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https://mathoverflow.net/questions/144432 | 4 | A *circular triangle* is a closed simple curve in the Euclidean plane $\mathbb{R}^2$ that can be expressed as the union of three circular arcs. Data naturally associated with such a figure includes:
* the edge lengths $\ell\_i$,
* the edge curvatures $\kappa\_i$, and
* the interior angles $\theta\_i$
(for $i \in {1... | https://mathoverflow.net/users/1557 | The space of circular triangles? | Actually the literature on circular triangles is enormous. Most of it is in German.
Here is a little sample:
A. Schonflies,
Ueber Kreisbogendreiecke und Kreisbogenvierecke,
Math. Ann., 44 (1894) 105--124.
F. Klein, Ueber die Nullstellen der hypergeometrischen Reihe,
Math. Ann., 37 (1890) 573--590.
F. Klein, Vorle... | 4 | https://mathoverflow.net/users/25510 | 144453 | 78,353 |
https://mathoverflow.net/questions/144448 | 3 | The nLab says in its [internal logic](http://ncatlab.org/nlab/show/internal+logic) article that the Completeness Theorem can be proven via a ``generic model'' of the theory. The model is generic in the sense that the only things true of it are those provable from the axioms. Since a model of $T$ is a functor $\text{Syn... | https://mathoverflow.net/users/41109 | Completeness theorem via syntactic categories | I think this is Proposition 1.5.1 in the Elephant. As Omar Antolín-Camarena says, one first shows that the generic model is complete. Then this extends to set models, not by picking a particular representable, but by showing that the collection of all representables is jointly conservative.
If I'm correct, the idea ... | 2 | https://mathoverflow.net/users/38258 | 144461 | 78,355 |
https://mathoverflow.net/questions/144408 | 1 | Let $\Sigma$ be a finite set of symbols with total order. Let $C\_k$ be the set of all $k$-multiset (unordered collection of $k$ elements from $S$, with repetition allowed). We can order all the elements of $C\_k$ in lexicographic order and index each element in that order i.e a bijective function $f: C\_k \to [ 1, |C\... | https://mathoverflow.net/users/26012 | Indexing combinations with repetition | @PietroMajer's comment essentially contained the answer already:
Define a bijection $\phi$ between $k$-multisubsets of $\{ 1,\ldots,n \}$ to $k$-subsets of $\{ 1,\ldots,n+k-1\}$ by sending $\{ c\_0 \leq c\_1 \leq \ldots \leq c\_{k-1} \}$ to $\{ c\_0 < c\_1+1 < c\_2 + 2 < \ldots < c\_{k-1}+k-1 \}$.
This map sends le... | 2 | https://mathoverflow.net/users/21291 | 144463 | 78,357 |
https://mathoverflow.net/questions/144464 | 14 | I was not able to find the origin of the name *Petrov* in the Petrov-Galerkin method for the numerical approximation of PDEs.
Wikipedia refers to a certain Alexander G. Petrov, but it is still not clear who he was: on the Internet I found two persons called Alexander G. Petrov, both born during the 40's, but I think... | https://mathoverflow.net/users/41123 | Who is Petrov of the Petrov-Galerkin method? | The Petrov you are looking for is: Georgii Ivanovich Petrov (1912-1987), biographies are [here](http://hit2003.narod.ru/persons/great/petrov/petrov_en.html) and [here](http://iopscience.iop.org/0038-5670/30/10/M11;jsessionid=86AE489FDBD558BBF96612B7300CF9ED.c3) and [here.](http://link.springer.com/article/10.1134/S0015... | 17 | https://mathoverflow.net/users/11260 | 144468 | 78,360 |
https://mathoverflow.net/questions/144419 | 15 | The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker ([1](http://prl.aps.org/abstract/PRL/v52/i18/p1575_1)), but the mathematicians found their proof insufficient, so that, FQS ([2](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-107/issue-4/Details-of-t... | https://mathoverflow.net/users/34538 | Is this error in this paper of Langlands fixable? | This morning I just discovered [these corrigenda](https://web.archive.org/web/20210506191547/http://math.univ-lyon1.fr/%7Eiohara/Corr/Corr-Chap11.pdf) of K. Iohara and Y. Koga (in which my name is cited in acknowledgement). In fact, three years ago, I have contacted K. Iohara (author, with Y. Koga, of the book *[Repres... | 8 | https://mathoverflow.net/users/34538 | 144472 | 78,361 |
https://mathoverflow.net/questions/144017 | 4 | What methods are there to show symmetry properties of the minimizer of a problem $\inf\_{u\in X}\mathcal{F}(u)$ in the calculus of variations? In general, the symmetry properties of $\mathcal{F}$ do not imply the symmetry of its minimizers. However, the minimizer may possess some weaker symmetry.
More precisely, I am ... | https://mathoverflow.net/users/19874 | Symmetry Properties of Minimizers - Calculus of Variations | Suppose that $\newcommand{\eF}{\mathscr{F}}$ $\newcommand{\bR}{\mathbb{R}}$ $\eF: C\to \bR$ is a convex functional defined on a closed convex subset $C$ of a say real Banach space $U$. (You can allow for more general topological spaces.) Suppose additionally that a compact Lie group $G$ acts on $C$. For any $u\in U$ we... | 2 | https://mathoverflow.net/users/20302 | 144486 | 78,362 |
https://mathoverflow.net/questions/144470 | 2 | We have this Mumford's theorem:
Let $X$ be a Riemann surface of genus $g$, $G$ a simple Lie group. We can consider a principal stable $G$-bundles over $X$ (say $\xi$), where $rk(\xi)=r$ and $deg(\xi)=d$. For fixed $X$, $r$ , $d$, there exists a connected moduli space $M^g$ of $S$-equivalence
classes of rank $r$ , degre... | https://mathoverflow.net/users/30618 | Moduli space of stable principal $G$-bundles | Yes. The standard reference is : A. Ramanathan: Moduli for principal bundles over algebraic curves, I and II, Proc. Indian
Acad. Sci. Math. Sci. 106 (1996) 301–328 and 421–449. The dimension of the moduli space is $(g-1)\dim G$ (note that the moduli space you seem to consider is that of $GL(r,\mathbb{C})$-bundles, and ... | 3 | https://mathoverflow.net/users/40297 | 144492 | 78,365 |
https://mathoverflow.net/questions/144493 | 6 | Let $M$ be an abelian group and let $R = \mbox{End}\_\Bbb{Z}(M)$. Under what conditions (on $M$), $\_RM$ is injective!?
| https://mathoverflow.net/users/nan | Abelian groups injective over their endomorphism | It seems that such modules where classified by Fred Richman and Elbert A. Walker, in *Modules over PIDs that are injective over their endomorphism rings*, Ring Theory (Proc. Conf., Park City, Utah, 1971), p.363-372, Academic Press, N.-Y., 1972.
The paper is available on [E. A. Walker's website](http://www.math.nmsu.e... | 7 | https://mathoverflow.net/users/39640 | 144495 | 78,366 |
https://mathoverflow.net/questions/144485 | 0 | I want an example to show that if $a,b$ are nilpotent elements of a ring $R$ with 1 and if $c$ is any element of $R$, then $abc=0\Rightarrow acb=0$ but $cab=0$ does not imply $acb=0$.
This is unlike symmetric ring, where we know that if $a,b,c\in R$ and $abc=0$ implies that $acb=0$.
Please help me to find a ring w... | https://mathoverflow.net/users/41137 | Example of a ring satisfying this variant definition of "symmetric" on nilpotent elements | If I understand correctly, there are two conditions on the ring $R$:
(1) if $a$ and $b$ are nilpotent elements of $R$, and if $c$ is arbitrary, then $abc=0$ implies $acb=0$;
(2) if $a$ and $b$ are nilpotent elements of $R$, and if $c$ is arbitrary, then $cab=0$ implies $acb=0$.
There exists a ring for which cond... | 6 | https://mathoverflow.net/users/39640 | 144512 | 78,369 |
https://mathoverflow.net/questions/144505 | 5 | I have the following question.
Suppose I have a simplicial set. Is there a way to detect if it actually is isomorphic to a nerve of a groupoid?
I've seen the fact that if you have a nerve $\mathcal{N}$ of a groupoid $\mathcal{G}$ then all homotopy groups $\pi\_n(\mathcal{N})$ for $n\geq 2$ must vanish. Is the conve... | https://mathoverflow.net/users/32741 | How to detect if a simplicial set is the nerve of a groupoid? | The answer is sort of well known. A simplicial set $X$ is the nerve of a groupoid if and only if any $n$-horn has a unique filler for all $n\geq 2$. The horn $\Lambda^k[n]$ is the simplicial subset of $\Delta[n]$ obtained by removing the non-degenerate $n$-simplex of $\Delta[n]$ and it's $k^{\text{th}}$ face. The previ... | 12 | https://mathoverflow.net/users/12166 | 144513 | 78,370 |
https://mathoverflow.net/questions/144365 | 2 | If we have a Laplacian matrix $\boldsymbol{A}$ such that
\begin{align}
&A\_{ii} >0 \\
&A\_{ii}=-\sum\_{j\neq i}A\_{ij}
\end{align}
with known eigenvalues $\lambda\_i$.
Define the matrix $\boldsymbol{B} = \boldsymbol{A}+\boldsymbol{A}^\top$. Is there some criteria for entries of $\boldsymbol{A}$ to ensure that $\bold... | https://mathoverflow.net/users/41064 | eigenvalue of Laplacian matrix | Let $z$ be the vector with all entries equal to 1. If $Az=0$ and $B=A+A^T$, then
$z^TBz=0$. Choose an orthogonal basis for $\mathbb{R}^n$ with $z$ as its first vector. If $C$ is the matrix representing $B$ relative to this basis, then $C\_{1,1}=0$. Also $B$ is positive semidefiniteif and only if $C$ is.
If $C$ is pos... | 1 | https://mathoverflow.net/users/1266 | 144514 | 78,371 |
https://mathoverflow.net/questions/144069 | 6 | I'll be working over the complex numbers.
Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down references for some facts about $K$-orbits on the flag variety $G/B$.
**Added later:** to clarify, $G$ is a complex algebraic... | https://mathoverflow.net/users/23907 | Topological properties of $K$ orbits in $G/B$ | 1. If $\theta$ is an involution of a complex linear algebraic group $G$ and if $K=G^\theta$ is its fixed-point set, then $K/K^0$ will always have exponent 2. This follows from a generalized "Cartan decomposition". The argument goes as follows. Let $\mathfrak p$ denote the (-1)-eigenspace of $d\theta$ on $\mathfrak g$. ... | 3 | https://mathoverflow.net/users/430 | 144517 | 78,374 |
https://mathoverflow.net/questions/144506 | 3 | The well known [Lebesgue's condition of Riemann integrability](http://en.wikipedia.org/wiki/Riemann_integral) says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost everywhere.
I vaguely remember that the same should be true for functio... | https://mathoverflow.net/users/16183 | Lebesgue's integrability condition in several variables | This fact is found in T. Apostol, *Mathematical Analysis*, section 10-5.
The proof is straightforward: if the set of points of discontinuity has positive measure, then the upper and lower sums are far apart. I wanted to include a reference for this in something I was writing (*Classics on Fractals*), but had to ask ... | 3 | https://mathoverflow.net/users/454 | 144519 | 78,375 |
https://mathoverflow.net/questions/144508 | 4 | It is known that when $\mu$ is $\sigma$-finte measure, then $L^\infty(\mu)$ is $1$-injective.
But I want to know whether it is right for any $L^\infty$ spaces.
| https://mathoverflow.net/users/41129 | When $L^\infty$ is 1-injective | A useful sufficient condition is that $(X,\Sigma,\mu)$ is a [localizable](http://ncatlab.org/nlab/show/localizable%20measure) measure space. See 363R of [Fremlin](http://www.essex.ac.uk/maths/people/fremlin/mt.htm), Volume 3, p.308.
| 6 | https://mathoverflow.net/users/nan | 144521 | 78,376 |
https://mathoverflow.net/questions/144507 | 5 | I have the following transform:
$$F(y) = \int\_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I\_0\left(xy\right)f(x)\;\mathrm{d}x$$
with the following conditions:
* **$f(x)$ and $F(y)$ must be real and positive** (they are continuous probability distributions)
* **$x,y$ must be real and positive** (the... | https://mathoverflow.net/users/41154 | Is this inverted integral transform valid? | define $G(y)=y^{-1}\exp(y^2/2)(Tf)(y)$ and $g(x)=x^{-1}\exp(-x^2/2)f(x)$, then you seek the solution to the integral equation
$$G(iq)=\int\_{0}^{\infty}g(x)J\_{0}(xq)xdx$$
This is a [Hankel transform.](http://en.wikipedia.org/wiki/Hankel_transform) The inverse is
$$g(x)=\int\_{0}^{\infty}G(iq)J\_{0}(xq)qdq$$
| 6 | https://mathoverflow.net/users/11260 | 144523 | 78,378 |
https://mathoverflow.net/questions/144529 | 27 | This question arises from an issue in my post on Ashutosh's excellent question on [Restrictions of the null/meager ideal](https://mathoverflow.net/q/130222/1946).
**Question 1.** Does every set of reals contain a measure-zero subset
of the same cardinality?
In other words, if $A\subset\mathbb{R}$, is there a meas... | https://mathoverflow.net/users/1946 | Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality? | It is consistent that both of the questions have a negative answer. Indeed, this happens if MA holds.
A set $E$ of reals is called a Luzin set if $E$ has size continuum and for every meager set $X$ the intersection $E\cap X$ has size less than continuum.
A set of reals $E$ is called a Sierpiński set if $E$ has size... | 29 | https://mathoverflow.net/users/1058 | 144538 | 78,382 |
https://mathoverflow.net/questions/144533 | 6 | I am seeing the proof of the Ax-Groethendieck theorem from commutative algebra and I have a problem. How can I prove that if $x\_1,...,x\_n$ are complex numbers and $I$ is a maximal ideal of $\mathbb{Z}[x\_1,...,x\_n]$, the quotient $\mathbb{Z}[x\_1,...,x\_n]/I$ is a finite field?
Thanks.
[Infinite fields, finite f... | https://mathoverflow.net/users/41166 | Quotient of $Z[x_1,...,x_n]$ by a maximal ideal is a finite field | Let $R$ be a finitely generated integral domain (over $\mathbb Z$) and let $I$ be a maximal ideal of $R$. We show $R/I$ is a finite field. Let $K$ be an algebraic closure of $R/I$. Let $p$ be the characteristic of $K$. Suppose $n$ elements generate $R$. Then we can write $R/I= \mathbb Z[x\_1,\ldots x\_n]/(f\_1,\ldots, ... | 5 | https://mathoverflow.net/users/15934 | 144542 | 78,384 |
https://mathoverflow.net/questions/144518 | 13 | The Kontsevich integral is known to be a universal Vassiliev invariant.
It is still an open question whether it is a complete knot invariant, *i.e.* whether it distinguishes a given knot from all other knots up to ambient isotopy and mirror image.
I am curious to know if there are some references giving the state of th... | https://mathoverflow.net/users/36625 | Kontsevich integral : state of the art | I don't think that there has been a tremendous amount of progress in understanding the Kontsevich Invariant of a knot in the last decade or so. It appears that essential new ideas may be needed in order to answer the fundamental questions.
> [Quantum Invariants: A study of knots, $3$-manifolds, and their sets](http... | 10 | https://mathoverflow.net/users/2051 | 144549 | 78,387 |
https://mathoverflow.net/questions/144526 | 1 | Let $\mathbb{E}\rightarrow\overline{M}\_{g,n}$ be the Hodge bundle. Let us cosider an $F$-curve of type $\overline{M}\_{1,1}\subseteq\overline{M}\_{g,n}$. **Is the degree of the restriction of $\mathbb{E}$ on such a curve negative?**
| https://mathoverflow.net/users/14514 | Hodge bundle on F-curves | No. It's easy to compute that the restriction of $\mathbb E\_g$ to $\overline M\_{1,1}$ is an extension of the Hodge bundle $\mathbb E\_1$ on $\overline M\_{1,1}$ and $g-1$ copies of the trivial line bundle. The Hodge bundle on $\overline M\_{1,1}$ has degree $1/24$.
More generally, it is actually true that $\mathbb... | 1 | https://mathoverflow.net/users/1310 | 144553 | 78,389 |
https://mathoverflow.net/questions/144562 | 0 | Consider the most classical form of the Mean Value Theorem: given a positive continuous function $f\in C([0,2])$ and a continuous function $g\in C([0,2])$, there exist $c\in(\frac{1}{2},\frac{3}{2})$ such that
$$
\int\_{\frac{1}{2}}^{\frac{3}{2}} f(t)g(t) dt =
g(c)\int\_{\frac{1}{2}}^{\frac{3}{2}} f(t) dt.
$$
Of cours... | https://mathoverflow.net/users/40120 | Regularity for the Mean Value Theorem | No. Take $f \equiv 1$ and define $g(x)$ as follows:
$$ g(x) = \begin{cases} 100 x & x \leq 1/2 \\
50 - (2x - 1) & 1/2 < x \leq 1 \\
49 + (2x - 2) & 1 < x< \leq 3/2 \\
50 - 50(2x - 3) \end{cases} $$
Since $g$ is piecewise linear and continuous, it is Lipschitz continuous.
We compute
$$ \int\_{1/2}^{3/2} g(t) \... | 1 | https://mathoverflow.net/users/3948 | 144565 | 78,392 |
https://mathoverflow.net/questions/70575 | 41 | For those who aren't familiar with the [Virasoro](http://en.wikipedia.org/wiki/Virasoro_algebra) or [Temperley-Lieb](http://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra) algebras, I include some definitions:
• The (universal envelopping algebra of the) **Virasoro algebra** is the $\star$-algebra $Vir\_c$ gene... | https://mathoverflow.net/users/5690 | Why is there such a close resemblance between the unitary representation theory of the Virasoro algebra and that of the Temperley-Lieb algebra? | **Overview of an explanation** :
**Jones-Wassermann subfactors for the loop algebra** :
Let $\mathfrak{g} = \mathfrak{sl}\_{2}$ be the Lie algebra, $L\mathfrak{g}$ its loop algebra and $\mathcal{L}\mathfrak{g} = L\mathfrak{g} \oplus \mathbb{C}\mathcal{L}$ the central extension :
$$[X^{a}\_{n},X^{b}\_{m}] = [X... | 16 | https://mathoverflow.net/users/34538 | 144572 | 78,395 |
https://mathoverflow.net/questions/144531 | 4 | Conjecturally, any two symplectic resolutions are related by sequences of Mukai flops. Consider different symplectic resolutions coming from Hamiltonian reductions with different stability conditions. For example, two Nakajima quiver varieties $\mathcal{M}\_{\theta\_1,\zeta}(v,w)$ and $\mathcal{M}\_{\theta\_2,\zeta}(v,... | https://mathoverflow.net/users/9096 | Nakajima quiver varieties with different stability conditions related by sequence of Mukai-flops? | Actually, Namikawa recently [proved the conjecture](https://arxiv.org/abs/1305.1698) you're referring to (at least for symplectic cones with a "good" $C^\*$ action), by establishing that every such symplectic resolution is a Mori dream space. For quiver varieties, I think one doesn't need to use Namikawa's results and ... | 2 | https://mathoverflow.net/users/66 | 144575 | 78,396 |
https://mathoverflow.net/questions/144541 | 4 | This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems.
I would like to know whether there exists a differentiable function from the (open or closed) unit interval to itself satisfying
$$1-x-f(f(x))-f(x)f'(f(x))=0$$
for all $x$.
Ideally, I'd also like a... | https://mathoverflow.net/users/10503 | A Differential Equation with Nested Functions | Write $$1-x=f(f(x))+f(x)f'(f(x)).$$
Note, that $f(x\_1)=f(x\_2)$ implies $x\_1=x\_2$ and thus the function is injective on $(0,1).$ Therefore, it is strictly monotone. This implies that $f(f(x))$ is strictly increasing.
If $f$ is increasing, then $f'(f(x))\ge 0$ and thus
$1-x\ge f(f(x)).$ Letting $x\to 1$ leads to a... | 6 | https://mathoverflow.net/users/17503 | 144578 | 78,397 |
https://mathoverflow.net/questions/144567 | 8 | Fix a homomorphism $f:A\rightarrow B$.
Choose $\{b\_1,\dots,b\_n\}$, $\{b'\_1,\dots,b'\_m\}$ subsets of elements in $B$. Suppose that $B$ is algebraic over $f(A)[b\_1,\dots,b\_n]$ and $\{b\_1,\dots,b\_n\}$ are algebraically independent over $f(A)$. Suppose that $\{b'\_1,\dots,b'\_m\}$ satisfies the same condition. Does... | https://mathoverflow.net/users/nan | Can transcedence degree be defined for arbitrary ring homomorphism? | The problem is actually the following. Let $A\subset B$ be a subring and suppose that the subrings $C:=A[b\_1,\dots,b\_n]$ and $C':=A[b'\_1,\dots,b'\_m]$ of $B$ are the rings of polynomials in the indicated variables such that $B$ is algebraic over both. Does it follow $m=n$ ?
If "$B$ is algebraic over $C$" means
$\f... | 10 | https://mathoverflow.net/users/40352 | 144580 | 78,399 |
https://mathoverflow.net/questions/144563 | 5 | Under the condition f continuous, integrable and:
$|f(t)| + |\hat{f}(t)| \le C (1+|t|)^{-1-a}$ (with a>0)
we have the twisted Poisson formula that holds (where $\chi(n)$ is a primitive Dirichlet character):
$\sum\_{-\infty}^{\infty} \chi(n) f(\frac{nx}{\sqrt{q}}) = \frac{A}{x} \sum\_{-\infty}^{\infty} \overline{\... | https://mathoverflow.net/users/38290 | Extension of Poisson Summation formula | The correct formula is, for $\chi$ primitive of conductor $q$,
$$ \sum\_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
\frac{A}{x}\sum\_{n\in\mathbb{Z}}\bar\chi(n)\hat f\left(\frac{n/x}{\sqrt{q}}\right). $$
Here $A:=\sqrt{q}/\tau(\bar\chi)$ is the so-called root number, it is of modulus $1$.
This formula... | 9 | https://mathoverflow.net/users/11919 | 144586 | 78,401 |
https://mathoverflow.net/questions/144593 | 4 | One of the main properties of the Laplace transform is given by the convolution theorem.
$$\mathcal{L}(f\*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$
**Question:** Is there a full characterization of the Laplace transform based on this property? I have in mind a theorem that reads: "Let $\mathcal{N}$ be an operator with the... | https://mathoverflow.net/users/18474 | Characterization of the Laplace Transform | This operational approach to a definition of Fourier/Laplace transforms has been developed by R.A. Kunze, [An operator theoretic approach to generalized Fourier transforms](http://www.jstor.org/stable/1970090).
>
> This is an attempt to give an intrinsic definition generalizing the
> conditions under which a pair ... | 4 | https://mathoverflow.net/users/11260 | 144598 | 78,406 |
https://mathoverflow.net/questions/144594 | 1 | Let $f:X\rightarrow Y$ be a finite morphism between normal varieties. Let $E$ be a vector bundle on $X$ and let us consider its pushforwad $f\_{\*}E$.
**Does anyone know an example where $E$ is nef but $f\_{\*}E$ is not nef ?**
| https://mathoverflow.net/users/14514 | Push-forward of a nef bundle | Take $f:X\rightarrow Y$ a double covering of smooth varieties, branched along an ample divisor $D$. Then there is a line bundle $L$ such that $D$ is the zero divisor of a section of $L^2$, and $f\_\*\mathcal{O}\_X=\mathcal{O}\_X\oplus L^{-1}$, which is certainly not nef.
| 5 | https://mathoverflow.net/users/40297 | 144599 | 78,407 |
https://mathoverflow.net/questions/143372 | 0 | Consider the family of distributions having the form $$f = \frac{1}{Z({\alpha,\beta,A,B})}\prod\_{j=1}^{n}\left(\frac{1}{1+e^{-x+A\_{j}}}\right)^{\alpha\_{j}}\prod\_{j=1}^{m}\left(\frac{1}{1+e^{x+B\_{j}}}\right)^{\beta\_{j}}$$
$Z$ is the normalizing constant. Typically $n$ and $m$ are both around 3. The $\alpha\_j$ a... | https://mathoverflow.net/users/8737 | Computing a particular expection for a family of distribution | With the change of variable $y = 1/(1+e^{-x})$, the integral is a special case of the Lauricella hypergeometric series
<http://en.wikipedia.org/wiki/Lauricella_hypergeometric_series#Integral_representation_of_FD>
| 0 | https://mathoverflow.net/users/8737 | 144611 | 78,409 |
https://mathoverflow.net/questions/144615 | 2 | Let $\Omega$ be a bounded region in $R^n$ and define
$W:=\{ u \in H^{1}(\Omega): u(x\_0)=0 \},$
where $x\_0 \in \partial \Omega$ is a fixed point. Is there a constant $C$ such that
$||u||\_{L^2(\Omega)} \leq C ||\nabla u||\_{L^2(\Omega)}$,
for all $u\in W$?
| https://mathoverflow.net/users/41214 | Sobolev Inequality | This inequality cannot hold. Here is a counterexample:
Let $\phi\_{\epsilon}$ be a bump function supported on $B\_{\epsilon}(0) \subset \mathbb{R}^n$ with value $1$ at $0$ and satisfying $|\nabla \phi\_{\epsilon}| < \frac{2}{\epsilon}$. Let $u\_{\epsilon} = 1-\phi(x-e\_n)$, the function which is 1 everywhere but dips... | 6 | https://mathoverflow.net/users/16659 | 144617 | 78,411 |
https://mathoverflow.net/questions/144603 | 16 | It is well known (see e.g. K. Brown, "Cohomology of groups") that a degree-3 cohomology class of a group G with coefficients in a module A can be thought of as an equivalence class of crossed modules, in the sense that every crossed module gives a double extension of G by A. Now, it is also well known that degree-3 coh... | https://mathoverflow.net/users/26536 | Crossed modules and degree-3 group cohomology | Here is a general procedure to construct a crossed module $\mathcal{G} = (G\_1 \stackrel{d}{\to} G\_0)$ from a normalized 3-cocycle $\alpha \in Z^3(G; \mathbb{R}/\mathbb{Z})$ which is additive in the third variable.
First, we get a cocycle $\omega \in Z^2(G; G^{\ast})$, where $G^{\ast}$ is the Pontryagin dual of $G$ ... | 15 | https://mathoverflow.net/users/396 | 144620 | 78,412 |
https://mathoverflow.net/questions/143775 | 10 | Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation
$f^\*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the usual topology of of uniform convergence on compact subsets of all partial derivatives.
It tuns out that sometimes $f... | https://mathoverflow.net/users/16183 | Pull-back of generalized functions | Recently I found a counterexample to a somewhat stronger version of the question I originally asked, but which I actually needed. Namely for a smooth map $f\colon X\to Y$ which is transversal to $\Lambda\subset T^\*Y\backslash 0$ (see e.g. Hormander's book for the definitions) the map $f^\*\colon C^{-\infty}\_{\Lambda}... | 1 | https://mathoverflow.net/users/16183 | 144632 | 78,415 |
https://mathoverflow.net/questions/133434 | 9 | This question is specifically related to the spectra $X(n)$ used in Devinatz, Hopkins and Smith's proof of the nilpotence conjectures, but any general answer in terms of the Thom isomorphism would also be appreciated.
The spectra $X(n)$ are Thom spectra coming from the maps $\Omega SU(n)\to\Omega SU\simeq BU\to BF$. ... | https://mathoverflow.net/users/11546 | Thom isomorphism's effect on module structure of n-oriented spectra | Okay, so maybe I'm being naive here, but I think this is probably a naive question. The answer might be a little complicated by the fact that I don't have any good way to drawcommutative diagrams on here. Also, this is all pretty much lifted directly from Mahowald's [paper](http://projecteuclid.org/DPubS?service=UI&ve... | 1 | https://mathoverflow.net/users/11546 | 144634 | 78,416 |
https://mathoverflow.net/questions/144627 | 0 | All
I have been looking around for a general way to solve the problem of $f(x+1) - f(x) = g(x)$, where $g(x)$ is given. Has this problem been studied before?
If there does not exist such a general way, could you please solve the following problem for me? I urgently want to know what $f(x)$ is.
$$ f(x+1) - f(x) = ... | https://mathoverflow.net/users/41220 | Generic way to solve f(x+1) - f(x) = g(x) when g(x) is given | Depending on the growth of $g$ at $\infty$ (if defined till there) you can try telescoping sums to obtain
$f(x+n)-f(x) = \sum\_{k=0}^{n-1} g(x+k)$
If the series in the right hand side converges as $k\to \infty$ then you obtain a solution $f\_0$ vanishing at $\infty$. This is so in your example (assuming your parame... | 2 | https://mathoverflow.net/users/24309 | 144636 | 78,418 |
https://mathoverflow.net/questions/144614 | 4 | Suppose we are given a smooth function $f\colon \mathbb{R}^n \rightarrow \mathbb{R}$ and some number $c$. What can be said about the preimage $f^{-1}(c)$.
There's the theorem on regular preimages, asserting that if $\nabla f$ is nowhere vanishing on $M=f^{-1}(c),$
then $M$ is a smooth submanifold of dimension $n-1$. ... | https://mathoverflow.net/users/35946 | Preimage of a smooth function | I do not mean to spoil the exercise, but note that if $\phi$ is any smooth function on a Banach space $E$, with bounded derivatives of any order (say $\|D^j\phi\|\_\infty < \infty$ for any $j\ge0$), then $f(x):=\sum\_{k=1}^\infty d\_k^k \phi\big(\frac{x-a\_k}{d\_k} \big)$ certainly defines a smooth function for any cho... | 5 | https://mathoverflow.net/users/6101 | 144647 | 78,421 |
https://mathoverflow.net/questions/144650 | 3 | Let $(C, O)$ be a ringed site -- i.e., $C$ is a small category with a grothendieck topology $\tau$ and $O$ a sheaf of rings on the site $(C,\tau)$. In this context, for any object $U$ of $C$ one can define the sheaf cohomology groups $\mathrm{H}^{\bullet}(U,O)$ and the cech cohomology groups $\check{\mathrm{H}}^{\bulle... | https://mathoverflow.net/users/5031 | 1st cech cohomology groups on ringed sites | First things first: $\check{H}{}^n(U, \mathscr{F})$ (resp. $H^n(U, \mathscr{F})$) are same whether you regard $\mathscr{F}$ as an $\mathscr{O}$-module or as an abelian sheaf, so we may simplify things by considering only abelian sheaves.
Let $\mathscr{F}$ be an abelian sheaf and write $\check{H}{}^\* (U, -)$ for the ... | 3 | https://mathoverflow.net/users/11640 | 144657 | 78,425 |
https://mathoverflow.net/questions/144643 | 5 | On page 4 of this [paper](http://arxiv.org/pdf/1302.6534.pdf) by H. Abbaspour, the author defines the **two-sided bar construction**
$$B(A,A,A):=A\otimes T(s\bar{A})\otimes A$$
of a differential graded algebra $(A,d\_A)$ (over a field).
The definition of the differential $d=d\_0+d\_1$ on $B(A,A,A)$ is unclear to me. Wh... | https://mathoverflow.net/users/29827 | Two-sided bar construction | I haven't looked at Abbaspour's paper, but here is what is going on,
in a bit greater generality. Let $N$ be a right, $M$ a left DG
$A$-module. Then $B=B(N,A,M)$ is defined and it is bigraded. The
grading with differentials that raise degree, which you apparently
have in mind, is a bit awkward, so regrade by $A\_n = ... | 6 | https://mathoverflow.net/users/14447 | 144659 | 78,427 |
https://mathoverflow.net/questions/144654 | 0 | The question is the title. For example, if we could show that $S$ is finite, then this would entail that every large enough integer $n$ is such that $\zeta(2n+1)$ is irrational and that, under RH, almost all non-trivial zeros of $\zeta$ have irrational imaginary part. Has this question been studied? Any reference?
T... | https://mathoverflow.net/users/13625 | What is known about the set $S$ of couples of rationals $(q,q')$ such that $\zeta(q+iq')$ is rational? | This set includes the non-positive integers (including the trivial zeros)
so the set is infinite.
I would be interested in another example.
| 3 | https://mathoverflow.net/users/12481 | 144664 | 78,429 |
https://mathoverflow.net/questions/144269 | 7 | Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim\_{n \to \infty} \tilde{f}^n(0)/n) + \mathbb{Z}$, where $\tilde{f}: \mathbb{R} \to \mathbb{R}$ is any lift of $f$. Intuitively, it measures the rate of circulation a... | https://mathoverflow.net/users/24792 | Rotation numbers for amenable group actions on the circle | The answer to my question, if $G$ is assumed to be a finitely generated torsion-free nilpotent group, is that *any* homomorphism $\psi: G \to \mathbb{R}/\mathbb{Z}$ can be realized as $\rho \circ \phi$ for $\phi: G \to Homeo\_+(S^1)$ 1-1. I'm sure this was already known, and the paper referenced by Dan Sălăjan definite... | 2 | https://mathoverflow.net/users/24792 | 144676 | 78,435 |
https://mathoverflow.net/questions/144544 | 13 | Suppose I have a form $$ f(x,y) = a x^2 + b x y + c y^2, $$ with $a,b,c$ integers, $\gcd(a,b,c)=1$ and $\Delta = b^2 - 4 a c > 0,$ but $\Delta \neq n^2$ for any integer $n.$
Do there exist (positive) primes $p,q$ such that $f$ integrally represents $p$ and $-q?$
I have most books on quadratic forms of which I've ev... | https://mathoverflow.net/users/3324 | primes represented by an indefinite binary quadratic form | Meyer (Über einen Satz von Dirichlet, Crelle 103 (1888)) proved that a primitive
binary quadratic form with nonsquare discriminant represents infinitely many primes that
lie in any given compatible residue class modulo a given integer N. He did so only for forms with even middle coefficient, and did not address the sig... | 8 | https://mathoverflow.net/users/3503 | 144677 | 78,436 |
https://mathoverflow.net/questions/141725 | 6 | Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence of holomorphic connections on $E$. We can refer to M, Kapranov's paper "Rozansky–Witten invariants via Atiyah classes" or... | https://mathoverflow.net/users/24965 | Is there a "by hand" proof on the symmetry of the Atiyah class of $TX$? | I'm not sure this answers your question since I think that "by hands" might have different meanings, but it seems to me that what you are asking for is precisely the content of Proposition 2.2 on page 14 of the paper of Roberts-Willerton you mentioned.
Let me explain what they do. The Atiyah class of a vector bundle... | 7 | https://mathoverflow.net/users/7031 | 144687 | 78,443 |
https://mathoverflow.net/questions/144689 | 12 | The product topology is the categorical product, and the disjoint union topology is the categorical coproduct. But the arrows in the characteristic diagrams for the subspace and quotient topologies point the same way as in the diagrams for the product and disjoint union topologies, respectively (but there are different... | https://mathoverflow.net/users/29961 | Categorical Construction of Quotient Topology? | Inclusions of subspaces are precisely the regular monomorphisms, and projections of quotients are precisely the regular epimorphisms.
| 24 | https://mathoverflow.net/users/2841 | 144692 | 78,445 |
https://mathoverflow.net/questions/144649 | 3 | If $X$ is a surface, projective and non-singular. Let $\mathbb{C}(X)$ be the function field of $X$. By a theorem of Siegel, we know that $trdeg\_{\mathbb{C}}\mathbb{C}(X)\leq 2$. But how to argue that $trdeg\_{\mathbb{C}}\mathbb{C}(X)= 2$？
| https://mathoverflow.net/users/40042 | How to determine the transcendence degree of function field | OK, make it an answer (which is just a piece of basic algebraic geometry). We do not assume that $X$ is smooth, just projective and irreducible. So, $X\subset{\mathbb P}\_{\mathbb C}^n$. If $n=2$, we a done as
$X={\mathbb P}\_{\mathbb C}^2$ and
$\text{tr}\,\text{deg}{\mathbb C}({\mathbb P}\_{\mathbb C}^2)=2$. Otherwise... | 3 | https://mathoverflow.net/users/40352 | 144716 | 78,450 |
https://mathoverflow.net/questions/144719 | 3 | The usual use of forcing begins with a "countable" and "transitive" ground model of $ZFC$ and reaches to a "countable" and "transitive" generic model of $ZFC$ with the "same ordinals". In a discussion a colleague told me about a special forcing by Solovay which begins with a countable transitive model of $ZFC$ and reac... | https://mathoverflow.net/users/nan | A Question on Special Forcings | I don't agree that the "usual" use of forcing uses only countable transitive models. Perhaps this used to be true, years ago, and forcing is sometimes still taught this way now, because it is somewhat easier to see where the generic filters come from, but neither hypothesis is actually needed to develop a full theory o... | 9 | https://mathoverflow.net/users/1946 | 144722 | 78,453 |
https://mathoverflow.net/questions/144715 | 1 | Let $X\_n$ be a random variable distributed on $A\_n:=\{1, \ldots, n\}$ and $g\_n\colon A\_n \to A\_n$ such that $\Pr\big(X\_n \neq g\_n(X\_n)\big) \to 0$. Putting $Y\_n=g\_n(X\_n)$, then by Fano's inequality $$\frac{H(X\_n\mid Y\_n)}{\log n} \to 0,$$ which can be written $$\frac{H(X\_n\mid Y\_n)}{H(X\_n)} \to 0 \qquad... | https://mathoverflow.net/users/21339 | order of convergence of the conditional entropy (2) | I think there's a similar counterexample to the previous one. Let $X\_n$ take the value 0 with probability $1-1/n$ and any value in $\{1,\ldots,N\_n\}$ with probability $1/(nN\_n)$.
Set $Y\_n=\min(X\_n,1)$. Now $H(X\_n|Y\_n)=(1/n)\log N\_n$ (the additional information in $X\_n$ comes if $Y\_n=1$ which has probability ... | 2 | https://mathoverflow.net/users/11054 | 144732 | 78,457 |
https://mathoverflow.net/questions/144727 | 4 | I am trying to learn something on the Brieskorn manifold (interested in the topological property)
Can the Mathoverflow Experts give me some good refencece (in English)?
By the way,is there an English translation of the following paper by Brieskorn:
Beispiele zur Differentialtopologie von Singularitäten
----------... | https://mathoverflow.net/users/41110 | good reference on brieskorn manifold | I would add Milnor's book ***"Singular points of complex hypersufaces"*** Ann of Math Studies, No. 61, Princeton University Press, 1968.
| 4 | https://mathoverflow.net/users/20302 | 144735 | 78,459 |
https://mathoverflow.net/questions/144701 | 2 | Does there exist a quasi-isometric embedding
$$MCG(S) \to (\mathrm{Teich}(S), d)$$
for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?
| https://mathoverflow.net/users/41219 | Quasi-isometric embeddings of the mapping class group into the Teichmuller space | A result of [Behrstock and Minsky](http://annals.math.princeton.edu/2008/167-3/p09) (cf. Hamenstadt too) implies that the rank of mapping class groups is the maximal rank of abelian subgroups, which is $3g+p-3$ for a connected hyperbolic surface of genus $g$ with $p$ boundary components. The rank of Teichmuller space w... | 5 | https://mathoverflow.net/users/1345 | 144739 | 78,462 |
https://mathoverflow.net/questions/144740 | 4 | This is [crossposted](https://math.stackexchange.com/questions/522826/wedderburn-decomposition-of-d-5) from MSE. The question:
>
> Find the Wedderburn decomposition of $D\_{5},$ the dihedral group of order 10, over the field $\mathbb{F}\_{3}.$
>
>
>
I have shown that the irreducible representations of $D\_{5}$ ... | https://mathoverflow.net/users/35005 | Wedderburn decomposition of $D_{5}$ | In the following paper the authors deal with the Wedderburn decomposition of group algebras of finite metacyclic groups over a finite field:
G.K. Bakshi - S. Gupta - I.B. Passi: Semisimple metacyclic group algebras, Proc. Indian Acad. Sci., Math. Sci. 121, No. 4, 379-396 (2011).
It is available at this link:
<htt... | 7 | https://mathoverflow.net/users/14653 | 144743 | 78,464 |
https://mathoverflow.net/questions/144733 | -2 | You have a block with a width of 3, depth of 3 and a height n
Given n, in how many ways can you fill this block with smaller blocks of 2 x 1 x 1?
if n is uneven, one 1x1x1 block will be unused. This is allowed.
| https://mathoverflow.net/users/41084 | Combination of 2 × 1 × 1 cubes inside a 3 × 3 × n cube | For simplicity, I'll assume $n$ is even. Each horizontal slice of the $3 \times 3 \times n$ block can have a finite number of "states". Each state consists of a packing of the $3 \times 3$ square with $2 \times 1$ blocks (in either orientation) and $1 \times 1$ blocks, where the $1 \times 1$ blocks are labelled either ... | 4 | https://mathoverflow.net/users/13650 | 144744 | 78,465 |
https://mathoverflow.net/questions/129447 | 6 | I want to study gradient Ricci solitons. Can anyone help me to find a simple and good reference about solitons and their applications?
Thanks
| https://mathoverflow.net/users/32817 | A simple and good reference about solitons | *I've combined two answers into one.*
**First answer: some results on gradient Ricci solitons including some history.** I like very much the answer Otis Chodosh gave. To this I would add the following:
Besides the physics literature (e.g., Friedan), the notion of gradient Ricci soliton first appeared in Hamilton's Ri... | 11 | https://mathoverflow.net/users/nan | 144751 | 78,470 |
https://mathoverflow.net/questions/144697 | 1 | For Witt Lie Algebras over field of characterestic $p>3$ we know that $\operatorname{dim}W(n;m):=np^{|m|}$ , such that $|m|=m\_1+⋯+m\_n$ . I would like to know what is the dimension of Witt algebras over $\mathrm {GF}(2)$. Why $\operatorname{dim}W(2,1)= ?$
| https://mathoverflow.net/users/40491 | Witt Lie algebras | I think there is only a problem of notation here. The Jacobson-Witt algebra $W(m; \underline{n})$ is known to be simple of dimension $mp^{\vert n\vert}$ (where $\vert n \vert=n\_1+n\_2 \cdots + n\_m$) except when $m = 1$ and the ground field has characteristic $p=2$. In the latter case the derived subalgebra $W(1; \und... | 5 | https://mathoverflow.net/users/14653 | 144758 | 78,473 |
https://mathoverflow.net/questions/109042 | 15 | I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics.
I think I have a firm grip on elliptic PDE after going through the first part of Gilbarg and Trudinger + some Monge-Ampere stuff. But that conclu... | https://mathoverflow.net/users/17965 | Reference request: parabolic PDE | I have to kindly dissent from Deane Yang's recommendation of the books that I coauthored. The reason being that the question by The Common Crane is about basic references for parabolic PDE and he/she is interested in Kaehler--Ricci flow, where many cases can be reduced to a single complex Monge-Ampere equation, and hen... | 20 | https://mathoverflow.net/users/nan | 144761 | 78,476 |
https://mathoverflow.net/questions/144763 | 9 | I'am reading the paper *Elementary submodels in infinite combinatorics* by Soukup ([arXiv link](https://arxiv.org/abs/1007.4309)) and there are a lot of proofs using elementary submodels, such as the proof of $\Delta$-system lemma and partitions theorems. However, I don't take the intuition and I would like more exampl... | https://mathoverflow.net/users/41166 | Elementary submodels in partitions theorems | Complementing Andres's excellent answer, let me simply try to help build your intuition for elementary submodels.
The basic situation is just like the familiar fact that if you
have finitely many group elements $g\_0,\ldots,g\_n$ in a large
group $G$, then they generate a countable subgroup of $G$. One
simply starts ... | 10 | https://mathoverflow.net/users/1946 | 144766 | 78,478 |
https://mathoverflow.net/questions/144770 | 1 | M. Reid has a theorem as following:
Let $X$ be a non-singular projective complex surface, $L \subset \Omega^1\_X$ be a line bundle. If $h^0(L^{\otimes n})\geq2$ for some $n\geq 1$, then there is a moephism $f:X\rightarrow C$ of $X$ onto a nonsingular curve $C$ of positive genus, and a ramification divisor $D$ (of the... | https://mathoverflow.net/users/40042 | How to estimate the $h^0(L^{\otimes n})$ | (a) The ramification divisor $D$ is contained in some fibers of $f$, so we can write $L=f^\*M(-E)$, with $M$ a line bundle on $C$ and $E$ effective. Therefore $h^0(L^{{\scriptscriptstyle\otimes} n })$ is bounded by $h^0(f^\*M^{{\scriptscriptstyle\otimes} n })=h^0(M^{{\scriptscriptstyle\otimes} n })$ (projection formula... | 1 | https://mathoverflow.net/users/40297 | 144779 | 78,484 |
https://mathoverflow.net/questions/144794 | 2 | Assume we showed that, in a certain transitive model of set theory, we have an isomorphism between two structures $M\_1$ and $M\_2$. Does the same result still holds in the real world?
| https://mathoverflow.net/users/38200 | Transfer of results from one model of set theory to another | The papers "Forcing isomorphism" and "Forcing isomorphism II" might be relevant.
**Review for the first paper:** As the authors explain in their introduction, for any theory $T$ it is easy to find two non-isomorphic models of $T$ that become isomorphic via a forcing which collapses cardinals. By contrast, if $T$ is c... | 4 | https://mathoverflow.net/users/11115 | 144797 | 78,490 |
https://mathoverflow.net/questions/144799 | 1 | Assume we have two objects $M\_1$ and $M\_2$ models of respective $L\_{\omega\_1,\omega}$-sentences $\Sigma\_1$ and $\Sigma\_2$.
Assume $M\_1$ and $M\_2$ are elementarily equivalent in some model of set theory. Is this property (of being elementarily equivalent) absolute between models of set theory?
Edit: Is the f... | https://mathoverflow.net/users/38200 | Is elementary equivalence absolute? | When the structures $M\_1$ and $M\_2$ are countable and the language is countable and you only want to compare the elementary equivalence between models of set theory having the same countable ordinals, then your argument is correct. Satisfying a given formula is uniformly $\Delta\_1$, since you can quantify over the (... | 1 | https://mathoverflow.net/users/1946 | 144803 | 78,492 |
https://mathoverflow.net/questions/144785 | 13 | Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$.
**Question: Why do we need a $G$-universe?**
A $G$-universe is defined to be a countably infinite-dimensional (real) representation of $G$ with an inner product such that
1. $\mathcal U$ contains the tri... | https://mathoverflow.net/users/5206 | Why do we need a $G$-universe? | Tyler, you are too fast: didn't give me a chance to answer first!
Of course, I agree with everything you say. I wrote the following
before seeing your answer (except for the last paragraph).
Since I introduced this choice, let me explain. But first, echoing
André, taking equivalence classes would be a wrong choice
ev... | 14 | https://mathoverflow.net/users/14447 | 144809 | 78,495 |
https://mathoverflow.net/questions/144749 | 7 | Let's fix an algebraically closed field $k$ of arbitrary characteristic and a connected nonsingular affine algebraic $k$-group $G$. Under what conditions can I assume that a connected nonsingular affine algebraic subgroup $H \subseteq G$ is a complete intersection in $G$? Or are such subgroups always complete intersect... | https://mathoverflow.net/users/1528 | Smooth affine algebraic subgroups as complete intersections | This answer is just to record that Will Sawin's example works: The Borel subgroup of $PGL\_2$ is not a complete intersection in $PGL\_2$. Recall that the coordinate ring of $GL\_2$ is $k[w,x,y,z,\Delta^{-1}]$ where $\Delta=wz-xy$. We are thinking of $w$, $x$, $y$, $z$ as entries of the matrix $\left( \begin{smallmatrix... | 6 | https://mathoverflow.net/users/297 | 144810 | 78,496 |
https://mathoverflow.net/questions/138949 | 4 | Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? If so I might also ask if it is possible that an exact admissible sequence $X \to Y \to Z$ of bounded morphisms of $k$-Ba... | https://mathoverflow.net/users/3396 | Spherical completions and flatness | Let me try to give an answer. In fact, I think that the statements you want always hold, i.e. for any complete valued field extension $K/k$. Which makes wonder whether I have missed something obvious... Anyway, writing things up clearly should help us start a discussion, so I will give it a try.
As regards the first ... | 2 | https://mathoverflow.net/users/4069 | 144811 | 78,497 |
https://mathoverflow.net/questions/144793 | 4 | **Def.** A group language is a recognizable language whose syntactic monoid is a group.
**q1.** Is it known a "nice" combinatorial characterization of group languages ?
**q1.1.** If no, is it well understood why it is so? Is there recent progress on this ?
**q2.** Let F be a proper family of group languages (for ... | https://mathoverflow.net/users/16758 | Progress on group languages characterizations | Usually it is deemed impossible to describe group languages completely since this involves understanding the word problem for each finite simple group for any generating set.
Nice descriptions are known for subclasses like p-groups, nilpotent groups, supersolvable groups and solvable groups. p-groups are done in Eil... | 4 | https://mathoverflow.net/users/15934 | 144814 | 78,498 |
https://mathoverflow.net/questions/144777 | 8 | Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm.
Is it possible to show a non-continuous linear form on $E$ exists without using a basis, i.e. without AC?
| https://mathoverflow.net/users/41060 | Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC | No. It's impossible.
In certain models of $\sf ZF+DC$ there is a property known as "automatic continuity" for Banach spaces, that means that every linear operator to a normed space is continuous.
Such models are, for example, Solovay's model where all sets of reals are Lebesgue measurable, and have the Baire proper... | 19 | https://mathoverflow.net/users/7206 | 144820 | 78,502 |
https://mathoverflow.net/questions/144790 | 7 | Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient
$$
\binom{2k\_1+3k\_2+4k\_3+...}{k\_1+2k\_2+3k\_3+...,k\_1,k\_2,k\_3,...}
$$
is divisible by $k\_1+2k\_2+3k\_3+...+1$.
Denoting the quotient by $C(k\_1,k\_2,k\_3,...)$, one may call these the mu... | https://mathoverflow.net/users/41291 | "MultiCatalan numbers" | An early reference is [W. T. Tutte, The number of planted plane trees with a given partition. Amer. Math. Monthly 71 (1964) 272–277](http://www.jstor.org/stable/2312183).
| 12 | https://mathoverflow.net/users/10744 | 144821 | 78,503 |
https://mathoverflow.net/questions/141741 | 17 | Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is necessarily continuous. The conjecture is equivalent to the statement that every algebra norm on C(X) is equivalent to the usual un... | https://mathoverflow.net/users/11115 | Kaplansky's conjecture and Martin's axiom | The answer is yes.
Recall that if $S$ is a Suslin tree, then PFA($S$) denotes the forcing axiom for the class of all proper posets which preserve $S$. PFA($S$) is consistent with ZFC relative to a supercompact cardinal, like PFA, and shares many of the consequences of PFA, but a model of PFA($S$) naturally does not s... | 11 | https://mathoverflow.net/users/11233 | 144828 | 78,508 |
https://mathoverflow.net/questions/144830 | 4 | I have a random variable $x \in (0,\infty)$ with distribution $P(x)$ falling off slowly $P(x) \sim 1/x^3$ for large $x$. So the expectation value $\bar{x}$ is finite but the second moment $\bar{x^2}$ is divergent.
If I remember correctly this class of distributions also falls into the Gaussian universality class, i.e... | https://mathoverflow.net/users/41312 | Central limit theorem for $P(x)\sim 1/x^3$ distribution | Indeed: For $P(x)\propto 1/x^3$ the estimated $\bar{x}$ will deviate from the true mean by an amount that decays with increasing $N$ as $(\log N/N)^{1/2}$, so only slightly less rapidly than the $1/\sqrt N$ decay expected from the central limit theorem. See chapter 2, *Elementary introduction to the theory of stable la... | 6 | https://mathoverflow.net/users/11260 | 144832 | 78,511 |
https://mathoverflow.net/questions/144228 | 3 | Let $(\Omega, (\mathcal F\_t), \mathbb P)$ denote the usual Wiener space where $\Omega = C[0,\infty)$, etc., and where $(W\_t)\_{t \geq 0}$ denotes the Wiener process.
Let $Z \in L^1(\mathbb P)$ with $Z > 0$, $\mathbb P$-almost surely. Let $Z\_t = \mathbb E [ Z | \mathcal F\_t ]$ denote the density process, which is a ... | https://mathoverflow.net/users/22157 | Example of Girsanov change of density with finite relative entropy, but with infinite integral over squared changed drift | I can now answer my own question. There exists no such counterexample. I can show that finite relative entropy in this setting implies $\mathbb E^{\mathbb Q} \int\_0^{\infty} \theta\_t^2 \ d t < \infty$. This can be proven by a localization argument using stopping times $(\tau\_n)$, such that $Z^{\tau\_n} \geq \frac 1 ... | 2 | https://mathoverflow.net/users/22157 | 144841 | 78,516 |
https://mathoverflow.net/questions/144843 | 12 |
>
> What are good English-language sources for reading about the Luzin affair?
>
>
>
I'm interested in the subject and am wondering about good historical sources.
| https://mathoverflow.net/users/658 | What are good English-language sources for reading about the Luzin affair? | S. S. Kutateladze, Roots of Luzin’s case,
Journal of Applied and Industrial Mathematics, September 2007, Volume 1, Issue 3, pp 261-267, <http://dx.doi.org/10.1134/S1990478907030015>.
Abstract: This is a brief overview of the so-called “case of Academician Luzin” as well as the mathematical and humanitarian roots of t... | 11 | https://mathoverflow.net/users/nan | 144847 | 78,518 |
https://mathoverflow.net/questions/144838 | 3 | Consider a 0-1 integer $n \times n$ matrix with coefficients chosen uniformly over $\{0,1\}$. The probability that it is singular is exponentially small, and so we expect that it has a well-defined [condition number](http://en.wikipedia.org/wiki/Condition_number).
What is the *expected* condition number — or better y... | https://mathoverflow.net/users/3723 | Condition number of a random 0-1 matrix | For matrices with i.i.d. cenetered normal entries the condition number was studied by Alan Edelman in [his thesis.](http://math.mit.edu/~edelman/thesis/thesis.pdf) For general subgaussian entries the state of the art is the work of Rudelson and Vershynin, as described in Vershynin's [excellent lecture notes](http://www... | 2 | https://mathoverflow.net/users/11142 | 144848 | 78,519 |
https://mathoverflow.net/questions/72780 | 17 |
>
> **Question:** *Does* **Con**($ZF$) *imply* **Con**($ZF$ + $DC$ + "*there is no paradoxical Banach-Tarski decomposition of the unit ball*")?
>
>
>
Here **Con**($X$) is the consistency of $X$; $DC$ is dependent choice.
**Motivation for the Question**: Since the "paradoxical" sets in the Banach-Tarski theorem... | https://mathoverflow.net/users/9269 | Paradoxical Decompositions | A positive answer is proved in S. Wagon's book *"The Banach-Tarski Paradox",* Theorem 13.2. Specifically, the statement proved there is:
Con(ZF) $\leftrightarrow$ Con(ZF + DC + GM),
where GM is the existence of an isometry-invariant measure on all subsets of $\mathbb R^n$ taking the value $1$ on the unit cube.
| 9 | https://mathoverflow.net/users/41274 | 144849 | 78,520 |
https://mathoverflow.net/questions/144822 | 5 | I am looking for some **introductory** reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?
| https://mathoverflow.net/users/40038 | Reference for Automorphisms of K3 surfaces | You may consult
Kondō, Shigeyuki Quadratic forms and K3. Enriques surfaces [translation of Sûgaku 42 (1990), no. 4, 346–360; MR1083944 (92b:14018)]. Sugaku Expositions. Sugaku Expositions 6 (1993), no. 1, 53–72.
| 3 | https://mathoverflow.net/users/37622 | 144851 | 78,522 |
https://mathoverflow.net/questions/144857 | 3 | Let $S\_{g,b}$ an orientable surface with genus $g$ and $b$ boundary components and $S\_g^b$ be an orientable surface with $b$ punctures.
Denote by $PMCG(S\_g^b)$ and $PMCG(S\_{g,b}) $ the pure mapping class groups, that is, the group of orientation preserving homeomorphisms of the surface fixing the punctures or th... | https://mathoverflow.net/users/41219 | Mapping class groups of a punctured surface vs. surface with boundary | No, it is not split (except in a few degenerate cases like $(g,n) = (0,1)$ or $(g,n)=(0,2)$; let's assume that $g \geq 2$ for the moment just to be careful). It is a central extension, so if it was split then the abelianization of the pure mapping class group of $S\_{g,b}$ would contain a copy of $\mathbb{Z}^b$; howeve... | 4 | https://mathoverflow.net/users/317 | 144859 | 78,526 |
https://mathoverflow.net/questions/144825 | -1 | Are there any examples of mathematical theories T1,T2 which satisfy the following conditions?
1. T1 and T2 have the same "vocabulary" and are both formalized in the classical first order predicate calculus (with identity).
2. T1 is a sub-theory of T2.
3. There is available a precise criterion for determining whether... | https://mathoverflow.net/users/4423 | A question about whether impredicative formulae always lead to (paradoxical) inconsistencies in formalized theories | As Andreas points out, condition 3 is problematic. However, I think the spirit of the question can be answered by a simple example. Let T be some weak subtheory of ZFC which is mutually interpretable with PA and let T1 be T plus the assertion Con(Z) for each finite fragment Z of ZFC. Let T2 be ZFC. Since ZFC proves Con... | 4 | https://mathoverflow.net/users/23141 | 144862 | 78,528 |
https://mathoverflow.net/questions/144874 | 1 | This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int\_{\Omega }d^2k\int\_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where $\mathbf{r},\mathbf{k},\mathbf{q}\in \mathbb{R}^2$ and $\Omega \subset\mathbb{R}^2$ is some finite region.
The question is: At large... | https://mathoverflow.net/users/37081 | What's the asymptotic behavior of this function at large distance? | As OP seems to be stuck, I give here the few lines needed to solve this problem as sugested in the comments. Given
$$
f(\mathbf{r})=\int\_{\Omega }d^2k\int\_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]=
\frac{1}{2}\int\_{\Omega }d^2ke^{i\mathbf{k}\cdot\mathbf{r}}\int\_{\Omega }d^2qe^{-i\mathbf{q}\cdot\mat... | 3 | https://mathoverflow.net/users/19520 | 144887 | 78,536 |
https://mathoverflow.net/questions/144773 | 68 | A [cumulant](https://en.wikipedia.org/wiki/Cumulant) is defined via the *cumulant generating function*
$$ g(t)\stackrel{\tiny def}{=} \sum\_{n=1}^\infty \kappa\_n \frac{t^n}{n},$$
where
$$
g(t)\stackrel{\tiny def}{=} \log E(e^{tX}).
$$
Cumulants have some nice properties, including *additivity*- that for statistically ... | https://mathoverflow.net/users/2051 | What is a cumulant really? | Cumulants have many other names depending on the context (statistics, quantum field theory, statistical mechanics,...): seminvariants, truncated correlation functions, connected correlation functions, Ursell functions...
I would say that the $n$-th cumulant $\langle X\_1,\ldots,X\_n\rangle^{T}$
of random variables $X\_... | 51 | https://mathoverflow.net/users/7410 | 144888 | 78,537 |
https://mathoverflow.net/questions/144904 | 1 | Let $\pi:\mathcal{X} \to U$ be a family of hypersurfaces (not necessarily smooth) in $\mathbb{P}^n$ for some $n \ge 3$. Assume that $U$ is simply connected (under analytic topology). For any pair $u,v \in U$ does there exist an isomorphism between their homology groups i.e., is $H\_k(X\_u,\mathbb{Z}) \cong H\_k(X\_v,\m... | https://mathoverflow.net/users/32151 | Isomorphism of homology groups under deformation | If $\pi$ is smooth, then the answer is yes, because by a theorem of Ehresmann, the fibres are diffeomorphic. In general, however, the answer is no. To see this, take the family of all cubics in $\mathbb{P}^2$, the ranks of $H\_1(X\_u)$ will take all possible values in the set $\{0,1,2\}$.
| 3 | https://mathoverflow.net/users/4144 | 144905 | 78,541 |
https://mathoverflow.net/questions/43915 | 17 |
>
> Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
>
>
>
The question closely related to [this one](https://mathoverflow.net/questions/43680).
| https://mathoverflow.net/users/10330 | Nonseparable example in dimension theory? | There are completely metrizable spaces $X$ with $\dim(X)=1$ and $\mathrm{ind}(X)=0$, where $\dim$ denotes the covering dimension and $\mathrm{ind}$ the small inductive dimension. The first such example is due to P. Roy (see *"nonequality of dimensions for metric spaces"*, TAMS 1968), but there are many others. My favou... | 5 | https://mathoverflow.net/users/17836 | 144909 | 78,542 |
https://mathoverflow.net/questions/144860 | 16 | Can somebody briefly introduce the mathematical aspects, in particular, those related to mathematical finance, of the three economists who were just awarded this year's [Nobel Memorial Prize in Economic Sciences](https://en.wikipedia.org/wiki/Nobel_Memorial_Prize_in_Economic_Sciences)?
According to the [New York Time... | https://mathoverflow.net/users/34483 | On mathematical aspects of the most recent Nobel Prize in economics winners' work | Just to add a little something to arsmath's very good answer: The mathematics in Fama's main idea that returns are impredictable are indeed simple, and moreover, not due to him. What Fama did is a huge empirical study to support that claim. For the mathematical argument itself, which is simple but not absolutely trivia... | 9 | https://mathoverflow.net/users/9317 | 144910 | 78,543 |
https://mathoverflow.net/questions/144906 | 1 | Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$.
Here $\Omega$ can be a subset of $\mathbb{R}^n$ with desired boundary, $\mathbb{R}^n$ or torus $\mathbb{T}^n$.
| https://mathoverflow.net/users/39756 | Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness | If $\Omega=B(0,1)$, take a radial function $f(r)=\frac{1}{r^{n/p}\left(\log r-1\right)^{2/p}}$ Then
$$
\int f^p dx = \omega\_n \int\_0^1 \frac{1}{r\left(\log r-1\right)^2} dr =\omega\_n,
$$
so $f\in L^p$, but not in $L^q$ for $q>p$, and therefore not in any W^{s,p}, $s>0$.
Now integrate $f$ k times in $r$, say $F\_k... | 6 | https://mathoverflow.net/users/40120 | 144916 | 78,546 |
https://mathoverflow.net/questions/144920 | 5 | Let A be the 4-dimensional algebra over Q with basis 1,i,j,k, and multiplication table
$$
i^2 = -1 \quad j^2 = -11 \quad k^2 = -11 \quad ij = k \quad jk = 11i \quad ki = j
$$
So, A is the unique "definite quaternion algebra of discriminant 11" over the rational numbers. I'm led to believe that there are two conjugacy... | https://mathoverflow.net/users/41368 | How do you find maximal orders in quaternion algebras? | For semisimple algebras over $\mathbb{Q}$, there is a general algorithm due to Gabor Ivanyos and Lajos Ronyai, described in *Finding maximal orders in semisimple algebras over $\mathbb{Q}$* and implemented in the computer algebra system [Pari/gp](http://pari.math.u-bordeaux.fr/dochtml/html-stable/Associative_and_centra... | 7 | https://mathoverflow.net/users/40821 | 144927 | 78,548 |
https://mathoverflow.net/questions/144954 | 2 | Suppose $\alpha$ is a fixed given irrational number with $\alpha\in [A, B]\subset [0,1]$, are there any (efficient) methods to compute the least integer $n$ such that the decimal part of $n\alpha$ lies in $[0,1]\setminus [A,B]$?
| https://mathoverflow.net/users/9305 | (efficient) method to test $\{n\alpha\}\not\in [A, B]\subset [0,1]$ | The multipliers which could be candidates for what you want form a very sparse subset of the integers. They can be found efficiently by
* Finding the (start of the) continued fraction expansion of $\alpha$
* Using it to find certain especially good rational approximants to $\alpha$
* Taking the denominators of these ... | 2 | https://mathoverflow.net/users/8008 | 144968 | 78,566 |
https://mathoverflow.net/questions/144898 | 0 | I am looking for a second proof of Jordan-Von Neumann theorem that characterizes inner product in normed spaces. The book "Inner Product Structures: Theory and Applications" talks about a second proof based in Frechet condition.
I would thank anyone who can help me to know how is exactly the second demonstration, the... | https://mathoverflow.net/users/41352 | Second proof of Jordan-Von Neumann theorem | The second proof is from
B. Reznick, Banach spaces which satisfy linear identities. Pacif. J. Math. 74, 221-233.
| 0 | https://mathoverflow.net/users/23007 | 144978 | 78,569 |
https://mathoverflow.net/questions/144977 | 2 | A classical property of the Gaussian distribution is that, if $\{Z\_i\}\_{1 \leq i \leq n}$ are i.i.d. standardised Gaussian distributions (i.e. $Z\_i \sim N(0,1)$) and $S = \sum\_{i=1}^n a\_i Z\_i$ where $a\_i \in \mathbb{R}$, then the law of $S$ is the same as the law of $\big( \sum a\_i^2 \big)^{1/2} Z\_1$.
Given ... | https://mathoverflow.net/users/18974 | Linear combination of i.i.d. $Z_i$ distributed as $Z_1$ | The distributions you're looking for are [stable distributions](http://en.wikipedia.org/wiki/Stable_distribution). Basically, the only such norms you can take are $\ell^p$ norms for $1 \le p \le 2$.
If you don't need an honest norm, you can also take the $\ell^p$-"norm" (really quasinorm) for $0 < p < 1$. But, up to ... | 5 | https://mathoverflow.net/users/1044 | 144979 | 78,570 |
https://mathoverflow.net/questions/144884 | 4 | I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to implement some software using a naive approach, but the results have been less than fruitful. As an example
Let $K = {\mathb... | https://mathoverflow.net/users/41319 | Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices | This is precisely what my [Magma package](http://www.normalesup.org/~page/software.html) does, if the base field $K$ has at most one complex place. The algorithm is described in [this paper](http://arxiv.org/abs/1206.0087).
In the example you give, the set $\{ -7 + (-4I + 5)i + (24I + 6)j + (17I + 4)k, -2I + 3Ij + 2I... | 5 | https://mathoverflow.net/users/40821 | 144984 | 78,574 |
https://mathoverflow.net/questions/144989 | 3 | Let $Y$ be a rationally connected variety over an algebraically closed field, and let
$$\phi:X\dashrightarrow Y$$
be a rational fibration such that the general fiber of $\phi$ is rationally chain connected. **Is it true that $X$ is rationally chain connected?**
**If we assume that the general fiber of $\phi$ is smoot... | https://mathoverflow.net/users/14514 | Rationally connected varieties and rational fibrations | Over $\mathbb{C}$, the answer to the second question is *yes*.
In fact, since rational connectedness is a birational property, one can solve the indeterminacy of the rational map $\phi$ in order to obtain a dominant morphism $f \colon Z \to Y$ (whose general fiber is birational to the general fiber of $\phi$) and the... | 3 | https://mathoverflow.net/users/7460 | 144992 | 78,578 |
https://mathoverflow.net/questions/144997 | 9 | Following Murphy's law for the published material:
"The paper you need is too old to be in the arXiv, it is not in any online database which your institution has subscription to, and... it not even in the library!!!"
I really need
Z. Ran. On Subvarieties of Abelian Varieties. $Invent. Math.$ **62** (1981) p. 459--47... | https://mathoverflow.net/users/41314 | Where to find digitized old papers on the internet? | Suppose the question is "How can I find old papers?" ... (so old that they are not on-line).
If it is "not even in the library" ... then see a librarian. He/she can get it by interlibrary loan.
| 8 | https://mathoverflow.net/users/454 | 145002 | 78,583 |
https://mathoverflow.net/questions/144958 | 9 | Let $G=(V,E)$ be a simple $2$-connected graph and $C$ is a cycle in $G$ satisfies:
For any vertex $v$ of $C$,there exists at least one vertex $u\in V(G)\backslash V(C)$ adjacent with $v$.
Is it true that there must exists a cycle in $G$ which is longer than $C$?
| https://mathoverflow.net/users/40096 | the length of cycles in a $2$-connected simple gragh | Here is a quick reduction. Hopefully someone else can finish it off. Since $G$ is 2-connected, it has an ear-decomposition starting with the cycle $C$. Next, when building the ear-decomposition, for as long as possible always choose ears $P$ such that both ends of $P$ are in $C$ and $P$ has two edges. Now consider the ... | 1 | https://mathoverflow.net/users/2233 | 145004 | 78,585 |
https://mathoverflow.net/questions/141133 | 1 | I have seen some attempt in considering topological pressure for Julia sets of exponential function, and elliptic function. However, there exists few reference according to my knowledge?
I want to ask whether there exist further general variational principle to make ergodic theory can been applied into transcendenta... | https://mathoverflow.net/users/11966 | general variational principle for the Julia sets of mermorphic function? | There is quite a lot of literature these days on the measurable dynamics of transcendental entire functions. You may be interested, in particular, in the works of Mariusz Urbanski and his co-authors (in particular, his papers with Volker Mayer come to mind).
There is a very general result (<http://arxiv.org/abs/1007... | 2 | https://mathoverflow.net/users/3651 | 145013 | 78,589 |
https://mathoverflow.net/questions/145005 | 6 | I am looking for a proof (or a reference to a proof) of the following theorem:
Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure $\mu$. Let $T\colon X\to X$ be a continuous map which is $\mu$-preserving. Then for $\mu$-almost every $x\in X$ there is... | https://mathoverflow.net/users/41401 | Poincare recurrence theorem and convergence on compact metric spaces | Let $\mathcal P\_1,\mathcal P\_2,\ldots$ be a sequence of refining partitions of decreasing diameter (converging to 0) of $X$. Let $\delta\_n$ be the diameter of $\mathcal P\_n$. Let $\mathcal P\_n(x)$ be the element of $\mathcal P\_n$ containing $x$.
For $\mu$-a.e. $x$, $\mu(\mathcal P\_n(x))\ne 0$.
Now the Poincar... | 8 | https://mathoverflow.net/users/11054 | 145018 | 78,590 |
https://mathoverflow.net/questions/144986 | 5 | This question arose from [my last question](https://mathoverflow.net/questions/144880/equivariant-normalization), which I considered answered - from the comments, however, it is obvious that the answer is only complete in characteristic zero, and I am trying to understand *why*.
Let $A$ be a $\Bbbk$-algebra1, where $... | https://mathoverflow.net/users/9947 | When does a group action on a k-algebra induce an algebraic action on the spectrum? | If I understand the question correctly, you have a map of schemes $G \times X \to X$, and the corresponding map of $k$-points is a group action (meaning that the obvious two maps $G(k) \times G(k) \times X(k) \to X(k)$ coincide), but you are not sure that it is a group action in the category of schemes. In other words,... | 2 | https://mathoverflow.net/users/297 | 145024 | 78,594 |
https://mathoverflow.net/questions/144880 | 10 | Let $G=\mathrm{Gl}\_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction field. Can $\tilde X$ be given the structure of a $\tilde G$-variety such that $\phi$ is equivariant?
| https://mathoverflow.net/users/9947 | Equivariant normalization? | Here is the sort of example I think Jason Starr was raising. (I looked at Brian's webpage, but it wasn't obvious which paper to read.) Take $k$ to be a perfect field of characteristic $p$, with $p \neq 0$, $2$. Let $A = k[x,y]/(y^2-x^p)$. The normalization of $A$ is $\tilde{A} = k[t]$, with $y=t^p$ and $x=t^2$.
Let ... | 7 | https://mathoverflow.net/users/297 | 145025 | 78,595 |
https://mathoverflow.net/questions/144618 | 13 | Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a\_{1}$ , $a\_{2}$,...$\rightarrow b$
so that $a\_{n} \in A$ but $b \notin A$.
Working in the topological category TOP, we assume the group $G$ is a sequential space, inversion... | https://mathoverflow.net/users/17029 | If G is a sequential topological group, must G x G be sequential? | This is consistently false.
It was proved by Malyhin and Shakhmatov (see *"Cartesian products of Frechet topological groups and function spaces"*, Acta Math. Hung. 1992) that in any model obtained by adding a Cohen real to a model of $MA + \lnot CH$, there is a sequential topological group $G$ such that $G \times G$... | 11 | https://mathoverflow.net/users/17836 | 145027 | 78,596 |
https://mathoverflow.net/questions/145030 | 2 | I'm studying some topics about topological games of length $\alpha\geq\omega$, where I came across the following statement about additively indecomposable ordinals (recall that $\alpha$ is additively indecomposable if $\beta+\gamma <\alpha$ whenever $\beta,\gamma<\alpha$):
If $\alpha\geq\omega$ is additively indecomp... | https://mathoverflow.net/users/41407 | A question about additively indecomposable ordinals | For the countable case, this is very easy: let $\alpha$ be countable (so $cf(\alpha)=\omega$) and additively decomposable. Then $\alpha$ is a limit, and we can decompose $\alpha$ into its "odd" and "even" bits: that is, there is a function $f: \alpha\rightarrow\alpha$ which is injective and order-preserving, such that ... | 4 | https://mathoverflow.net/users/8133 | 145031 | 78,598 |
https://mathoverflow.net/questions/144729 | 6 | I'm reading section 2.1 of Lawson's book, Spin Geometry. The book states the following fact. Let $X$ be a manifold and $E$ a vector bundle over it. Equip $E$ with a Riemannian structure. Let $P\_O$ be the bundle of orthonormal frames in $E$ which is a principal $O\_n$ bundle. The fibration $O\_n \rightarrow P\_O(E) \ri... | https://mathoverflow.net/users/41259 | Exact sequences of the cohomology induced by fiber bundle | What you are asking about is a consequence of a more general statement about fibrations. Let $p: E \to B$ be a fibration with $B$ path connected and based. Set $F = p^{-1}(\*)$.
Assume $B$ is $r$-connected and $p$ is $s$-connected. Then there's a exact sequence
$$
0 \to H^0(B) \to H^0(E) \to H^0(F) \to H^1(B) \to \cdo... | 8 | https://mathoverflow.net/users/8032 | 145038 | 78,601 |
https://mathoverflow.net/questions/145047 | 7 | The permanent $P(M)$ of a matrix $M$ of size $n$ is defined to be:
$$
P(M) := \sum\_{\sigma \in S\_n}\prod\_{i=1}^n M\_{i\sigma(i)}
$$
If you have a matrix of the form
$$
M\_{ij} := A\_i + B\_j
$$
where $A$ and $B$ are indexed sets of numbers and if $x\_k(S)$ is the sum of all products of $k$ elements taken from the ... | https://mathoverflow.net/users/7043 | Permanent identities for special classes of matrices | The permanent of the biadjacency matrix of a bipartite graph counts the number of perfect matchings of that graph. By the biadjacency matrix, I mean the matrix with rows indexed by one of the independent sets and the columns indexed by the other. An entry is 1 if the vertex represented by the row and the vertex represe... | 7 | https://mathoverflow.net/users/41283 | 145048 | 78,605 |
https://mathoverflow.net/questions/144974 | 6 | The unpublished preprint:
D. G. A. Jackson, The irreducibility of a cubic over $\mathbb{F}\_q$, Research Report 98-17, Univ. of Sydney (1998)
gives necessary and sufficient conditions (when ${\rm char}({\mathbb F}\_q)\neq2,3$) for a cubic polynomial over $\mathbb{F}\_q$ to be irreducible.
I have conditions (below... | https://mathoverflow.net/users/23827 | Irreducibililty tests for cubic and quartic polynomials over finite fields | These results have been known for many years. Here are some references and further developments; full bibliographic details are at the end of this answer.
Reducibility criteria for cubics over finite fields of characteristic at least $5$ were given already in 1906 by Dickson. In fact he gave conditions for a cubic to... | 10 | https://mathoverflow.net/users/30412 | 145055 | 78,608 |
https://mathoverflow.net/questions/145026 | 28 | It is well known that the alternating group $A\_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the group $SO(n)$ is simple for odd $n$ and the group $SO(n)/\{\pm I\}$ is simple for even $n\neq 4$).
My question is: are th... | https://mathoverflow.net/users/33757 | The non-simplicity of $SO(4)$ and $A_4$ | Here are some general remarks which hold for arbitrary semisimple Lie algebras ${\mathfrak g}$ relating its algebraic properties to that of its Weyl group $W$.
1. ${\mathfrak g}$ is simple if and only if the standard linear action of its Weyl group $W$ is irreducible. However, $W$ itself might still split nontrivial... | 8 | https://mathoverflow.net/users/21684 | 145064 | 78,611 |
https://mathoverflow.net/questions/145070 | 3 | Put $X := \mathbb A^{n+1}\!-\lbrace0\rbrace$. Let $G=\mathbb C^\*$ act on $X$ with (positive) weights $w\_0,\dots,w\_n$. The quotient stack $[X/G]$ is called the weighted projective stack.
Each vector bundle on $[X/G]$ corresponds to a $G$-equivariant vector bundle $E$ on $X$, and vice versa. So I want to understand... | https://mathoverflow.net/users/5206 | Vector bundles on a weighted projective stack | No, that is not true. The easiest counterexample is when $n$ equals $2$ and the weights are all $1$, i.e., the quotient stack is actually the scheme $\mathbb{P}^2$. For the tangent sheaf on $\mathbb{P}^2$, the pullback sheaf on $\mathbb{A}^3\setminus \{0\}$ is not a trivial locally free sheaf (even without considering ... | 6 | https://mathoverflow.net/users/13265 | 145072 | 78,614 |
https://mathoverflow.net/questions/145073 | 4 | I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
>
> The derivation of the conditions of exact integrability of an ordinary
> differential equation of the nth. order (or of a differential expression involving
> derivatives of a single depe... | https://mathoverflow.net/users/38721 | Exact Differential Equations of Order n via Pfaffian Differential Equations? | I'll assume all along you're referring to ODE with real-analytic/holomorphic coefficients. You're looking for something called "non-linear differential Galois theory". This is related to this question [Solution of linear ODE](https://mathoverflow.net/questions/140849/solution-of-linear-ode/140851#140851) in the linear ... | 3 | https://mathoverflow.net/users/24309 | 145074 | 78,615 |
https://mathoverflow.net/questions/145077 | 34 | Why are optimization problems often called *programs*?
* linear programming
* geometric programming
* convex programming
* Integer programming
* ...
| https://mathoverflow.net/users/41438 | Why are optimization problems often called "programs"? | It may be that this question had been answered here before, but I couldn't find the answer.
Anyway, the answer is given by the person who coined the name itself: George Dantzig wrote in "[LINEAR PROGRAMMING](https://www2.informs.org/History/dantzig/LinearProgramming_article.pdf)":
>
> Here are some stories about ... | 56 | https://mathoverflow.net/users/9652 | 145079 | 78,617 |
https://mathoverflow.net/questions/145081 | 2 | **Update:** Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers.
**Definition:** Let $\kappa$ be a measurable cardinal. Define:
$\mathbb{M}\_{\kappa}:=\lbrace \mu:P(\kappa)\rightarrow \lbrace 0,1\rbrace~|~\mu~\text{is a non-trivial}~\kappa~\text{-additi... | https://mathoverflow.net/users/nan | Are measures of a measurable cardinal measurable? (Edited and Updated Version) | If you mean only to consider the measures on $\kappa$, in the sense of a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, then of course there are at most $2^{2^\kappa}$ many of them, and when $\kappa$ is measurable one can prove that every measure has at least $2^\kappa$ many isomorphic copies. So the number of... | 4 | https://mathoverflow.net/users/1946 | 145086 | 78,619 |
https://mathoverflow.net/questions/145045 | 3 | A generalized Moore graph - as defined by Cerf, Cowan, Mullin and Stanton - is one for which the girth G and diameter D satisfy G ≥ 2D - 1. These graphs retain many of the optimal properties of Moore graphs, but are considerably more plentiful (though still quite rare - it has been conjectured that there are finitely m... | https://mathoverflow.net/users/4336 | Generalized Moore Graphs | You are right that there is very little published on the subject.
There are 6 cubic GMGs of order 60 and none of order 62 or 64. This is known from a computational study I did with Catherine Menon that is long overdue for publication (my fault). Order 66 is too hard to do exhaustively by our method. We also have some... | 3 | https://mathoverflow.net/users/9025 | 145089 | 78,621 |
https://mathoverflow.net/questions/145085 | 4 | Is it known what the next-to-leading order term is in the variance of the central limit distribution for the average of $N$ variables each of which is distributed according to $P(x) \sim 1/x^{1+\alpha}$ for $1 < \alpha$?
So we have $\bar{x} = N^{-1} \sum\_{i=1}^N x\_i$ where $x\_i$ are independent and distributed acc... | https://mathoverflow.net/users/41312 | Variance of central limit distribution for $P(x) \sim 1/x^{1+\alpha}$ for finite but large $N$? | You are looking for a generalization of the [Edgeworth expansion](https://en.wikipedia.org/wiki/Edgeworth_series) to distributions that have slowly decaying tails.
For $\alpha>2$ this can be found here:
[Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities](http://arxiv.org/abs/1103.4306... | 5 | https://mathoverflow.net/users/11260 | 145091 | 78,622 |
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