns only converges in distribution; it does not converge in probability to ... | 7 | https://mathoverflow.net/users/81488 | 230459 | 107,256 |
https://mathoverflow.net/questions/230457 | 3 | If $(E,<)$ is a linear order, let $s(E,<)$ denote the least ordinal which doesn't embed in $(E,<)$.
I am trying to prove the following:
* If $(M,+,.,0,1)$ is a model of open induction, (or equivalently, the set of positive elements of an integer part of a real closed field) then $s(M,<)$ where $x < y$ is defined by... | https://mathoverflow.net/users/45005 | Least ordinal not embedded in a total order | Your claim isn't true.
For a counterexample, let's construct a $\kappa$-like model. A model of arithmetic (or indeed any ordered structure) is $\kappa$-like, if it has size $\kappa$, but every proper initial segment of it has size less than $\kappa$.
One can construct a $\kappa$-like model $M\models\text{PA}$ for ... | 2 | https://mathoverflow.net/users/1946 | 230463 | 107,257 |
https://mathoverflow.net/questions/230412 | 5 | Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}\_p[[x]]$, where $\mathbb{C}\_p$ is the completion of the algebraic closure of $\mathbb{Q}\_p$, the field of $p$-adic numbers. Given two power series $f(x), g(x) \in \mathbb{C}\_p[[x]]$, it is known that the r... | https://mathoverflow.net/users/40086 | When does the radius of convergence of the product of two $p$-adic power series increase? | Here is a counterexample to your question at the end, for each $p$. Let $f\_u(x) = x + ux^p/p$ for $u \in \mathbf C\_p$ with $|u|\_p = 1$ and $|u-1|\_p = 1$. (Such $u$ can be taken in $\mathbf Z\_p^\times$ if $p > 2$, but you need to go outside $\mathbf Q\_p$ if $p = 2$ to an extension with residue field of size greate... | 8 | https://mathoverflow.net/users/3272 | 230479 | 107,259 |
https://mathoverflow.net/questions/230424 | 7 | On p. 76 of the 1996 edition of Serre's *A Course in Arithmetic*, one reads the following (inline) remark:
>
> One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and is equal to $k$.
>
>
>
Here, $A$ is a subset of $\bf P$ (the set of all positive rational primes), and the na... | https://mathoverflow.net/users/16537 | If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal | UPDATED: As Franz suspected, `es steht schon bei Landau.'
On p. 118 of the first volume of Landau's *Handbuch*, one finds the following theorem: Let $f(s)=\sum\_{n\ge 1} a\_n/n^s$ be a Dirichlet series (with real coefficients $a\_n$) that converges for $s>1$. Let $S(x)=\sum\_{n \le x} a\_n$. Then $$\limsup\_{s\downar... | 8 | https://mathoverflow.net/users/16510 | 230499 | 107,263 |
https://mathoverflow.net/questions/230498 | 1 | There is a saying "Do you read the masters?"
I want to read some basic papers in Topology/geometry...
I can not clearly state what is basic as of now...
My back ground includes course in
* Category theory, Some group Cohomology
* Algebraic topology
* Differential forms, deRham cohomology
* Representation theor... | https://mathoverflow.net/users/nan | research articles in topology/geometry | It is worth noting that in $K$-theory one can read `the masters' without that necessarily meaning reading research papers. Specifically, if you are interested in topological $K$-theory, then [Atiyah's book](http://www.amazon.co.uk/K-theory-Advanced-Classics-Michael-Atiyah/dp/0201407922/ref=sr_1_1?ie=UTF8&qid=1454877913... | 4 | https://mathoverflow.net/users/31603 | 230500 | 107,264 |
https://mathoverflow.net/questions/228871 | 3 | Is there an invertible measure-preserving transformation (preferably a nice one) admitting every irrational rotation as a factor ? I guess the spectrum is the relevant tool to address this question but I do not master spectral theory. Of course, if it is true, I am interested in a "minimal" such transformation. It is u... | https://mathoverflow.net/users/21339 | Transformation extending all ergodic rotations | There is an ergodic system $\mathbf Z = (Z,m,R)$, such that $\mathbf Z$ admits every irrational rotation as a factor.
As mentioned in the comments, such a system $\mathbf Z$ cannot have $L^2(Z,m)$ separable, since the eigenspaces are mutually orthogonal, and there are uncountably many eigenspaces. However, $\mathbf Z... | 1 | https://mathoverflow.net/users/10457 | 230503 | 107,265 |
https://mathoverflow.net/questions/230460 | 12 | Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular representation.
*Notation*: the word "generating" does not mean "generating exactly", but as a direct factor.
*Question*:... | https://mathoverflow.net/users/34538 | An inequality for the minimal number of generators of a finite group | Let $P$ be a $p$-group. Then the smallest number of irreducible characters of
$P$ whose kernels intersect trivially is at least $n(Z(P))$, and thus
$m(P) \ge n(Z(P))$. So to find an example where $n(P) < m(P)$, it suffices to find $P$ such that $n(P) < n(Z(P))$. An example in the Magma or GAP data
base of small groups ... | 4 | https://mathoverflow.net/users/9694 | 230506 | 107,267 |
https://mathoverflow.net/questions/230504 | 8 | Again, this question is related (\*\*) to a [previous one](https://mathoverflow.net/questions/101700/large-cardinals-without-the-ambient-set-theory?rq=1):
in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: they are special ordi... | https://mathoverflow.net/users/15293 | Direct axiomatization of ordinal and cardinal numbers | Are these papers of Takeuti the sort of thing you want?
MR0086751 (19,237e) 02.0X
Takeuti, Gaisi,
On the theory of ordinal numbers.
J. Math. Soc. Japan 9 (1957), 93–113.
MR0099918 (20 #6354) 02.00
Takeuti, Gaisi,
On the theory of ordinal numbers. II.
J. Math. Soc. Japan 10 1958 106–120
MR0197302 (33 #5467) 02.18
... | 9 | https://mathoverflow.net/users/6794 | 230513 | 107,270 |
https://mathoverflow.net/questions/230511 | 3 | I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given choice of invariant polynomial, and the construction of Chern-Simons forms is a functor on the category of principal bundl... | https://mathoverflow.net/users/56938 | Differential characters, Chern-Simons forms, and differential cohomology | The simple beginning of this story is that the curvature of a $\mathrm{U}(1)$ connection does not tell you the bundle it's a connection on — not even up to isomorphism. Differential cohomology is designed to fix this 'problem'.
Say we have a connection $A$ on a $\mathrm{U}(1)$ bundle $P$ over a smooth manifold $M$. ... | 12 | https://mathoverflow.net/users/2893 | 230514 | 107,271 |
https://mathoverflow.net/questions/230489 | 8 | Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My question is what kind of concentration inequalities do we have for the random variable $|A+A|$?
| https://mathoverflow.net/users/50426 | Does $|A+A|$ concentrate near its mean? | If $\epsilon>1/6$ then $|A+A|=N$ with high probability. Suppose that $\epsilon<1/6$. Call a pair $(x,y) \in {\mathbb Z}/N{\mathbb Z} \times {\mathbb Z}/N{\mathbb Z} $ "bad" if $(x,y) \in A \times A $ and there is another pair $(z,w) \in A \times A $ (also different from $(y,x)$), such that $z+w=x+y$. The probability th... | 8 | https://mathoverflow.net/users/7691 | 230516 | 107,272 |
https://mathoverflow.net/questions/230477 | 7 | $\newcommand{\C}{\mathcal C}\newcommand{\I}{\mathcal I}\newcommand{\D}{\mathcal D}\newcommand{\J}{\mathcal J}$Let $(\C, \otimes, I, \multimap)$ be a complete closed monoidal category and $\I$ a small category. Then the functor category $[\I, \C]$ is closed monoidal with the pointwise tensor product and its right adjoin... | https://mathoverflow.net/users/25527 | When do powers and ends in functor categories act pointwise? | Yes, this is correct. It is generally true that the evaluation functors $\text{ev}\_Z \colon [\mathcal{J},\mathcal{D}] \to \mathcal{D}$ for $Z \in \mathcal{J}$ jointly create limits (and colimits). So a limit in $\mathcal{C} = [\mathcal{J},\mathcal{D}]$ can be computed pointwise if and only if the corresponding pointwi... | 6 | https://mathoverflow.net/users/57405 | 230517 | 107,273 |
https://mathoverflow.net/questions/230292 | 4 | Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A\_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A\_i)\geq \epsilon$ for all $i\in\mathbb{N}$.
Is it always true that there are indices $i<j$ such that $\mu(A\_i\cap A\_j)\geq \epsilon^2$ ? Is it possible to classify... | https://mathoverflow.net/users/57519 | Measure of intersections in probability spaces | Take Borel measure on $[0,1]$ as an example. Cut off disjoint intervals $I\_1,I\_2,\dots$ where $I\_i$ has length $2^{-i}\epsilon$. That's length $\epsilon$ altogether. In the remaining $1-\epsilon$, take independent events $B\_1,B\_2,\ldots$ with $B\_i$ of measure $(1-2^{-i})\epsilon$. Define $A\_i=I\_i\cup B\_i$. The... | 5 | https://mathoverflow.net/users/9025 | 230519 | 107,275 |
https://mathoverflow.net/questions/228697 | 5 | The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted [here](https://mathoverflow.net/questions/228389), in the context of classical logic. The weak König's lemma in constructive mathematics has been extensively studied as noted [he... | https://mathoverflow.net/users/28128 | Constructive compactness for countable models? | I share Vladimir Kanovei's view that the syntactical approach might be a good oppportunity to constructive NSA in much the same way as it was for Nelson's Internal Set Theory.
| 1 | https://mathoverflow.net/users/85036 | 230532 | 107,277 |
https://mathoverflow.net/questions/230530 | 10 | Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau\_1 \neq \tau$ on $X$ such that the collection of dense subsets of the space $(X,\tau\_1)$ also equals ${\cal D}$?
| https://mathoverflow.net/users/8628 | Collection of dense subsets as a "fingerprint" for Hausdorff topologies? | The standard topology and [the lower limit topology](https://en.wikipedia.org/wiki/Lower_limit_topology) on $\mathbb{R}$ have the same dense subsets. They are two different topologies(even up to homeomorphism) on the real line.
So the next question could be "Is the collection of **open** dense subsets a “fingerprint”... | 11 | https://mathoverflow.net/users/36688 | 230536 | 107,278 |
https://mathoverflow.net/questions/230531 | 1 | Suppose $T$ is a tree on $\omega \times \omega \times \delta$ for some ordinal $\delta$ is a homogeneous tree (with some coherent set of measures witnessing homoegeneity). ($T$ can have additional properties if it is helpful for getting the tree $S$ with the property I want below.)
Under the appropriate large cardina... | https://mathoverflow.net/users/43354 | Absoluteness and Tree Representations | Assuming $\lambda$ is a limit of Woodin cardinals, $\delta\_0,...,\delta\_n,...$, then the pointclass $Hom\_{<\lambda}$ is closed under $\forall^{\mathbb{R}}$. This indeed follows from Martin-Steel.
I think the kind of absoluteness you're looking for is the "tree production lemma" of Woodin. The tree production lemm... | 1 | https://mathoverflow.net/users/3859 | 230543 | 107,281 |
https://mathoverflow.net/questions/230527 | 19 | [Birkhoff's representation theorem](https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices) implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for [modular lattices](https://en.wikipedia.org/wiki/Modular_lattice)?
For example,... | https://mathoverflow.net/users/2841 | Representation theorem for modular lattices? | There are lots of relations satisfied in lattices of submodules besides the ones implied by modularity. For example, there is the Desarguesian identity mentioned [here](https://ncatlab.org/nlab/show/Mal%27cev+variety#_is_a_desarguesian_lattice) (which holds in any lattice of congruence relations on an algebra of a Mal'... | 19 | https://mathoverflow.net/users/2926 | 230544 | 107,282 |
https://mathoverflow.net/questions/230550 | 3 | This is primarily in reference to [this](https://mathoverflow.net/questions/222740/brownian-motion-in-n-dimensions) question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the sphere $S^{n - 1}\_r = \{x \in \mathbb{R}^n... | https://mathoverflow.net/users/86386 | Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification) | Your intutive reasoning is leading you astray because you are thinking of Brownian motion as behaving like a smooth curve, for which there is a well-defined "direction" in which it is heading. Brownian motion isn't like that. (As you probably know, it's almost surely nowhere differentiable, so its "velocity vector" is ... | 9 | https://mathoverflow.net/users/4832 | 230562 | 107,288 |
https://mathoverflow.net/questions/230565 | 11 | In Jeffrey Weeks book ["The Shape of Space"](https://books.google.de/books/about/The_Shape_of_Space.html?id=Lurp6nB4LtQC&redir_esc=y) he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
* A non-trivial connected sum $M\_1\# M\_2$ admits a geometric structure if and o... | https://mathoverflow.net/users/84120 | Random links and $3$-manifolds | There is no universally accepted model of random three-manifolds (or random knots/links) for that matter, however, hyperbolicity is pervasive in all known models. The most popular (but not really satisfying) model of three-manifolds is the Dunfield-Thurston model:
```
Finite covers of random 3-manifolds.
Nathan M. ... | 19 | https://mathoverflow.net/users/11142 | 230570 | 107,290 |
https://mathoverflow.net/questions/230568 | 4 | Suppose $Gr\_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F\_{q}$, we can calculate the number of $Gr\_{\mathbb F\_q}(k,n)(\mathbb F\_q)$.
Could someone give me a direct reference of this r... | https://mathoverflow.net/users/37096 | A reference about Grassmannian over finite fields | Stanley's *Enumerative Combinatorics Volume I* (2nd Edition), Proposition 1.7.2. But like Ben I think not including a proof or reference would probably be fine.
| 4 | https://mathoverflow.net/users/290 | 230579 | 107,293 |
https://mathoverflow.net/questions/230556 | 1 | Let $u$ and $v$ be the weak solutions of
$$u\_t - \Delta u = f$$
$$u(0)=u\_0$$
and
$$-\Delta v = f$$
$$|\Omega|^{-1}\int\_\Omega v =0$$
on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous Neumann BCs. Here we may take $f$ and $u\_0$ to have spacial mean values zero.
I'm trying to prove an estimate of ... | https://mathoverflow.net/users/84288 | $L^1$ convergence to equilibrium of solutions of heat equation | Here is an idea. Put $w=u-v$, $w\_0=u\_0-v$. The required inequality takes form $\lVert{w}\rVert\_{L^1(\Omega)} \leq C(t)\lVert{w\_0}\rVert\_{L^1(\Omega)}$. Denote $G(x,y,t)$ the Green function of the Neumann problem for the heat equation in $\Omega$. There is an integral representation of the solution:
$$
w(x,t)=\int\... | 1 | https://mathoverflow.net/users/14551 | 230585 | 107,296 |
https://mathoverflow.net/questions/230466 | 13 | Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists a point $z\_0\in Z$ such that $f(g^{-1}(z\_0))\subseteq Y$ has dimension $k$ is it true that $f(g^{-1}(z))\subseteq Y$... | https://mathoverflow.net/users/nan | Generalization of the rigidity lemma in birational geometry | **EDIT:** *I've just realized that this holds under somewhat weaker assumptions. It is not necessary that the fibers of $g$ are connected.*
**EDIT#2:** *Apparently, in my previous edit I weakened the conditions too far... properness of $g$ is back as it is needed, but connectivity of fibers is not as it is not. I think... | 11 | https://mathoverflow.net/users/10076 | 230604 | 107,303 |
https://mathoverflow.net/questions/230601 | 5 | Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m\_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C\_{\mu,k}$ of probability measures $\nu$ on $[0,1]$ such that $m\_j(\nu) = m\_j(\mu)$ for $j = 1,\dotsc,k$.
We are interested in whether $\mathcal C = \mathca... | https://mathoverflow.net/users/41900 | Does the truncated Hausdorff moment problem admit absolutely continuous solutions? | Here is a positive answer, for the interior points of the set $M\_k:=\mathcal M\_k$. Let indeed $c=(c\_1,\dots,c\_k)$ be any point in the interior of $M\_k$.
Let $P$ stand for the set of all probability measures on $[0,1]$.
Let us show that then there is an absolutely continuous measure $\nu\in P$ such that $m\_j(\nu... | 2 | https://mathoverflow.net/users/36721 | 230617 | 107,307 |
https://mathoverflow.net/questions/230610 | 7 | I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsváth-Szàbo's original definitions:
>
> A pointed Heegaard diagram is called *strongly admissible for the ${Spin}^c$ structure $\mathfrak{s}$* if for every... | https://mathoverflow.net/users/86413 | Admissibility in Heegaard Floer, especially with torsion Spin^c structures | Yes, this is correct. It is spelled out explicitly on page 20 of András Juhász' [*A survey of Heegaard Floer homology*](http://arxiv.org/abs/1310.3418).
---
**Edit**: it was not clear to me that you were also asking why, for torsion spin$^c$ structures, strong admissibility implies weak admissibility. Here's a po... | 4 | https://mathoverflow.net/users/13119 | 230625 | 107,309 |
https://mathoverflow.net/questions/230633 | 15 | Wikipedia [claims (permanent link)](https://en.wikipedia.org/w/index.php?title=Dual_graph&oldid=694744772) without reference:
>
> Testing whether one planar graph is dual to another is NP-complete.
>
>
>
Another [claim](https://en.wikipedia.org/wiki/Medial_graph) with reference:
>
> For any plane graph G, ... | https://mathoverflow.net/users/12481 | Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)? | The dual graph and medial graph depend on the choice of an embedding in the plane. The Wikipedia claim seems to be that testing whether *there are choices of embeddings* for which two graphs are dual is NP-complete.
It looks as though Wikipedia makes a distinction between a planar graph (can be embedded in the plane)... | 23 | https://mathoverflow.net/users/22989 | 230639 | 107,313 |
https://mathoverflow.net/questions/230564 | 3 | Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that $$I=\sum\_{l\_{1}\geq1}\sum\_{l\_{2}\geq1}\sum\_{\gamma>0}\gamma^{-k-3/2}\int\_{0}^{\gamma}e^{-N\left(l\_{1}^{2}+l\_{2}^{2}\right)v^{2}... | https://mathoverflow.net/users/68301 | Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros | Notation: write $ A \approx B$ for $(A\ll B) \wedge (B\ll A)$.
It's very tempting to apply Poisson sum, so we set up the following formalism:
We'll apply the identity (Poisson summation formula)
$$\sum\_{\ell\in\mathbb{Z}} f(\ell) = \sum\_{k\in\mathbb{Z}} \hat{f}(k)$$ to the Gaussian $f(x) = \exp(-zx^2)$ with Fouri... | 6 | https://mathoverflow.net/users/327 | 230654 | 107,319 |
https://mathoverflow.net/questions/230652 | 3 | In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that
$$\lVert e^{-Ht}f \rVert\_2 \leq c\_1t^{-\mu/ 4}\lVert f \rVert\_1$$
where the norms are $L^p$ norms. He states
>
> by duality, it follows that
> $$\lVert e^{-Ht}f \rVert\_\infty \leq c\_1t^{-\mu/ 4}\lVert f \rVert\_2$$
>
>
>... | https://mathoverflow.net/users/84288 | Duality argument to get $L^\infty-L^2$ inequality | You have shown that $e^{-Ht}\colon L^1\to L^2$ is bounded with the given operator norm. It follows that the dual operator
$$\left(e^{-Ht}\right)^\*\colon (L^2)^\* \to (L^1)^\*$$
is bounded with the same operator norm, because of the characterisation
$$\|T\|\_{L^1\rightarrow L^2}=\|T^\*\|\_{L^\infty\rightarrow L^2}=\sup... | 5 | https://mathoverflow.net/users/327 | 230657 | 107,320 |
https://mathoverflow.net/questions/230646 | 9 | For cell complexes${}^1$ $X$ we have an isomorphism
$$
K^\*(X)\otimes \mathbb{Q}\cong H^{\*}(X;\mathbb{Q}),
$$
which is induced by the Chern character.
What is the analogous statement for $KO(X)$?
${}^1$:Hatcher states finite, but I've seen arbitrary CW-complexes stated as well.
edit: The footnote seems wron... | https://mathoverflow.net/users/12156 | $K$ theory and singular cohomology | In the following, $X$ is a finite complex. The Adams operator $\psi^{-1}$ (complex conjugation) acts on $K^0(X)$. After inverting $2$, the group $KO^0(X) \otimes \Bbb Z[1/2]$ maps isomorphically to the subset of $K^0(X) \otimes \Bbb Z[1/2]$ fixed by $\psi^{-1}$.
We can then tensor this with $\Bbb Q$. Then $K^0(X) \ot... | 12 | https://mathoverflow.net/users/360 | 230660 | 107,321 |
https://mathoverflow.net/questions/230574 | 11 | Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $\pi:TS^n\to S^n$ into horizontal and vertical parts as $TTS^n = V\oplus H$, where each summand is canonically isomorphi... | https://mathoverflow.net/users/86401 | Solutions of equations characterizing a complex structure | The answer is 'Yes, there are many other solutions, even global ones, and, when $n>1$, the local solutions depend on one holomorphic function of n complex variables.' (Of course, when $n=1$, $J\_{\delta,\beta}$ is always integrable.)
For a geometric context of this question, see Remark 3 below.
Briefly, here is a des... | 11 | https://mathoverflow.net/users/13972 | 230666 | 107,323 |
https://mathoverflow.net/questions/230656 | 6 | We can concluded that $\mathcal{D}(\Omega):=\bigcup\_{K \in \mathcal{K}(\Omega)} \mathcal{D}\_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset $\Omega \subset \mathbb{R}^n$) it's a countable union of Fréchet spaces, and $\mathcal{D}(\Omega)$ is a locally convex spac... | https://mathoverflow.net/users/86432 | Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$ | In Schwartz's *Théorie des distributions*, chapter III, Théorème VII : $\mathcal D$ is a Montel space, where bounded sets are relatively compact. Then the weak and strong topologies, *restricted to bounded sets*, coincide, and convergent *sequences* are the same in these two topologies (and also in weaker Hausdorff top... | 5 | https://mathoverflow.net/users/75422 | 230673 | 107,327 |
https://mathoverflow.net/questions/230643 | 2 | For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, $$0<\lim\_{m\to \infty} \sup\_{n>m}\frac {|S\cap n|}{n}.$$ Suppose $(x\_n)\_{n\in N} $ is a decreasing sequence of positive re... | https://mathoverflow.net/users/81583 | Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent? | **Summary.** The answer to your question is "No". But it is "Yes" under the additional condition that $\liminf\_n n x\_n > 0$. All of this follows from work of T. Šalát in the 1960s. You find some details below.
---
Earlier this morning, I had posted another answer (now deleted). But on my way to the chocolate sh... | 2 | https://mathoverflow.net/users/16537 | 230674 | 107,328 |
https://mathoverflow.net/questions/230675 | 2 |
>
> For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times 1}$, describe the set of solutions of this system which have minimal support (or the highest number of zero elements) in $\mat... | https://mathoverflow.net/users/85776 | Minimal Support Solutions of a Linear System (Dissertation) | The problem of determining a minimum support solution of a linear system is indeed NP-hard. Here is a reduction that works over the binary field $\mathbb{F}\_2$. Given a graph $G$, an *odd dominating set* is a set $S \subseteq V(G)$ such that $|N\_G[v] \cap S|$ is odd for all $v \in V(G)$. Here $N\_G[v]$ means the set ... | 4 | https://mathoverflow.net/users/2233 | 230686 | 107,329 |
https://mathoverflow.net/questions/230632 | 1 | I want perform a simple check for total unimodularity.
**Question:**
what, if anything, can be concluded from the fact, that $$det(A)=1,\ a\_{ij}\in\{-1,0,+1\}\ \wedge\ a\_{ij}^{-1}\in\{-1,0,+1\}$$
where $a\_{ij}^{-1}$ denotes the entries of $A^{-1}$?
| https://mathoverflow.net/users/31310 | Pragmatic Test for Total Unimodularity | There won't be any counterexamples to total unimodularity for $n=3$
since when $\det A=1$ the entries of $A^{-1}$ are up to sign the
$2\times 2$ minors of $A$. On the other hand, for $n=4$ let
$$ A=\left[ \begin{array}{rrrr} 1 & 0 & 1 & 1\\ 0 & 1 & -1 & 1\\
0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right]. $$
Then $... | 1 | https://mathoverflow.net/users/2807 | 230704 | 107,331 |
https://mathoverflow.net/questions/230690 | 6 | Let $\kappa$ be an infinite cardinal. Consider the following example to $2^\kappa\nrightarrow (3)^2\_\kappa$.
$V$ is a set of vertices, each of which is an element of $2^\kappa$. Color the edge between two vertices $f,g\in 2^\kappa$ by the least ordinal on which $f,g$ disagree. It follows that there is no homogeneous... | https://mathoverflow.net/users/13694 | Example to $2^\kappa\nrightarrow (3)^2_\kappa$, plus closed walks of odd length? | The answer is yes. In fact, we can get an example by slightly modifying your example.
Let $V = 2^\kappa \cup \{a,b,c,d,e\}$. Color the edges $ab$, $bc$, $cd$, $de$, and $ea$ with some ordinal $< \kappa$, say $0$. Color all other edges from $a$ with a different color, say $A$. Color all other edges from $b$ with a dif... | 3 | https://mathoverflow.net/users/70618 | 230709 | 107,333 |
https://mathoverflow.net/questions/230710 | 6 | Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: $$p\_{12}p\_{34}+ p\_{23}p\_{14}- p\_{13}p\_{24}=0,$$ $$p\_{12}p\_{35}+p\_{23}p\_{15}- p\_{13}p\_{25}=0,$$ $$p\_{12}p\_{45}+p... | https://mathoverflow.net/users/86315 | Number of Plücker relations for a Grassmannian | "The number of Plucker relations" is a little ambiguous, but there is no sense in which it is $k(n-k)$.
The number of Plucker coordinates is $\binom{n}{k}$, so the number of degree $2$ monomials in Plucker coordinates is $\tfrac{1}{2} \left( \binom{n}{k}^2 + \binom{n}{k} \right)$. The vector space they span inside th... | 6 | https://mathoverflow.net/users/297 | 230711 | 107,334 |
https://mathoverflow.net/questions/230706 | 1 | Reading [this](https://math.stackexchange.com/questions/1476568/strange-definitions-about-basic-probability-need-clarification/1580395#comment3335933_1580395) post, I realize that is possible to have *another* type of PDF (probability density function) in the special case when the sample space is an Euclidean space.
... | https://mathoverflow.net/users/40591 | If the sample space is an Euclidean Space, we can use a different type of PDF | Your formula
$$\int\_{\{X\in A\}}f\ dx = P[X\in A]\tag{1}$$
(I guess you wanted to say it should hold for all Borel $A\subseteq\mathbb R$) can be rewritten as $\int\_B f\ dx = P(B)$ for all $B$ in $\sigma(X)$, the smallest sigma-algebra with respect to which the random variable (r.v.) $X$ is measurable. So, condition ... | 1 | https://mathoverflow.net/users/36721 | 230713 | 107,335 |
https://mathoverflow.net/questions/230651 | 7 | If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, where $\pi^1$ can be taken to be the topological fundamental group or the étale one (as it is the profinite completion o... | https://mathoverflow.net/users/85535 | Lefschetz on étale fundamental group for quasi-projective varieties | This won't work for arbitrary smooth quasiprojective varieties. Take $X = \mathbb A^n$, then certainly any $H = \mathbb A^{n-1}$,.
But $\pi\_1(\mathbb A^{n-1}) \to \pi\_1(\mathbb A^n)$ is not an isomorphism in characteristic $p$ because there are nontrivial finite etale coverings of $\mathbb A^1$, which when pulled b... | 8 | https://mathoverflow.net/users/18060 | 230720 | 107,339 |
https://mathoverflow.net/questions/230726 | 3 | I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not understand the Holonomic splitting proposition. What does it exactly say? Can one put this in mathematical language? I think... | https://mathoverflow.net/users/86463 | Holonomic splitting | The theorem says that there does not only exist a trivialisation, but even a holonomic trivialisation. In the end, the holonomic sections are the ones that carry geometric meaning, so they are the ones you are interested in. The theorem says that once you have one holonomic section, you get a lot of them (via the holon... | 4 | https://mathoverflow.net/users/62434 | 230734 | 107,344 |
https://mathoverflow.net/questions/230735 | 0 | Let $\mathbb{N}$ be the set of positive integers and for $A\subseteq {\mathbb{N}}$ set $$m(A) = \text{lim sup}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
Does every ultrafilter ${\cal U}$ on $\mathbb{N}$ have the property that there is $U\in{\cal U}$ with $m(U) = 0$? If not, is it possible that $\inf\{m(U):U\in{... | https://mathoverflow.net/users/8628 | Does every ultrafilter contain sets of sup-measure $0$? | The answer to your second question is no. For any $n$, $\mathcal U$ contains exactly one of the sets $n\mathbb N+k$ for $k=0,\dots,n-1$. These sets have density $\frac{1}{n}$, hence $\inf\{m(U):U\in\mathcal U\}\leq\frac{1}{n}$, hence $\inf\{m(U):U\in\mathcal U\}=0$.
The answer to the first question is no as well: Let... | 14 | https://mathoverflow.net/users/30186 | 230738 | 107,345 |
https://mathoverflow.net/questions/230749 | 3 | Assuming $\lvert x\lvert<1$ and $0<a<c$, the following formula holds true
$$F(a,b,c;x)=\sum\_{n=0}^{+\infty} \frac{(a)\_n(b)\_n}{(c)\_n(1)\_n} x^n=\frac{\Gamma ( c )}{\Gamma(a)\Gamma(c-a)}\int\_0^1 t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b} dt\, , $$
where $(d)\_n:=d(d+1)\cdots(d+n-1)$ for every $d$ and every $n\in \mathbb N$.
... | https://mathoverflow.net/users/40381 | First proof of the integral representation of the hypergeometric function $F(a,b,c;\cdot)$ | The following paper discusses the hypergeometric functions and its history in mathematics (for the integral representation cf. p. 26 and references given there):
* J. Dutka. The early history of the hypergeometric function. Archive for History of Exact Sciences Vol. 31, No. 1 (1984), pp. 15-34
[JSTOR link](http://... | 4 | https://mathoverflow.net/users/50846 | 230754 | 107,350 |
https://mathoverflow.net/questions/230753 | 1 | Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?
| https://mathoverflow.net/users/41562 | Four Sphere Fibrations | For connected $H$ the long exact homotopy sequence implies $\pi\_1M=0$, which by dimension reason leaves only the possibilities $M=S^2$ or $M=S^3$. But $S^4$ is neither an $S^1$-bundle over $S^3$ (because it is simply connected) nor a $T^2$-bundle over $S^2$ (for example because it is not symplectic, or again because o... | 1 | https://mathoverflow.net/users/39082 | 230761 | 107,353 |
https://mathoverflow.net/questions/230737 | 5 | [A145722](http://oeis.org/A145722) is
`Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.`
Using the pari program and offset 0, up to $2000$,
$$A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$$
>
> Q1 Is this congruence true?
>
>
>
@Gjergji Zaimi proved similar... | https://mathoverflow.net/users/12481 | Up to $2000$, $A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$ | Yes, the congruence is true. Here's a modular forms proof. Let $\eta(z) = q^{1/24} \prod\_{n=1}^{\infty} (1-q^{n})$, where $q = e^{2 \pi i z}$. Define
$$
g(z) = \frac{\eta(16z) \eta(40z)^{3}}{\eta(4z) \eta(8z) \eta(20z) \eta(80z)} = \sum\_{n=0}^{\infty} {\rm A145722}(n) q^{4n+1}.
$$
This is a modular form of weight zer... | 9 | https://mathoverflow.net/users/48142 | 230791 | 107,365 |
https://mathoverflow.net/questions/230790 | 12 | Are the stable homotopy groups $\pi^s\_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.
| https://mathoverflow.net/users/14233 | Stable homotopy groups of $RP^{\infty}$ | The following paper contains a list of stable homotopy of projective spaces in dimensions $\leq 8$:
* A. Liulevicius. A theorem in homological algebra and stable homotopy projective spaces. Transactions of the American Mathematical Society
Vol. 109, No. 3 (Dec., 1963), pp. 540-552
[JSTOR link](http://www.jstor.org/... | 18 | https://mathoverflow.net/users/50846 | 230795 | 107,367 |
https://mathoverflow.net/questions/230778 | 8 | Quoting from Wikipedia [article](https://en.wikipedia.org/wiki/Quaternionic_projective_space) on quaternionic projective space:
>
> Therefore the quotient manifold
> $$
> \mathbb{HP}^{2}/\mathrm{U}(1)
> $$
> may be taken, writing $U(1)$ for the circle group. It has been shown that this quotient is the $7$-sphere,... | https://mathoverflow.net/users/38254 | Circle Action on Quaternionic Projective Space | $\mathbb{HP}^n\cong \mathrm{Sp}(n+1)/(\mathrm{Sp}(n)\times \mathrm{Sp}(1))$ is a symmetric space, so every one-parameter subgroup of $\mathrm{Sp}(n+1)$ acts on it. As was noted in the comments, such an action always has at least $n+1$ fixed points. Let me restrict to the action $z\cdot [x\_0:\cdots: x\_n] = [x\_0z:\cdo... | 9 | https://mathoverflow.net/users/35687 | 230796 | 107,368 |
https://mathoverflow.net/questions/230798 | 1 | I want to prove that in a sequence W of length n, consisting of 1s and 0s, $P$( in $W$ there is at most $\frac{\log\_2n}2$ consecutive zeroes ) $\leq \frac{K}{n} $ for some constant K. Can anyone help me start on the problem or refer me to some literature that could be of help.
Thank you very much.
| https://mathoverflow.net/users/86500 | Probability of at most $K$ consecutive zeroes in a sequence of 0s and 1s | Let $a<1<b$. I prove that for large $n$:
1) the probability that a random sequence of length $n$ has at least $B:=b\log\_2 n$ consecutive zeroes is at most $n^{1-b}$;
2) the probability that a random sequence of length $n$ does not contain $A:=\lfloor a\log\_2 n\rfloor$ consecutive zeroes is at most $e^{-n^{1-a+o(... | 1 | https://mathoverflow.net/users/4312 | 230802 | 107,370 |
https://mathoverflow.net/questions/230799 | 6 | According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to \mathbb{C}$ and almost all $x\_0 \in X$, the function from $\mathbb{Z}$ to $\mathbb{C}$ defined by $n \mapsto f(T^nx\_0)$ is... | https://mathoverflow.net/users/21339 | Discrete spectrum and almost periodicity | It follows from ergodic theorem that for a.e. $x \in X$ and any $f \in L^1(X)$ the limit $\lim\limits\_{N\to \infty}\frac{1}{2N+1}\sum\limits\_{k =-N}^N |f(T^kx)|$ exists and coincides with $\|f\|\_{L^1(X)}$. For $k \in \mathbb{Z}$ we can use this for the function $f(T^k\cdot)-f(\cdot)$ and obtain that for a.e. $x \in ... | 6 | https://mathoverflow.net/users/86505 | 230807 | 107,372 |
https://mathoverflow.net/questions/230794 | 3 | I believe that the following is true, and I'd like to make sure that it is and to have a reference. Suppose that $\mu\_N$ are a sequence of measures on $\mathbb{R}$. Let $m\_{N,k}$ be the $k$-th moment of $\mu\_N$. Suppose that, for each $k$, $m\_{N,k}$ converges to a limit $m\_k$ as $N\rightarrow \infty$; no uniformit... | https://mathoverflow.net/users/83174 | Converging to moments obeying Carleman's condition | Yes, this works. It's convenient to view the $\mu\_n$ as measures on the compact space $\mathbb R\_{\infty}$, with $\mu\_n(\{\infty\})=0$. By a diagonal process, we then find a subsequence (which I'll write as the original sequence) such that $x^{2k}\, d\mu\_n(x) \to d\rho\_k(x)$ weak $\*$ for certain measures $\rho\_k... | 1 | https://mathoverflow.net/users/48839 | 230809 | 107,373 |
https://mathoverflow.net/questions/230683 | 14 | In <http://arxiv.org/pdf/1410.6240.pdf> M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the cone which it is resolving. However, I could not find the motivation for this conjecture in the above article. Is there ... | https://mathoverflow.net/users/12395 | Why should intersection cohomology and quantum cohomology be related for a symplectic resolution? | I'll try to explain the genesis of the paper, then some intuition we developed later.
**The original motivation was a coincidence**. For $X$ hypertoric, Tom Braden and Nick Proudfoot had defined a ring structure on $IH^\*(X^{aff})$ whereas Daniel Shenfeld and I found a generators & relations presentation of $QH\_{\ma... | 8 | https://mathoverflow.net/users/78703 | 230812 | 107,375 |
https://mathoverflow.net/questions/225154 | 2 | Let $\chi\_0$ be the unique Dirichlet character $\text{mod }1$ (i.e. $\chi\_0(n) = 1$ for all $n$), $\zeta\_p$ be a primitive $p$th root of $1$, and for any $a \in \mathbb{F}\_p^\times$ and any Dirichlet character $\chi$ modulo $p$ (or $\chi\_0 \text{ mod }1$) define the Gauss sum$$g\_a(\chi) = \sum\_{x \in \mathbb{F}\... | https://mathoverflow.net/users/nan | Property of Dirichlet character | By Galois Theory, it is enough to show that
$$
{{g\_1(\chi)^b}\over{\sigma\_b(g\_1(\chi))}}, \ \ g\_1(\chi)^m $$
are fixed by all $\tau\_a$, which is defined for $(a,p)=1$, $\tau\_a(\zeta\_p)=\zeta\_p^a$, $\tau\_a(\zeta\_m)=\zeta\_m$.
In fact,
$$
\tau\_a({{g\_1(\chi)^b}\over{\sigma\_b(g\_1(\chi))}})={{g\_a(\chi)^b... | 0 | https://mathoverflow.net/users/21090 | 230840 | 107,385 |
https://mathoverflow.net/questions/230848 | 3 | I found in the book of Murphy, C\*- Algebras and Operator Theory, the Theorem 7.1.2 :
>
> Let A be an unital C\* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
> Neumann equivalence) in $M\_∞(A)$ is cancellative.
>
>
>
For the proof he proceeds as follow :
* take some projection... | https://mathoverflow.net/users/86526 | $V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative? | Theorem 7.1.2 in Murphy actually says that for $A$ a unital C$^\*$-algebra the semigroup $K\_0(A)^+$ is cancellative. But $K\_0(A)^+$ is the semigroup of *stably equivalent* projections in $M\_\infty(A)$ not just Murray von Neumann equivalent.
As a reminder $P,Q$ projections in $M\_\infty(A)$ are stably equivalent if... | 2 | https://mathoverflow.net/users/76593 | 230872 | 107,392 |
https://mathoverflow.net/questions/230885 | 5 | For $H=\left( \begin{smallmatrix} 0 & I\_n \\ -I\_n & 0 \end{smallmatrix} \right)$ and a commutative ring $F$, the symplectic group $Sp(2n,F)$ is the set of all matrices $M\in F^{2n\times 2n}$ such that $MHM^T = H$. (Here, $M^T$ means the transposed matrix.)
My question is: for $\mathbb{F}\_p$ a finite field of prime... | https://mathoverflow.net/users/62593 | Symplectic group over integers and finite fields | Yes: $\operatorname{Sp}(2n, \mathbb F\_p)$ is generated by its root subgroups. Each root subgroup is cyclic, and generated by an element that lifts in an obvious way to $\operatorname{Sp}(2n, \mathbb Z)$.
More concretely, if I haven't messed up the calculations in your form (I'm used to a different one), then, writin... | 9 | https://mathoverflow.net/users/2383 | 230889 | 107,395 |
https://mathoverflow.net/questions/230887 | 3 | Let $(F^\bullet,d\_F)$ and $(G^\bullet,d\_G)$ be two complexes in an abelian category $\mathbf{A}$.
The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ is defined as
$$Cone(\varphi)^i=G^i\oplus F^{i+1},$$
and its differential is
$$d(g^i,f^{i+1})=(d\_G(g^i)+\varp... | https://mathoverflow.net/users/4096 | Definition of the differential of the Cone of a morphism of complexes | Short answer: Otherwise it wouldn't depend on $\phi$!
Longer answer: Think about it this way: write $F$ and $G$ vertically side by side (in the 0-th and 1st column, respectively), with horizontal maps $\phi$. Since $\phi$ commutes with $d$, you get a double complex, call it $C$. The projection to $F$ and the inclusio... | 7 | https://mathoverflow.net/users/3847 | 230890 | 107,396 |
https://mathoverflow.net/questions/230891 | 3 | Define $N\_k \geq 6$ to be the $k-th$ primorial number and let $\sigma(n)$ be the divisor function.
It *seems* that $u\_k = \dfrac{\sigma(N\_k)}{N\_k \log\log N\_k}$ is a decreasing function ?
By computation, we find that the first few values of $u\_k$ (to 2dp) are
$u\_1 = 3.42, u\_2 = 1.96, u\_3 = 1.63, u\_{4} =... | https://mathoverflow.net/users/85379 | A decreasing sequence involving the divisor function? | The answer is probably no. Assume that $u\_{k+1}<u\_k$ for all $k$. Then
$$ \frac{1}{p\_{k+1}}<\frac{\log\log N\_{k+1}}{\log\log N\_k}-1=\frac{\log\frac{\log N\_{k+1}}{\log N\_k}}{\log\log N\_k}<\frac{\frac{\log N\_{k+1}}{\log N\_k}-1}{\log\log N\_k}=\frac{\log p\_{k+1}}{\log N\_k\log\log N\_k}.$$
In particular,
$$ \th... | 8 | https://mathoverflow.net/users/11919 | 230898 | 107,399 |
https://mathoverflow.net/questions/230131 | 2 | Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]\_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix
$$\tilde{Q}\,=\,(1-\alpha)Q+\alpha J$$
Though $Q$ is reducible meaning it has a eigenspace of dimension more than $1$ corresponding to the Per... | https://mathoverflow.net/users/27249 | Eigenvectors of a perturbed reducible stochastic matrix |
>
> So this transition from reducibility of Q to irreducibility of $\tilde Q$ is a function of α.
>
>
>
It is easy to show that for any $\alpha > 0$ $\tilde Q$ will be irreducible. See Lemma 3.1 from [A note on perturbations of stochastic matrices](http://fma2.math.uni-magdeburg.de/~willems/papers/pert.ps) by B... | 3 | https://mathoverflow.net/users/31830 | 230901 | 107,401 |
https://mathoverflow.net/questions/230902 | 2 | Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let $f:X\rightarrow Y$ be a morphism. Assume that $D\_Y = f(D)\subset Y$ is a divisor. Could we say that $h^0(Y,D\_Y) = 1$ as well?
... | https://mathoverflow.net/users/nan | Rigid effective divisors | No. Let $X$ be the blow-up of $\mathbb P^2$ at two points, and let $f : X \to Y$ be the map down to $\mathbb P^2$. Let $D$ be the strict transform on $X$ of the line between the two points you blew up. This is a $(-1)$-curve, hence $h^0(X,D) = 1$. But $D\_Y$ is a line in $\mathbb P^2$, which has larger $h^0$.
In fact... | 7 | https://mathoverflow.net/users/nan | 230903 | 107,402 |
https://mathoverflow.net/questions/230876 | 5 | Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let
$$
\mathfrak{J}\_{3}=\{A\in\mathcal{M}\_{3}(\mathbb{O}):A\text{ is Hermitian}\},
$$
that may be considered as a $K$-vector space of dimension $27$. We are going to denote $\mathbb{P}^{26}=\mathbb{P}... | https://mathoverflow.net/users/86397 | Looking for Severi varieties | Partial answer:
Let $A = \left( \begin{array}{ccc}
a & b & c \\
\overline{b} & e & d \\
\overline{c} & \overline{d} & f \end{array} \right) $ be the generic hermitian matrix with octonionic coefficients. This means that $b,c,d$ can be written as $X\_1.1 +X\_2.i\_1 + \cdots + X\_8.i\_7$, where the $X\_p$ are abstract ... | 4 | https://mathoverflow.net/users/37214 | 230915 | 107,407 |
https://mathoverflow.net/questions/230826 | 4 | Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}\_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of modules over another $\mathcal{E}\_2$-ring spectrum $A'$? Is there standard notation for $A'$ in terms of $A$?
| https://mathoverflow.net/users/86514 | Opposite of an E2-algebra | You should call $A'$ the "opposite" of $A$. The $E\_2$-operad has an automorphism given by reflecting in the plane in which the operad is defined (in terms of disks or squares or whatever). Your algebra $A'$ is the pullback of $A$ along this automorphism.
Does the choice of reflection matter? Sort of. The ratio of tw... | 5 | https://mathoverflow.net/users/78 | 230922 | 107,410 |
https://mathoverflow.net/questions/230913 | 4 | In [a video](https://youtu.be/W1OkVkq2vFM) I watched last night on [nuking](https://math.stackexchange.com/questions/798215/) [mathematical mosquitos](https://mathoverflow.net/questions/42512), Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The [Green-Tao theor... | https://mathoverflow.net/users/4177 | Use of infinitude of primes in the Green-Tao theorem | This was in fact answered by [Thomas Bloom](https://mathoverflow.net/users/385/thomas-bloom) in [this comment](https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts?page=3&tab=active#comment100823_42513) in response to exactly my question above (posed by Qiaochu Yuan):
>
> [Green and... | 7 | https://mathoverflow.net/users/4177 | 230936 | 107,414 |
https://mathoverflow.net/questions/230912 | 9 | I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is *almost*, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step currently is writing down the edge maps on the $E\_2$ page.
Can someone point me to a reference that derives the Leray... | https://mathoverflow.net/users/56878 | Topological Derivation of Leray Spectral Sequence | [Bott and Tu](http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf) do this in their book *Differential forms in algebraic topology*, see Section 14, ``Leray's construction" (starting on page 179).
| 5 | https://mathoverflow.net/users/8103 | 230939 | 107,415 |
https://mathoverflow.net/questions/230942 | 7 | Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ and suppose that ${\cal U}$ is a free ultrafilter on $\omega$. We write $f \leq\_{\cal U} g$ if $$\{n\in\omega: f(n) \leq g(n)\}\in{\cal U}.$$
Similar to the [usual bounding and dominating numbers](https://en.wikipedia.org/wiki/Cardinal_characte... | https://mathoverflow.net/users/8628 | Bounding and dominating numbers ${\frak b}, {\frak d}$ via ultrafilters | For any two functions $f,g$ the sets $\{n\in\omega:f(n)\leq g(n)\}$ and $\{n\in\omega:g(n)\leq f(n)\}$ cover $\omega$, so one of them must be in $\mathcal U$. Hence we have a dichotomy $f\leq\_{\mathcal U}g$ or $f\leq\_{\mathcal U}g$.
It follows that every unbounded family is dominating: if $B$ is an unbounded with r... | 11 | https://mathoverflow.net/users/30186 | 230948 | 107,416 |
https://mathoverflow.net/questions/230951 | 4 | The title might be misleading, but whether such a function exists is what boggles me about the following problem:
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all $a<b $ satisfying $f(a)=f(b)$, there exists $c$ in $(a,b)$ such that $f(a)=f(c)=f(b)$.
Prove that $f$ is monotonous on... | https://mathoverflow.net/users/86573 | Continuous non-constant function with infinite intersections with horizontal line on a compact interval? | Preimage $f^{-1}(v)$ of any value $v$ is a closed set, hence its complement $U(v)$ is open. This open set $U(v)$ is a disjoint union of intervals. If some interval is finite, say $(a,b)$, then $f(a)=f(b)=v$, but $f(c)\ne v$ for $a<c<b$. So, all intervals in $U(v)$ are infinite. Hence preimage of $v$ is connected: it is... | 9 | https://mathoverflow.net/users/4312 | 230952 | 107,417 |
https://mathoverflow.net/questions/230937 | 6 | Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f\_\*\mathcal{O}\_X=\mathcal{O}\_Y$, and $X$ is smooth.
We know that the general fiber of $f$ is not smooth in general. But can we say that the general fiber is an integral scheme?
Maybe... | https://mathoverflow.net/users/80473 | Generic Smoothness Type of Results in Positive Characteristic | **Correction.** I just realized that there are examples where the geometric generic fiber is **NOT** generically reduced. In all of my comments and the answer below, I was assuming that the geometric generic fiber is generically reduced. When that is true, then the geometric generic fiber is integral. However, without ... | 7 | https://mathoverflow.net/users/13265 | 230962 | 107,419 |
https://mathoverflow.net/questions/230967 | 3 | **Definitions.**
* By an $\mathbb{N}$-*graded set*, I mean a set $X$ together with a function $|\Box|\_X:\mathbb{N} \leftarrow X,$ called the *grading.* These will simply be called *graded sets* hereafter.
* If $Y$ and $X$ are graded sets, then a function $f : Y \leftarrow X$ is said to be a *morphism of graded sets*... | https://mathoverflow.net/users/26080 | Seeking more information regarding the "rigoidal category" of $\mathbb{N}$-graded sets | If $M$ is any monoidal category, the presheaf category $[M^{op}, \text{Set}]$ inherits a monoidal structure given by [Day convolution](https://ncatlab.org/nlab/show/Day+convolution). It is uniquely determined by the condition that it restricts to the given monoidal structure on $M$ and that it preserves colimits in bot... | 5 | https://mathoverflow.net/users/290 | 230977 | 107,423 |
https://mathoverflow.net/questions/230928 | 2 | My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove?
It seems like it should be easy because the disk is compact and contractible, but I sill don't know what to do about it. I am working on problems in ma... | https://mathoverflow.net/users/86561 | Isotopy class of closed 2-ball embedded in R^3 | The answer to your updated question is yes. In fact more is true:
any two PL embeddings $f,g$ of the 2-disk in any connected 3-manifold $M$ are isotopic.
One way to see this is to use the following statement, which follows from
Guggenheim's theorem: any two PL embedded $n$-balls in a connected PL
$n$-manifold are iso... | 3 | https://mathoverflow.net/users/85994 | 230985 | 107,426 |
https://mathoverflow.net/questions/230960 | 7 | The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see <http://oeis.org/A000688>), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for $a(n)$ itself (it is much too irregular), but there should an asymptotic for $\sum\_{n\leq x} a(n),$ but I can't seem to ... | https://mathoverflow.net/users/11142 | Asymptotics for the number of abelian groups of order at most $x.$ | The problem was studied quite a bit; a complete summary will be complicated to give (in any case I cannot). A standard reference for classical results on this is **A. Ivić "The Riemann Zeta-function: Theory and Applications"** (1985); it seems there is a recent Dover edition. Ivić has various papers on this problem, to... | 14 | https://mathoverflow.net/users/nan | 230987 | 107,427 |
https://mathoverflow.net/questions/230989 | 15 | I'm new here. I hope to do it right!
I am interested in studying mixed Hodge structures over complex algebraic surfaces and their generalizations.
Let us take a smooth complex variety $X$ and a variation of Hodge structures $\mathbb{V}$ over $X$ of weight $k$. That is, $\mathbb{V}$ is a local system of complex vect... | https://mathoverflow.net/users/86596 | Mixed Hodge structure on sheaf cohomology of a variation of Hodge structures | The answer to your question is yes (provided the VHS is polarized). This is due to Morihiko Saito. It is implicitly contained his two long papers on (mixed) Hodge modules, and there is an explicit statement in his note
"Mixed Hodge modules and admissible variations" Compte Rendus 1989.
When the base is a curve, it goe... | 20 | https://mathoverflow.net/users/4144 | 230992 | 107,428 |
https://mathoverflow.net/questions/230853 | 8 | For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F\_\omega)$ on $P$ which descends to a characteristic form on the base $M$ from the curvature $F\_\omega$. This form is exact when considered as a form ... | https://mathoverflow.net/users/56938 | Chern-Simons forms, characteristic numbers, and boundary terms? | You might have heard the following. I am actually referring to the second meaning of Chern-Simons classes $\tilde p(\nabla^0,\nabla^1)\in\Omega^\bullet(M)$ satisfying $d\tilde p(\nabla^0,\nabla^1)=p((\nabla^1)^2)-p((\nabla^0)^2)$. They can of course be recovered from the classes you described.
Given a vector bundle $... | 4 | https://mathoverflow.net/users/70808 | 230999 | 107,430 |
https://mathoverflow.net/questions/230745 | 2 | I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following :
> Let $K$ be an algebraically closed field. A connected (affine) algebraic subgroup of $(K,+)^n$ having dimension $1$ is isomorphic to $(K,+)$.
**Any ideas fo... | https://mathoverflow.net/users/18583 | One-dimension Algebraic groups | Maybe I missed something, but in the case where the characteristic is zero, it seems to be simple as follow: if $G$ is a closed subgroup of $(K,+)^n$ which is not trivial, then it contains an element $g$, and thus contains all powers of $g$, corresponding to $m\cdot g$, $m\in \mathbb{Z}$ (view $(K,+)^n$ as a vector spa... | 2 | https://mathoverflow.net/users/23758 | 231008 | 107,434 |
https://mathoverflow.net/questions/231007 | 16 | I asked this on stackexchange with no answer.
The negation would be the obvious generalization of Gödel's second incompleteness from r.e. extensions of PA to any arithmetically definable extension of PA.
I see that there is a complete consistent $\Sigma^0\_2$ extension of PA. Said theory therefore contains either t... | https://mathoverflow.net/users/84846 | Is there a consistent arithmetically definable extension of PA that proves its own consistency? | Surprisingly, the answer is yes! Well, let me say that the answer
is yes for what I find to be a reasonable way to understand what
you've asked.
Specifically, what I claim is that if PA is consistent, then there
is a consistent theory $T$ in the language of arithmetic with the
following properties:
1. The axioms of... | 22 | https://mathoverflow.net/users/1946 | 231011 | 107,437 |
https://mathoverflow.net/questions/230996 | 24 | In a recent [article](http://arxiv.org/pdf/1602.02705.pdf), Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element
$$ S = \prod\_{k=1}^{\frac{N-1}2} k^k $$
is a $p$-th power modulo $N$, and observed that he did not know an elementary pro... | https://mathoverflow.net/users/3503 | Elementary congruences and L-functions | **1.** First we show that
$$\left(\frac{T}{N}\right) = \begin{cases}
- (-1)^{(h-1)/2} & \text{ if } N \equiv 3 \bmod 16, \\
(-1)^{(h-1)/2} & \text{ if } N \equiv 11 \bmod 16. \\
\end{cases}$$
Consider the sets
$$A\_0:=\{1,2,\dots,\tfrac{N-1}{2}\},\quad A\_1:=\{1,3,\dots,\tfrac{N-1}{2}\},\quad A\_2:=\{2,4,\dots,\tfrac... | 22 | https://mathoverflow.net/users/11919 | 231020 | 107,441 |
https://mathoverflow.net/questions/230941 | 11 | The following integral
$$\int\limits\_0^\infty \frac{\cos{\left(\frac{1}{2}\sqrt{3}s\right)}}{\sqrt{\cosh{s}-\cos{\theta}}}\,ds$$
can be found in the paper
>
> Tevian Dray and Gerard 't Hooft, *The gravitational shock wave of a massless particle*, Nuclear Physics B **253** (1985) 173--188, doi:[10.1016/0550-3213(8... | https://mathoverflow.net/users/32389 | Calculation of the integral related to the gravitational shock wave | A closed form exists in Gradshteyn and Ryzhik, 8.842.1.
>
> As a comment to this answer:
>
>
> Referring to the Gradshteyn and Ryzhik, the following result is given in <http://arxiv.org/abs/hep-th/9408169> (On Gravitational Shock Waves in Curved Spacetimes, by K. Sfetsos):
> $$\int \limits\_0^\infty \frac{\cos{(... | 7 | https://mathoverflow.net/users/86616 | 231034 | 107,443 |
https://mathoverflow.net/questions/231036 | 10 | I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?
| https://mathoverflow.net/users/17644 | Rational homology sphere that is not Seifert manifold | By Thurston, all but finitely many $(p,q)$-surgeries on a hyperbolic knot in $S^3$ result in hyperbolic rational homology spheres for $p\neq 0$. In particular there are infinitely many integral homology spheres among them.
| 13 | https://mathoverflow.net/users/52936 | 231039 | 107,445 |
https://mathoverflow.net/questions/231041 | 9 | First Rickard (in [Splendid Equivalences: Derived Categories and Permutation Modules](http://plms.oxfordjournals.org/content/s3-72/2/331) ) and then Rouquier ([Block theory via stable and Rickard equivalences](http://www.math.ucla.edu/~rouquier/papers/charl.pdf), Appendix A.1) define splendid equivalences between (prin... | https://mathoverflow.net/users/86393 | Why do we want $p$-permutation modules in splendid equivalences? | The motivation for the definition was an attempt to explain structurally the phenomenon of an "isotypy". This makes sense for arbitrary blocks, but let's stick to principal blocks for simplicity.
Suppose $G$ is a finite group with abelian Sylow $p$-subgroup $P$, and $H=N\_G(P)$ is the normalizer of $P$. Then Broué's ... | 14 | https://mathoverflow.net/users/22989 | 231043 | 107,446 |
https://mathoverflow.net/questions/231048 | 1 | Let's work over an algebraically closed field $K$. A $1$-dimensional Zariski-closed connected subgroup of ${\mathbf G}\_{\mathbf a}^n$ is isomorphic to ${\mathbf G}\_{\mathbf a}^1$. If $K$ has characteristic $0$, then a Zariski-closed subgroup $G\subset{\mathbf G}\_{\mathbf a}^n$ is simply a $K$-vector space since for ... | https://mathoverflow.net/users/18583 | Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$ | The answer to your question is yes, assuming your algebraic groups are smooth (equivalently reduced, since your ground field $K$ is algebraically closed).
Indeed, every smooth connected commutative $p$-torsion linear algebraic group over $K = \overline{K}$ of characteristic $p>0$ is a vector group (i.e., direct prod... | 4 | https://mathoverflow.net/users/81332 | 231057 | 107,454 |
https://mathoverflow.net/questions/231058 | 16 | As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?
| https://mathoverflow.net/users/86626 | What is a Futaki invariant, what is the intuition behind it, and why is it important? | The best reference is the nice survey paper of Tian about Futaki invariant and CM polarization <http://bicmr.pku.edu.cn/~tian/?page_id=31> and another paper of Tian with Ding <http://www.maths.ed.ac.uk/cheltsov/cambridge/pdf/tian92.pdf>
The Futaki invariant is a Lie algebra character on the space of
holomorphic vecto... | 12 | https://mathoverflow.net/users/nan | 231060 | 107,455 |
https://mathoverflow.net/questions/227288 | 4 | Do there exist non-isomorphic finitely generated groups, $G$ and $H$, along with epimorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$, such that every generating set of these groups is an image of a generating set, that's mean, for every generating sets $\{g\_1,\dotsc,g\_m\}$ and $\{h\_1,\dotsc,h\_n\}$ of $G$ a... | https://mathoverflow.net/users/84700 | A question about generating set of groups and epimorphism | I only want to explain Yves's example in details.
Put $\mathbf Z:=\mathbb Z[t,t^{-1}]$, the ring of Laurent polynomials. Let $B$ be a group of matrices
[\begin{pmatrix}
1&P&R\\
0&D&Q\\
0&0&1
\end{pmatrix}
]
where $P$, $Q$ and $R$ belongs to $\mathbf{Z}$, and $D\in\langle t\rangle$. The group $B$ is easily checked to... | 1 | https://mathoverflow.net/users/84700 | 231061 | 107,456 |
https://mathoverflow.net/questions/231053 | 9 | Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}\_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the formula $g.x=\chi(g)^nx$ where $\chi:G\to \mathbb{Z}\_\ell^{\times}$ is the cyclotomic character.
**Question:** Is it true... | https://mathoverflow.net/users/10707 | Weight filtration on certain Galois representations | No, life is not so easy I'm afraid. For instance, the group $\mathrm{Ext}^1\_{G\_{\mathbf{Q}}}(\mathbf{Z}\_\ell, \mathbf{Z}\_\ell(n)) = H^1(\mathbf{Q}, \mathbf{Z}\_\ell(n))$ has positive rank for all odd integers $n$, whatever the sign; this is easy to see from Tate's global Euler characteristic formula.
The point i... | 8 | https://mathoverflow.net/users/2481 | 231070 | 107,459 |
https://mathoverflow.net/questions/230587 | 4 | It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
\sup\_{t,s\in[0,1]}\frac{|W\_t-W\_s|}{|t-s|^{\alpha}}<\infty.
$$
On the other hand, Brownian motion is clearly **not** globally $\alpha$-Holder. Indeed, it follows fro... | https://mathoverflow.net/users/7646 | Weighted global Holder property for Brownian motion paths | Let $\alpha,\gamma >0$. As Nate notes, for the inequality
$$
\sup\_{t,s \ge 0}\frac{|W\_t-W\_s|}{(t^\gamma\vee s^\gamma\vee 1)|t-s|^{\alpha}}<\infty
$$
to hold almost surely, it is necessary that $\gamma+ \alpha>1/2$. Of course $\alpha<1/2$ is also needed.
Claim: These conditions are also sufficient.
As observed b... | 2 | https://mathoverflow.net/users/7691 | 231076 | 107,461 |
https://mathoverflow.net/questions/231077 | 14 | Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq\_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$.
We say that $A\subseteq \omega$ has *measure 1* if $$\text{lim inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n+1\}|}{n+1} ... | https://mathoverflow.net/users/8628 | Are these two quotients of $\omega^\omega$ isomorphic? | Very nice question!
They are not isomorphic.
What I claim is that when we take the quotient with respect to density, there is a countably infinite antichain above $0$ having a minimal upper bound, but when we take the quotient modulo finite, there is no such antichain. This is a property that distinguishes the isom... | 18 | https://mathoverflow.net/users/1946 | 231088 | 107,467 |
https://mathoverflow.net/questions/229593 | 4 | Is there a finite dimensional Banach algebra $A$ for which $K\_{0}(A)$ is a finite group?
I asked this question in MSE but I received no answer
<https://math.stackexchange.com/questions/1624250/k-theory-of-finite-dimenional-banach-algebras>
| https://mathoverflow.net/users/36688 | $K$-Theory of finite dimensional Banach algebras | The definition of K$\_0 (A)$ (where $A$ is a finite dimensional Banach algebra), that it be the kernel of $K\_0(\tilde A) \to K\_0(C) \cong Z$ leads immediately to it being a free abelian group, possibly zero (free on no generators); the latter occurs iff $A$ is nilpotent; $\tilde A$ denotes the unitification.
Since... | 5 | https://mathoverflow.net/users/42278 | 231105 | 107,475 |
https://mathoverflow.net/questions/230254 | 0 | Given a traceless matrix $C\in M\_n(\mathbb{F})$, i.e., tr$(C)=0$, what is the relationship between tr$|\mathbb{I}+C|$ and tr$|C|$? The two matrices are of dimension $n$.
| https://mathoverflow.net/users/86258 | Matrix inequality between a traceless matrix and identity | I will assume that $C$ is hermitian, as you say in the comments. I will also assume $n\ge2$.
Summary
-------
There are three cases, depending on the value of $\mathrm{tr}|C|$:
1. For $\mathrm{tr}|C|\le2(n-2)$, we have
$$
0\le \mathrm{tr}|\mathbb I+C|\le n +\mathrm{tr}|C|\ .
$$
2. For $2(n-1)\ge\mathrm{tr}|C|>2(n-... | 0 | https://mathoverflow.net/users/16710 | 231120 | 107,480 |
https://mathoverflow.net/questions/231115 | 1 | Let $G$ be a finite simple group of Lie type and $x$ be a central involution (that is, an involution which is contained in the center of a Sylow $2$-subgroup).
Is it true that, if $y$ is another involution in $G$, then $C\_{G}(x)$ involves a group isomorphic to $C\_{G}(y)$?
| https://mathoverflow.net/users/21566 | Centralizer of a central involution in a simple group of Lie type | Just to add a bit more detail to my comment, let $G = {\rm PSL}(4,3)$, which has order $6065280 = 2^7.3^6.5.13$.
The image $x$ of the diagonal matrix $t$ with entries $(-1, -1, 1, 1)$ is a central involution whose centralizer is the image of a subgroup of index $2$ in ${\rm GL}(2,3) \wr C\_2$. (Note that elements tha... | 4 | https://mathoverflow.net/users/35840 | 231125 | 107,484 |
https://mathoverflow.net/questions/187274 | 23 | Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K\_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be zero.
Since the paper is from 2004 I was wondering if anything new is known about those groups.
I'm interested on an... | https://mathoverflow.net/users/43108 | References for $K_{4k}(\mathbb{Z})$ | Well, the consensus seems to be that this is an open problem, for $k >1$.
This is a quote from A. Raghuram's paper on the volume "[The Bloch–Kato Conjecture for the Riemann Zeta Function](http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/blochkato-conjecture-riemann-zeta-function?format=PB#bookP... | 3 | https://mathoverflow.net/users/43108 | 231126 | 107,485 |
https://mathoverflow.net/questions/231117 | 18 |
>
> **Definition.** $$\mathbb{J} = \{1,2,3,\ldots\}.$$
>
>
> We can refer to the elements of $\mathbb{J}$ as "joiners."
>
>
> * The product of joiners is inherited from $\mathbb{Z}$.
> * The sum of joiners will be defined by $$a \oplus b = a+b-1.$$
>
>
>
The motivation is that if we're thinking of the elemen... | https://mathoverflow.net/users/26080 | Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication? | I believe this could be related to an algebraic structure useful for studying set-theoretical solutions of the Yang-Baxter equation, the so-called ***braces***. At least the funny distributive property appears in the theory of set-theoretical solutions of the Yang-Baxter equation.
**Definition.** An abelian group $(A... | 19 | https://mathoverflow.net/users/17845 | 231129 | 107,487 |
https://mathoverflow.net/questions/231141 | 8 | Inspired by this [question](https://mathoverflow.net/questions/82083/when-is-the-tensor-product-of-two-fields-a-field) we ask: Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$?
1. A field $K$ with the property that $K\otimes\_{\mathbb{Z}} K$ is a field... | https://mathoverflow.net/users/36688 | Tensor product of fields over integers | I already wrote this in the comments but I think this might be worth of an answer. I think we can classify all fields $K$ such that $K\otimes K$ is a field.
**Claim** If $K$ is a field such that $K\otimes\_\mathbb{Z}K$ is a field then the multiplication map $K\otimes\_\mathbb{Z} K\to K$ is an isomorphism
In fact th... | 20 | https://mathoverflow.net/users/43054 | 231143 | 107,489 |
https://mathoverflow.net/questions/227751 | 4 | I'm not familiar enough with the auction theory to know where to look, but this seems close to what seems to be known as the "standard auction model".
Say an asset is up for auction.The true value of this asset follows a normal distribution $tv \sim N(0,\eta^2)$. All bidders share that prior.
On top of that, all bidd... | https://mathoverflow.net/users/22620 | standard auction model | This problem is known as the "mineral rights" model, which is a special case of pure common value auctions. A simple and classical result by Milgrom and Weber (see Proposition 6.1 in p.89 of Krishna's "Auction Theory" book) states that the SECOND-price equilibrium is for each bidder to bid $E[V| X\_i=x, max\_{X\_{-i}}=... | 1 | https://mathoverflow.net/users/87677 | 231156 | 107,493 |
https://mathoverflow.net/questions/227171 | 42 | Arthur's long-awaited book project [is now published](http://bookstore.ams.org/COLL-61) (*The endoscopic classification of representations: orthogonal and symplectic groups*). However, in the book he takes some things for granted:
1. The stabilization of the twisted trace formula for GL($N$) and SO($2n$).
2. Orthogon... | https://mathoverflow.net/users/6518 | What is the status of Arthur's book? | Now I think the answer is **yes, Arthur's work is now unconditional** for quasi-split special orthogonal and symplectic groups. Wee Teck Gan kindly directed me to the relevant papers of Waldspurger and Moeglin addressing the assumptions 2-4 above, though I was not able to verify with certainty that their stated results... | 18 | https://mathoverflow.net/users/6518 | 231159 | 107,494 |
https://mathoverflow.net/questions/231160 | 21 | I couldn't find a list of open problems in Hopf algebras. So my question is the following:
>
> In the theory of Hopf algebras, what are the (big) open problems?
>
>
>
Any kind of problem/question will be welcome. I am interested in any kind of problem involving Hopf algebras, say going from combinatorics or to... | https://mathoverflow.net/users/87680 | Open problems in Hopf algebras | There had been a workshop on Hopf algebras and related areas in September 2015. Its report (<https://www.birs.ca/workshops//2015/15w5053/report15w5053.pdf>) includes a large list of open problems and conjectures in Hopf algenbras, for example "Is the antipode of a noetherian Hopf algebra bijective ?".
| 22 | https://mathoverflow.net/users/17734 | 231162 | 107,496 |
https://mathoverflow.net/questions/231174 | 3 | A uniform lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact.
My question is whether a uniform lattice exists in the group
$$
G={\mathbb R}^2\rtimes SL\_2({\mathbb R}).
$$
This group is unimodular, so an obvious criterion is satisfied. It also admits a lattic... | https://mathoverflow.net/users/nan | Uniform lattice in semidirect product | No, there's no uniform (=cocompact) lattice in $\mathbf{R}^n\rtimes\mathrm{SL}\_n(\mathbf{R})$ for any $n\ge 2$. Up to the action by automorphisms of $\mathrm{GL}\_n(\mathbf{R})$, all lattices are contained in $\mathbf{R}^n\rtimes\mathrm{SL}\_n(\mathbf{Z})$, which is not cocompact.
Indeed, let $\Gamma$ be a lattice. ... | 6 | https://mathoverflow.net/users/14094 | 231178 | 107,502 |
https://mathoverflow.net/questions/231151 | 2 | There are many examples of potentials $V(x)$ for which Schrodinger's equation for a single particle in one dimension is exactly solvable, in the sense that we can give "nice" expressions for the eigenfunctions and eigenvalues of the operator $-\frac{1}{2}\partial\_x^2+V(x)$; I do not know if there is a technical defini... | https://mathoverflow.net/users/83174 | Exactly solvable examples of diffusion equation with variable diffusivity? | This would be the Schrödinger equation with a position dependent mass. Some exactly solvable examples are presented in
* [Analytic results in the position-dependent mass Schrödinger problem](http://arxiv.org/abs/1306.0933)
* [Explicit solutions for N-dimensional Schrödinger equations with position-dependent mass](ht... | 1 | https://mathoverflow.net/users/11260 | 231183 | 107,505 |
https://mathoverflow.net/questions/231144 | 3 | Let $S=E\times C$ be a product of two curves, where $E$ is an elliptic curve and $C$ is a curve of genus at least two. Consider a foliation on $S$ generated by a global holomorphic 1-form $p\_1^\*(\omega\_1)+p\_2^\*(\omega\_2)$, where $p\_i$ is a projection map and $\omega\_1$ and $\omega\_2$ are nonzero holomorphic 1-... | https://mathoverflow.net/users/87669 | algebraic leaves of foliation on a product of two curves | The foliation $\mathcal F$ defined by $p\_1^\* \omega\_1 + p\_2^\* \omega\_2$ is everywhere transverse to the fibration $p\_2 : S \to C$. One can therefore lift paths
from $C$ to leaves of $\mathcal F$ in order to obtain a representation
$$
\rho: \pi\_1(C,b) \to \mathrm{Aut}(p\_2^{-1}(p)) \simeq \mathrm{Aut}(E) \, .
$... | 6 | https://mathoverflow.net/users/605 | 231207 | 107,510 |
https://mathoverflow.net/questions/231196 | 0 | It is well known that the spectrum is continuous as function of operator. More precisely, let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of linear operators acting on $\mathcal{H}$, then one has
$$(\forall A,B\in\mathcal{B}(\mathcal{H}))(\forall \epsilon>0)(\exists \delta... | https://mathoverflow.net/users/56553 | Uniform continuity of spectrum as function of operator | The "well-known" fact is, of course, false. According to a theorem of C. Apostol and B. Morrel (On uniform approximation of operators by simple models, *Indiana Univ. Math. J.* **26** (1977), 427–442), if $K$ is a nonempty compact subset of $\mathbb{C}$ and $A$ is a normal operator such that $\sigma(A)$ is connected an... | 5 | https://mathoverflow.net/users/23141 | 231222 | 107,516 |
https://mathoverflow.net/questions/231168 | 1 | Suppose you have two polytopes $P\_1, P\_2 \in \Bbb{R}^n$ given by
$$ P\_1 = \lbrace x: A\_1 x \le b\_1\rbrace$$
$$ P\_2 = \lbrace x: A\_2 x \le b\_2\rbrace $$
I wish to find their convex hull, that is a matrix $A\_3$ and vector $b\_3$ such that
$$ A\_3 x \le b\_3$$
Is the convex hull of $P\_1$ and $P\_2$. Now... | https://mathoverflow.net/users/46536 | Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces | In general, computing the "H-representation" (your "$Ax \leq b$") of the convex hull of the union of two polytopes given by their H-representations is NP-hard:
>
> Tiwary. [On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes](http://dx.doi.org/10.1007/s00454-008-9097-3). Discrete & Comp... | 2 | https://mathoverflow.net/users/38434 | 231224 | 107,517 |
https://mathoverflow.net/questions/231155 | 3 | Given free modules $N \le M$ of finite rank over a PID $R$, it's well-known that there is a basis $\{x\_1,\ldots,x\_n\}$ of $M$ and there are $e\_1,\ldots,e\_n \in R$ such that $\{e\_ix\_i\mid e\_i \neq 0\}$ is a basis of $N$.
Now my question is, if this property also holds for chains of submodules.
Formally: Let ... | https://mathoverflow.net/users/17734 | Elementary divisors for chains of submodules | No. For example, let $R=\mathbb{Z}$, $M=\mathbb{Z}^2$, $N$ the subgroup generated by $(4,0)$ and $(2,1)$, and $L$ the subgroup generated by $(8,0)$ and $(0,2)$.
Then $M/N$ and $N/L$ are both isomorphic to $\mathbb{Z}/4\mathbb{Z}$, so if there were bases as described in the question, $M/L$ would have to be isomorphic ... | 5 | https://mathoverflow.net/users/22989 | 231249 | 107,526 |
https://mathoverflow.net/questions/231245 | 10 | I have some questions regarding the dynamics of elements of $GL\_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL\_n(\mathbb{Z})$, and given an integer column vector $v \in \mathbb{Z}^n$ which is not equal to the zero vector, I want to know about the asymptotics of th... | https://mathoverflow.net/users/20787 | Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$ | Showing that 4 implies 5 reduces to showing:
>
> Let $A\in\mathrm{GL}\_n(\mathbf{Z})$. Let $V\_-,V\_{\mathrm{ru}}\subset\mathbf{C}^n$ be the sum of characteristic subspaces of $A$ relative to eigenvalues of modulus $\le 1$ (resp., to eigenvalues that are roots of unity). Then $V\_-\cap\mathbf{Q}^n=V\_{\mathrm{ru}}\... | 13 | https://mathoverflow.net/users/14094 | 231251 | 107,528 |
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