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https://mathoverflow.net/questions/228861 | 2 | I have a question regarding exercise 2.1.5 on page 19 in this book:
<http://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf>
I would like a reference or help on this exercise.
The exercise asks the following:
Consider symmetric random matrices $X\_N$, with the zero mean independent random varibles $\{ X\_N(i,j) \... | https://mathoverflow.net/users/13904 | A question from Zeitouni's Introduction to Random Matrices | I think the essence is the central limit theorem. If you compute the traces of powers of your random matrix, they will be the sum of many independent random variables and will be Gaussian distributed when $N$ is large. So you should be able to show that these traces are not too far from the traces of the simpler case i... | 3 | https://mathoverflow.net/users/78061 | 228864 | 106,681 |
https://mathoverflow.net/questions/228856 | 4 | In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.
As pointed out below, the functor does not preserve all epimorphisms in the category of Lie groups, but only those which are also epimorph... | https://mathoverflow.net/users/56938 | Lie functor preserves "surjections" in synthetic differential geometry? | The inclusion $\mathbb Q\to \mathbb R$ is an epimorphism in the category of Lie groups.
Its image in the category of Lie algberas is not an epimorphism.
| 4 | https://mathoverflow.net/users/5690 | 228891 | 106,688 |
https://mathoverflow.net/questions/227227 | 5 | Let $f(x)$ and $g(x)$ be coprime monic polynomials in $\mathbf{Z}[X]$ of positive degrees $m$ and $n$ respectively. It seems that in this case their reduced resultant can be obtained from the expression $uf + vg = 1$ over $\mathbf{Q}[X]$ with $\deg u < n$ and $\deg v < m$. Namely - reduced resultant in this case is the... | https://mathoverflow.net/users/76284 | Reduced resultant of monic polynomials | The ideal generated by $f$ and $g$ in ${\bf Z}[x]$ is the set of all $uf+vg$ with $u,v$ in ${\bf Z}[x]$.
Now suppose $uf+vg=d$ for some integer $d$. Assuming $f$ and $g$ are monic, we have $$v=fq\_1+r\_1,\qquad u=gq\_2+r\_2$$ with $q\_i$ and $r\_i$ in ${\bf Z}[x]$, $\deg r\_1<\deg f$, $\deg r\_2<\deg g$. Then $$fg(q... | 8 | https://mathoverflow.net/users/3684 | 228896 | 106,691 |
https://mathoverflow.net/questions/222084 | 0 | Fix $n \in \mathbb{N}$, and let $\mathfrak{h}\_n$ denote the Heisenberg Lie algebra of dimension $2n+1$ (over any given field $k$). Namely, $\mathfrak{h}\_n$ is the Lie algebra with basis $x\_1, \dots, x\_n, y\_1, \dots, y\_n, c$ and with the Lie bracket defined by$$[x\_i, y\_j] = \delta\_{ij}c,\text{ }[x\_i, x\_j] = [... | https://mathoverflow.net/users/nan | Maximal possible dimension of abelian Lie subalgebra of Heisenberg Lie algebra of dimension $2n+1$? | As already answered [here on MathSE](https://math.stackexchange.com/questions/1414661/), the answer is $n+1$, and furthermore all maximal abelian subalgebra have this dimension. This is a basic exercise, following the standard fact that in an $n$-dimensional symplectic vector space, maximal isotropic subspaces have dim... | 1 | https://mathoverflow.net/users/14094 | 228901 | 106,692 |
https://mathoverflow.net/questions/228900 | 8 | Consider a measure $\mu$ on a finite set, and let $x\_1, \ldots, x\_n$ be i.i.d samples from $\mu$. Then the expression $S\_n = -\frac{1}{n} \sum\_{i=1}^n \log \mu(x\_i)$ converges by a.s. to the entropy $H(\mu)$.
What concentration inequalities exist for finite $n$? In other words, what upper bounds are known for t... | https://mathoverflow.net/users/52834 | concentration inequality for entropy from sample | Actually, Bernstein's inequality does not really require boundedness of the i.i.d. random summands; a finite exponential moment of the absolute value of a random summand will suffice. However, here we can just use Markov's inequality.
Let $X,X\_1,\dots,X\_n$ be independent identically distributed random variables (i... | 5 | https://mathoverflow.net/users/36721 | 228907 | 106,694 |
https://mathoverflow.net/questions/198149 | 6 | Let $G$ be a complex reductive algebraic group (connected, simply connected etc), viewed as a real group. We study the representations of $G$, and we follow the notations in the paper of Barbasch and Vogan:
"Unipotent representations of complex semisimple groups", Annals of Math. Vol.121, 41-110, 1985.
Consider the... | https://mathoverflow.net/users/68384 | Wavefront sets of irreducible representations with non-integral infinitesimal characters | 1) An example is the metapletic representation in the complex symplectic group. They have half-integral infinitesimal characters, and the wavefront set is the minimal orbit which is non-special.
2) Check out the book by Monty McGovern ([here](http://bookstore.ams.org/MEMO-108-519)). In Section 5. He mentioned how to ... | 3 | https://mathoverflow.net/users/14226 | 228915 | 106,699 |
https://mathoverflow.net/questions/228916 | 2 | Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$.
We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ equals $\deg(f)$.
Does $C$ admit a finite map $f \colon C \to \mathbb P^1$ which is totally ramified at $x$ and $y$?
| https://mathoverflow.net/users/85533 | Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points? | The answer is in general **no**. More precisely, the following holds.
>
> A finite map $f \colon C \to \mathbb P^1$ as in the question exists if and only if $\mathcal{O}\_C(x-y)$ is a point of finite order in $\textrm{Pic}^0\, C$.
>
>
>
In fact, if $f$ exists, calling $n$ its degree we have $\mathcal{O}\_C(nx... | 11 | https://mathoverflow.net/users/7460 | 228919 | 106,700 |
https://mathoverflow.net/questions/228928 | 0 | Let $\varphi\_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets
$$S\_\ell:=\left\{\left[\begin{matrix}x\\x\varphi\_k^\ell\end{matrix}\right]\in\mathbb{C}^{2n}\;\middle|\;x\in\mathbb{R}^n\right\}$$
with $\ell=1,\ldots,k\leq 2n$. Taking randomly (independent and identically random with some continu... | https://mathoverflow.net/users/81838 | Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent? | **Edit:** In response to asker comment, edited significatively the answer as it failed to take into account the case $n<k$
---
Here is my argument why the probability of linear independence is 1:
Case $k\leq n$:
Notice $y\_1,...,y\_k$ linearly dependent implies $x\_1,...,x\_k$ are linearly dependent. So the ... | 1 | https://mathoverflow.net/users/81368 | 228938 | 106,709 |
https://mathoverflow.net/questions/228918 | 3 | It is known that weak convergence implies norm convergence in $\ell^1(\mathbb{N})$, see e.g. [here](http://www.math.ucla.edu/~nickcook/l1weakstrong.pdf).
Because of the typical analogies of the Schatten ideals $C\_p \subset B(H)$ (where $H$ is a Hilbert space), it seems natural to ask whether **weak convergence in $C... | https://mathoverflow.net/users/16702 | Weak convergence implies norm convergence for trace class operators? | It's still false, however. Let $T\_n: v \mapsto \langle v, e\_1\rangle e\_n$ where $(e\_n)$ is an orthonormal basis of $H$. Then $\|T\_n\|\_1 = 1$ for all $n$, but the sequence converges weakly to zero. For any $A \in B(H)$, the sequence ${\rm Tr}(AT\_n)$ reads off the entries of the first row of $A$, which has to lie ... | 5 | https://mathoverflow.net/users/23141 | 228939 | 106,710 |
https://mathoverflow.net/questions/226090 | 3 | I am interested in the isolated singularity defined over $\mathbb{C}$ by
$$
x\_1^2+\cdots + x\_n^2+x\_{n+1}^k=0,
$$
where $n>2$ and $k>2$.
I would like to know whether this singularity is rational, as in the surface case. Any reference is welcome.
| https://mathoverflow.net/users/48866 | Rationality of higher dimensional du Val singularities | The answer is **yes**.
In the sequel, I will refer to S. Ishii's book *[Introduction to Singularities](http://www.springer.com/gp/book/9784431550808)* ([Is]). We have the following
>
> **Proposition.** An $n$-dimensional isolated $1$-Gorenstein singularity $(X, \, x)$ is rational if and only if $\kappa\_{\delta}... | 1 | https://mathoverflow.net/users/7460 | 228940 | 106,711 |
https://mathoverflow.net/questions/228941 | -2 | Does the inequality $\int\_2^{\infty} \dfrac{\sqrt x(\log x)^3 + (1+ \log x^2) x}{x(\log x)^2(x^2 - 1)} \,\mathrm {d}x > \ln \dfrac{17}{10}$ hold ?
| https://mathoverflow.net/users/85379 | Is this intergral inequality valid? | Mathematica says that
```
NIntegrate[ (Sqrt[x] Log[x]^3 + x (1 + 2 Log[x]))/(
x Log[x]^2 (x^2 - 1) ), {x, 2, Infinity}] - Log[17/10] // N
```
is about $1.08424$,
so yes.
| 1 | https://mathoverflow.net/users/1056 | 228942 | 106,712 |
https://mathoverflow.net/questions/228943 | 0 | Theorem 4.1 of [this](http://projecteuclid.org/download/pdf_1/euclid.lnms/1215465631) paper says that there exist distance matrices that are not conditionally negative definite (CND). How do I construct an example of a distance matrix that is not CND? Do you know an example?
| https://mathoverflow.net/users/84393 | Example distance metric that is not conditionally negative definite | You need a finite metric space $(X,d)$ such that the same space with the square root of the distance $(X,\sqrt{d})$ is not isometrically embeddable into any Euclidean space. An example is given by the metric space defined as the 0-skeleton of the graph with vertices $A,B,C,D,E$, and edges $\{AB,AC,AD,BE,CE,DE\}$ (so al... | 4 | https://mathoverflow.net/users/14094 | 228948 | 106,714 |
https://mathoverflow.net/questions/228954 | -2 | Let $G$ be a simple $3$-regular (every vertex has degree $3$) $2$-edge connected graph. Does $G$ contain a perfect matching?
| https://mathoverflow.net/users/85554 | Does every 3-regular bridgeless graph have a perfect matching? | Every bridgeless cubic graph contains a perfect matching according to [Petersen's Thereom](https://en.wikipedia.org/wiki/Petersen%27s_theorem).
| 5 | https://mathoverflow.net/users/1098 | 228958 | 106,718 |
https://mathoverflow.net/questions/228953 | 6 | The (a?) Kleene tree is a computable (a.k.a. decidable) sub-tree of the full binary tree with no computable path. It is well-known.
I need a variant. (For those in the know, I need a c-bar which is not a D-bar.) What I need is also based on a computable labeling of the binary tree. With the Kleene tree, once a binary... | https://mathoverflow.net/users/42160 | a variant of the Kleene tree | Let $K\_s$ be a computable monotone sequence of finite sets whose union is $K$, the halting set. Let $T$ be the tree of all $\{0,1\}$-sequences $\tau$ such that for some $s \geq |\tau|$, $\tau$ is the characteristic function of $K\_s \cap \{0,\ldots,|\tau|-1\}$. The tree $T$ is of the type you want and its only infinit... | 2 | https://mathoverflow.net/users/2000 | 228968 | 106,721 |
https://mathoverflow.net/questions/228964 | 6 | I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert\_2 = \lVert y \lVert\_2$, ie with an orthogonal Jacobian.
I can only think of trivial functions (like $f(x) = Ox + c$ for $O$ o... | https://mathoverflow.net/users/85560 | functions with orthogonal Jacobian | Such maps are conformal. A theorem of Liouville says that if $n\geq 3$,
the only conformal maps (defined in some region in $R^n$) are Mobius. A Mobius map is a composition of inversions
in spheres. For example $x\mapsto x/|x^2|$ is the inversion in the unit sphere.
Inversions in all spheres generate the Mobius group.
... | 12 | https://mathoverflow.net/users/25510 | 228971 | 106,723 |
https://mathoverflow.net/questions/228791 | 8 | The wikipedia gives us a formula for the [determinant of a circulant matrix](https://en.wikipedia.org/wiki/Circulant_matrix#Determinant). That is:
$$\mathrm{det}(C)
= \prod\_{j=0}^{n-1} (c\_0 + c\_{n-1} \omega\_j + c\_{n-2} \omega\_j^2 + \dots + c\_1\omega\_j^{n-1})= \prod\_{j=0}^{n-1} f(\omega\_j),$$
with $\omega... | https://mathoverflow.net/users/45564 | Exact determinant of a circulant matrix | We need to pick a prime $p$ and integer $k \ge 2$ such that $p = nk + 1$. (Alternatively you can choose a positive integer $p > 2n$ (not necessarily prime) such that $n$ divides $\phi(p)$, where $\phi$ is the Euler Totient function. However, I am not sure you can use FFT in the subsequent steps, and may have to resort ... | 3 | https://mathoverflow.net/users/36779 | 228993 | 106,729 |
https://mathoverflow.net/questions/228994 | 11 | I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x\_1,\ldots,x\_k)\mapsto p(1-x\_1,\ldots,1-x\_k)$ and re-expand in the Schur basis. Is there some nice combinatorial description of what happen... | https://mathoverflow.net/users/5281 | Plugging $1-x$ into Schur polynomials | Let $t$ be an indeterminate. Let
$\vartheta:\Lambda
\rightarrow \Lambda[t]$
be the specialization (homomorphism) defined by
$$ \vartheta(p\_k)=t+\sum\_{i=1}^k {k\choose i}p\_i, $$
where $p\_i$ is a power sum summetric function. According to item 75 of
<http://math.mit.edu/~rstan/ec/ch7supp.pdf>, we have
... | 14 | https://mathoverflow.net/users/2807 | 228999 | 106,732 |
https://mathoverflow.net/questions/228698 | 2 | Does $1^n + 2^n + \cdots + m^n$ divide $(1+2+ \cdots +m)^n$ for any even integers $m, n\geq 2$ ?.
For $n\leq 4$, the solution easily follows from the relevant identities. For $n\geq 6$, i suspect that Bernoulli polynomials might be the best approach, but haven't found how ?
| https://mathoverflow.net/users/85379 | On the divisibility of a certain power sum | Let $n$ be even, fixed, large.
Sketch of proof that $\{m|m>0, 1^n+\dots+m^n \mbox{ divides } (1+\dots+m)^n \}$ has zero density in $\mathbb{N}$.
Let $f(n)$ be the degree of the largest irreducible factor of $B\_{n+1}(x)/(n+1)$ in $\mathbb{Q}[x]$. Let $S$ be the set of primes $p$ for which this factor splits into li... | 3 | https://mathoverflow.net/users/85586 | 229008 | 106,734 |
https://mathoverflow.net/questions/228963 | 7 | A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the braiding, they are called modular tensor categories (MTCs) (cf. Chapter 4 of the book <http://www.math.ucsb.edu/~zhenghwa/d... | https://mathoverflow.net/users/45600 | How to make a premodular category a modular tensor category? | A premodular category is always spherical, so you can take the Drinfel'd center:
<http://arxiv.org/pdf/math/0111205v1.pdf>
EDIT: I probably should have pointed out, as Marcel does in the comments, "that the original category naturally embeds as a braided category into its center."
| 7 | https://mathoverflow.net/users/68910 | 229010 | 106,735 |
https://mathoverflow.net/questions/151095 | 1 | Suppose a vector-valued diffusion process X satisfies the stochastic differential equation
$$dX\_t = b(X\_t)dt + \sigma(X\_t) dW\_t,$$
in which $W$ is a Brownian motion and $b,\sigma$ are such that strong existence and uniqueness of solution to the SDE hold. Assume also that $X$ has a stationary distribution. Sufficien... | https://mathoverflow.net/users/34483 | On numerical approximation to stationary distribution of diffusion process | I don't have that book on me (I am saving up to get it!), but if you are mentioning [this algorithm](http://ac.els-cdn.com/S0377042704001657/1-s2.0-S0377042704001657-main.pdf?_tid=1313c1de-c0dc-11e5-a47f-00000aacb361&acdnat=1453448895_9872cb1c501ba1155fc880982e5cb57d) then it seems that the convergence rate is simply t... | 0 | https://mathoverflow.net/users/85578 | 229014 | 106,737 |
https://mathoverflow.net/questions/229011 | 5 | Vaidyanathaswamy calls an arithmetic function *rational* if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely multiplicative function. Since multiplicative functions are closed under convolution and (Dirichlet) inversion, rational function... | https://mathoverflow.net/users/6043 | Are there multiplicative functions which are not rational? | Not all multiplicative functions are rational. For simplicity take arithmetic functions with complex values. It is easy to show that $f$ is rational of order $(m,n)$ (meaning the Dirichlet product of $m$ completely multiplicative and and $n$ inverse completely multiplicative functions) if and only if for every prime $q... | 5 | https://mathoverflow.net/users/17218 | 229015 | 106,738 |
https://mathoverflow.net/questions/226196 | 1 | In at least two papers ([here](http://www.math.vanderbilt.edu/~ual/papers/mar-zen-001.pdf) and [here](http://www.math.vanderbilt.edu/~ual/papers/mar-zen-002.pdf)) Jorge Martínez and Eric R. Zenk say that Zorn's Lemma implies that all algebraic frames are spatial. However, I haven't been able to find an actual explanati... | https://mathoverflow.net/users/14257 | "Zorn's Lemma guarantees that all algebraic frames are spatial." Why? | Let $L$ be a complete lattice with top element $1$. Let ${\cal M}(L)$ be the collection of meet-irreducible elements of $L$.
Recall that a frame is said to be *spatial* if for all $x\in L$ with $x<1$ we have $x=\bigwedge\{z\in {\cal M}(L): z\geq x\}$.
>
>
> >
> > **Lemma**: Every complete, algebraic lattice $L$... | 1 | https://mathoverflow.net/users/8628 | 229020 | 106,741 |
https://mathoverflow.net/questions/215501 | 6 | Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n\_{1},n\_{2},n\_{3})$
where these integers are the number of roots in the interior of unite circle, on the unit circle and out of unit disc, respectively.
$\ell^{2}$ ... | https://mathoverflow.net/users/36688 | An equivalence relation on the space of polynomials in one complex variable | Much more is true. According to [the answer](https://mathoverflow.net/a/229154/4312) by Alexandre Eremenko (I do not have this paper, but I completely trust him), from $P(\mathbb{S}^1)=Q(\mathbb{S}^1)$ (which are essential spectra of $P(T)$, $Q(T)$ as noted in the comments) it follows that $P=f(z^n)$, $Q=f(wz^m)$ for s... | 5 | https://mathoverflow.net/users/4312 | 229023 | 106,743 |
https://mathoverflow.net/questions/83304 | 4 | I have a sequence of continuous time random variables $X\_n(t)$ where $t \in [0,1]$. Suppose that there is a filtration $F\_t$ such that for each $n$, $X\_n$ is a martingale with respect to this filtration. Note that the filtration does not depend on $n$. Also assume that $$\sup\_n E[\sup\_{0 \leq t \leq 1 } |X\_n(t)|]... | https://mathoverflow.net/users/14581 | When is the limit of Martingales a Martingale? | Posting the answer already given in comments:
All you really need is that $X\_n(t) \to X(t)$ in $L^1$ for each $t$. Conditional expectation with respect to any $\sigma$-field is continuous with respect to $L^1$ convergence; this is an elementary property of conditional expectation which follows from the inequality $\... | 5 | https://mathoverflow.net/users/4832 | 229040 | 106,747 |
https://mathoverflow.net/questions/229039 | 0 | Let $(\mathbb{R}^2,\langle .,.\rangle)$ be the Euclidean space and define the almost complex structure $J\_{\delta,\beta}:TT\mathbb{R}^2\longrightarrow TT\mathbb{R}^2$ with
\begin{align}
J\_{\delta,\beta}(X^h)=\beta X^h +\alpha X^v\\
J\_{\delta,\beta}(X^v)=-\beta X^v -\delta X^h,
\end{align}
where $X^h... | https://mathoverflow.net/users/85606 | Is there any solution for this PDE system? | Your system is inconsistent. For simplicity, fix $x\_2, y\_2$ and just consider the dependence on $x\_1, y\_1$ which I'll write as $x,y$. From equation (2),
$$ \delta(x, y) = \dfrac{1}{y + c(x)}$$
Then from equation (1),
$$ \beta(x,y) = \dfrac{d(x)}{y + c(x)}$$
From equation (4) we get
$$ d(x) = - c'(x)$$
and then equ... | 3 | https://mathoverflow.net/users/13650 | 229045 | 106,750 |
https://mathoverflow.net/questions/224325 | 6 | Let $G$ denote a reductive group over a local field $F$. Suppose that $G$ is split over $F$ and fix a maximal (split) torus $A$. Let $A^+$ denote a Weyl chamber in $A$ and let $K$ be a suitable maximal compact subgroup of $G$.
Then it is known that a Cartan decomposition holds:
$$
G=KA^+K.
$$
Hence there is a weight fu... | https://mathoverflow.net/users/nan | Cartan integral formula for a p-adic group? | In response to Anton for a uniform proof of your formula: note that $K\backslash KaK\cong(a^{-1}Ka\cap K)\backslash K$. Let $\pi:K\rightarrow G(\mathbb{F}\_q)$ be the natural quotient map, $K\_1:=\ker(\pi)$ and $P:=\pi(a^{-1}Ka\cap K)$. Then $P$ is the $\mathbb{F}\_q$-points of the parabolic subgroup of $G$ determined ... | 4 | https://mathoverflow.net/users/31327 | 229053 | 106,752 |
https://mathoverflow.net/questions/229052 | 3 | Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks.
My guess is that $G$-gerbes for $G$ an abelian group are classified as usual by $H^2\_{an}(\mathbb C,G)$ and that this group is trivial.
What about non-abelian gro... | https://mathoverflow.net/users/85613 | Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$ | In fact you have the following general result: Let $G$ be a Lie group, $M$ a manifold, and $p:P\rightarrow M$ a gerbe such that for every open subset $U$ of $M$, the objects of the fibre of $U$ are principal $G$-bundles. Suppose that the following condition (T) is satified:
There exists a covering $(U\_i)\_{i\in I}$ ... | 4 | https://mathoverflow.net/users/80891 | 229054 | 106,753 |
https://mathoverflow.net/questions/229056 | 2 | Assume that $\Gamma$ is a group with neutral element $e$. We associate to $\Gamma$ the following groupoid $G$:
$G=\Gamma \times \Gamma,\;\;\;G^{(0)}=\Gamma \times \{0\},\;\;s(a,b)=(a,e),\;\;\; r(a,b)=(ba, e)$
If $\phi:\Gamma\_{1}\to \Gamma\_{2}$ is a group isomorphism, then $\tilde{\phi}:G\_{1} \to G\_{2}$ with $\t... | https://mathoverflow.net/users/36688 | Groupoid isomorphism vs. group isomorphism | Look at the group $Aut(x)$ where $x$ is an object of the groupoid. You find that this group is trivial. Therefore your groupoid is THE groupoid with trivial automorphism groups and its isomorphy class only depends on the cardinality of the set of objects.
Hence any two groups of the same cardinality will give isomorphi... | 8 | https://mathoverflow.net/users/nan | 229059 | 106,755 |
https://mathoverflow.net/questions/229057 | 21 | in $C^\*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$
Now, of course there is also in classical physics and quantum mechanics the definition of a state.
In classical physics this is either a point in phase space or more generally a probabil... | https://mathoverflow.net/users/85616 | States in C*-algebras and their origin in physics? | 1.) Yes. In the commutative case, this is the statement of the Riesz representation theorem (any linear functional on $C(K)$ is an integral against a measure, which has to be positive by the positivity condition on the state).
In the non-commutative case, the answer may be Yes or No, depending on how you phrase the q... | 9 | https://mathoverflow.net/users/2622 | 229080 | 106,761 |
https://mathoverflow.net/questions/229094 | 5 | Suppose that a set of sentences of a 1st order language has an infinite model $M$.
Under what conditions is there is a proper class-sized elementary extension of $M$?
How does the answer change if we begin with a *proper class* of sentences?
| https://mathoverflow.net/users/83742 | When does an infinite model have a proper class-sized elementary extension? | The answer to your main question is that in ZFC there is always such a proper-class elementary-extension.
**Theorem.** In ZFC, every set-sized model in a set-sized
first-order language has a proper-class elementary extension.
**Proof.** This is easiest to see in the case that the global
axiom of choice holds, in o... | 8 | https://mathoverflow.net/users/1946 | 229096 | 106,766 |
https://mathoverflow.net/questions/229095 | 3 | I am attempting to do a calculation of a colimit in $Cat$, the category of small categories. To this end, people have suggested that I do this by calculating coproducts and using coequalizers. I have no idea how to do this. I have seen the [definition for coproducts in Cat](https://ncatlab.org/nlab/show/coproduct) but ... | https://mathoverflow.net/users/10007 | colimits in Cat via coproducts and coequalizers | Firstly, there exists a general method to construct colimits in arbitrary category via coproducts and coequalizers. I will point it briefly. Let $A$ and $B$ be categories, $T\colon A\to B$ be a functor. If $B$ has coproducts of all families indexed by objects and morphisms of $A$ and all binary coequalizers, then such ... | 6 | https://mathoverflow.net/users/35349 | 229103 | 106,769 |
https://mathoverflow.net/questions/229104 | 1 | Let $n=15r$ where $r>5$ is an odd prime number. If $r\!\!\! \mod 15 \equiv w$ then is it true that $\Phi\_{n}(x)$ is not flat whenever $2<w<13$ ? In other words, are the flat ones necessarily among those satisfying $r\equiv \pm 1 \!\!\! \mod 15$ and $r\equiv \pm 2 \!\!\! \mod 15$. I am looking for a Yes or No answer, p... | https://mathoverflow.net/users/83515 | Ternary cyclotomic polynomials with $n=15r$ | **Yes**
More precisely, the flat ones are exactly those where $r$ is $\pm 1$ modulo $15$, all others (including those $\pm 2$) are not flat.
The following result is due to Kaplan (Theorems 2 and 3 in the paper linked below):
>
> Let $p<q<r$ be primes, and let $s > q$ be prime such that $s$ is $r$ or $-r$ modulo... | 3 | https://mathoverflow.net/users/nan | 229108 | 106,771 |
https://mathoverflow.net/questions/229065 | 3 | If we define $$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \cos(2\pi{x})}$$ how would one go about evaluating
$$ \int\_1^R \frac{1}{x} \log{f(xe^{i\alpha})} dx$$ for some parameter angle $\alpha$ and an arbitrarily large $R$? I know I'm supposed to show some progress but I really have no idea how to ... | https://mathoverflow.net/users/74600 | Integral involving the gamma function | I understand that the OP is after the imaginary part of the integral in the small-$\alpha$ limit. To evaluate this limit, I note that [Wilson's theorem](https://en.wikipedia.org/wiki/Wilson%27s_theorem) implies that, for real positive $x$, the function
$$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \c... | 5 | https://mathoverflow.net/users/11260 | 229122 | 106,774 |
https://mathoverflow.net/questions/229078 | 12 | Let $X\_1,\ldots, X\_n$ be i.i.d. Rademacher random variables. That is, $\operatorname{Pr}(X\_i = 1) = \operatorname{Pr}(X\_i = -1) = 1/2$. I was wondering if the following argument is true:
$$
\mathbb{E} \exp\biggl( C\cdot \left(\sum\_{i=1}^n X\_i\right)^4\big/n^3 \biggr) = 1 + O(1/n),
$$
where $C \geq 0$ is a constan... | https://mathoverflow.net/users/81633 | Asymptotics of functional of i.i.d. Rademacher random variables | I believe the conjecture is true for sufficiently small $C$. Previously I was trying to disprove it using Large deviation theory. But I missed a sign at the last step. But the same argument can be turned around.
In compact form, one gets
$$ \lim \frac{1}{n} \log P(\sum\_i X\_i > \alpha n) = -I(\alpha),$$
where $I(\al... | 4 | https://mathoverflow.net/users/4923 | 229124 | 106,776 |
https://mathoverflow.net/questions/229116 | 10 | Let $k$ be a field, with $F,k'$ field extensions of $k$. The ring $k' \otimes\_k F$ is denoted by $F\_{k'}$. In Borel's *Linear Algebraic Groups*, it is claimed (I believe erroneously) that "each of [$F\_{k'}$'s] prime ideals is minimal." Indeed, by a result due to Grothendieck, the Krull dimension of $F\_{k'}$ is the ... | https://mathoverflow.net/users/38145 | What are the basic possibilities for a tensor product of two fields? | Looks like nfdc23 has explanations for (a),(b), and (c).
But: indeed, primes of $F\_{k'}$ are not in general minimal if $F/k$ is not algebraic. Let $k'=k(x)$ and $F=k(y)$ be transcendental extensions of $k$.
Then $F\_{k'}$ identifies with a subalgebra of the field $k(x,y)$, hence is an integral domain [assertion ... | 8 | https://mathoverflow.net/users/4653 | 229128 | 106,779 |
https://mathoverflow.net/questions/229112 | 2 | If $\pi(x) > \operatorname{Li(x)},$ is $\vartheta(x) > x$? Are the two inequalities (solutions to both of which are known to exist but not known exactly) equivalent, similar, or mostly unrelated?
$\vartheta(x)$ is the [Chebyshev Theta Function](https://en.wikipedia.org/wiki/Chebyshev_function).
| https://mathoverflow.net/users/74600 | Bounds on $\pi(x)$ vs. bounds on $\vartheta(x)$ | One can address this problem using tools from comparative prime number theory ("prime number races"). Assuming RH, the (logarithmic) limiting distribution of the vector-valued error function
$$
\bigg( \frac{\pi(t)-\mathop{\rm Li}(t)}{\sqrt t/\log t} , \frac{\theta(t)-t}{\sqrt t} \bigg)
$$
is supported on the diagonal $... | 6 | https://mathoverflow.net/users/5091 | 229134 | 106,782 |
https://mathoverflow.net/questions/229113 | 7 | Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional $f\mapsto f(a)$ is bounded on $\mathcal{H}$. By Riesz Representation Theorem, there exists an element $K\_a\in\mathcal{H}... | https://mathoverflow.net/users/85652 | Orthonormal bases on Reproducing Kernel Hilbert Spaces | The error is in this line:
>
> The standard argument shows that $\widetilde{\mathcal{M}}$ is an RKHS of functions on $X$.
>
>
>
In fact, this is not generally true. The completion $\widetilde{\mathcal{M}}$ may not be naturally identified with a space of functions on $X$.
The "obvious" way that one would try ... | 8 | https://mathoverflow.net/users/4832 | 229138 | 106,784 |
https://mathoverflow.net/questions/229130 | -1 |
---
**Background**
I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and assumes the reader be familiar with the general definition.
---
**Question:**
So what is the precise de... | https://mathoverflow.net/users/36886 | Definition: Grigelionis Process? | A Grigelionis process is a special semimartingale with *absolutely continuous* integral characteristics (in time). This is insofar a generalization of a Lévy process, as Lévy processes can be characterized as special semimartingales with *linear* integral characteristics (in time).
A nice starting point seems to be [... | 1 | https://mathoverflow.net/users/20026 | 229140 | 106,786 |
https://mathoverflow.net/questions/229068 | 0 | I would like to know more about divisibility among power-divisor functions. Put $\sigma\_k(n) = \sum\_{d \mid n} d^k$ for all positive integers $k$ and $n$.
**My question here is** : for which positive integers $x$ and $y$ do we have that $\sigma\_x(n) $ divides $\sigma\_y(n) $ for all $n$?
**EDIT01 :** For instanc... | https://mathoverflow.net/users/74330 | For which $x$ and $y$ does $\sigma_x(n) $ divide $\sigma_y(n)$ for all $n$? | The answer is when $x=y$. Using primes $p$ for $n$, one sees that $y$ has to be an odd multiple of $x$, say $y=kx$ for $k$ odd. Now if $k>1$, use $n= p^{k-1}$, set $q=p^x$, and note that $\sigma\_x(n)$ divides $q^k - 1$, so $\sigma\_y(n)$ is $k$ mod $\sigma\_x(n)$. So we do not have (for $k > 1$) $\sigma\_x(n)$ dividin... | 1 | https://mathoverflow.net/users/3402 | 229141 | 106,787 |
https://mathoverflow.net/questions/229132 | 31 | Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). Does it follow that there exist polynomial $f(z)$, positive integers $m,n$ and complex number $w\in S$ such that $P(z)=f(z... | https://mathoverflow.net/users/4312 | Polynomials with the same values set on the unit circle | This is a special case of the main theorem in the paper by
I. N. Baker, J. A. Deddens, and J. L. Ullman,
A theorem on entire functions with applications to Toeplitz operators,
Duke Math. J.
Volume 41, Number 4 (1974), 739-745.
They proved a similar statement for arbitrary entire functions.
| 27 | https://mathoverflow.net/users/25510 | 229154 | 106,790 |
https://mathoverflow.net/questions/229046 | 3 | Does there exist a prime $p$ and a smooth genus 2 curve $C / \mathbf{F}\_p$ such that the characteristic polynomial of Frobenius on the Tate module of $J(C)$ is given by $(T^2 - p)^2$?
More generally, for a curve of arbitrary genus, it possible that both $\sqrt{p}$ and $-\sqrt{p}$ can occur as eigenvalues of the Frob... | https://mathoverflow.net/users/2481 | Curve with given Frobenius polynomial | In its action on the Tate module, $\operatorname{Frob}\_q$ is an element of $GSP\_{2g}(\mathbb Q\_\ell)$ whose action on the symplectic form is multiplication by $q$. This is due to the Weil pairing, or Poincare duality for etale cohomology.
Every such matrix has eigenvalues $\lambda\_1, \dots, \lambda\_{2g}$ whose w... | 5 | https://mathoverflow.net/users/18060 | 229160 | 106,792 |
https://mathoverflow.net/questions/225930 | 7 |
>
> Let $X$ be a projective variety and let $D$ be a simple normal
> crossings divisor on $X$
>
>
> Does $$IH^\*(X;\mathbb C)\cong H\_{(2)}^\*(X\setminus D;\mathbb C)$$ hold
> true for each Kähler metric on $X\setminus D$? Is there any
> counterexample?
>
>
> What about the Fubini-Study metric on $X\setminus D... | https://mathoverflow.net/users/nan | A conjecture of Cheeger about intersection cohomology and $L^2$- cohomology | I'm not an expert, but you don't seem to be getting any answers.
First of all, I would be surprised if holds for *any* Kähler metric (e.g. for one with really bad singularities along $D$) but I don't have a counterexample\*. Regarding the case of Fubini-Study metric, for isolated singularities, I believe it was
settl... | 6 | https://mathoverflow.net/users/4144 | 229165 | 106,793 |
https://mathoverflow.net/questions/229173 | 0 | I have to referee a paper not really in my field and need some answers concerning the prime radical of a ring and nilpotent ideals.
The definition of a strong nilpotent element already have appeared in this question:
[strong nilpotent elements](https://mathoverflow.net/questions/97002/strong-nilpotent-elements)
R... | https://mathoverflow.net/users/85682 | characterization of strong nilpotent elements | 1. I believe that the counterexample given in the accepted answer to the question you link ([strong nilpotent elements](https://mathoverflow.net/questions/97002/strong-nilpotent-elements)) shows that there are strongly nilpotent elements for which $RxR$ is not nilpotent.
2. One precise reference for the statement that ... | 1 | https://mathoverflow.net/users/1306 | 229178 | 106,796 |
https://mathoverflow.net/questions/229184 | -6 | I would like to know whether the notion of automorphism of the set of partitions of a positive integer $n$ has been considered so far or not. To make things clearer, I say that a partition of $n$ in $k$ summands $s\_1,s\_2,...s\_k$ sorted in decreasing order has signature $(a\_1,a\_2,...,a\_m)$ if and only if $\forall ... | https://mathoverflow.net/users/13625 | Automorphisms of partitions | Let $P(n,\alpha)$ be the set of partitions of $n$ with signature $\alpha$. Such a set is invariant under your group, and any permutation of its elements is an automorphism. Your group is just $$G=\Pi\_\alpha S\_{|P(n,\alpha)|}.$$
| 1 | https://mathoverflow.net/users/78061 | 229186 | 106,797 |
https://mathoverflow.net/questions/229191 | 7 | Let $X \stackrel{\pi}{\to} \mathbb{D}$ be a proper holomorphic family with fibres $X\_t = \pi^{-1}(t)$. Siu proved, when the $X\_t$'s are projective, that the plurigenera $h^0(X\_t, mK\_{X\_t})$ are constant. It is conjectured that the same will hold true when the fibres $X\_t$ are Kähler.
Are there examples known w... | https://mathoverflow.net/users/47692 | Example of a non-Kähler manifold with varying plurigenera | The first example of this phenomenon was discovered by Iku Nakamura in his 1975 paper [*Complex parallelisable manifolds and their small deformations*, J. Differential Geom. 10 (1975), 85-112.](https://projecteuclid.org/euclid.jdg/1214432677)
The manifold $X$ is a $3$-dimensional solvmanifold, and he writes down its ... | 6 | https://mathoverflow.net/users/13168 | 229206 | 106,803 |
https://mathoverflow.net/questions/229192 | 6 | I don't understand the following as I read along a proof in a paper (Page 66, "Asymptotic Behaviour of some interacting systems", by Sylvie Meleard):
>
> We denote by $\mathcal{P}({M})$ the space of probability measures on a metric space $M$, equipped with the weak topology.
>
>
> Let $E$ be a metric space. Let $... | https://mathoverflow.net/users/62049 | Weak convergence in random measures | Let $F : \mathcal{P}(E) \to \mathbb{R}$ be bounded and continuous, and let $M := \sup |F|$. In adjusted notation, we wish to show $\mathbb{E} F(\mu\_n) \to \mathbb{E} F(\delta\_{\mathbb{Q}}) = F(\mathbb{Q})$. To save me some typing, let's suppose without loss of generality that $F(\mathbb{Q}) = 0$.
Fix $\epsilon > 0$... | 4 | https://mathoverflow.net/users/4832 | 229208 | 106,804 |
https://mathoverflow.net/questions/229213 | 2 | What are the groups $G$ and fields $\Bbb K$ for which $\Bbb K[G]\cong\Bbb K^{|G|}$ holds?
For example $\Bbb R[\Bbb F\_2^n]\cong\Bbb R^{2^n}$ holds.
| https://mathoverflow.net/users/nan | A group algebra isomorphism problem | This is true iff $G$ is finite and abelian, the characteristic of $K$ does not divide $G$, and $K$ has all $n^{th}$ roots of unity whenever $G$ has an element of order $n$. Hopefully it is clear why $G$ must be finite and abelian. The characteristic and root of unity conditions follow from writing $G$ as a product of c... | 11 | https://mathoverflow.net/users/290 | 229214 | 106,808 |
https://mathoverflow.net/questions/229171 | 10 | I’m studying the paper of [(Baum-Connes-Higson, ex 4.25)](http://www.mmas.univ-metz.fr/~gnc/bibliographie/BaumConnes/Baum-Connes-Higson.pdf), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial representation of the compact circle group $K... | https://mathoverflow.net/users/83246 | Construct discrete series of SL(2,R) as kernel of twisted Dirac operators | For $G=Spin(2n,1)$ (the double cover of $SO(2n,1)$) and $K=Spin(2n)$, the fact that Dirac induction $R(K)\rightarrow K\_0(C^\*\_r(G))$ is an isomorphism, is checked by hand, explicitly, in section 3 of my old paper "K-theory for the reduced C\*-algebra of a semi-simple Lie group with real rank 1 and finite centre", Qua... | 4 | https://mathoverflow.net/users/14497 | 229234 | 106,811 |
https://mathoverflow.net/questions/229231 | 2 | I saw the question:
[Abelian varieties with CM](https://mathoverflow.net/questions/228310/abelian-varieties-with-cm)
and though I know that there are rare CM elliptic curves, I wonder
what kind of curves with higher genus have the CM Jacobians?
| https://mathoverflow.net/users/85711 | Curves of higher genus | I am not sure whether this answers your question: it is a conjecture of Coleman that for a fixed genus $g$ sufficiently high, there should be only finitely many CM Jacobians of genus $g$. In fact Coleman stated the conjecture for $g\geq 4$, but by now there are counter-examples for $g\leq 7$ (at least). See for instanc... | 6 | https://mathoverflow.net/users/40297 | 229236 | 106,812 |
https://mathoverflow.net/questions/229185 | 4 | Consider a set of linearly independent vectors $\{x\_1,\dots,x\_n\}$ in some finite-dimensional Hilbert space $H$. For any subset $S \subset [n]$, let $P\_S$ be the (orthogonal) projection (operator) onto the span of $x\_S := \{x\_i, \;i \in S\}$. Let us also write $P\_j = P\_{\{j\}}$.
We would like to study the coll... | https://mathoverflow.net/users/36687 | Collection of projection operators in finite dimension and algebraic techinques | I don't know about question 2, but question 1 can indeed be answered using a general result about the projection lattice $P$ (ordered by $p\leq q\Leftrightarrow p=pq$) of a von Neumann algebra $A$.
>
> $Q=\{q\in P:pa=qa\}$ is a complete sublattice of $P$, for any $a\in A$ and $p\in P$
>
>
>
Proof: Let $[b]$ de... | 4 | https://mathoverflow.net/users/38085 | 229239 | 106,813 |
https://mathoverflow.net/questions/228955 | 10 | Everyone of us had sometimes this awful feeling that some sign is lost in a calculation and that this sign is perturbing some fundamental understanding of what is going on. I feel the same has happened for me today and I can't figure this sign problem out, so I count on you.
A Calogero-Moser system is defined as a Ha... | https://mathoverflow.net/users/21800 | Sign problem in a Calogero-Moser system: proof of integrability? | The answer to my question (provided by BS) is the following:
We have to change the action by looking at the group $G=U(n, \mathbb{C})$ and its action by conjugacy on pairs of Hermitian matrices. The space of pairs of such matrices can be identified with $T^\* (Lie U(n, \mathbb{C})^\*)$ because $Lie U(n, \mathbb{C})$ ... | 4 | https://mathoverflow.net/users/21800 | 229241 | 106,814 |
https://mathoverflow.net/questions/229217 | 7 | Let $f:X\to Y$ be a morphism of algebraic stacks.
If the **geometric** fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces.
I'm wondering about analogues of this fiberwise criterion in the context of gerbes and DM stacks.
For instance:
Q1. If the **geometric** fibres of $f$ are DM ... | https://mathoverflow.net/users/85707 | Fiberwise criterion for a stack to be a gerbe | As for **Q2**, I don't think so. For example, consider $(BG \times (\mathbb{A}^1\setminus \{ 0 \}))\amalg BG\to \mathbb{A}^1$, everything over $\mathbb{C}$.
**Added** [Edit: this one doesn't work, see comments below]: for a flat, but perhaps more contrived, example, I think one can take $\mathscr{X}:=B\mu\_{3,\mathbb... | 3 | https://mathoverflow.net/users/4721 | 229251 | 106,818 |
https://mathoverflow.net/questions/229268 | 6 | Let $G$ be a Lie group with a **left** invariant metric $g$.
Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is **right**-$H$-invariant. (That is $d(R\_h)\_e:T\_eG \to T\_hG$ is an isometry for every $h \in H$). Note that this is equivalent to the statement, that for every $h \in H$ the map of right multipl... | https://mathoverflow.net/users/46290 | Totally geodesic subgroups in Lie groups | **Edit.** I totally misunderstood the question, maybe now its better.
Assume that $g$ is left-$G$ and right-$H$-invariant. Then one can construct a Riemannian submersion $p\colon G\to G/H$. This is now a family of Riemannian manifolds with fibre $H$ and compact structure group $H$, because you could also write it as a ... | 3 | https://mathoverflow.net/users/70808 | 229272 | 106,823 |
https://mathoverflow.net/questions/229200 | 10 | Is there an $m$-dimensional simplicial complex $S$ with the following properties:
* *The cone over $S$ is homeomorphic to $\mathbb{E}^{m+1}$.* Here $\mathbb{E}^{m+1}$ denoes the $(m+1)$-dimensional Euclidean space.
* *There is a vertex $v$ in $S$ such that the complement $S\backslash\{v\}$ is not simply connected.*
... | https://mathoverflow.net/users/1441 | Wild half-line in a Euclidean space | I have got the following answer from Alexander Lytchak:
An example can be constructed the following way.
Start with a nontrivial homology sphere,
pass to its spherical suspension.
Now shrink one of the meridians of suspension to the point, which we denote by $v$.
The obtained space $S$ is the example;
it admits a nat... | 4 | https://mathoverflow.net/users/1441 | 229277 | 106,824 |
https://mathoverflow.net/questions/229265 | 1 | I know there's no general formula for all the roots of a polynomial with a degree greater than 4, but is there some sort of limit (or other) definition to calculate the roots (particularly the largest root)?
| https://mathoverflow.net/users/74600 | Is there a limit definition for the roots of a polynomial with arbitrary degree? | For computing the largest root, see the [Graeffe Method.](https://www.wikiwand.com/en/Graeffe's_method)
For bounding the largest root, there are many bounds, see [the Wikipedia article](https://www.wikiwand.com/en/Properties_of_polynomial_roots) (which is incomplete, but since you don't tell us what you want, maybe t... | 3 | https://mathoverflow.net/users/11142 | 229279 | 106,825 |
https://mathoverflow.net/questions/229282 | 6 |
>
> Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a closed self-adjoint subalgebra of $C(X)$ which contains the constants. Then $\mathcal{A}$ is the collection of continuous functions on $X$ which are constant on the sets of $\prod\_\mathcal{A}$ where
> $$
> \prod\_\mathcal{A}=X/\sim
> $$
> with $x\sim ... | https://mathoverflow.net/users/nan | Who gave the generalized Stone-Weierstrass Theorem? | This (for real-valued rather than complex-valued functions) was in [Stone's original paper](http://www.ams.org/journals/tran/1937-041-03/S0002-9947-1937-1501905-7/home.html) that proved the Stone-Weierstrass theorem, as Theorem 84. The statement is a bit funny, since he defines the equivalence relation $x\sim y$ not in... | 11 | https://mathoverflow.net/users/75 | 229286 | 106,829 |
https://mathoverflow.net/questions/58495 | 71 | I have recently run into [this Wikipedia article on mereology](https://en.wikipedia.org/wiki/Mereology). I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set theory, which is founded on the idea of set membership, mereology is built upon wh... | https://mathoverflow.net/users/12976 | Why hasn't mereology succeeded as an alternative to set theory? | I have long found this question interesting, and in some recent joint work with Makoto Kikuchi, now available, we consider various aspects of the question of whether a set-theoretic version of mereology can form a foundation of mathematics. In particular, for our main thesis we argue that the particular understanding o... | 41 | https://mathoverflow.net/users/1946 | 229302 | 106,835 |
https://mathoverflow.net/questions/222375 | 5 | Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.
I have seen a lot work related to minimal $(n-1)$-blockings set.
[Covering finite fields with cosets of subspaces](http://www.sciencedirect.com/science/article/pii/0097316577900012).
[T... | https://mathoverflow.net/users/42586 | $(n-2)$-blocking sets in $AG(n,2)$ | **Not much is known for the general case.**
Let $m(k, n, q)$ denote the minimum size of an $k$-blocking set in $AG(n, q)$. Trivially we have $m(0, n, q) = q^n$ and $m(n, n, q) = 1$. By Jamison/Brouwer-Schrijver we get $m(n-1, n, q) = 1 + n(q-1)$ as you have mentioned. To at least give bounds on other values we can p... | 5 | https://mathoverflow.net/users/34180 | 229325 | 106,842 |
https://mathoverflow.net/questions/229328 | 11 | Physicists routinely wrote all 3 Pauli spin matrices as a vector.
$$ \sigma\_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right) \hspace{0.25in}
\sigma\_2 = \left( \begin{array}{cc} 0 & -i \\ i & 0\end{array} \right)\hspace{0.25in}
\sigma\_3 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1\end{array} \right) $$... | https://mathoverflow.net/users/1358 | What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$? | The Pauli spin vector $\vec\sigma$ relates a two-component spinor $s={\alpha\choose \beta}$ to its corresponding three-component vector $\vec{v}=(a,b,c)$. In Dirac bra-ket notation the relation is written as
$$\vec{v}=\langle s|\vec\sigma|s\rangle,$$
in components $v\_i=s^\ast\sigma\_i s$. A pair of unitary operations... | 6 | https://mathoverflow.net/users/11260 | 229331 | 106,844 |
https://mathoverflow.net/questions/229332 | 3 | Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation $\bar{\rho}$ which is $p$-distinguish and minimal and such that the restriction of $\bar{\rho}$ to $G\_{\mathbb{Q}(\sqrt{(-1)^{... | https://mathoverflow.net/users/46460 | p-adic modular forms, Hecke algebra, deformation theory and modular curves. | To give a reference: in the article by Böckle referenced below, he proves an $R^{\mathrm{ord}}=T$ type theorem (Thm. 3.9) without assuming that the tame level is square free. In fact, he starts with the residual representation $\bar\rho$, then associates some conductor $N$ to $\bar\rho$ (which may not be square free) a... | 2 | https://mathoverflow.net/users/33820 | 229343 | 106,848 |
https://mathoverflow.net/questions/229352 | 3 | Does anyone know of a closed formula for the function
$f\_k(x)=\sum\_{n=1}^{\infty}{n^k x^n}$ ? That is, the generating function of the sequence $1^k,2^k,3^k...$.
It is not hard to see that $f\_k(x)=\frac{P(x)}{(1-x)^{k+1}}$, where $P(x)$ is a monic polynomial of degree $k$ (this follows from the identity $f\_k(x)=x... | https://mathoverflow.net/users/85783 | Closed formula for the generating function of the sequence of powers | Your polynomials are $x$ times the [Eulerian polynomials](http://oeis.org/wiki/Eulerian_polynomials).
| 14 | https://mathoverflow.net/users/13650 | 229356 | 106,854 |
https://mathoverflow.net/questions/229358 | 14 | Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M\_{/A}$. This category has a natural monoidal structure induced by taking fibered products over $A$. However, it should admit another monoidal structure coming from the product on $A$. In particular, give... | https://mathoverflow.net/users/11546 | Non-Cartesian Monoidal Model Structure on a Slice Category | This construction came up in an Australian Category Seminar talk given by Ross Street last month, from which I will copy for **1.** and **2.** below. I'm afraid I don't know a reference.
**1.** (*monoidal structure*) If $\mathscr{F}$ is a monoidal category and $T$ is a monoid in $\mathscr{F}$, then $\mathscr{F}/T$ b... | 13 | https://mathoverflow.net/users/57405 | 229371 | 106,859 |
https://mathoverflow.net/questions/229368 | 0 | Find the unique cases when ${t}^{2} - 4$ is a perfect square say, ${n}^{2}$, with height bound $|t| \le N$ for positive integer $N \ge 1$, when $t$ is a rational where $t = p/q$ and integers $p$ an $q$ are relatively prime, $|p| \le N$ and $1 \le q \le N$. I am looking for a counting like solution of the unique cases, ... | https://mathoverflow.net/users/62471 | Find the rational cases where ${t}^{2} - 4$ is a perfect square with height bound $|t| \le N$ for positive integer $N \ge 1$ | From $t^2-4=s^2$ we get
$$
t^2-s^2=4~~ \Longrightarrow ~~ (t+s)(t-s) = 4
$$
hence the general rational solution $(t,s)$ is, putting $2\lambda = t+s$:
$$
\left( \lambda+\frac{1}{\lambda}, \lambda-\frac{1}{\lambda} \right).
$$
(It is easy to check that this indeed solves your equation.) So now we need to find the height ... | 4 | https://mathoverflow.net/users/17907 | 229372 | 106,860 |
https://mathoverflow.net/questions/229370 | 8 | In [HHR](http://www.math.rochester.edu/people/faculty/doug/mypapers/Hill_Hopkins_Ravenel.pdf), an important part is the periodicity theorem. For proving the theorem, they invert a carefully defined class $D \in \pi^{C\_8}\_{19\rho\_8}(N^8\_2MU\_{\mathbb{R}})$ and they can find an element in $x \in \pi\_{256}^{C\_8}D^{-... | https://mathoverflow.net/users/17440 | (Non)-equivariant equivalence in $G$-spectra | The condition required on $X$ which makes this work is that $X$ is cofree (Definition 10.1 in the linked paper): the map $X \to F(EG\_+,X)$ is an equivalence. For any equivariant map $X \to Y$ of cofree $G$-spectra which is an equivalence on the underlying spectra, the resulting maps $X^H \to Y^H$ of fixed-point sets a... | 6 | https://mathoverflow.net/users/360 | 229378 | 106,863 |
https://mathoverflow.net/questions/229376 | 6 | It is well known that adding a truth predicate to arithmetic in the most natural way leads to a contradiction.
Suppose as usual that we add a one place relation *T* to the language of arithmetic, and define some system of Godel numbering $\ulcorner \cdot \urcorner$ for this expanded language. Given a set of axioms A ... | https://mathoverflow.net/users/85789 | Adding a truth-like predicate to PA |
>
> Obviously, adding rules of inference that tell that us we can go from $\phi$ to $T(\ulcorner\phi\urcorner)$ and back, and $\neg\phi$ to $\neg T(\ulcorner\phi\urcorner)$ and back, would be too strong and lead to contradictions in well known ways.
>
>
>
Actually, that's not correct. According to [a theorem of ... | 5 | https://mathoverflow.net/users/23141 | 229381 | 106,864 |
https://mathoverflow.net/questions/228687 | 11 | The irreducible representations of the Symmetric group $S\_5$ are classified by the partitions of $5$. For the standard representation which corresponds to the partition (4,1) the ring of invariants is generated by the elementary symmetric polynomials and hence is a polynomial ring. For other irreducible representation... | https://mathoverflow.net/users/84990 | Invariant ring of $S_5$ | In fact at least in the case you seem most interested in, the (4-dimensional) standard representation tensored by the sign representation, it is possible to compute the invariants by computer, using the standard tools in MAGMA. The result is a minimal system of generating invariants of degrees 2, 4, 6, 8, 10, 13, and 1... | 6 | https://mathoverflow.net/users/82616 | 229391 | 106,868 |
https://mathoverflow.net/questions/229344 | 6 |
>
> Given a nonempty set of integers, and given that there exists a subset of this set whose elements sum to zero, is finding the *smallest* such subset NP-complete?
>
>
>
**Disclaimer:** The above question involves part of ongoing research for my dissertation.
To flesh this out, this question is derived fro... | https://mathoverflow.net/users/85776 | Variation on the Subset Sum Problem | This is merely a simplification of Tony Huynh's answer, but still more than a comment imho. Note that throughout, I work with multi sets (or sequences of integers) rather than sets. Reduce SUBSET SUM to SMALLEST SUBSET SUM as follows:
Given an instance $(a\_1, \ldots, a\_n)$ of SUBSET SUM, let $b := -(a\_1 + \ldots + a... | 5 | https://mathoverflow.net/users/37432 | 229392 | 106,869 |
https://mathoverflow.net/questions/229382 | 0 | **Edit 01**:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma\_y(n)\bmod \sigma\_x(n)=0$, by [wolfram alpha](http://www.wolframalpha.com/input/?i=Table[sigma_3%28n%29+mod+sigma_2%28n%29]%3D0+%2Cn%3D2+to+5000)... | https://mathoverflow.net/users/74330 | Is there an example of integers ($x,p, q ,y$ ) which satisfies the below conditions in this claim? | This is possible, as expected (usually if there are no obvious reasons why not, the answer in such a problem is yes.)
Try $p=3$, $x=2$. Then we need $10|q^b+1$, $q^2+1|3^c+1$ for some odd $b,c$. First relation is possible for $q=10k-1$, $b=1$. The second holds if the order of $-3$ modulo $q^2+1$ is odd. Computations ... | 4 | https://mathoverflow.net/users/4312 | 229405 | 106,873 |
https://mathoverflow.net/questions/229402 | 3 | Here is a basic technique in logic which seems well-known in folklore, but which I haven’t managed to find written down anywhere. $\newcommand{\T}{\mathbf{T}}$
**Fact.** Let $\Sigma$ be a signature (in the sense of predicate logic; i.e. sets of “function symbols” and “predicate symbols”, equipped with natural-number ... | https://mathoverflow.net/users/2273 | Reference request: eliminating function symbols in predicate logic | The techique is in Bell & Machover: A course in mathematical logic, ch 2 §10 as theorem 10.5.
It states…
Select an $n$-ary function symbol $\mathbf f$ of $\mathcal L$, and let $\mathcal L'$ be obtained from $\mathcal L$ by excluding $\mathbf f$ and introducing a new $(n+1)$-ary predicate symbol $P$. We prove:
The... | 4 | https://mathoverflow.net/users/35779 | 229407 | 106,875 |
https://mathoverflow.net/questions/229399 | 2 | I know that the p-adic representaions from geometries are de Rham representations and hence they are Hodge-Tate
representations. Then, are there (more than 2-dimensional) Hodge-Tate representations not coming from geometries? If there are, let me know
the references. Sorry if it is trivial.
| https://mathoverflow.net/users/85711 | Hodge-Tate representations | Every de Rham representation is Hodge--Tate, but the converse is not true -- there are Hodge--Tate representations which are not de Rham, and thus cannot appear in geometry. (Examples of these arise in dimension 2 via the theory of p-adic modular forms.)
I don't think anyone knows how to characterise, even conjectura... | 10 | https://mathoverflow.net/users/2481 | 229410 | 106,876 |
https://mathoverflow.net/questions/229403 | 13 | $\newcommand{\omegaoneck}{\omega\_1^{\text{CK}}}$
Pardon the extremely basic question - this isn't quite my area - but I'm confused about the definition of proof theoretic ordinals.
The proof theoretic ordinal of a theory is defined to be the smallest ordinal that the theory cannot prove is well founded. In other wo... | https://mathoverflow.net/users/70015 | How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$? | Just slightly expanding the comment of Emil Jeřábek: on one hand, in ZFC we can define some objects we call ordinals and prove transfinite induction of each of them. This is *not* what we mean by the ordinal of ZFC. The latter is defined roughly as follows (although there are several nonequivalent definitions):
1. Yo... | 14 | https://mathoverflow.net/users/57888 | 229411 | 106,877 |
https://mathoverflow.net/questions/229349 | 6 | For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology classes we now that an indued on the cohomology ring $H(M)$, which is usually denoted again by $\ast$.
Now one could also ... | https://mathoverflow.net/users/51325 | Hodge map and the Cohomology Ring of a Riemannian Manifold | You've probably seen this before, but in case you haven't:
Let $M$ be a compact oriented manifold. Let $\Omega^k$ be the smooth $k$-forms, let $Z^k$ be the closed $k$-forms and let $B^k$ be the exact $k$-forms. So $H^k=Z^k/B^k$. As Johannes's answer shows, $\ast$ does not carry $Z^k$ to $Z^{n-k}$, nor $B^k$ to $B^{n-... | 2 | https://mathoverflow.net/users/297 | 229413 | 106,879 |
https://mathoverflow.net/questions/229422 | 0 | Briefly: A hint (if this is easy), reference or derivation would be of great help.
The question
------------
Let $C\_n$ be the directed cycle with loops in each of its $n$ vertices, and consider the random walk that at each step stays put with probability $\frac{1}{2}$ and moves clockwise with probability $\frac{1}... | https://mathoverflow.net/users/74773 | Mixing time of lazy random walk on the directed cycle $C_n$ | Let $A\_n$ be the adjacency matrix of the directed $n$-cycle without loops. That is for instance
$$A\_4=\begin{bmatrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\\ 1& 0 & 0 &0 \end{bmatrix}.$$
The transition matrix $P\_n$ of your random walk is then given by
$P\_n=\frac{1}{2}A\_n+\frac{1}{2}I\_n$ where $I\_n$ is... | 2 | https://mathoverflow.net/users/85651 | 229427 | 106,881 |
https://mathoverflow.net/questions/229435 | 2 | I am interested in the ergodic (invertible) transformations $T$ such that $T\times R\_\theta$ is ergodic where $R\_\theta$ is the rotation on $S^1$ with a given irrational angle $\theta$ (not all $R\_\theta$, only this one).
This includes all weakly mixing transformations, because of the two following results which c... | https://mathoverflow.net/users/21339 | Transformations whose product with a given rotation are ergodic | $T\times R\_\theta$ is ergodic if and only if $T$ is ergodic and $e^{2\pi im\theta}$ is not an eigenvalue of $T$ for any $m\in \mathbb Z\setminus\{0\}$.
For an idea of the proof, let $T\colon X\to X$. Then $L^2(X)$ can be decomposed as an orthogonal direct sum $V\_c\oplus V\_d$, the functions with continuous spectrum... | 5 | https://mathoverflow.net/users/11054 | 229437 | 106,884 |
https://mathoverflow.net/questions/229242 | 6 | This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try.
At the risk of repeating well known stuff I tried to make the question as precise as possible.
Let $\pi: P \to M$ be a $G$-bundle with connection form $\omega \in \Ome... | https://mathoverflow.net/users/22810 | Transferring connection information to associated bundles and back | Ad 1.: Since every vector can be decomposed in its horizontal and vertical part. Thus it is enough to consider the case a) where all vectors are horizontal (this is trivial) and b) where at least one is vertical and the rest is horizontal (this has to be calculated explicitly but is relatively simple since one can use ... | 4 | https://mathoverflow.net/users/17047 | 229438 | 106,885 |
https://mathoverflow.net/questions/227734 | 5 | A lot of research has been devoted to the study of first order language of graphs (FO). Formulas in this language are constructed using variables $x, y,\dots$ ranging over the vertices of a graph, the usual quantifiers ∀, ∃, the usual logical connectives ¬, ∨, ∧, etc., parentheses and the binary
relations =, ∼, where x... | https://mathoverflow.net/users/46573 | first order languages over graphs (and other discrete models) | To complement J.-E. Pin's answer, many other structures have been considered in the literature, like words [1], discrete metric spaces [2], maps [3] or simplicial complexes [4].
In [5] the relations between some of these results are investigated as instances of a more abstract theorem.
The case of $k$-uniform hypergra... | 3 | https://mathoverflow.net/users/85130 | 229445 | 106,887 |
https://mathoverflow.net/questions/217947 | 14 | It is well known that the fundamental class of a compact Lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable equivalence $G\simeq A\vee S^n$ for some subcomplex $A$, with $n$ being the dimension of $G$.
My question relates to exac... | https://mathoverflow.net/users/54788 | How stable is the top cell of a Lie group? | I don't know the minimal number of suspensions required, but for the classical groups $O(n)$, $U(n)$ and $Sp(n)$ the existence of a bound quadratic in $n$ follows from Miller's stable splittings:
* H. Miller. Stable splittings of Stiefel manifolds. Topology 24 (1985) 411-419.
The result is that there is a splitting... | 6 | https://mathoverflow.net/users/50846 | 229451 | 106,888 |
https://mathoverflow.net/questions/229425 | 10 | Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum\_{i=1}^3 \lambda\_{i}(n\_{i}\otimes n\_{i}),
$$
where $ n\_{i} $ and $ \lambda\_{i} $ are the unit eigenvectors and eigenvalues of $B$, and ... | https://mathoverflow.net/users/85818 | Derivative of eigenvectors of a matrix with respect to its components | If $B$ depends on a single parameter $t$ then derivating with respect to $t$ the equality
$$ B n\_i =\lambda\_i n\_i $$
we deduce
$$\dot{B} n\_i +B\dot{n}\_i=\dot{\lambda}\_i n\_i +\lambda\_i\dot{n}\_i. $$
Here we assume that $\Vert n\_i\Vert =1$. Hence $\dot{n}\_i\perp n\_i$, $\forall i$. Taking the inner prod... | 18 | https://mathoverflow.net/users/20302 | 229467 | 106,898 |
https://mathoverflow.net/questions/229439 | 13 | A user on MSE, @martin , asked <https://math.stackexchange.com/questions/1611411/pell-equations-upper-bound> about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears reasonable to guess that
$$ x < p^{\sqrt p} $$
when $p > 2.$ I had the computer solve by Lagrange's method, no continue... | https://mathoverflow.net/users/3324 | Upper bound on answer for Pell equation | Let $d$ be a positive fundamental discriminant, $\epsilon\_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi\_d$ the primitive character associated to the discriminant $d$. The class number formula gives
$$
\log \epsilon\_d = \sqrt{d} L(1,\chi\_d)/h(d) \le \sqrt{d} L(1,\chi\_d),
$$
since the class ... | 19 | https://mathoverflow.net/users/38624 | 229468 | 106,899 |
https://mathoverflow.net/questions/229452 | 2 | Let $k$ be an algebraically closed field and let $p$ be the characteristic of the field. Let $f : X \to S$ be a projective morphism such that the fibres are generically reduced, pure of dimension $1$. We suppose $X$ and $S$ are both irreducible and quasi-projective varieties. For every $s \in S(k)$, does there exist a ... | https://mathoverflow.net/users/85020 | Embedding a family of curves in projective space. | No, that is not always possible. For instance, if $p$ equals $2$, if $S$ is the dense, Zariski open subscheme of $\mathbb{P}^5$ parameterizing smooth plane conics in $\mathbb{P}^2\_k$, and if $f:X\to S$ is the restriction to $S$ of the universal conic, then the index of $f$ equals $2$.
The simplest way to see this is... | 2 | https://mathoverflow.net/users/13265 | 229471 | 106,901 |
https://mathoverflow.net/questions/229357 | 1 |
>
>
> >
> > It is known that if $D$ is a continuous derivation on a commutative Banach algebra $\mathcal{A}$, then for any nonzero character $\theta$ on $\mathcal{A}$ we have $D(\mathcal{A})⊆ker\theta $.
> >
> >
> >
>
>
>
Please help me with these questions or give some references.
How this can be restated... | https://mathoverflow.net/users/85784 | The image of a derivation on a Banach algebra is contained in the kernel of a character | I shall assume that in your question, you are asking about *continuous* derivations on a Banach algebra $A$. Results for everywhere-defined-but-not-continuous derivations were studied intensively 30-40 years ago but my understanding is that the remaining open problems are thought to be inaccessible.
So, let $A$ be a ... | 3 | https://mathoverflow.net/users/763 | 229478 | 106,903 |
https://mathoverflow.net/questions/229473 | 2 | If $A$ is a $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$. If ever possible, what would be an efficient way of extracting a full rank $k\!\times\!k$ sub-matrix of $A$ by removing columns and rows of the same indices (if row $i$ is to be removed so is column $i$) ?
| https://mathoverflow.net/users/85834 | Extracting a full rank matrix from a 0-1 matrix | There are many search strategies that might be tried. Note that if the coefficient of $X^{n-k}$ in the characteristic polynomial $P\_A(X)$ of $A$ is nonzero, such a submatrix does exist. This is not an if and only if, but if the submatrix exists, then for most $t = (t\_1,\ldots, t\_n)$, all $t\_i$ nonzero, it will work... | 2 | https://mathoverflow.net/users/13650 | 229480 | 106,904 |
https://mathoverflow.net/questions/229488 | 6 | Toen in [The homotopy theory of dg-categories and derived Morita theory](http://arxiv.org/abs/math/0408337) Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
RHom(C,D)=Int((C\otimes^{\mathbb{L}}D^{op})-Mod^{rqr}).
$$
where "rqr" stands for right q... | https://mathoverflow.net/users/24965 | Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories? | Toen proved that $RHom$ provides the internal hom in the homotopy category of dg-categories. For what you want to be true, you need something more than this: you need to know that $RHom$ is actually the right-derived functor of the strict internal hom $DG$. Apparently, this was proved in [arXiv:1202.3359](http://arxiv.... | 6 | https://mathoverflow.net/users/2503 | 229494 | 106,906 |
https://mathoverflow.net/questions/229073 | 2 | Suppose that $X$ is an extremally disconnected topological space (meaning that the closure of an open set is still open). Then $X$ has the following property: the family of all sets $S$ such that $S$ differs from a clopen (open and closed simultaneously) set by the set of first category is a $\sigma$-algebra which cont... | https://mathoverflow.net/users/24078 | Extremally disconnected spaces and a measure theoretic property | I claim that if such a space is a Baire space then it is extremally disconnected. Suppose to the contrary that $X$ is a Baire space which is not extremally disconnected. Furthermore, I claim that if $U$ is an open set where $\overline{U}$ is not clopen, then $U\not\in S$.
Suppose that $U$ is open but $\overline{U}$ ... | 2 | https://mathoverflow.net/users/22277 | 229495 | 106,907 |
https://mathoverflow.net/questions/229487 | 2 | I'm having difficulty understanding a fact stated in a research paper I'm reading. Namely, let $T$ be a tree with all nodes of degree $4$ (ie, the root has $4$ daughter nodes and all other nodes have $3$ daughter nodes). The paper states without explanation that the spectrum of the Laplacian on $T$ is in $[4-2\sqrt{3},... | https://mathoverflow.net/users/85847 | Spectrum of Laplacian matrix of an infinite tree graph | A very lucid explanation is given in [Luca Trevisan's blog.](https://lucatrevisan.wordpress.com/2014/08/20/the-spectrum-of-the-infinite-tree/)
| 1 | https://mathoverflow.net/users/11142 | 229499 | 106,908 |
https://mathoverflow.net/questions/229446 | 1 | If $X=\operatorname{Spec} A$, where $A$ is a noetherien, complete local ring, with a finite residual field $\mathbb{F}\_p$. We can associate to $A$ a rigid analytic space with two different ways, we can use Berthelot functor or the method of Raynaud. These methods rise to same rigid space when the scheme is proper over... | https://mathoverflow.net/users/46460 | Berthelot functor, rigid analytic space | Let me say something related to your first question (the local one) in a slightly different setting. Let $\mathscr{A}$ be a $\mathbb{Q}\_p$-affinoid algebra and consider its reduction $\tilde{\mathscr{A}} = \mathscr{A}^\circ/\mathscr{A}^{\circ\circ}$. (The ring $\mathscr{A}^\circ$ is close to your $A$.) We have a reduc... | 2 | https://mathoverflow.net/users/4069 | 229500 | 106,909 |
https://mathoverflow.net/questions/229308 | 2 | Let the integers $n\geq 2$, $k\geq 1$, $v=0$ or $1$ and $n\_1,\cdots,n\_k\geq 1$ such that
$$
\sum\_{i=1}^k n\_i+v=n.
$$
Define $P\_a^b=0$ if both $a,b$ are odd and $P\_a^b={{[a/2]}\choose {[(a+b)/2]}}$ otherwise.
Given a fixed $n$, consider the free $\mathbb{Z}$-module generated by
$$
c[v,n\_1,\cdots,n\_k]
$$... | https://mathoverflow.net/users/41075 | programming to compute kernel quotient image of a $\mathbb{Z}$-module endomorphism | Ok, not really beautiful, but the lines below are a simple SAGE implementation of the map $\partial$, computing both the representing matrix and the elementary divisors. In the implementation I assumed that $P^b\_a$ is actually the binomial coefficient $\binom{\lfloor (a+b)/2\rfloor}{\lfloor a/2\rfloor}$ (the order in ... | 3 | https://mathoverflow.net/users/50846 | 229503 | 106,911 |
https://mathoverflow.net/questions/229511 | 6 | Hello mathoverflow community,
I am a little stucked working on my master thesis. For a representation on $\mathbb{Z}\_p\ltimes\mathbb{Z}\_p^\*$ induced from the additive character $\chi$ of $\mathbb{Z}\_p$ given by $\chi(g)=e^{\frac{2\pi i g}{p}}$, I obtain a complex matrix form. That is for $(a,b)\in\mathbb{Z}\_p\lt... | https://mathoverflow.net/users/85860 | How do I determine a real matrix form for a group representation? | Here's one way: Note that your semidirect product has a normal subgroup $\mathbb{Z}\_{p} \mathbb{Z}\_{2}$ which is dihedral with $2p$ elements ( I assume the semidirect product you want to work with is the holomorph of $\mathbb{Z}\_{p}$).
It is easy to see a real faithful representation (of degree $2$) of that dihedr... | 6 | https://mathoverflow.net/users/14450 | 229513 | 106,912 |
https://mathoverflow.net/questions/229514 | 11 | It is well known that $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and this is the only solution to $\frac{1}{x\_1}+\frac{1}{x\_2}+\frac{1}{x\_3}=1$
with $2\leq x\_1<x\_2<x\_3$.
My question is:
>
> Let $n\in \mathbb{N}$ and $x\_i\in \mathbb{N} ,1\leq i\leq n$.
>
>
> *How many* $n$-tuples $(x\_1,x\_2,...,x\_n)$... | https://mathoverflow.net/users/38851 | How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have? | N. Burshtein has published several papers on this problem, [On distinct unit fractions whose sum equals 1](http://ac.els-cdn.com/0012365X73901362/1-s2.0-0012365X73901362-main.pdf?_tid=2e819cfe-c5bc-11e5-b495-00000aacb35f&acdnat=1453984953_26680765f7196bbca82cbd943b861d18), and [a more recent paper,](http://www.scienced... | 7 | https://mathoverflow.net/users/11260 | 229517 | 106,913 |
https://mathoverflow.net/questions/229505 | 5 | Let $F(n)$ be free group of rank $n\geq 2$. Denote by $F\_d(n)$ the d-th derived subgroup, that is $F\_d(n)=[F\_{d-1}(n),F\_{d-1}(n)]$ where $F\_0(n)=F(n)$. The free solvable group of rank $n$ and solvable class $d$ is $S\_{n,d}=F(n)/F\_d(n)$.
By Corollary 2.14. in "FINITELY PRESENTED WREATH PRODUCTS AND DOUBLE
COSET... | https://mathoverflow.net/users/60483 | finitely presented subgroup and free solvable group of class 3 | No, it does not exist.
It's not too hard to check that a fp subgroup of a free metabelian group is abelian. A consequence (since any quotient of a fp solvable group is fp, by Bieri-Strebel) is that any fp subgroup of a free 3-solvable group has abelian image in the metabelianization, and in particular, has to be meta... | 5 | https://mathoverflow.net/users/14094 | 229544 | 106,924 |
https://mathoverflow.net/questions/175875 | 8 | Fix an odd prime $p$.
For concreteness, let $N$ be coprime to $p$, and let $2 \leq k \leq p$. Let $S^+(N,k)$ be the newforms in $S\_k(\Gamma\_1(N))$.
Let $f = \sum a\_n q^n \in S^+(N,k)$. We say that $f$ is $p$-ordinary if $v\_p(a\_p)=0$. (If $p$ splits in the field of coefficients of $f$, we require that $a\_p$ no... | https://mathoverflow.net/users/10547 | Density of p-ordinary modular forms | An ordinary modular form needs to be $p$-stable (by definition). If you have a modular form of level $N$ co-prime to $p$, you need to associate a $p$-stabilization to this latter to get an ordinary modular form with level $Np$.
If you take $T$ the Hecke algebra acting on Katz p-adic modular forms, it is known that th... | 3 | https://mathoverflow.net/users/46460 | 229545 | 106,925 |
https://mathoverflow.net/questions/229433 | 4 | $\newcommand{\al}{\alpha}$
Let $M\_n$ be the space of $n \times n$ real matrices.
**Question:**
For which $n$, is there an inner product on $M\_n$ which satisfies:
$$(\*) \, \, \langle Q^TXQ,Q^TYQ \rangle = \langle X, Y\rangle $$
For every $Q \in SO(n)$,
but does **not** satisfy $(\*)$ for every $Q \in O(n)... | https://mathoverflow.net/users/46290 | Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic | Disclaimer: this answer is at least 50% due to @Holonomia. If you like it, why don't you upvote some of her posts?
The groups $O(n)$ and $SO(n)$ act by conjugation on $M\_n(\mathbb R)$.
There is an equivariant decomposition
$$M\_n(\mathbb R)\cong\mathbb R E\_n\oplus\mathrm{Sym}\_0^2(\mathbb R^n)
\oplus\Lambda^2(\math... | 6 | https://mathoverflow.net/users/70808 | 229549 | 106,927 |
https://mathoverflow.net/questions/224585 | 1 | Was "arithmetical translation" (that is, coding in the Goedel sense) ever a part of Hilbert's Program? I ask this question for several reasons:
i) it gives the numerals |, ||, |||,.... an ersatz 'meaning' in themselves and Hilbert (at least in his paper "On the Infinite") states that "These numerals, which are the ob... | https://mathoverflow.net/users/20597 | Was "arithmetical translation" (coding in the Goedel sense) ever a part of Hilbert's Program? | The following quote from Dan Willard's preprint "On the Broader Epistemological Significance of Self-Justifying Axiom Systems" (found on his Homepage) is, I believe, of some small significance:
"In any case some years after he wrote \*'s initial statement
[\* 'It must be expressly noted Proposition XI represents no... | 0 | https://mathoverflow.net/users/20597 | 229551 | 106,929 |
https://mathoverflow.net/questions/229546 | 1 | first of all: english is not my native language, so there might be differences between what I meant and what you understood. Sorry for that in advance.
As a research project, I try to comprehend and re-implement an algorithm from a paper. Included in this algorithm is an interpolation of a function that is only known... | https://mathoverflow.net/users/85877 | Interpolation of a series of data points via Chebyshev approximation? | Numerical Recipes, 3rd edition, Chapter 3 describes a vast number of interpolation methods with source code. It seems from what you are telling us that something is confused in the paper: Chebyshev interpolation has to do with picking points at which to minimize error of the ensuing result, but perhaps something else w... | 2 | https://mathoverflow.net/users/6084 | 229576 | 106,937 |
https://mathoverflow.net/questions/228386 | 1 | It is well known that one can use Goedel's primitive recursive functionals of finite type to prove the consistency of $PA$ (Peano Arithmetic). As such, one can certainly use them to prove the consistency, say, of Primitive Recursive Arithmetic ($PRA$), but does one need the full 'power' (so to speak) of Goedels primiti... | https://mathoverflow.net/users/20597 | What restriction(s) of Goedel's primitive recursive functionals is (are) necessary and sufficient to prove the consistency of $PRA$ | Sorry for taking a bit longer to answer: Everything I say here is from Jeremy Avigad and Sol Feferman's article in the Handbook of Proof Theory, Gödel’s functional (“Dialectica”) interpretation: <http://www.andrew.cmu.edu/user/avigad/Papers/dialect.pdf>
First let me note that PRA itself is an answer to the question a... | 4 | https://mathoverflow.net/users/2004 | 229591 | 106,943 |
https://mathoverflow.net/questions/229482 | 0 | Background:
I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist:
Let $$L(f)(\eta)= \sum\_{x \in \mathbb{Z}}\frac{1}{2}\left(1\_{\eta(x+1) \neq \eta(x)}+ 1\_{\eta(x-1)\neq \eta(x)} \right)(f(\eta\_x)-f(\eta)), $$ where $\eta \in S:=\{0,1\}^{\mathbb{Z}}$ an... | https://mathoverflow.net/users/85616 | Weak convergence of process | Functions that depend on only finitely many co-ordinates are dense in continuous functions on that function space, as is plausible but also follows from Stone-Weierstrass. If f is such a function that depends only on the co-ordinates in the interval $[-N, N]$ then $$ \mathbb{E}^0 f(\eta\_{X\_t}) = \mathbb{E}^0 (\mathbb... | 0 | https://mathoverflow.net/users/nan | 229608 | 106,947 |
https://mathoverflow.net/questions/229601 | 7 | Let $V\_2$ and $V\_3$ be the two hypersurfaces of $\mathbb P^3$ defined by
\begin{equation\*}
V\_2:={x\_2x\_3 + r(x\_0, \, x\_1)=0}, \quad V\_3:={x\_2^3+x\_3^3+s(x\_0, \, x\_1)=0},
\end{equation\*}
where $r, \, s \in \mathbb{C}[x\_0, \, x\_1]$ are general homogeneous forms of degree $2$ and $3$, respectively.
Then $... | https://mathoverflow.net/users/7460 | Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action | Yes. Start from a genus 2 curve $C\_2$, and choose a point of order 3 in $JC\_2$, giving rise to an étale $\mathbb{Z}/3$-covering $C\_4\rightarrow C\_2$.
Then $C\_4$ cannot be hyperelliptic: a $g^1\_2$ on $C\_4$ would be stable under the covering automorphism $\sigma $, hence descend to $C\_2$, which is impossible for... | 14 | https://mathoverflow.net/users/40297 | 229610 | 106,948 |
https://mathoverflow.net/questions/229611 | 6 | Is there a general criterion for which partial orders can be realized by the prime ideals of commutative rings (like we have for topological spaces - <https://en.wikipedia.org/wiki/Spectral_space>)?
And in general, what constraints on the partial order are created by additional classical properties of rings - such as... | https://mathoverflow.net/users/59012 | Partial Orders realized by Prime Ideals on commutative rings | The following characterization follows easily from the general theory of spectral spaces, though it isn't exactly the most explicit criterion to apply in practice.
>
> **Theorem** (Hochster, Proposition 12 of [this paper](http://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0251026-X/)): Let $X$ be a poset.... | 8 | https://mathoverflow.net/users/75 | 229616 | 106,950 |
https://mathoverflow.net/questions/229569 | 15 | Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$.
The Cartan connection is supposed to formalize what it means to "roll without slipping" the homogeneous space $G/H$ on the manifold $M$. I am wondering if the following is one correct way to i... | https://mathoverflow.net/users/56938 | Intuition for the Cartan connection and "rolling without slipping" in Cartan geometry | You don't need to be 'handwavy' at all, and, in the standard modern way to understand this, one does not need one to choose local sections and talk about infinitesimal motions. Here's the standard approach:
First, a Cartan connection of type $(G,H)$ on an $n$-manifold $M$ is, as you know, a principal right $H$- bundl... | 14 | https://mathoverflow.net/users/13972 | 229633 | 106,956 |
https://mathoverflow.net/questions/229615 | 2 | I am asking the question in a purely topological setting; a zero-set of some topological space $X$ is a subset which can be realized as the counterimage of a single point through a continuous real-valued function.
Is there any workable characterization of zero-sets which talks only about $X$ and does not deal with ot... | https://mathoverflow.net/users/18889 | Is there any workable internal characterization of zero-sets? | A simple such characterization exists when one is working with proximity spaces instead of topological spaces. Suppose that $(X,\delta)$ is a proximity space with complete containment relation $\ll$. Then we say that a subset $Z\subseteq X$ is a proximally zero set if there is some proximity map $f:(X,\delta)\rightarro... | 2 | https://mathoverflow.net/users/22277 | 229635 | 106,957 |
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