parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/154542 | 2 | Let $F(x,u,Du,D^2u)=0$ for $x\in\Omega$ be our equation of interest. One way of defining a viscosity subsolution is this:
We say $u$ is a viscosity subsolution iff for any test function $\phi\in C^2(\Omega)$ such that $u-\phi$ has a LOCAL maximum at $x\_0$ then $F(x\_0,D\phi(x\_0),D^2\phi(x\_0))\leq 0$.
However in ... | https://mathoverflow.net/users/32325 | Does replacing 'local' with 'global' in the definition of viscosity sub(super)solution change the definition? | The answer is no, and how to solve the analysis question is explained in exercise 3.5 in [Albert Fathi's lecture notes](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&ved=0CEcQFjAC&url=http://www.ceremade.dauphine.fr/~pbernard/enseignement/m2/fathi.pdf&ei=_HzWUuzuG86r0gWLl4GADQ&usg=AFQjCNF6bx0iQF... | 2 | https://mathoverflow.net/users/40120 | 154616 | 82,133 |
https://mathoverflow.net/questions/154612 | 4 | How does one compute the group of biholomorphisms of $\mathbf{D}^n = \{(z\_1, \ldots, z\_n) \in \mathbb{C}^n: \forall\_i \; |z\_i| \leq 1\}$, i.e., the unit polydisk, and of the unit ball $B^n = \{(z\_1, \ldots, z\_n) \in \mathbb{C}^n: |z\_1|^2 + \ldots |z\_n|^2 \leq 1\}$, and show these automorphism groups are not iso... | https://mathoverflow.net/users/30081 | automorphism groups of unit disk $\mathbf{D}^n $ and unit ball $ B^n $ | All this is explained in the book:
MR1192135
Shabat, B. V.
Introduction to complex analysis. Part II.
Functions of several variables. AMS, Providence, RI, 1992.
And I am sure it is in many other textbooks.
The group of automorphisms of the ball $B$ consists of fractional-linear transformations
$$w\_j=\frac{a\_{j,0}+... | 9 | https://mathoverflow.net/users/25510 | 154633 | 82,140 |
https://mathoverflow.net/questions/154636 | 2 | I have a simple question I've managed to get myself quite confused about.
Given a Hilbert space H, what do we know about the cardinality of
(a) the set $P(H)$ of projection operators onto $H$ (equivalently, the set of all closed subspaces of $H$),
and
(b) the set of all Boolean subalgebras of $P(H)$?
I imag... | https://mathoverflow.net/users/45570 | Cardinality of the set of Boolean subalgebras of the lattice of projections on a Hilbert space | Up to a unitary isomorphism, a Hilbert space is uniquely determined by its dimension, and closed subspaces are Hilbert spaces in their own right. So $P(H)$ is the disjoint union, over all cardinal numbers $\alpha \leq \dim(H)$, of closed subspaces of dimension $\alpha$. And there as many of the latter as unitary operat... | 2 | https://mathoverflow.net/users/10368 | 154639 | 82,141 |
https://mathoverflow.net/questions/154642 | 0 | Let $0\in U \subset \mathbb{C}$
be a small neighborhood of origin in the complex plane and $f\_1,f\_2\colon U\to \mathbb{C}$ be two complex valued functions such that
$$f\_1(0)=f\_2(0)=0$$
$$\bar\partial{f}\_1(0)= \bar\partial f\_2(0) =0$$
$$df\_1 ,df\_2\neq 0$$.
Let $f\_2/f\_1\colon U\setminus \{0\} \to \mathbb{C}$... | https://mathoverflow.net/users/5259 | quotient of holomorphic functions at a point | No. Writing as usual $z=x+iy$, take $f\_1=z$ and $f\_2=z+x^2$. Then your conditions are satisfied but $\dfrac{f\_2}{f\_1}=1+\dfrac{x^2}{z} $ does not extend to $0$.
| 2 | https://mathoverflow.net/users/40297 | 154645 | 82,142 |
https://mathoverflow.net/questions/154415 | 3 | Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.
I know that the interior of $K'$ is exactly the set $\tilde{K} = \{ \phi\ |\ \phi(x) > 0,\ \ \forall x \in K\backslash0\}$. (see for example [this q... | https://mathoverflow.net/users/45468 | Interior of a dual cone | The bad news is that many closed convex cones in infinite dimensions have empty interior, e.g., the natural cone of positive functions in $L^p(\Omega)$ for all $\Omega \subset \mathbb{R}^n$; or in $W^{1,p}(\Omega)$ for $\Omega \subset \mathbb{R}^n$ and $p < n$.
The positive cone in $L^2(\Omega)$ is self-dual (if you ... | 6 | https://mathoverflow.net/users/32507 | 154647 | 82,143 |
https://mathoverflow.net/questions/154371 | 4 | I asked this question on math.stackexchange [here](https://math.stackexchange.com/questions/524058/on-continuous-replicative-functions), but it did not receive much attention. Thus, I was suggested to post it here.
Knuth, in The Art of Computer Programming Vol. 1, defines a replicative function as a function $f$ such... | https://mathoverflow.net/users/45448 | On Continuous Replicative Functions | Functions satisfying such replicative identites are the cotangent,
Bernoulli polynomials, and the logarithmic
derivative of the Gamma function, to name a few.
The equation for n=2 is part of the Herglotz trick (see Aigner and Ziegler,
the BOOK).
Replicative functions are discussed by
Schroth (Aequ. Math. 20 (1980... | 3 | https://mathoverflow.net/users/3503 | 154656 | 82,147 |
https://mathoverflow.net/questions/154654 | 3 | In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf $\mathscr O\_X/\mathscr I$ for an analytic set $X\subset\mathbb C^n$ and a *finitely generated* sheaf of ideals on it. Inde... | https://mathoverflow.net/users/45582 | Definition of a complex space | If X is a scheme and $\mathcal{I} \subset \mathcal{O}\_X$ is a sheaf of ideals, then in general $\mathcal{I}$ does not define a closed subscheme. For example, suppose that $X = \mathbf{A}^n\_k$ where $k$ is a field and $\mathcal{I} = j\_!\mathcal{O}\_U$ where $j : U \to \mathbf{A}^n\_k$ is the inclusion of the compleme... | 5 | https://mathoverflow.net/users/44817 | 154667 | 82,149 |
https://mathoverflow.net/questions/154382 | 13 | I'm trying to calculate the following integral:
$\int\_0^\infty \mathrm{BesselJ}[l\_0,k\_0r] \cdot \mathrm{BesselJ}[l\_1,k\_1r] \cdot \mathrm{BesselJ}[l\_0-l\_1,kr] \cdot r\,dr$
($\mathrm{BesselJ}[n,x]$ is the Bessel function of the first kind of order $n$)
I assume that $l\_0,l\_1$ are integers, and that $k\_0,k\_1,... | https://mathoverflow.net/users/45454 | $\mathrm{Bessel}^3$ Integral | In fact, precisely your integral has been computed in closed form, in: Annie Gervois and Henri Navelet, [*Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities*](http://www.ams.org/mathscinet-getitem?mr=761861), J. Math. Phys. 25 (1984), no. 11, 3350–3356. Their result i... | 18 | https://mathoverflow.net/users/19276 | 154672 | 82,153 |
https://mathoverflow.net/questions/154662 | 15 | A Suslin line is a linear order $L$ which is dense with no endpoints, complete, and ccc but not separable. I'm wondering what kind of order-preserving maps there are from $L$ into $L$. Specifically,
**Question**: Can there be a Suslin line $L$ such that for every one-to-one, monotonic function $f$ from an uncountable... | https://mathoverflow.net/users/11233 | Can a Suslin line be 2-entangled? | Your property is too strong; there is no Suslin line like that.
The reason is that every Suslin line contains a copy of the real line, and we can define a counterexample function $f$ that concentrates only on this copy of $\mathbb{R}$. Specifically, suppose that $L$ is a Suslin line. Since the order is dense, we may... | 7 | https://mathoverflow.net/users/1946 | 154676 | 82,154 |
https://mathoverflow.net/questions/154668 | 5 | For any natural number $N$ and $0\le n\le N$ define
$$
f(n) = f(n,N) = \frac{1}{(N+1)!} \sum\_{\substack{{S\subset \{1,\ldots,N\}} \\ {|S|=n}}} \prod\_{s\in S} s.
$$
(The empty product is interpreted as $1$.) It is easy to see that
$$
\sum\_{n=0}^{N} f(n) = 1,
$$
so that $f$ may be thought of as a probability d... | https://mathoverflow.net/users/45587 | summation of products of combinatorials | The unnormalized sums are [unsigned Stirling numbers of the first kind](http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind), the coefficient of $x^{N-n}$ in $(x+1)(x+2)\cdots(x+N).$
For $p=1/2$, this is the central Stirling number of the first kind $3,35,735,22449, ..., S\_1(2n-1,n), ...$. See [A129505](... | 2 | https://mathoverflow.net/users/2954 | 154682 | 82,157 |
https://mathoverflow.net/questions/154681 | 3 | Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l\_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of $\theta$ can the projection $P\_{l\_\theta}$ of $X$ onto $l\_\theta$ have smaller order of magnitude? Say $|P\_{l\_\theta}(X... | https://mathoverflow.net/users/35673 | Number of small projections | Yes; indeed the number of such $\theta$ can grow as $|X|^{1-2\epsilon}$,
which is unbounded for all $\epsilon < 1/2$. For large $N$
let $X$ be the set of integer points $(x,y)$ with $x^2 + y^2 < N$,
so $|X| \sim \pi N$.
If $l\_\theta$ has rational slope $a/b$ then $P\_{l\_\theta}(x,y)$
depends only on the integer $bx+a... | 6 | https://mathoverflow.net/users/14830 | 154684 | 82,158 |
https://mathoverflow.net/questions/154629 | 24 | For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times m/n$ rectangle, which is of square-rational proportion $m^2/n^2$. Is there any essentially different way to rectangularl... | https://mathoverflow.net/users/7458 | Can a unit square be cut into rectangles that tile a rectangle with irrational sides? | Suppose that the unit square has been cut into $n$ rectangles: $S=\cup\_{i=1}^nR\_i,$ where each rectangle $R\_i$ has dimensions $s\_{2i-1}\times s\_{2i}$. Also let $r$ be an irrational number and let $R^\*$ be a $1/r \times r$ rectangle. We will define an additive "weight" (which can take negative values) on certain p... | 8 | https://mathoverflow.net/users/8008 | 154687 | 82,159 |
https://mathoverflow.net/questions/154685 | 1 | Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g\_x$ defines a quadratic form on $T\_xM$ such that $g\_x(v,v)\ge0$, but $g\_x(v,v)=0$ not necessarily imply $v=0$.
Also, and additional hypothesis (\*) is: if $x\in A$, $g\_x$ defines an inner product on $T\_... | https://mathoverflow.net/users/45597 | Normal tubular neighborhood theorem for semi(or pseudo)-riemannian manifolds | For a pseudo-Riemannian manifold, the local version of the exponential slice theorem is the Luna slice theorem,
* MR0342523 (49 #7269)
Luna, Domingo
Slices étales. (French) Sur les groupes algébriques, pp. 81–105. Bull. Soc. Math. France, Paris, Memoire 33 Soc. Math. France, Paris, 1973.
There are singular points ... | 2 | https://mathoverflow.net/users/26935 | 154691 | 82,160 |
https://mathoverflow.net/questions/154655 | 52 | Given discriminant $d$ and [*j-function*](http://mathworld.wolfram.com/j-Function.html) $j(\tau)$, I was looking at,
$$F(\tau) = \sqrt{\big(j(\tau)-1728\big)d}$$
which appears in Ramanujan-type pi formulas. Let $C\_d$ be the odd prime factors of the ***constant term*** of the minimal polynomial for $F(\sqrt{-d})$. ... | https://mathoverflow.net/users/12905 | There's something strange about $\sqrt{\big(j(\tau)-1728\big)d}$ | "Numerology" such as you've observed is explained in the paper
>
> Gross, B.H., and Zagier, D.: [On singular moduli](https://doi.org/10.1515/crll.1985.355.191), *J. reine angew. Math.* **355** (1985), 191$-$220. MR772491 (86j:11041)
>
>
>
which gives more generally the factorizations of the constant terms
of t... | 65 | https://mathoverflow.net/users/14830 | 154692 | 82,161 |
https://mathoverflow.net/questions/154051 | 5 | Does anyone know of papers that mention the classification of non-singular Morse-Smale (NMS) flows up to topological equivalency? I am particularly interested in the flows on manifolds of dimension 3. But all I can find so far is the paper by Bin Yu on the classification of the NMS flows defined on the three-sphere.
| https://mathoverflow.net/users/45288 | Topological classification of Morse-Smale flows | Generally, if you hope a very clean list (like topological classification of surfaces) to completely classify NMS flows on 3-manifolds (even in three sphere) up to topological equivalence. It seems hopeless. One reason is that heteroclinic trajectories
connecting saddle orbits will lead a complete list quite wild.
I... | 5 | https://mathoverflow.net/users/19051 | 154702 | 82,163 |
https://mathoverflow.net/questions/60288 | 20 | In his book "The Mathematical Theory of Context-Free Languages" (1966), Ginsburg mentioned the following open problem:
```
Find a decision procedure for determining if an arbitrary semilinear set
is a finite union of linear sets, each with stratified periods.
```
Does anyone know if any progress has been made ... | https://mathoverflow.net/users/12640 | Status of an open problem about semilinear sets | See the paper below.
"On the open problem of Ginsburg concerning semilinear sets and related problems" by Ibarra and Seki, TCS 501, pp.11-19, 2013
link: <http://dl.acm.org/citation.cfm?id=2527409>
Hope it can help.
| 10 | https://mathoverflow.net/users/45611 | 154703 | 82,164 |
https://mathoverflow.net/questions/154705 | 5 | Need some help / ideas to proceed. Stuck for a while on this.
In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda\_{\max}(A)$ where $\lambda\_{\max}(A)$ is the largest eigenvalue of the adjacency matrix $A$, thus implying that the outbreak is basically dependent on the spectral... | https://mathoverflow.net/users/45615 | Epidemic threshold | The problem is addressed in
>
> Peng, Chengbin, Xiaogang Jin, and Meixia Shi. "Epidemic threshold and immunization on generalized networks." Physica A: Statistical Mechanics and its Applications 389.3 (2010): 549-560. [Link](http://www.researchgate.net/publication/233918441_PHYSA12267_published/file/9fcfd50ced21b2e... | 2 | https://mathoverflow.net/users/34050 | 154714 | 82,165 |
https://mathoverflow.net/questions/154715 | 4 | Let $X$ be a compact metric space, $T:X\rightarrow X$ a homeomorphism and $\mu$ be a $T$-invariant probability measure on $X$ such that the set of points with dense orbit in $supp(\mu)$ has full measure. Is the following statement true?
if $\mu$ is the only measure which satisfies that condition, then $\mu$ is ergodi... | https://mathoverflow.net/users/43741 | A question about ergodicity | Let $Y$ be the support of $\mu$ and suppose that $\mu$ is not ergodic. Then there exists a $T$-invariant measurable set $A\subset Y$ such that $0<\mu(A)<1$. The measures $\mu\_1$, $\mu\_2$ defined by $$\mu\_1(B):=\mu(B \cap A)/\mu(A),$$ $$\mu\_2(B):=\mu(B \setminus A)/\mu(X \setminus A)$$ are distinct and invariant, an... | 4 | https://mathoverflow.net/users/1840 | 154717 | 82,166 |
https://mathoverflow.net/questions/154716 | 2 | Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the symbol of $A$. I'm interested in such symbols $\sigma$ that for any function $u$ in the domain of definition of $A$ we have $... | https://mathoverflow.net/users/17896 | Support-preserving pseudodifferential operators | A classical result due to Peetre (Math. Scand. 8, 1960) says that if for all $u$,
$$\text{supp}\ Au\subset \text{supp}u,$$
then $A$ is a differential operator. On the other hand if a Fourier multiplier $a(D)$ satisfies your property, then $\hat a$ must be compactly supported. It seems that the following condition is re... | 4 | https://mathoverflow.net/users/21907 | 154726 | 82,169 |
https://mathoverflow.net/questions/154730 | 4 | Special case of [Bunyakovsky conjecture](https://en.wikipedia.org/wiki/Bunyakovsky_conjecture)
Let $f(x)$ be non-constant irreducible polynomial with integer
coefficients, no fixed prime factor and positive
leading coefficient. Let $S$
be a finite set of primes.
>
> Q1 Is it possible $f(n)$ to be divisible by a m... | https://mathoverflow.net/users/12481 | Can a polynomial be almost always divisible by a member of a finite set of primes? | The answer to Q1 is "no". The value of $f(n)$ mod $p$ depends only on the value of $n$, mod $p$.
Let $N$ be the product of the primes in $S$. Assuming "almost all" means "all but finitely many", then that means that for $n$ sufficiently large, the set $n+1,\ldots,n+N$ are all multiples of primes from $S$. But since a... | 7 | https://mathoverflow.net/users/14901 | 154732 | 82,172 |
https://mathoverflow.net/questions/154751 | 4 | Let $p$ be a prime, and $\alpha$ a positive integer. How do you compute the space of $p$-ordinary $p$-adic modular forms (in the sense of Serre) of weight 2 on $\Gamma\_0(p^\alpha)$? I'm really only concerned with the cases where $\Gamma\_0(p^\alpha)$ is genus zero and $\alpha = 1, 2$.
[Edit: Changed from space of $... | https://mathoverflow.net/users/32344 | computing spaces of $p$-adic modular forms | It depends rather what you mean by "compute". These are **very big** spaces (infinite-dimensional p-adic Banach spaces, and without a good theory of Hecke eigenforms). Can you be a bit more specific what it is you want to compute about them?
EDIT: You've edited your question to focus attention on the ordinary cusp fo... | 3 | https://mathoverflow.net/users/2481 | 154752 | 82,182 |
https://mathoverflow.net/questions/154725 | 1 | There are many examples of non-finitely based varieties. In a finite signature, is there an example of such variety with a known explicit set of identities?
| https://mathoverflow.net/users/44949 | Example of a non-finitely based variety with explicit set of defining identities | You can look at "Bases for Equational Theories of Semigroups" by P Perkins, J Algebra 11, 298-314 (1968). Theorem 2: the identities
$xyzw=xzyw$ and $yx^ky=xyx^{k-2}yx$ for $k=2,3,\dots$
define a non-finitely based variety of semigroups.
| 3 | https://mathoverflow.net/users/32831 | 154753 | 82,183 |
https://mathoverflow.net/questions/154749 | 5 | I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f\_t + f\_{xx}/2 = 0$ for the function $f$ of real variables $t$ and $x$, it is asked whether any solution exists, defined for all $x$ and in a neighbo... | https://mathoverflow.net/users/45630 | Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation? | I think the question you stated is not the one you intended, since the Schrodinger evolution is indeed a smooth flow in the Schwartz class, as can be seen by Fourier analysis, and so smooth compactly supported data $f\_0$ always leads to a smooth global solution which is also Schwartz in space.
My guess is that you a... | 8 | https://mathoverflow.net/users/766 | 154759 | 82,187 |
https://mathoverflow.net/questions/154765 | 1 | I have two questions regarding universal algebra, and also its ordered version.
1. If a variety $\mathcal{V}$ is generated by a specific two element algebra $2 = \{0,1\}$, then is that the only subdirectly-irreducible algebra in $\mathcal{V}$?
2. If an ordered variety $\mathcal{V}$ is generated by a specific two elem... | https://mathoverflow.net/users/5152 | Varieties generated by a two element algebra | Question 1 amounts to asking whether $HSP(2)=ISP(2)$ (i.e., the quasivariety generated by $2$ is a variety), and this can fail.
For instance, the algebra $\langle\{0,1\},\neg\rangle$ generates the variety of involutive unary operations, however nontrivial subdirect products of this algebra are additionally fixpoint-f... | 2 | https://mathoverflow.net/users/12705 | 154770 | 82,190 |
https://mathoverflow.net/questions/154738 | 3 | In reading Gauduchon's notes (cannot link them, anyway it is standard material) I ran into the following construction.
Let $(M, \omega\_0)$ be a compact symplectic manifold which is fixed.
An almost complex structure $J$ on $M$ is said to be compatible with $\omega\_0$ if $\omega\_0( \cdot, J \cdot)$ is a Riemannian ... | https://mathoverflow.net/users/19545 | Integrable compatible complex structures | The space of integrable $J$ is acted on by the diffeomorphism group, making it infinite dimensional. Symplectomorphisms will preserve the compatibility, so the orbit of an integrable $J$ under the symplectomorphism group is still infinite dimensional. The quotient by the symplectomorphism group is not known to be even ... | 3 | https://mathoverflow.net/users/13268 | 154779 | 82,198 |
https://mathoverflow.net/questions/154781 | 3 | Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g\_k(x)h\_k(y)$ for all $(x,y)$ in the domain of $f$ (the series could be infinite, in that case the convergence is just pointwise convergence)? Are there... | https://mathoverflow.net/users/18137 | series representation of bivariate functions | For example, $C^\infty([-a,a])\hat \otimes C^\infty([-b,b]) = C^\infty([-a,a]\times [-b,b])$
for the completed tensor product, and all tensor product topologies between the projective and the inductive one coincide, because the spaces are nuclear.
See for example,
* MR2296978 Trèves, François: Topological vector spac... | 3 | https://mathoverflow.net/users/26935 | 154795 | 82,205 |
https://mathoverflow.net/questions/154741 | 8 | Suppose we have an instance of Erdős-Renyi $G(n,p)$ graph with $p = d/n$. Thus the expected node degree is $d$ which we will fix, while letting $n \to \infty$. Then, there will be more than one connected component (CC) for large $n$.
Suppose that we pick the largest CC and relabel the nodes randomly from $1,\dots,s$... | https://mathoverflow.net/users/36687 | Erdős-Renyi graph restricted to largest connected component | If $d>1$, the paper [Anatomy of the giant component: the strictly supercritical regime](http://research.microsoft.com/en-us/um/people/eyal/papers/giant_super.pdf) by Ding, Lubetzky and Peres might be what you are looking for. The paper also has references for the cases $d<1$ and $d=1$.
| 5 | https://mathoverflow.net/users/14302 | 154806 | 82,210 |
https://mathoverflow.net/questions/154830 | 1 | Let $f(X)\geq 0$ be a nonconvex $C^\infty$ function: $\mathbb R^3\to \mathbb R$.
Give any fixed $X\_0$ such that $f(X\_0)=\epsilon^+$, and the level set:
${L}=\{X\in \mathbb R^3:f(X)\leq \epsilon^+\}$ is a convex set.
And $\forall X\_0$, $L$ is always a convex set.
Then what is the necessary and sufficient condit... | https://mathoverflow.net/users/43296 | How to find the necessary and sufficient conditions for a non-convex function to be locally convex? | Equivalent to convexity is that the Hessian matrix is positive semidefinite for all $x \in L$. This can be done by checking if all eigenvalues of this symmetric matrix are nonegative. The eigenvalues in turn can be calculated by the zeros of the characteristic polynomial. They are real. In your special case $L\subset \... | 1 | https://mathoverflow.net/users/45644 | 154832 | 82,215 |
https://mathoverflow.net/questions/154814 | 4 | My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution $f\in\mathcal{S}(G)^\times$ can be expressed as an integral of Dirac deltas
$$f=\int\_X \mathrm{d}\mu(x)\,\, f(x)\, \delta\... | https://mathoverflow.net/users/12793 | Is every distribution a linear combination of Dirac deltas? | The short answer is that the $\delta\_x$'s alone are not enough for a "basis". But it is enough to simply include all possible derivatives $\partial\delta\_x$, $\partial\partial \delta\_x$, ..., as well.
| 7 | https://mathoverflow.net/users/2622 | 154833 | 82,216 |
https://mathoverflow.net/questions/154762 | 2 | Consider a compact set $K \subset\mathbb{R}^d$ and a $\mathscr{C}^1$-diffeomorphism $\varphi:\mathbb{R}^d\rightarrow\mathbb{R}^d$.
Fix $\varepsilon>0$. Since $K$ is compact, we may define
\begin{align\*}
N(\varepsilon,K) := \min\left\{n\in\mathbb{N}\,:\, \exists x\_1,\dots,x\_n \in K\text{ s.t. } K\subseteq\bigcup\... | https://mathoverflow.net/users/27767 | $\varepsilon$-covering number after diffeomorphism | (Since the OP appears satisfied with my comment, I am reproducing it here as an answer.)
Fix $\epsilon > 0$. Since $\phi$ is $\mathscr{C}^1$, it must also be Lipschitz-continuous on compact subsets of its domain. Let $C$ be the Lipschitz constant of $\phi$ on the set $K^\epsilon$ which equals the union $\bigcup\_{x \... | 1 | https://mathoverflow.net/users/18263 | 154837 | 82,217 |
https://mathoverflow.net/questions/154831 | 4 | Let $k$ be a perfect field, let $n$ and $m$ be two positive integers.
Consider $X=\mathbb P\_k^n\times \mathbb P\_k^m$. Let $x\_0=(1,0,\cdots,0;1,0,\cdots,0)\in X$ be fixed.
For any pair of integers $(d\_1,d\_2)$, let $\mathcal O(d\_1,d\_2)=pr\_1^\ast \mathcal O(d\_1)\otimes pr\_2^\ast \mathcal O(d\_2)$.
Let $A$ ... | https://mathoverflow.net/users/42572 | A problem on Jets in algebraic geometry | Let $\mathfrak{m}$ (resp. $\mathfrak{m}\_0$) be the maximal ideal sheaf at $x$ (resp. $x\_0$). Put $L:=\mathcal{O}(d\_1,d\_2)$.
If I understand correctly, you are asking whether the restriction map
$$H^0(X,L)\rightarrow (L/\mathfrak{m}^rL)\times (L/\mathfrak{m}\_0^rL)$$ (or maybe $\mathfrak{m}^{r+1}$, this depends o... | 6 | https://mathoverflow.net/users/40297 | 154842 | 82,219 |
https://mathoverflow.net/questions/154825 | 2 | I'm wondering what the precise relationship is between an algebraic stack being locally of finite presentation and being limit preserving. Under some mild hypotheses on the diagonal (in force throughout the book), Prop. 4.18 of the book by Laumon--Moret-Bailly shows that a stack locally of finite presentation over some... | https://mathoverflow.net/users/45657 | Algebraic stacks: limit preserving versus locally of finite presentation | Suppose that $\mathcal{X}$ is an algebraic stack which is limit preserving on objects over $S$. Suppose that $U \to \mathcal{X}$ is a smooth surjective map from a scheme. Then $U \to S$ is limit preserving by the results of [Section Tag 06CT](http://stacks.math.columbia.edu/tag/06CT). Thus $U$ is locally of finite pres... | 5 | https://mathoverflow.net/users/44862 | 154844 | 82,221 |
https://mathoverflow.net/questions/154816 | 0 | Let $G=(G\_1, G\_0, s, t, u, i,\circ)$ be a groupoid, where $s, t$ are source and target maps, $i$ is the inverse, $u$ is the unit, and $\circ$ is the composition.
Denote $\underline{G\_1}, \underline{G\_0}$ the trivial groupoid on $G\_1, G\_0$ respectively. There are morphisms of Lie groupoids $s:\underline{G\_1}\to... | https://mathoverflow.net/users/7341 | Groupoid as a 2-coequaliser | Your claim is incorrect because you truncated the simplicial diagram too much. Indeed, if what you said were true, then the isomorphism class of a group would be determined by its cardinality, but this is obviously not true.
What you are missing is the data that tells us about composition. We need to look at the dia... | 3 | https://mathoverflow.net/users/11640 | 154849 | 82,225 |
https://mathoverflow.net/questions/128736 | 2 |
>
> Is it possible to construct the midpoint of a segment in the hyperbolic plane
> using the *set square* only?
>
>
>
With the *set square* one can
* draw the line through the given two points and
* drop the perpendicular from the given point to the given line.
---
The following construction produce ... | https://mathoverflow.net/users/10330 | Mid point with set square? | Finally, I realized that the answer is NO.
Let us think of hyperbolic plane as about subset of projective plane for $\mathbb R^{2,1}$.
Note that in this model, the set square tool is the same as hyperbolic-cross-product tool;
i.e. for any two vectors $u$ and $v$
you can construct the vector
$$w=J(u\times v),$$
wh... | 1 | https://mathoverflow.net/users/10330 | 154857 | 82,227 |
https://mathoverflow.net/questions/154864 | 1 | What is knowns about Weierstrass points on modular curves? Are there any explicit formulas of them, or any information about Weierstrass gaps? I am interested in (compactifications of) the quotients of the upper half plane by $\Gamma(n)$, $\Gamma\_1(n)$ or $\Gamma\_0(n)$. The level $n$ may be assumed to be prime, if th... | https://mathoverflow.net/users/38468 | Weierstrass points on modular curves | You might try google or MathSciNet yourself before asking on MathOverflow. It took 1 minute to find the following three articles:
MR2059755 (2006b:11056) Ahlgren, Scott . The arithmetic of Weierstrass points on modular curves X0(p).
Galois theory and modular forms,
3--12, Dev. Math., 11, Kluwer Acad. Publ., Boston... | 7 | https://mathoverflow.net/users/11926 | 154865 | 82,231 |
https://mathoverflow.net/questions/154773 | 1 | Let $\mathcal{D}(m)$ be the set of [phase-type distributions](http://en.wikipedia.org/wiki/Phase-type_distribution) constructed from $m+1$-state Markov chains. Recall that the *coefficient of variation* of a distribution $D$ is the ratio of the standard deviation to the mean, which we will denote by $C(D)$.
Let
$$ F(m)... | https://mathoverflow.net/users/8938 | Minimal variance for phase-type distributions? | I haven't checked the details, but Aldous and Shepp's paper ["The least variable phase-type distribution is Erlang"](http://ftp.stat.berkeley.edu/~aldous/Papers/me32-scan.PDF) seems to show that $F(m)=\frac{1}{\sqrt{m}}$ is the minimum. The standard deviation over expectation is the square root of the coefficient of va... | 1 | https://mathoverflow.net/users/5963 | 154872 | 82,233 |
https://mathoverflow.net/questions/154852 | 1 | Perhaps this is a trivial question, but I have no idea how to justify it.
Call a pair of groups $(G\_1, G\_2)$ ring-compatible if $G\_1$ is abelian and there exists a ring $R$ with addition and multiplication $(+, \ast)$ say satisfying the condition that $(R, +)$ is isomorphic to $G\_1$ and the group of units of $R$,... | https://mathoverflow.net/users/10898 | A basic question about rings | The answer for your question is NO, not every pair of groups is ring compatible, even if the first group is abelian. There are abelian groups $G\_1$ (called Nil groups), in which only zero multiplication turns $G\_1$ into a ring. A complete discussion of your question can be find in
"L. Fuchs, Infinite abelian groups"
... | 5 | https://mathoverflow.net/users/44949 | 154882 | 82,238 |
https://mathoverflow.net/questions/154887 | 1 | Is there an example of two groups $G\_{1}, G\_{2}$ such that there are
two non isomorphic ring $R\_{1}$ and $R\_{2}$ such that the additive group of both rings is isomorphic to $G\_{1}$ and their unit groups is isomorphic to $G\_{2}$?
Lets generalize this question as follows:
Is there an example of two groups $G\_{... | https://mathoverflow.net/users/36688 | A question in ring theory | Take $\mathbb{R}$ and $\mathbb{R}[x]$. All units in the latter are constant polynomials, so the unit groups are isomorphic. The additive groups of both have continuum dimension as rational vector spaces, so they are isomorphic. But clearly they are not isomorphic.
It seems the same argument shows that we could take ... | 7 | https://mathoverflow.net/users/2926 | 154888 | 82,240 |
https://mathoverflow.net/questions/154861 | 6 | Let $\mathbb F$ be a finite field with characteristic 2 and let $S \in M(2k, 2k, \mathbb F)$ be the matrix defined as follows
$$ S=\left[\begin{array}{ccccccc}
0 & \cdots & & & 0 & 1 & s\_1 \\
0 & \cdots & & & 0 & s\_1 & s\_2 \\
0 & \cdots & & 1 & s\_1 & s\_2 & s\_3 \\
0 & \cdots & 0 & s\_1 & s\_2 & s\_3 & s\_4 \... | https://mathoverflow.net/users/45664 | On the rank of a matrix $S$ with coefficients in $\mathbb F_{2^m}$ | The result is true indeed. Here is a rather technical solution. I work by induction over $k$, thus denoting by $S\_k$ the above $(2k)\times(2k)$ matrix. Let us transform the matrix through the following series of row and column operations:
$C\_{2k} \leftarrow C\_{2k}+s\_1 C\_{2k-1}$ and
then, for *odd* $j$ from $3$ to ... | 5 | https://mathoverflow.net/users/34951 | 154914 | 82,252 |
https://mathoverflow.net/questions/145630 | 0 | I have to calculate analytically this integral:
$$
{\rm J}\left(q\right)
=
\int\_{0}^{\infty}{{\rm d}x \over x^{q}\left({\rm e}^{kx}-1\right)}
$$
where $-1\le q\le N$
with:
$N\in\mathbb{N}$ and $q\in\mathbb{N}$, $k\le 5\times10^{-5}$
I didn't find anything on the Gradshteyn Ryzhik and Mathematica isn't able to integrat... | https://mathoverflow.net/users/21258 | Definite integral of a function containing an exponential | $\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}%
... | 2 | https://mathoverflow.net/users/45677 | 154917 | 82,255 |
https://mathoverflow.net/questions/154928 | 5 | I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question.
For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ be its root system.
**Question.** Can one "roughly" classify, up to isomorphism, all rank $32$ even unimodular latti... | https://mathoverflow.net/users/4333 | Even unimodular lattices with root system $32 A_1$ | Even unimodular lattices of rank $n$ with root system $n\cdot A\_1$
correspond bijectively with Type II codes of length $n$ and minimal
weight at least $8$. ("Type II" = self-dual and doubly even.)
The results you quote for $n=8,16,24$ correspond to the fact that
there's no such code for $n=8$ and $n=16$,
and a unique ... | 11 | https://mathoverflow.net/users/14830 | 154931 | 82,261 |
https://mathoverflow.net/questions/154933 | 9 | In representation theory, there are the related concepts of *weights* and *roots*. Since both are kinds of generalised eigenvalues, and eigenvalues are roots of e.g. the characteristic polynomial, the word "root" makes sense to me (at least, the question is reduced to why zeros of polynomials / equations are called "ro... | https://mathoverflow.net/users/27465 | Origin of the term "weight" in representation theory | Robert Bryant's comment motivates me to mention the "weighty" historical monograph *Emergence of the Theory of Lie Groups* (Springer, 2000) written by Thomas Hawkins. As usual with terminology such as "weight", the history reaches back into nineteenth century's invariant theory (Cayley, G. Kowalewski) but becomes most ... | 6 | https://mathoverflow.net/users/4231 | 154947 | 82,267 |
https://mathoverflow.net/questions/154964 | 5 | From the theorem of classification of finitely generated abelian groups, we can see that every finitely generated commutative group can be considered as the additive structure underlying (at least) one unitary and commutative ring. My question is about possible generalization of
this result.
Question: Is it true that w... | https://mathoverflow.net/users/30395 | Is every commutative group structure underlying at least one (unitary, commutative) ring structure | As Sasha Anan'in already mentioned, there are counterexamples like $\mathbb{Z}/p^\infty = \mathbb{Z}[\frac{1}{p}]/\mathbb{Z}$ and $\mathbb{Q}/\mathbb{Z}$. The common feature of these groups is that there are divisible *and* torsion and therefore $A\otimes\_\mathbb{Z}A=0$. A ring $A$ has a surjective and hence non-zero ... | 16 | https://mathoverflow.net/users/3041 | 154971 | 82,275 |
https://mathoverflow.net/questions/154963 | 2 | Let $G$ be a (non-abelian) group, and let $G\_2$ denote its commutator subgroup. Then the abelianization $G\_2^{ab} = H\_1(G\_2,\mathbb{Z})$ is a module over the group ring $\mathbb{Z}[G^{ab}]$. The action of $G^{ab}$ on $G\_2^{ab}$ is induced by the conjugation action of $G$ on $G\_2$. I am interested in understanding... | https://mathoverflow.net/users/24525 | Module structure of the abelianization of the commutator subgroup | To answer your **Q1**, the group in question is $H\_1$ of the complex
$$0\to\Lambda a\_1\oplus\ldots\oplus\Lambda a\_n\overset\partial\to\Lambda\to0,$$
where $\Lambda$ is your ring of Laurent polynomials and $\partial\colon a\_i\mapsto(x\_i-1)$. (Clearly, the $a\_i$ are in a correspondence with the generators of $F\_n... | 1 | https://mathoverflow.net/users/44953 | 154973 | 82,276 |
https://mathoverflow.net/questions/154884 | 8 | The following statement can be proven using elementary submodels and sufficiently generic conditions:
"If $S \subseteq cof(<\kappa) \cap \kappa^+$ is stationary, and $\kappa^{<\kappa} =\kappa$, then the stationarity of $S$ is preserved by $\kappa$-closed forcing."
If we just assume $\kappa$ is regular, do we need t... | https://mathoverflow.net/users/11145 | Preservation of some stationary sets by sufficiently closed forcing | No: your cardinal arithmetic assumption can be dropped when $\kappa$ is regular.
This follows from $I[\lambda]$ analysis.
$S \subseteq \lambda \cap \mathrm{cof}(\kappa)$ (where $\kappa < \lambda$ are regular) is said to be $\textit{in $I[\lambda]$}$ if there is a sequence of sets $\langle a\_i \mid i < \lambda \ran... | 7 | https://mathoverflow.net/users/45416 | 154996 | 82,283 |
https://mathoverflow.net/questions/154989 | 1 | Let $ A $ be an $ \mathcal{H}^1$-measurable subset of $ \mathbb{R} $ and $ \gamma \colon A \subseteq \mathbb{R} \to \ell^\infty $ be a Lipschitz mapping with the Lipschitz constant $ L $. Also, assume that for all $ n \in \mathbb{N} $ and $ \mathcal{H}^1$-a.e. $ t \in A $,
$$
\gamma\_n'(t) = 0,
$$
where $ \gamma\_n $ i... | https://mathoverflow.net/users/3124 | Is the speed of a curve in $ \ell^\infty $ zero a.e. if the derivative of each component is zero a.e.? | Is ${\cal H}^1$ one-dimensional Hausdorff measure? So this is just Lebesgue measure?
Then I think the answer is yes, specifically the desired limit is zero at every Lebesgue point of $A$. If $t$ is a Lebesgue point then for any $\epsilon > 0$ we can find $r > 0$ such that $\frac{\mu(A \cap I)}{\mu(I)} \geq 1-\epsilon... | 2 | https://mathoverflow.net/users/23141 | 154998 | 82,284 |
https://mathoverflow.net/questions/155008 | 3 | A set $\Sigma$ of group identities is called bounded if there is $n\geq 1$ such that for any $(w\approx 1)\in \Sigma$, we have $w\in F(x\_1, \ldots, x\_n)$. A variety $\mathbf{V}$ is called bounded defined if $\mathbf{V}=Mod(\Sigma)$ for some bounded set $\Sigma$.
Question: Is there an example of a bounded defined n... | https://mathoverflow.net/users/44949 | Non finitely based varieties of groups defined by finitely many variables | Yes. It is usually called a *non-finitely based variety of finite axiomatic rank*.
Actually, one of the first examples of non-finitely based variety of groups has axiomatic rank two; [Adian proved](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=2442&option_lang=eng) that the following set of ide... | 5 | https://mathoverflow.net/users/24165 | 155018 | 82,296 |
https://mathoverflow.net/questions/154961 | 11 | The Manin-Drinfeld theorem asserts that a divisor on the compact modular curve $X\_0(N)$ which is supported on the cusps is torsion.
Equivalently, if $Y\_0(N)$ is the open modular curve, the mixed Hodge structure $H^1(Y\_0(N), \mathbb{Q})$ splits.
Are there some kind of generalizations of this theorem to higher-d... | https://mathoverflow.net/users/45693 | higher dimensional analogues of the Manin-Drinfeld theorem | This is perhaps more of a comment than an answer.
In various texts by Günter Harder he uses the terminology that *the Manin-Drinfeld principle holds* for the Shimura variety $X$, the coefficient system $\mathbb V$ and the cohomology theory $H(-)$ (usually $\ell$-adic cohomology or mixed Hodge theory) if there is a di... | 5 | https://mathoverflow.net/users/1310 | 155019 | 82,297 |
https://mathoverflow.net/questions/154990 | 7 | Let $G\_1$ and $G\_2$ be countable abelian groups, and let $\iota\colon G\_1\to G\_2$ be an injective group homomorphism (so that we may regard $G\_1$ as a subgroup of $G\_2$). Suppose that for every finitely generated subgroup $K$ of $G\_2$, there is a map $\pi\_K\colon K\to G\_1$ such that $\iota \circ \pi\_K\circ \i... | https://mathoverflow.net/users/29566 | Does an injective group homomorphism between countable abelian groups that splits over every finitely generated subgroup, necessarily split? | Here is another concrete example constructed as in Yves Cornulier's comment. Let $G\_2$ be the group defined by the (abelian group) presentation
$G\_2=\langle\ x,y\_i\ (i \ge 0) \mid y\_i^2 = y\_{i-1}x\ (i \ge 1)\ \rangle$
and $G\_1 = \langle x \rangle$.
Any finitely generated subgroup is contained in $\langle y... | 4 | https://mathoverflow.net/users/35840 | 155020 | 82,298 |
https://mathoverflow.net/questions/154851 | 2 | The question [Getting a bound on the coefficients of the factor polynomial](https://mathoverflow.net/questions/108726/getting-a-bound-on-the-coefficients-of-the-factor-polynomial?rq=1) got very nice answers on Gelfond's theorem. But for work on proof theory of arithmetic I want a proof in arithmetic. The published proo... | https://mathoverflow.net/users/38783 | Elementary proof of bounds on factor polynomials | The Henselian strategy of Zassenhaus uses $p$-adic bounds rather than the integer exponential bounds of Gelfond's theorem. Weinberger and Rothschild develop this approach over algebraic number fields, by an algorithm explicit enough to be its own proof, in ``Factoring Polynomials Over Algebraic Number Fields,'' ACM Tra... | 1 | https://mathoverflow.net/users/38783 | 155036 | 82,301 |
https://mathoverflow.net/questions/154630 | 4 | Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 2}(\partial\Omega)$ to be space of functions $u \in L^2(\partial\Omega)$ such that
$$|u|\_{H^{\frac 12}(\partial\Omega)} ... | https://mathoverflow.net/users/43959 | Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous? | Lions and Magenes define $H^\frac12$ by complex interpolation, hence your desired property holds by assumption; but they always assume $\Omega$ to have smooth boundary.
In the books of Adams and Grisvard you will find some related results, but - as far as I have (quickly) seen - not exactly what you look like.
| 2 | https://mathoverflow.net/users/26039 | 155072 | 82,330 |
https://mathoverflow.net/questions/155092 | 1 | I read the paper "Kurosh rank of intersections of subgroups of free products of orderable groups", which can be found here: <http://arxiv.org/abs/1109.0233>.
The proposition 2.3 states that: Let $H$ be a subgroup of $G$. Let $T'$ be a $H$-subtree of $T$. Then $κ\_{T}(H)$ = $κ\_{T′}(H)$.
In particular, $T\_{H}/H$ is fini... | https://mathoverflow.net/users/42001 | Bass- Serre theory exercise | First you need that every vertex of $T$ whose $H$-stabilizer is nontrivial is contained in the $H$-minimal subtree $T\_H \subset T$, and so each such vertex is contained in $T'$. This implies that $c\_T(H)=c\_{T'}(H)$ where $c$ is the number of orbits of vertices with nontrivial stabilizer.
Next you need that the eq... | 2 | https://mathoverflow.net/users/20787 | 155096 | 82,335 |
https://mathoverflow.net/questions/154764 | 2 | Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s\_\delta(x), W^u\_\delta(x)$ for sufficiently small $\delta>0$. its well known that $W^s(x)=\cup\_{n\geq0}f^{-n}(W^s\_\delta(f^n(x))) ... | https://mathoverflow.net/users/43741 | the union of local stable manifolds along local unstable manifolds | The local product structure says that for hyperbolic diffeos (it works also for flows), given two points $z$ and $z'$ in a small neighbourhood of $y$, then $W^s\_\delta(z)$ and $W^u\_\delta(z')$ intersect at exactly one point, often denoted by $[z,z']$.
Apply this property on $B(y,r)$ for $r$ small enough to any $z\i... | 4 | https://mathoverflow.net/users/30691 | 155106 | 82,339 |
https://mathoverflow.net/questions/155083 | 4 | I need to prove the following inequality: for any $Z=(z\_1,\dots,z\_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$
\begin{equation}
\left|\left|\sum\_{j=1}^l z\_j\right|^p-\sum\_{j=1}^l\left|z\_j\right|^p\right|\leq C\_{p,l}\sum\_{i\neq j}\left|z\_i\right|\left|z\_j\right|^{P-1}.
\end{equation}
| https://mathoverflow.net/users/45729 | A homogeneous but slightly asymmetric inequality | Here is a somewhat silly way to do it. You need to prove that:
$$\frac{\left|\sum\_{j=1}^l z\_j\right|^p-\sum\_{j=1}^l\left|z\_j\right|^p}{\sum\_{i\neq j}\left|z\_i\right|\left|z\_j\right|^{P-1}}$$
is bounded function on the unit sphere $S^{l-1}$. $S^{l-1}$ is compact, and this function is continuous away from the ... | 4 | https://mathoverflow.net/users/18060 | 155110 | 82,342 |
https://mathoverflow.net/questions/155085 | 5 | There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: how can one recognize those metric spaces that are isometrically embeddable into Hilbert spaces?
Later edit:
I have r... | https://mathoverflow.net/users/54780 | Isometric embeddings of metric spaces in Hilbert spaces | The answer was given in the papers of I. Schoenberg and von Neumann,
MR1501980, MR1503439, MR0004644.
| 4 | https://mathoverflow.net/users/25510 | 155115 | 82,344 |
https://mathoverflow.net/questions/155112 | 4 | Given a topological group, say $G$, it is well known that the classifying space $BG$ classifies $G$ principal bundles. Under some mild assumption on $G$, one model of $BG$ is the geometric realization of the nerve of the one object groupoid $G⇉\*$.
Let $P\to M$ be a $G$ principal bundle. We can form the action groupo... | https://mathoverflow.net/users/7341 | Classifying map of a principal bundle | If $M$ is paracompact, or more generally $P$ is trivialised over a [numerable open cover](http://ncatlab.org/nlab/show/numerable+open+cover), this map is indeed the classifying map for $P\to M$. Note that the universal bundle $EG \to BG$ is given by the geometric realisation of $G\rtimes G \to \ast \rtimes G$, since yo... | 4 | https://mathoverflow.net/users/4177 | 155117 | 82,346 |
https://mathoverflow.net/questions/155111 | 2 | Let $P\_n$ be the space of polynomials in $n$ variables over a field of characteristic 0.
I'm very sure that the space spanned by powers of linear functions is the whole space $P\_n$.
Anyone can come up with a proof, or indicate a reference for it?
Thanks a lot.
| https://mathoverflow.net/users/5082 | Powers of linear functions span the space of polynomial functions? | Yes, actually, the space of homogeneous degree $d$ polynomials in $(n+1)$ variable is generated by
$$(x\_0+i\_1x\_1+\ldots+i\_nx\_n)^d,\quad 0\le i\_1,\ldots,i\_n\le d.$$
For proof, write it as a binomial
$$[(x\_0+i\_1x\_1+\ldots+i\_{n-1}x\_{n-1})+i\_nx\_n]^d$$
and, using Vandermonde, get all products $(x\_0+i\_1x\_1+... | 8 | https://mathoverflow.net/users/44953 | 155120 | 82,348 |
https://mathoverflow.net/questions/155101 | 8 | I am taking a course this semester on QFT, which deals much with constructive quantum field theory. Some of its topics so far involve relationships between non-Gaussian probability measures,Feynman path integrals and so on. While I am totally new to the subject of QFT (I only have some basic ideas of classical mechanic... | https://mathoverflow.net/users/27040 | References request: constructive quantum field theory | The standard reference for constructive QFT is the classic book by J. Glimm and A. Jaffe, [*Quantum Physics: a Functional Integral Point of View*](http://rads.stackoverflow.com/amzn/click/0387964770) (2nd. ed., Springer-Verlag, 1988). It is certainly more than satisfactory from the viewpoint of mathematical rigor, it h... | 16 | https://mathoverflow.net/users/11211 | 155122 | 82,349 |
https://mathoverflow.net/questions/155114 | 1 | EDIT: This is a case of being too wrapped up in a formulation
($e\_j,p\_i,$ and the like) to try something simple. It did not
occur to me to pull exp to the outside in the weeks I have
stared at this. Thanks to Aaron Meyerowitz and Brendan McKay for
a humbling revelation. I leave it to others to decide if this is
an o... | https://mathoverflow.net/users/3402 | Truncated sums of symmetric polynomials; reference request for an algebraic derivation | Try this:
$$\prod\_{1 \leq j}^{\infty}(\sum\_{0 \leq l}^{\infty} [ - \sum\_{i=0}^k\frac{x\_i^j}{j} ]^l/l! ])=$$
$$\prod\_{1 \leq j}^{\infty}\mathop{exp}\left( - \sum\_{i=0}^k\frac{x\_i^j}{j}\right)=$$
$$\mathop{exp}\left(\sum\_{j=1}^{\infty}\left( - \sum\_{i=0}^k\frac{x\_i^j}{j}\right)\right)=$$
$$\mathop{exp}\left(\s... | 2 | https://mathoverflow.net/users/8008 | 155123 | 82,350 |
https://mathoverflow.net/questions/155102 | 2 | Studying some problem I've arrived to the following notion.
Let a $2r$-regular graph $G$ be called *neighbour-matching* if $N(v) = rK\_2.$ In other words, the neighbourhood of any vertex induces a matching in $G.$
Doing some computation, it appears that such graphs are quite rare. There is one such graph on 9 vert... | https://mathoverflow.net/users/1737 | Regular graphs whose neighbourhoods induce matchings | This answer just suggests a couple of directions that you can continue searching, rather than being definitive.
Brouwer, Cohen and Neumaier use the term *edge regular* for the property that every pair of adjacent vertices have the same number of common neighbours (in other words, like strongly regular but dropping th... | 6 | https://mathoverflow.net/users/1492 | 155128 | 82,352 |
https://mathoverflow.net/questions/155121 | 6 | Does anyone have a reference for a good description of representations of $S\_{n}$ obtained by inducing up from $C\_{S\_{n}}(\pi)$, for some element $\pi$ of $S\_{n}$? (I'd prefer an efficient combinatorial description if there is one.)
This seems like it should be doable, since I'm fairly sure that centralizers of e... | https://mathoverflow.net/users/45745 | Representations of S_n induced from centralizers of elements | I gave an answer for inducing the trivial representation in a comment at [Decomposing the conjugacy representation of Sym$(n)$ for small $n$](https://mathoverflow.net/questions/153991). It is in terms of a plethysm that seems to be just as intractable as the notorious $s\_m[s\_n]$.
| 7 | https://mathoverflow.net/users/2807 | 155132 | 82,356 |
https://mathoverflow.net/questions/155126 | 11 | What is the easiest construction of the Casson invariant? The original construction using representation spaces (as found, for instance, in Akbulut-McCarthy) is very technical since you have to perturb things to make them transverse.
One possibility would be to do some kind of Kirby calculus construction. Such a thin... | https://mathoverflow.net/users/45747 | Construction of the Casson invariant | Hoste has a combinatorial formula for the Casson invariant for integral homology $3$-spheres. Let $\Sigma$ be an integral homology $3$-sphere obtained via surgery on a framed link $L = K\_1 \cup \dots \cup K\_n$ in $S^3$ with framings $1/q\_i$. Additionally, we assume $\mathrm{lk}(K\_i, K\_j) = 0$ for $i \neq j$ (this ... | 21 | https://mathoverflow.net/users/21375 | 155134 | 82,358 |
https://mathoverflow.net/questions/155080 | 0 | Lindemann's prove of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative.
In the following, *trisectable* always means with straightedge and compass alone.
I could however find nothing but some vague statements abou... | https://mathoverflow.net/users/31310 | Characterization of Angles Trisectable with Straightedge and Compass | There is an aspect of the word "construct" that comes off a bit sour here. There are infinitely many angles such that $\theta$ and $\theta/3$ are both constructable. However, the set of constructable angles $\theta$ for which $\theta/3$ is not constructable is dense. By continuity, it follows that there is no procedure... | 7 | https://mathoverflow.net/users/3324 | 155136 | 82,359 |
https://mathoverflow.net/questions/154514 | 6 | This question was posed by my colleague Torbjörn Lundh in his paper [Which Ball is the Roundest? A Suggested Tournament Stability Index](http://www.math.chalmers.se/~torbjrn/finalmanuscript_tournament.pdf), Journal of Quantitative Analysis in Sports 2(3), 2006. We have discussed it a number of times without finding a s... | https://mathoverflow.net/users/14302 | Best ranking in tournament: polynomial time algorithm? | The problem you are looking for is called the Feedback Arc Set problem for tournaments (FAST). It was proved to be NP-hard by Alon:
<http://www.tau.ac.il/~nogaa/PDFS/paley.pdf>
and separately Charbit, Thomassé and Yeo:
<http://www.liafa.jussieu.fr/~charbit/Recherche/publis/minfastour.ps>
| 3 | https://mathoverflow.net/users/6325 | 155164 | 82,367 |
https://mathoverflow.net/questions/155163 | 8 | If $P$ is a notion of forcing in $M$, then $G$ is a $P$-generic filter over $M$ if $G\subseteq P$ is a filter, and for every $D\in M$ which is a dense subset of $P$, $G\cap D\neq\varnothing$.
Equivalently we can replace $D$ being dense by being pre-dense, open dense, or a maximal antichain.
Given such a generic fil... | https://mathoverflow.net/users/7206 | Genericity by names | Your question is fairly open-ended, but here is one way to do it. For any antichain $A\subset P$, consider the name $\dot a\_A=\{\langle\check a,a\rangle\mid a\in A\}$, which is a mixture of the elements of $A$ on the antichain $A$. This is the name that is trying to be an element of $A$, with values determined by $A$ ... | 6 | https://mathoverflow.net/users/1946 | 155169 | 82,369 |
https://mathoverflow.net/questions/155157 | 2 | I have a set of Bernoulli random variables $X\_1,X\_2,\ldots, X\_n$ and I would want to bound the joint probability $P(X\_1=0,X\_2=0,\ldots, X\_n=0)$ using the norm $\lvert \lvert \mathbf{p}\rvert \rvert$, where $$\mathbf{p}:=[P(X\_1=1)\quad P(X\_2=1)\ldots P(X\_n=1)]^\top$$
For the lower bound, I use Bonferroni's in... | https://mathoverflow.net/users/45764 | Relating joint probability to norm of vector of probabilities | $\mathsf{P}\{X\_1 = 0, \dots, X\_n = 0\} \le \min\_i \mathsf{P} \{X\_i = 0\} = 1 - \max\_i p\_i \le 1 - N^{-1/2} \Vert p \Vert\_2$
| 1 | https://mathoverflow.net/users/22758 | 155174 | 82,371 |
https://mathoverflow.net/questions/155175 | 1 | In Andreas Blass's famous paper `Seven Trees in One', the existence of a natural bijection between binary trees and 7-tuples of binary trees is related to the equation $T^7 = T$ being satisfied by a complex number which satisfies the equation $T = 1 + T^2$ (the categorification of which is satisfied by the set of binar... | https://mathoverflow.net/users/39521 | Natural bijection between sets with coloured elements? | If I take an $m$-coloured set and an $n$-coloured set, their disjoint union (using disjoint sets of colours) is an $(m+n)$-coloured set. Conversely, if I have an $(m+n)$-coloured set, it has a subset which uses the first $m$ colours, and another subset which uses the other $n$...
While this answer is (perhaps unexpec... | 6 | https://mathoverflow.net/users/14901 | 155176 | 82,372 |
https://mathoverflow.net/questions/155137 | 2 | The genus $g$ Torelli group $I\_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface.
The first paper I am aware of that uses the name "Torelli group" for this group is
Johnson, Dennis A survey of the Torelli group. Low-dimensional topology (Sa... | https://mathoverflow.net/users/45747 | Origin of the name "Torelli group" | As abx noted, the real object that algebraic geometers and complex analysts considered was not the Torelli group per se, but rather *Torelli space*, which is the quotient of Teichmuller space by the Torelli group. It's always hard to trace the origins of a name like this, but I believe that the terminology was introduc... | 6 | https://mathoverflow.net/users/317 | 155184 | 82,375 |
https://mathoverflow.net/questions/155180 | 9 | Where do people essentially use the reductive groups in the theory of GIT? Or how does reductive groups simplify the constructions in GIT?
I found that the property of completely reducible of reductive groups (in character 0) is fascinating (see Mumford's GIT book page 26-27), but I don't know if it is this property ... | https://mathoverflow.net/users/29730 | Why people usually consider reductive groups in GIT? | Since the comments are already getting long, I'll add this in community-wiki format to clarify a few points. I should emphasize that I'm not at all a specialist in GIT but have dealt with neighborhing problems involving algebraic groups.
First, there are fundamental differences between characteristic 0 (where Mumfor... | 10 | https://mathoverflow.net/users/4231 | 155191 | 82,377 |
https://mathoverflow.net/questions/155078 | 7 | Let $X$ be a CW complex with given cohomologies $H^n(X; \mathbb{Z}) = G$, $H^m(X; \mathbb{Z}) = H$ and other reduced cohomologies are zero. Which additional algebraic information/structures do I need to identify the homotopy type of $X$? Similar question for three or more nonzero cohomologies. Can the answer be similar... | https://mathoverflow.net/users/45731 | Moore decomposition, dual to Postnikov tower | This isn't really an answer but it's a bit long for a comment.
Do you really mean cohomology rather than homology? The dual of a Postnikov tower is usually considered to be a homology decomposition, building a space up one homology group at a time. A space with only two nonzero cohomology groups could have up to four... | 6 | https://mathoverflow.net/users/23571 | 155197 | 82,380 |
https://mathoverflow.net/questions/155189 | 4 | I am thankful of Anton Klyachko who introduced axiomatic rank to me: the axiomatic rank of a variety is the minimum number of variables which we need to define that variety by identities.
It seems clear that the axiomatic rank of the variety of groups is equal to three. But why?
| https://mathoverflow.net/users/44949 | Why the axiomatic rank of the variety of groups is equal to three? | Let $V$ be the variety axiomatized by
\begin{align\*}
x+y&=y+x,\\
x+(x+y)&=y,\\
x+0&=x,\\
-x&=x.
\end{align\*}
(This also implies $x+x=0$.) Straightforward induction on the length of a term shows that every $2$-variable term is equal over $V$ to one of $0,x,y,x+y$, which implies that $V$ axiomatizes all $2$-variable id... | 4 | https://mathoverflow.net/users/12705 | 155200 | 82,382 |
https://mathoverflow.net/questions/154968 | 4 | If we have nonnegative $V \in L^1\_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for example, E.B. Davies in *Heat Kernels and Spectral Theory*. He also proves that $H$ is the infinitesimal generator of an ultr... | https://mathoverflow.net/users/12968 | (Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation | There is an old [paper by Ball](http://people.maths.ox.ac.uk/~ball/Papers/Ball_AMS_77.pdf) where he discusses the abstract connections between weak solutions and semigroup orbits. He proves that the mild solution is essentially always the weak solution for strongly continuous semigroups.
Theorem 3.1.7 from the [monog... | 2 | https://mathoverflow.net/users/12898 | 155202 | 82,383 |
https://mathoverflow.net/questions/155209 | 5 | Let $E$ be a vector bundle (i.e. locally free $\mathcal{O}\_X$-module) on some smooth algebraic variety $X$ and let $\nabla: E \to E \otimes \Omega^1\_X$ be an integrable connection.
I have seen that $\nabla$ induces a connection $\nabla^\vee$ on the dual vector bundle $E^\vee=\mathcal{Hom}\_{\mathcal{O}\_X}(E, \mat... | https://mathoverflow.net/users/45791 | how is the dual connection defined? | 1. The dual connection $\nabla^\vee$ on $E^\vee$ is defined by
$$\langle \nabla^\vee \phi, s\rangle = d\langle \phi, s\rangle - \langle\phi, \nabla s\rangle,\quad \forall \phi\in E^\vee, s\in E,$$
where $\langle-,-\rangle:E^\vee\otimes E\rightarrow \mathcal{O}$ is the pairing. One can check that the formula is $\mathca... | 8 | https://mathoverflow.net/users/18512 | 155216 | 82,387 |
https://mathoverflow.net/questions/155237 | 8 | The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For instance, in $L$, there is a definable well-ordering of the real numbers (since there is a definable well-ordering of the uni... | https://mathoverflow.net/users/24611 | In $L$, does there exist a definable non-principal ultrafilter on $\mathbb{N}$ | Yes, but it's not particularly nice: since $L$ has a definable well-ordering of the sets of reals (in fact, of all of $L$) coming from the $L$-hierarchy itself, there is a formula defining the "least" (in that well-ordering) ultrafilter, $U$.
(This $U$ isn't the only naturally definable ultrafilter in $L$; we could a... | 12 | https://mathoverflow.net/users/8133 | 155238 | 82,393 |
https://mathoverflow.net/questions/155229 | 14 | I've often heard that Lord Kelvin was one of the first people to study knot theory, as he hypothesized that atoms were knots in the ether. I assume that he had some compelling evidence for this fact.
What properties of knots did Lord Kelvin consider similar to atomic properties?
| https://mathoverflow.net/users/27933 | What properties of knots lead Lord Kelvin to hypothesize that atoms were knots in the ether? | It wasn't the properties of knots, but rather the hydrodynamical properties of closed fluids. It stems from the most basic facts in fluid mechanics:
Kelvin proved (assuming inviscid flows) that a closed curve $C$ of fluid particles (velocity field $u$) has its circulation $\oint\_Cu\cdot dl$ independent of time. His... | 20 | https://mathoverflow.net/users/12310 | 155241 | 82,395 |
https://mathoverflow.net/questions/155247 | 13 | We know from Hartshorne's "Algebraic geometry" that if a curve $C$ is smooth then there exists a closed immersion of $C$ into $\mathbb{P}^3$. Does this still hold if $C$ is not generically reduced or singular at finitely many points?
| https://mathoverflow.net/users/45397 | Can any curve be embedded into $\mathbb{P}^3$? | No. If a curve embeds into $\mathbb{P}^3$, its tangent space at every point has dimension $\leq 3$. This is a strong restriction on the possible singularities. For instance,
a curve locally isomorphic to the union of the lines $x=y=z=0$, $x=y=t=0$, $x=z=t=0$, $y=z=t=0$ in $\mathbb{A}^4$ cannot be embedded in $\mathbb{... | 25 | https://mathoverflow.net/users/40297 | 155249 | 82,398 |
https://mathoverflow.net/questions/155258 | 8 | Let $\pi:\mathcal{X} \to B$ be a family (flat, projective, surjective morphism) of projective curves (not necessarily reduced) where $B$ is smooth, irreducible. Suppose that for some closed point $b\_0 \in B$, $\pi^{-1}(b\_0)=\mathcal{X}\_{b\_0}$ can be embedded into $\mathbb{P}^n$ for some integer $n$. Does there exis... | https://mathoverflow.net/users/45397 | Projective embedding in families of curves | No, that is not true. I am sure that somebody else has answered this on MO before. I guess the simplest example is a family of genus 6 curves, where some fibers are embeddable as plane quintics. If memory serves, the canonical image of a general genus 6 curve (non-hyperelliptic) is the intersection of a Pfaffian quinti... | 13 | https://mathoverflow.net/users/13265 | 155259 | 82,401 |
https://mathoverflow.net/questions/155235 | 12 | By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem is that subalgebras of $R$ seem to be 'usually' singular.
Upd. 1. I realized that $R$ actually contains plenty of smo... | https://mathoverflow.net/users/2191 | Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem | One can prove Popescu's theorem directly in this special case (due to the two strong assumptions -- characteristic $0$ and one-dimensionality -- present here). The basic point is that any 'singular' subalgebra $A \subset R$ may be resolved by a proper birational map $X \to \mathrm{Spec}(A)$ by Hironaka, and that the ma... | 5 | https://mathoverflow.net/users/45704 | 155279 | 82,410 |
https://mathoverflow.net/questions/155281 | 4 | In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel \_{p}$.
Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$.
>
> Assume that $S$ is a subvector space of $L^{q}[0, 1]$ such that the identity operator $\text{Id.}: (S, \parallel . \parallel \_{p}) \to (S, \parallel . \parallel \... | https://mathoverflow.net/users/36688 | A generalization of a theorem of Grothendieck | No. The condition implies that the subspace is isomorphic to a Hilbert space. In fact, Kadec and Pelczynski proved that a subspace of $L\_p$, $2<p<\infty$, is closed in $L\_r$ for some $r<p$ if and only if the subspace is isomorphic to a Hilbert space.
Kadec, M. I.; Pełczyński, A. Bases, lacunary sequences and comple... | 12 | https://mathoverflow.net/users/2554 | 155286 | 82,413 |
https://mathoverflow.net/questions/155275 | 1 | Given $d<k$. Let ${\cal M}\_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}\_{d\times k}(\mathbb{R})$ is a continuous matrices-valued function such that $H(x)$ is full rank for every $x \in \mathbb{R}^k$.
I'd like to construct a continuous... | https://mathoverflow.net/users/45305 | Constructing a continuous matrix valued function | I got an answer that requires an additional hypothesis:
* $\forall x$, every *principal submatrix* of $H(x)$ has to be non singular.
Here, we define the principal submatrices of a generic matrix $A\in\mathbb{R}^{d \times k}$ as the matrices $A^{(1)},\ldots,A^{(d)}$ given by $A^{(m)}\in\mathbb{R}^{m \times m}$ and $... | 1 | https://mathoverflow.net/users/41123 | 155300 | 82,418 |
https://mathoverflow.net/questions/155269 | 3 | Let $F\_n$ be a free profinite group of finite rank $n$ and let $V\_k$ denote the intersection of all open subgroups of $F\_n$ of rank at most $k$ ($k \in \mathbb{N}$).
My questions are:
>
> Can I explicitly compute the index of $V\_k$ in $F\_n$ for all $k \in \mathbb{N}$?
>
>
> Or if not, maybe for some specia... | https://mathoverflow.net/users/45818 | Intersection growth of free profinite groups | Ok, I think I understand the question now. To clarify, the free profinite group of rank $n$ may be regarded the profinite completion $\hat{K}\_n$ of $K\_n=\mathbb{Z}^{\ast n}$ the free group of rank $n$. An open subgroup $H< F\_n$ must be finite index since $F\_n$ is compact, and the cosets of $H$ cover $F\_n$. Thus, $... | 3 | https://mathoverflow.net/users/1345 | 155303 | 82,419 |
https://mathoverflow.net/questions/155304 | 4 | Let $H$ be an infinite dimensional (separable if necessary) complex Hilbert space, and denote by $K(H)$ the ideal (in $B(H)$) of compact operators on $H$. Let $G\_c=\{I+K\in B(H): I+K \text{ is invertible and } K\in K(H)\}$, and $U\_c=\{I+K\in B(H): I+K \text{ is unitary and } K\in K(H)\}$.
$G\_c$ and $U\_c$ act by l... | https://mathoverflow.net/users/16107 | Does the group of compact perturbations of the identity act transitively on the compact operators? | No. Let $e\_n$ be a basis of the Hilbert space, and let $\alpha\_n$ be any sequence converging to zero. Then you have a compact operator $K\_0$ defined by $K\_0e\_n=\alpha\_ne\_n$. Now let $K\_1$ be any other compact operator. Since compact operators are continuous weak to strong, $K\_1e\_n$ converges to zero. Conseque... | 7 | https://mathoverflow.net/users/12120 | 155311 | 82,421 |
https://mathoverflow.net/questions/155288 | 5 | If I'm attempting to mutate one arbitrarily chosen binary string $s\_a$, to another arbitrarily chosen binary string $s\_b$, in the smallest number of steps (i.e. with the smallest number of mutations) via a procedure where I decide to either:
(1) Randomly and with uniform probability choose a "0" bit and flip it to ... | https://mathoverflow.net/users/45728 | An intutive reason why a "distance" metric may be a poor one for a procedure where we attempt to modify a string (mutating 0 OR 1 bits) | I'll appeal to the linked question's answers for computational details.
Intuition only: (as requested here)
It is "easier" using your procedure to move from $00110^{n-4}$ to $11000^{n-4}$ than it is to go from $0000^{n-3}$ to $1110^{n-3}$, despite the Hamming distance being smaller in the latter case. You have to "... | 4 | https://mathoverflow.net/users/45255 | 155328 | 82,428 |
https://mathoverflow.net/questions/155205 | 22 | I raised this question in my answer to [On the rank of a matrix $S$ with coefficients in $\mathbb F\_{2^m}$](https://mathoverflow.net/questions/154861). Let $s\_1,s\_3,s\_5,\dots$ be indeterminates over the field $\mathbb{F}\_2$, and recursively set $s\_{2n}=s\_n^2$, with $s\_0=1$. Let
\begin{eqnarray\*} F(x) & = & \... | https://mathoverflow.net/users/2807 | Number of terms in certain polynomials over $\mathbb{F}_2$ | The claim is true. Lets start by simplifying the expression as follows:
$$x\left(1+\frac{1}{1+s\_1x+s\_2x^2+s\_3x^3+\cdots}\right)=\frac{\sum\_{n\geq 1}s\_{2n-1}x^{2n}}{\sum\_{n\geq 0} s\_{2n} x^{2n}}.$$
This follows because $\sum\_{n\geq 0}s\_{2n}x^{2n}=\left(\sum\_{n\geq 0}s\_nx^n\right)^2$, and some basic manipulati... | 11 | https://mathoverflow.net/users/2384 | 155330 | 82,429 |
https://mathoverflow.net/questions/155334 | 8 | Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want to investigate the question of whether $P$ is a $G$-torsor (in the fppf topology).
If I understand correctly, then si... | https://mathoverflow.net/users/8080 | Torsors and the fpqc topology | One way to define an fppf (or etale, or...) torsor is to require that $G \times\_S P \rightarrow P \times\_S P$ given by $(g, p) \mapsto (gp, p)$ is an isomorphism and that $P$ has a section fppf (resp., etale, etc.) locally. The first condition can be checked even fpqc locally: use fpqc descent for the property 'an is... | 17 | https://mathoverflow.net/users/5498 | 155337 | 82,430 |
https://mathoverflow.net/questions/155329 | 12 | I seem to remember having read that the proof-theoretic ordinal (sup of ordinals the theory can prove well-ordered) of $\mathsf{PA} + \mathsf{Con}(\mathsf{PA})$ is the same as that of $\mathsf{PA}$, namely $\epsilon\_0$. However, the one particular source where I remember reading something like this is [The Realm of Or... | https://mathoverflow.net/users/8929 | What is the proof-theoretic ordinal of PA + Con(PA), PA + Con(PA + Con(PA)) etc., and why? | This is an instance of a general phenomenon: adding true $\Pi\_1$ sentences to a reasonable theory doesn't change its proof-theoretic ordinal. $Con(T)$ is $\Pi\_1$, so if $T$ is any consistent theory, $PA+Con(T)$ still has proof-theoretic ordinal $\epsilon\_0$.
This is a standard fact in the area, though I'm not sure... | 16 | https://mathoverflow.net/users/8991 | 155338 | 82,431 |
https://mathoverflow.net/questions/155293 | 8 | Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:
* as a quotient of a semisimple real Lie group $G$ by a parabolic subgroup (subgroup containing a Borel),
* as an orbit of the isotropy representation (action of the poi... | https://mathoverflow.net/users/17064 | Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation | A real flag manifold is $M=G/P$ where $G$ is a real semisimple Lie group and $P$ is a parabolic subgroup of $G$. Here parabolic means that the Lie algebra
$\mathfrak p$ contains a minimal parabolic subalgebra $\mathfrak p\_{min}$ of $\mathfrak g$.
In order to define this, let $\mathfrak g=\mathfrak k+\mathfrak s$ be... | 9 | https://mathoverflow.net/users/15155 | 155342 | 82,434 |
https://mathoverflow.net/questions/155318 | 7 | There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in analogy to the usual Grothendieck fibration over commutative rings in discrete algebra. One then can ask which morphisms of c... | https://mathoverflow.net/users/11546 | Which morphisms of ring spectra are of effective descent for modules? | In DAG7, Theorem 6.1, Lurie proves faithfully flat descent for modules: the functor $A \mapsto \mathrm{Mod}\_A$ is a hypercomplete sheaf for the fpqc topology on $E\_\infty$-rings. This, in particular, gives effective descent for module spectra along faithfully flat maps.
| 7 | https://mathoverflow.net/users/45704 | 155343 | 82,435 |
https://mathoverflow.net/questions/155240 | 5 | The program `nauty` comes with `gtools` which contains, among others, several generation programs like geng, genbg, ... I was wondering whether there is some article or other source describing the details of the algorithm used. I assume that the technique described in 'Isomorph-free exhaustive generation' is used, but ... | https://mathoverflow.net/users/45807 | Details of generation programs supplied with nauty | MathOverflow is not a good place for questions like this. The best place for technical questions about *nauty* is the [mailing list](http://mailman.anu.edu.au/mailman/listinfo/nauty).
Anyway, the parent of a graph $G$ is a graph $G-v$ where $v$ is some vertex. In *geng* you can assume $v$ has the maximum degree in $... | 8 | https://mathoverflow.net/users/9025 | 155370 | 82,444 |
https://mathoverflow.net/questions/155305 | 3 | Let $F$ be a local field (whose residue field is $q$) and $E$ its quadratic extension. Let $\pi$ be a irreducible principal series representation $\pi(\chi\_1, \chi\_2)$ of $GL\_F(2)$ especially where $\chi\_1, \chi\_2$ are unitary characters. Then I know $L\_F(s,\pi)=\frac{1}{1-\chi\_1(\varpi)q^s} \cdot \frac{1}{1-\ch... | https://mathoverflow.net/users/29422 | The effect of base change on the L-function of GL(2)? | If $\pi = \pi(\chi\_1, \chi\_2)$ is an irreducible principal series, then $BC(\pi)$ is the irreducible principal series attached to the unramified characters $\chi\_i \circ N\_{E/F}$ of $E^\times$. So if the $\chi\_i$ are unramified then we have
$$L\_E(s, BC(\pi)) = (1 - \chi\_1(\varpi)^2 q^{-2s})^{-1}(1 - \chi\_2(\v... | 5 | https://mathoverflow.net/users/2481 | 155374 | 82,447 |
https://mathoverflow.net/questions/155290 | 2 | We work over an algebraically closed field of characteristic 0.
Let $\mathfrak{g}$ be a reductive Lie algebra and let $\mathfrak{p}\supset\mathfrak{m}$ be a parabolic subalgebra, respectively a Levi subalgebra.
There is the adjoint action of $\mathfrak{m}$ on $\mathfrak{g}$ and I would like to know how it decomposes in... | https://mathoverflow.net/users/1328 | adjoint action of a Levi subalgebra | Not sure that there would be a closed formula as Jim has pointed out. You can easily calculate it in examples because you can determine highest weight vectors in $g$ for $m$. Indeed, if $\alpha\_1, \ldots \alpha\_k$ are simple rooots for $m$, you just need to find those roots $\beta$ of $g$ such that no $\beta + \alpha... | 5 | https://mathoverflow.net/users/5301 | 155377 | 82,449 |
https://mathoverflow.net/questions/155367 | 12 | One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?
| https://mathoverflow.net/users/15482 | subsets of groups which have to be closed no matter what | Subsets of a group that are closed with respect to any Hausdorff group topology
are called *unconditionally closed*.
Clearly, all *algebraic* sets are unconditionally closed, where a subset of a group $G$ is called *algebraic* if it is an intersection of finite unions of the sets of solutions to some equations with ... | 25 | https://mathoverflow.net/users/24165 | 155378 | 82,450 |
https://mathoverflow.net/questions/155372 | 2 | I'm reading the paper:
SGA 7 II, Intersections sur les surfaces regulieres.
In Papge 6 , I cannot understand why there is sign $-1$ in the formula (1.10.4):
Let $S$ be a trait, for any $\mathcal O\_S$-module $M$ concentrated at the closed point of $S$, then we have
\begin{equation}
(1.10.4)\quad\quad\quad\quad \chi... | https://mathoverflow.net/users/45861 | A problem in intersection theorem | For a module of finite length $M$ over a principal ideal domain $A$, we have $\mathrm{Hom}(M,A)=0$, and $\mathrm{Ext}^1(M,A)$ (the *dual* of $A$, see Bourbaki, *Algebra* VII, §4, No. 9) is non-canonically isomorphic to $M$. Therefore $\chi (\mathrm{RHom}(M,A))=-\mathrm{length}\,(\mathrm{Ext}^1(M,A))=-\chi (M)$.
| 6 | https://mathoverflow.net/users/40297 | 155386 | 82,452 |
https://mathoverflow.net/questions/155362 | 2 | Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space
$T\_xM=E^s(x)\oplus E^u(x)$
I want to know what is the definition of angle between $E^s(x)$ and $E^u(x)$?
| https://mathoverflow.net/users/43741 | Angle between two subspaces | I suppose you're referring to statements in hyperbolic dynamics of the type "the angle between $E^s(x)$ and $E^u(x)$ is bounded away from zero".
I haven't seen a formal definition, but I think the typical implicit assumption is that a Riemannian metric is chosen, and under this metric the minimum angle between tangen... | 6 | https://mathoverflow.net/users/3928 | 155389 | 82,454 |
https://mathoverflow.net/questions/155210 | 3 | Let $A$ be a finitely generated torsion $\mathbb{Z}\_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}\_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB \rightarrow A/pA \rightarrow C/pC \rightarrow 0$ is a short exact sequence of abelian groups. Now $A/pA, B/pB, C/pC$ are f... | https://mathoverflow.net/users/44637 | Splitting as $\mathbb{F}_p[[X]]$-modules | Better not to prove it, as it is wrong. Writing $M[p] = \lbrace m \in M: pm = 0 \rbrace$ for the exact $p$-torsion of a module $M$, I understand your question as follows: *If* the exact sequence
$$0 \rightarrow A[p] \rightarrow A \rightarrow A/A[p] \rightarrow 0$$
remains exact modulo $p$ (which is not always the case:... | 3 | https://mathoverflow.net/users/27465 | 155396 | 82,458 |
https://mathoverflow.net/questions/155380 | 10 | Let $3Cob$ be the category whose objects are closed surfaces and whose morphisms are diffeomorphism classes of cobordisms.
By sending a diffeomorphism $\phi$ of a surface $X$ to its associated cobordism cylinder $M\_\phi$ we get a homomorphism
$$
M : \pi\_0 (Diff(X)) \rightarrow Aut\_{3Cob}(X)
$$
which in general dim... | https://mathoverflow.net/users/401 | Mapping class group vs automorphism group in cobordism category | Injectivity is not hard to prove: (I'm quoting an email from Bruce.) Let f,g be pseudo-isotopic diffeomorphisms of a surface X. Then they are homotopic. Therefore they are isotopic. (see eg. Farb and Margulit, A primer on mapping class groups, pg 43).
Here is how you prove surjectivity; this amounts to proving that ... | 5 | https://mathoverflow.net/users/3460 | 155400 | 82,462 |
https://mathoverflow.net/questions/155369 | 3 | There is a theorem by Sullivan of the following form:
>
> **Theorem:** There is an equivalence of $H$-spaces
> $$ G/PL[\tfrac{1}{2}] \simeq BO\_{\otimes}[ \tfrac{1}{2} ]\ . $$
>
>
>
It can be found for example in the book by Madsen and Milgram (Theorem 4.34 on page 97). I have two questions about this:
* Is... | https://mathoverflow.net/users/3995 | Sullivan's $H$-space equivalence between $G/PL[1/2]$ and $BO[1/2]$ | Here is Dennis Sullivan's answer:
"Yes to the first.
For the second question I interpret it to mean does the H space map really mean a map of loop spaces? [this is synonymous with the A/infinity operad]
I read a paper by Milgram [from 81] where he is assuming these sorts of things for calculations
This question ... | 3 | https://mathoverflow.net/users/732 | 155412 | 82,466 |
https://mathoverflow.net/questions/155403 | 5 | In Proper and Improper forcing, VI.5; Claim 5.1 part 1 is the following:
If $F$ is a P-point in $V$, $P$ is a proper forcing notion and
$\Vdash\_P `` F$ generates an ultrafilter"
Then the ultrafilter generated by $F$ is a P-point in $V^P$.
The proof starts like this: let $p\Vdash \{\tilde A\_n : n<\omega\}$ be su... | https://mathoverflow.net/users/4241 | Shelah's proof that proper forcing preserves P-points | In more details: Let $N$ be a countable elementary submodel of some sufficiently large $H(\chi)$ containing $p$, $P$, $F$, and the sequence of names $\{\dot A\_n:n<\omega\}$. Then the countable set $N\cap F$ works for the collection of $A\_{n, m}$, and any $(N,P)$-generic extension of $p$ will work for $q$.
| 5 | https://mathoverflow.net/users/18128 | 155421 | 82,470 |
https://mathoverflow.net/questions/98515 | 8 | Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^{p})^{1/p} \le O(p) \cdot (\mathsf{E} \langle X \rangle ^{p/2})^{1/p}$ for large $p$.
Can the growth rate be improved... | https://mathoverflow.net/users/22758 | What is the optimal growth of the constant in BDG? | I know a version which exactly gives the constant $O(p^{1/2})$ for $p\ge 2$. It is contained in a lecture note by D. Khoshnevisan on SPDE.
| 7 | https://mathoverflow.net/users/34483 | 155432 | 82,473 |
https://mathoverflow.net/questions/155428 | 10 | Are very large cardinal axioms like $I\_0$, $I\_1$, $I\_2$ consistent with $CH$ and $GCH$?
| https://mathoverflow.net/users/nan | Very Large Cardinal Axioms and Continuum Hypothesis | It is known in the folklore (I could never find a source for the results) that $I\_0$, $I\_1$, and $I\_2$ cardinals are all indestructible by small forcing, meaning that we can force ${\rm CH}$ over a universe with such a large cardinal without destroying it. I have the argument for $I\_0$ written up in lecture notes [... | 19 | https://mathoverflow.net/users/5984 | 155441 | 82,477 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.