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/102059 | 0 | Let $f \in C^p[0,2\pi]$ and periodic. Denote $\omega\_p$ as the moduli of continuous of $f^{(p)}$. Then
$
|f - S\_Nf| \le K \frac{\log{N}}{N^p}\omega\_p(2\pi/N),
$
where $S\_N$ is the Fourier partial sum of order $N$, and $K$ is some constant.
I could not prove this result. Can someone help me please?
I found a s... | https://mathoverflow.net/users/25056 | Convergence of Fourier series for $C^p$ functions | There is a theorem of Lebesgue that says that for a continuous periodic $f$,
$$
\|f - S\_N f\|\_\infty \le C \log N \|f - f^\* \|\_\infty.
$$
This appears as Theorem 2.2 in Rivlin's book. Combined with the result you already know, you get what you want.
| 1 | https://mathoverflow.net/users/1044 | 102363 | 59,401 |
https://mathoverflow.net/questions/102357 | 7 | Let $G=PSL(2,q)$ where $q$ is prime power. What is Aut$(G\times G)$ and Aut$(G\times G\times G)$? Also if $G=A\_{n}$ where $A\_{n}$ is the alternating group of degree $n$, then what is Aut$(G\times G)$?
Thanks in advance
| https://mathoverflow.net/users/24936 | Automorphism group | In general, if $S$ is a finite non-Abelian simple group, and $E$ is a direct product of $n$ copies of $S$, then ${\rm Out}(E) = {\rm Aut}(E)/E$ is isomorphic to ${\rm Out}(S) \wr S\_{n}.$ This is because every minimal normal subgroup of $E$ is isomorphic to $S$ (in fact, is one of the obvious simple direct factors of $... | 13 | https://mathoverflow.net/users/14450 | 102365 | 59,403 |
https://mathoverflow.net/questions/102376 | 6 | Is anything known about the consistency strength of the statement:
"There is a normal measure (on a cardinal) that is not ordinal-definable"?
In particular, is it consistent relative to the existence of a measurable cardinal?
It looks like it's consistent relative to the existence of a supercompact cardinal.
If $\k... | https://mathoverflow.net/users/1682 | Measures that are not OD | It was proved, long ago, that one can force over a model with a measurable cardinal $\kappa$ and get a model with lots ($2^{2^\kappa}$, I think) of normal measures. I believe that (1) most of those measures won't be OD in that model and (2) the relevant paper is
Kenneth Kunen and Jeff Paris, Boolean extensions and m... | 7 | https://mathoverflow.net/users/6794 | 102377 | 59,408 |
https://mathoverflow.net/questions/102386 | 31 | One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not so clear.
Let $\Sigma \subset 2^\mathbb{R}$ be the sigma-algebra of all Lebesgue measurable sets. Give $2^\mathbb{R}$ t... | https://mathoverflow.net/users/8410 | Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable? | The answer to your second question (assuming the axiom of choice, to dodge Asaf's comment) is that $2^{\mathbb R}/\Sigma$ has dimension $2^{\mathfrak c}$, where $\mathfrak c=2^{\aleph\_0}$ is the cardinality of the continuum. The main ingredient of the proof is a partition of $[0,1]$ into $\mathfrak c$ subsets, each of... | 23 | https://mathoverflow.net/users/6794 | 102388 | 59,413 |
https://mathoverflow.net/questions/102317 | 6 | I recently came across an interesting result of [Kobayashi](http://www.ams.org/mathscinet-getitem?mr=1016093) [Corollary 5.5], a special case of which is the following: Suppose $\Gamma$ is a discrete torsion free subgroup of $SL\_n(\mathbb{R})$ which acts properly discontinuously on the homogeneous space $X=SL\_n(\math... | https://mathoverflow.net/users/25135 | cohomological dimension of a group acting on a product | I think [Kobayashi's approach](http://www.kurims.kyoto-u.ac.jp/~toshi/texpdf/pro-the.pdf) also works in the OP's situation:
>
> If $M$ is an oriented compact connected manifold then $cd\_\mathbb{R}(G) \le n$.
>
>
>
*Proof:* As in Kobayashi the base ring is the field of real numbers. Let $X = M \times \mathbb... | 6 | https://mathoverflow.net/users/10194 | 102397 | 59,417 |
https://mathoverflow.net/questions/102384 | 13 | Hi everybody, for my research I am dealing with the following function:
$$\alpha\_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|\_{z=0},\quad n\in \mathbb{N},$$
It is possible to show that
$$\alpha\_n(x)=\frac{P\_n(x)}{x^{n+1}},$$ where $P\_n(\cdot)$ is a polynomial of or... | https://mathoverflow.net/users/25157 | Showing that a family of polynomials has positive and real roots. | I'm noticing a pattern in the numerators. To avoid getting indices wrong, I'll write out an example:
$$(22680, 18900, 4410, 255, 1) = (36 \times 630, 16 \times 630+21 \times 420, 9 \times 420 + 10 \times 63, 4 \times 63 + 3 \times 1, 1)$$
Here $(36, 21, 10, 3)$ is the odd indexed triangular numbers and $(16, 9, 4,1)$ a... | 16 | https://mathoverflow.net/users/297 | 102404 | 59,422 |
https://mathoverflow.net/questions/102328 | 0 | Let $H$ be a Hilbert space, and $S\subseteq \mathcal{B}(H)$. We denote
$\bar S$ the ultraweak closure of $S$, and $B\_r$ the closed ball of center 0
and radius $r>0$ of the normed space $\mathcal{B}(H)$.
If $S$ is a subalgebra of $\mathcal{B}(H)$, Do we have $\overline{S\cap B\_r} = \bar S \cap B\_r$ ?
| https://mathoverflow.net/users/25139 | Ultraweak closure inside a closed ball | The answer is NO for general non-selfadjoint subalgebras. With a subspace $X\subset \mathcal{B}(H)$, one can associate a subalgebra
$$S\_X= \langle \begin{pmatrix} \alpha & x \cr 0 & \alpha \end{pmatrix} : \alpha \in \mathbb{C}I\_H,\ x \in X \rangle \subset{\cal B}(H\oplus H).$$
Now choose a state $\omega$ which elimi... | 3 | https://mathoverflow.net/users/7591 | 102411 | 59,427 |
https://mathoverflow.net/questions/102391 | 0 | Let $\pi:X \longrightarrow C$ be a smooth projective morphism onto a smooth projective curve. If the general fibers are of nonnegative Kodaira dimension, is $\pi\_{\ast} \mathcal{O}(k K\_{X/C})$ nonzero for sufficiently divisible $k$? If it is, is there an algebraic proof? That is, without using invariance of plurigene... | https://mathoverflow.net/users/24445 | On direct image of the relative pluri-canonical divisor | The stalk of $\pi\_\* (\omega\_{X/C}^{\otimes k})$ at the generic point $\eta$ of $C$ equals $H^0(X\_\eta, \omega\_{X\_\eta/\kappa(\eta)}^{\otimes k})$. If the generic fiber $X\_\eta$ has nonnegative Kodaira dimension, then there exists an integer $k$ such that $H^0(X\_\eta,\omega\_{X\_\eta/\kappa(\eta)}^{\otimes mk})$... | 3 | https://mathoverflow.net/users/13265 | 102419 | 59,430 |
https://mathoverflow.net/questions/102406 | 3 | I have the following question: If $E$ is a ring spectrum, then a complex orientation of $E$ is an element of $E^2(\mathbb{C}P^{\infty})$ that is mapped to $1$ in $E^2(\mathbb{C}P^{1})$.
I have read that there is a natural orientation on $MU$ the spectrum of complex bordism. Can you tell me, how this orientation is defi... | https://mathoverflow.net/users/20818 | Orientation of complex bordism spectrum | A complex orientation on a homotopy commutative ring spectrum $E$ is sometimes defined to be an element in $\widetilde{E}^2(\mathbb{CP}^\infty)$ which restricts to the canonical element of $\widetilde{E}^2(S^2) = \pi\_0 E $.
In fact, there is another definition of an orientation, which generalizes better to other se... | 7 | https://mathoverflow.net/users/344 | 102420 | 59,431 |
https://mathoverflow.net/questions/102394 | 2 | Let $S=\mathrm{Spec}(R)$, $s=\mathrm{Spec}(k)$ and $\eta=\mathrm{Spec}(K)$, where $R$ is a d.v.r. with fraction field $K$. Let $j:\eta\rightarrow S$
Now how to compute the sheaf $R^1j\_\*(\mathbb{G}\_{m,\eta})$ in the fppf topology?
The case for etale topology is zero by considering the stalks and use the Hilbert 9... | https://mathoverflow.net/users/3848 | How to compute the first derived direct image along an open immersion, for the fppf sheaf represented by a multiplicative group? | It seems to me that (equation) 5.2 and Appendice 11.7 of "Le Groupe de Brauer, III" imply that the cohomology for the \'etale topology equals the cohomology for the fppf topology.
| 1 | https://mathoverflow.net/users/13265 | 102422 | 59,433 |
https://mathoverflow.net/questions/102299 | 3 | In Bollobas' proof of the chromatic number of random graphs with constant $p$, he computes the value $r \in \mathbb{R}$ for which the expected number of $r$-cliques in $G(n,p)$ is exactly one:
Given $\binom{n}{r} p^{\binom{r}{2}} = 1$, I want to obtain an expression for $r$.
In particular, applying Stirling approxi... | https://mathoverflow.net/users/25133 | Computations on Bollobas' proof of the $\chi(G(n,p))$ for constant $p$ | Go back to the original equation: $\binom nr p^{r(r-1)/2}=1$ and rewrite it as
$b^{r(r-1)/2}=n^r/r!\left[\binom nr/(n^r/r!)\right]$.
Notice that $b^{r(r-1)/2}$ becomes larger than $n^r$when $b^{(r-1)/2}>n$. That is when $r=O(\log n)$. Notice that the term in brackets is very close to 1 in that range (certainly betwee... | 1 | https://mathoverflow.net/users/11054 | 102427 | 59,437 |
https://mathoverflow.net/questions/102438 | 5 | A famous conjecture of Looijenga states that the moduli space of curves $M\_{g,n}$ is the union of $g- \delta\_{0,n}+ \delta\_{0,g}$ open affine subsets, where $g,n$ are non-negative integers satisfying $2g-2+n>0$, and $\delta$ is the Kronecker delta.
I know of proofs of this conjecture in the case $(g,0)$ for $2 \le... | https://mathoverflow.net/users/23194 | Is $M_{1,n}$ affine? | Yes, $M\_{1,n}$ is affine. More generally, $M\_{g,n+1}\to M\_{g,n}$ is an affine morphism for any $n \geq 1$.
| 9 | https://mathoverflow.net/users/1310 | 102439 | 59,441 |
https://mathoverflow.net/questions/102423 | 31 | (Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.)
I saw this amusing derivation on the blackboard at MSRI a few months ago (I'm paraphrasing and reformatting slightly, though my attempts at formatting... | https://mathoverflow.net/users/3621 | solving linear equations made difficult | Actually this calculation has a formal sense in every ring, by working in the ring of formal power series in $a$ (here $1-a$ is invertible with inverse $\sum\_{i \geq 0} a^i$). There are many "pseudo-analytic" proofs in ring theory (one was discussed [here](https://mathoverflow.net/questions/31595/how-would-you-solve-t... | 5 | https://mathoverflow.net/users/2841 | 102441 | 59,442 |
https://mathoverflow.net/questions/102184 | 3 | Let $f: S \to \Bbb R$ be a Morse function on a Riemann surface. Let $x\_0$ be a saddle point of $f$. Since $x\_0$ is a critical point of $f$, it makes sense to talk about the bilinear forms $f\_{z\overline{z}}(x\_0)$, $f\_{zz}(x\_0)$ and $f\_{\overline{z}\overline{z}}(x\_0)$ as components of the real Hessian of $f$ at ... | https://mathoverflow.net/users/17965 | Special Morse function on a Riemann surface | First, let me point out that only the differential form $\newcommand{\pa}{\partial}$ $\newcommand{\bpa}{\bar{\partial}}$ $\pa\bpa f$ is well defined globally on $S$. Locally
$$ \pa\bpa f= f\_{z\bar{z}} dz\wedge d\bar{z}, $$
but the coordinate $z$ is only locally defined. If you choose different a different local co... | 1 | https://mathoverflow.net/users/20302 | 102455 | 59,451 |
https://mathoverflow.net/questions/102453 | 1 | Recently I came up with the following problem.
Suppose $U$ is an open subset of $\mathbb{R}^n$ and we are given a continuous map $M:U\to GL(n;\mathbb{R})$. Does anybody know if there are conditions for the existence of a $C^1$ map $f:U\to \mathbb{R}^n$ such that $$Jf(x)=M(x)\quad \forall x \in U$$
(here $Jf(x)$ is th... | https://mathoverflow.net/users/16494 | Map with prescribed Jacobian | Suppose we can find such an $f$. If $f$ is continuously differentiable, then the i'th row and j'th column of the Jacobian (in the standard basis) is the j'th partial derivative of the the i'th component of $f$. Indeed, the components of $f$ are continuously differentiable if and only if $f$ is. So it suffices to consid... | 6 | https://mathoverflow.net/users/25179 | 102458 | 59,453 |
https://mathoverflow.net/questions/102412 | 3 | Hello, I apologize in advance if this question is misguided somehow, since my algebraic geometry is pretty shaky.
I am wondering if there is a way to understand all the conics in a generic quadric line complex $X$. Remember that $X$ is defined as the intersection of the image of the complex Grassmanian $G(2,4)$ in $... | https://mathoverflow.net/users/492 | Conics in the quadric line complex | One interpretation of the quadric line complex $X$ is as the moduli space of stable, rank $2$, degree $1$ vector bundles on a genus $2$ curve $C$ (whose associated Kummer is the Kummer in G&H), cf. Newstead, "Topological Properties of Some Spaces of Stable Bundles". From this point of view, Ana-Maria Castravet has desc... | 5 | https://mathoverflow.net/users/13265 | 102462 | 59,456 |
https://mathoverflow.net/questions/102440 | 11 | It is a well-known result of Griffiths that the pieces of Hodge filtration of a smooth hypersurface $X:= (f=0)$ of degree $d$ in $\mathbb{P}^{n}$ are isomorphic to graded pieces of the Jacobian ring associated to $X$ i.e the ring $R= \mathbb{C}[x\_{1},...x\_{n}]/J$ where $J$ is the ideal generated by the partial deriva... | https://mathoverflow.net/users/18000 | Deformations of smooth projective hypersurfaces and the Jacobian ring | For $d\geq 5$, for a smooth plane curve of degree $d$, it is simply not true that the natural map $R\_d \to H^1(X,T\_X)$ is surjective: there are plenty of abstract deformations which are not plane curves. I am including below the proof that the natural map is surjective when $n\geq 4$, when $n=3$ and $d\leq 3$, and wh... | 5 | https://mathoverflow.net/users/13265 | 102465 | 59,457 |
https://mathoverflow.net/questions/102415 | 9 | I understand that in the limit that $\hbar$ goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation for the propagator which is a function of the classical trajectory (see [this document,](http://www.blau.itp.unibe.ch/l... | https://mathoverflow.net/users/25136 | Classical Limit of Feynman Path Integral | Things stay in this way. Consider the action of a given particle that appears in the path integral. We consider the simplest case
$$
L=\frac{\dot x^2}{2}-V(x)
$$
and so, a functional Taylor expansion around the extremum $x\_c(t)$ will give
$$
S[x(t)]=S[x\_c(t)]+\int dt\_1dt\_2\frac{1}{2}\left.\frac{\delta^2 S}{\delta... | 11 | https://mathoverflow.net/users/19520 | 102467 | 59,459 |
https://mathoverflow.net/questions/102398 | 20 | Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution
$$
\mu: T^\*\mathcal{B}\rightarrow \mathcal{N}
$$
is the moment map, if we identify $\mathfrak{g}$ with $\mathfrak{g}^\* $ by the Killing form and consider $\mathcal{N} \subset \mat... | https://mathoverflow.net/users/24965 | What is the significance that the Springer resolution is a moment map? | One reason to emphasize the Springer resolution's role as a moment map is that it is the semiclassical shadow of Beilinson-Bernstein localization. More precisely passing to functions, the moment map description asserts that the Springer map is describing the Hamiltonian functions on the cotangent to the flag variety wh... | 20 | https://mathoverflow.net/users/582 | 102477 | 59,465 |
https://mathoverflow.net/questions/102288 | 16 | **Motivation:** According to the "chromatic" picture of stable homotopy, we should think of the moduli stack $M\_{FG}$ of formal groups as a "good approximation" to the stable homotopy category (more precisely, we should think of the category of quasi-coherent sheaves on the stack $M\_{FG}$ as the "good approximation")... | https://mathoverflow.net/users/344 | Is there an algebro-geometric description of $\nu$? | The answer seems to be yes: given an even-periodic cohomology theory $E$, then $E^\*(\mathbb{HP}^\infty)$ can be described as *even* functions on the formal group of $E$ (just as $E^\*(\mathbb{CP}^\infty)$ corresponds to functions on the formal group of $E$).
In fact, there is a natural map $\mathbb{CP}^\infty \to \... | 4 | https://mathoverflow.net/users/344 | 102483 | 59,467 |
https://mathoverflow.net/questions/102486 | 9 | Two equivalent formulations of the axiom of choice are:
1. Every family $(X\_i)\_{i \in I}$ of pairwise disjoint nonempty subsets of a set $X$ has a choice function.
2. Every family $(X\_i)\_{i \in I}$ of nonempty subsets of a set $X$ has a choice function.
However, the usual proof of the implication (1) → (2) repl... | https://mathoverflow.net/users/2000 | Distinguishing two local versions of the axiom of choice | Suppose that $X$ is a strongly amorphous set, that is an amorphous set that every partition has only finitely many non-singletons parts. It is clear that there is no choice function from every family of non-empty subsets of $X$, as $X$ cannot be split into two disjoint infinite sets.
If $\cal F$ is a pairwise-disjoin... | 7 | https://mathoverflow.net/users/7206 | 102489 | 59,468 |
https://mathoverflow.net/questions/102479 | 2 | I have been reading Pippenger and Spencer's paper "Asymptotic behavior of the chromatic index for hypergraphs" and they comment that their result is applicable to the family of random k-uniform hypergraphs $G\_k(n,p)$ whenever $p(n) = o(1)$ and $\frac{\log n}{n^{k-1}} = o(p(n))$.
I'm looking for a proof of this.
L... | https://mathoverflow.net/users/22236 | matchings in hypergraphs | I'm assuming that the degree of a vertex is the number of edges incident with it (other definitions are possible). The degree of a vertex in $G\_k(n,p)$ has a binomial distribution
$$\operatorname{Bin}\left( \binom{n-1}{k-1}, p\right).$$ Find values $N\_1,N\_2$ above and below the mean so that the total weight of this ... | 3 | https://mathoverflow.net/users/9025 | 102513 | 59,478 |
https://mathoverflow.net/questions/102374 | 4 | Suppose I have a central extension $1 \to U(1) \to \hat{G} \to G \to 1$ of a topological group $G$ by the circle group $U(1)$ in such a way that $\hat{G} \to G$ is a principal $U(1)$-bundle. Moreover, suppose that $G$ is $3$-connected. In particular, $\hat{G} \cong G \times U(1)$ as topological spaces.
>
> Is the ... | https://mathoverflow.net/users/3995 | Extensions of topological groups | No, $G=({\mathbb R}^2 ,+)$ and a certain quotient $\widehat{G}$ of the Heiseberg group is an example. Namely, $\widehat{G}={\mathbb R}^2\times S^1$ with the product $(x,a)⋅(y,b)=(x+y,ab(\omega(x,y)+{\mathbb Z}))$ where $\omega$ is a nonzero bilinear form.
| 7 | https://mathoverflow.net/users/5301 | 102516 | 59,480 |
https://mathoverflow.net/questions/102518 | 2 | Hi,
I am wondering if anything is known about irreducible representations of a semidirect product over $C\_p:=\mathbb{Z} / p \mathbb{Z}$ in general or at least in special cases. For example of $C\_q \rtimes C\_l$ over $C\_p$, where $p,q,l$ are some primes. Is there any well-known literature?
Thank you for hints!
| https://mathoverflow.net/users/25200 | Representations of semidirect product over $C_p$ | Look at Weintraub "Representation theory ... " for Mackey Machine (pg 120 or so).
Edit in response to [Mackey Maschine for semidirect product](https://mathoverflow.net/questions/104087/mackey-maschine-for-semidirect-product). The reference does not give a solution to the problem, but studying the Mackey machine/Cliff... | -1 | https://mathoverflow.net/users/10400 | 102522 | 59,482 |
https://mathoverflow.net/questions/102524 | 0 | I am currently working on complex networks. I consider a matrix $\cal N$ with random entries $\delta\_{ik}$. These entries are varying randomly in time and so I have a sequence of random matrices that determines a stochastic process.
My question is quite simple: Does it exist a generalization of standard stochastic d... | https://mathoverflow.net/users/19520 | Stochastic processes with random matrices | Yes. A quick search turns up papers like:
<http://arxiv.org/pdf/math-ph/0402061.pdf>
and
<http://arxiv.org/pdf/1004.0301v2.pdf>
| 1 | https://mathoverflow.net/users/6269 | 102526 | 59,484 |
https://mathoverflow.net/questions/102062 | 12 | Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^\*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can only find full and faithful appearing together in the literature, and I need to extricate the two conditions. Does anyone... | https://mathoverflow.net/users/1040 | Fullness of pullback functor in algebraic geometry | In case $X, Y$ are smooth,
$ f^\* $
is full if and only if it is full and faithful. This is explained in the introduction of [arXiv:1101.5931](http://arxiv.org/abs/1101.5931) (by Canonaco-Orlov-Stellari), which also studies when this implication holds more generally. Thus the pull-back is full and faithful if and only ... | 11 | https://mathoverflow.net/users/7437 | 102528 | 59,485 |
https://mathoverflow.net/questions/102506 | 14 | Fix an integer $d > 1$ and $2d-2$ points $P\_1, \ldots, P\_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, 1983), we know that there are only a finite number of rational functions $\phi(z)$ of degree $d$ with complex coefficients t... | https://mathoverflow.net/users/13410 | Determining rational functions by their critical points | One answer is that there are **NO** rational expressions in $P\_1, \dots, P\_{2d-2}$ that allow you to single out one of the functions $\phi\_i$, or that even allow you to single out a nonempty, proper subset of the set of all $\rho(d)$ functions. The proof of this is the usual argument of this type (also in Harris's p... | 6 | https://mathoverflow.net/users/13265 | 102543 | 59,488 |
https://mathoverflow.net/questions/102536 | 9 | Are there any set of axioms that completely characterize the Riemann zeta function?
i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.
| https://mathoverflow.net/users/nan | Axioms for Riemann $\zeta$ function | Perhaps you are looking for something like *Hamburger's Theorem*?
It states, essentially, that the only Dirichlet series with a finite number of singularities satisfying the same functional equation as the zeta-function is the zeta-function. You can find the details in Titchmarsh's book.
Googling I found the foll... | 11 | https://mathoverflow.net/users/3659 | 102547 | 59,490 |
https://mathoverflow.net/questions/102546 | 12 | Consider a noetherian ring $R$ and a collection $m\_i$, $i\in I$ of maximal ideals of $R$. Let $P$ be a prime ideal of $R$. It is well-known that if the collection is finite (i.e. the index set $I$ is finite), then $P\subseteq \cup\_{i\in I}m\_i$ if and only if $P$ is contained in one of the $m\_i$. (This is true under... | https://mathoverflow.net/users/11599 | union of infinitely many prime ideals | I fixed your notation so that there wasn't any equivocation. I changed $P\_i$ to $m\_i$, changed the ideal $I$ to $P$, and kept the set $I$ as $I$.
There is indeed a counterexample. Take $R=\mathbb C[x,y]$, the prime ideal $P=(x,y)$ is not contained in any of the ideals $m\_{a,b}=(x-a,y-b)$ for $(a,b)$ not both $0$, ... | 18 | https://mathoverflow.net/users/18060 | 102548 | 59,491 |
https://mathoverflow.net/questions/102562 | 0 | Let us assume that we have $n$ polynomials in $n$ variables, $p\_1(\vec{x}), p\_2(\vec{x}),\ldots p\_n(\vec{x})$.
Jacobian, $J(p)$ of $\vec{p}$ is a matrix with $i,j$ entry $\frac{\partial p\_i}{\partial x\_j}$
Assume that $p\_i$ are algebraically dependent, i.e., there exists polynomial $f$ such that $f(p\_1,\ldots, ... | https://mathoverflow.net/users/4246 | Jacobian and Algebraic independence | I think perhaps this problem is not appropriate here. At any rate, here is a counterexample: $n$ equals $2$, $p\_1(x,y) = x^2$, $p\_2(x,y) = x^2$ and $\vec{v}(x,y) = (x,-x)$. Plugging in $\vec{p}(x)$ into any polynomial results in a polynomial in $x^2$. So the entries of $\partial f(\vec{p})$ would be polynomials in $x... | 1 | https://mathoverflow.net/users/13265 | 102565 | 59,496 |
https://mathoverflow.net/questions/102544 | 1 | I'm looking to find a root $(x\_1,\dots,x\_n)$ of a polynomial $p \in {\mathbb R}[x\_1,\dots,x\_n]$ such that $0 \leq x\_i < 1$ for all $i$. Further, I know in advance that setting $x\_1 = \cdots = x\_n$ is a root of $p$, but wish to avoid this root. How can I find at least one of these roots, preferably using a comput... | https://mathoverflow.net/users/4167 | Non-Uniform Root of Polynomial in Open Cube | If you can find points $u, v \in [0,1]^n$ where $p(u) < 0$ and $p(v) > 0$, there will be
a solution of the form $tu + (1-t) v$ for some $t \in [0,1]$, and that can be found by solving a polynomial in one variable. If $n > 2$, generically the line $tu+(1-t)v$ will not intersect the diagonal. With a bit of luck, you migh... | 2 | https://mathoverflow.net/users/13650 | 102567 | 59,497 |
https://mathoverflow.net/questions/102557 | 7 | Let $A$ be a commutative ring with 1, $I$ an ideal in $A$, $B$ an $A$-algebra. I am trying to prove the following isomorphism of $A$-algebras:
$$ \big( A^\* \otimes \_A B \big) ^\* \cong B^\* $$
"$^\*$" denotes the $I$-adic completion: every $A$-algebra $X$ may be endowed with the $I$-adic topology, defined by the idea... | https://mathoverflow.net/users/25209 | Completion and Tensor Product of Algebras | It's true if $A$ is Noetherian.
For any $A$-algebra $C$ and any ideal $J$ in $A$, note that $C/JC$ is isomorphic to the tensor product algebra $C \otimes\_A A/J$.
Now for any $n \geq 0$, $A/I^n$ is a module over $A^\*$, and the multiplication map $A/I^n \otimes\_A A^\* \to A/I^n$ is an isomorphism, since $A$ is Noe... | 3 | https://mathoverflow.net/users/6827 | 102571 | 59,499 |
https://mathoverflow.net/questions/102549 | 2 | Is it possible to have a function with the following properties?
(i) The function maps a bounded $n$-dimensional space $\mathcal{X}$ (say $\left[0,1\right]^n$) onto a bounded interval $\mathcal{Y}$ (say $\left[0,1\right]$);
(ii) The function has exactly $k$ minima in $X$, whose locations are distributed randomly, a... | https://mathoverflow.net/users/25207 | A Function with Exactly $k$ Minima in a Bounded Space | On $\mathbb{R}^{n}$ define $f(x)=e^{-\frac{1}{1-|x|^{2}}}$ for $|x|<1$ and $f(x)=0$ for $|x|\geq 1$. This is a smooth function with a maximum at the origin that vanishes outside of the unit ball. Given a point $a\in\mathbb{R}^{n}$ and $\epsilon>0$ define $g\_{a,\epsilon}(x)=f((x-a)/\epsilon)$ this has a single maximum ... | 3 | https://mathoverflow.net/users/25179 | 102576 | 59,502 |
https://mathoverflow.net/questions/102573 | 3 | Consider group algebra k[G] of finite group G.
If k is alg.closed then every irrep lives there with multiplicity equal to dimension.
(More conceptually as bimodule over GxG it is multiplicity free and all irreps live there).
Now let k=F\_p what is known then ?
Is true that all irreps of G live in k[G] ? (Seems obviou... | https://mathoverflow.net/users/10446 | Structure of F_p[G], for finite group G ? | It seems natural to work with an algebraically closed field of characteristic $p$, or, less restrictively, a splitting field of characteristic $p$ for $G$. For example, any field containing the primitive $m$-th roots of unity, where $|G| = p^{a}m$ and $p$ dos not divide $m$, so I assume now that $k$ is algebraically cl... | 10 | https://mathoverflow.net/users/14450 | 102577 | 59,503 |
https://mathoverflow.net/questions/102568 | 7 | My question is somewhat related to [this one](https://mathoverflow.net/questions/98600/k-theory-of-schemes-monoidal-vs-exact). However I think it adds something new to the table so I decided to post it sperately.
There is a construction of K-theory for symmetric monoidal categories (smc) and one for exact categories.... | https://mathoverflow.net/users/18744 | K-theory, monoidal vs. exact | As you note, the classifying space of any exact category is contractible because of the presence of an initial object, so your second K(D) (which I take to be the group completion of the classifying space of D under direct sum) is trivial. But the other two (namely K(D), meaning the Q-construction K-theory of D, and (i... | 7 | https://mathoverflow.net/users/3931 | 102583 | 59,506 |
https://mathoverflow.net/questions/102540 | 9 | Consider the counting function
$$
f(x)=|\{n\le x:n\text{ is a product of Fibonacci numbers}\}|
$$
so for example $f(4)=4=|\{1,2,3,4\}|$ since 1, 2, and 3 are Fibonacci numbers and $4=F\_3\cdot F\_3.$ (See [A065108](https://oeis.org/A065108).)
What is known, asymptotically, about the growth of $f$?
It's clear that f... | https://mathoverflow.net/users/6043 | Product of Fibonacci numbers | I think the following is related to Pietro Majer's comment:
**Upper bound:** The first $2$ Fibonacci numbers are $1$, so we can leave them out of products. The $n$th Fibonacci number is greater than $\phi^{n-2}$ for $\phi = \frac{1+\sqrt 5}{2}$. So, for any product of Fibonacci numbers less than $x$, there is a corre... | 9 | https://mathoverflow.net/users/2954 | 102599 | 59,515 |
https://mathoverflow.net/questions/87734 | 0 | Sorry for the vague title but I couldn't find a better one.
I want to compute the sum $S = \frac{1}{N}\sum\_{i=1}^N c\_i x\_i$ where $c\_i$s are known positive constants. The problem is that computing the value of each $x\_i$ takes a huge amount of time. That's why I want to estimate the sum $S$ by using only a smal... | https://mathoverflow.net/users/5223 | Estimating a sum | This isn't an answer, but it occurs to me that a relatively simple (and timely) toy example of the type of calculation the OP is describing might help clarify what kinds of assumptions are needed in order to provide an actual answer.
Suppose you'd like to know how many of the 538 total votes in the Electoral College ... | 0 | https://mathoverflow.net/users/15837 | 102601 | 59,516 |
https://mathoverflow.net/questions/102593 | 1 | In its Algebraic number theory (second edition), Lang uses in the proof of Theorem 1, chapter XI, paragraph 2, page 215-216, this group index identity:
$\displaystyle{\prod\_{v\in S}(k^\ast\_v:k^{\ast n}\_v)=n^{2s}}$
in which:
1. $n>2$ is an integer;
2. $k$ is a number field containing $n$-th roots of unity;
3. $S$ i... | https://mathoverflow.net/users/20529 | Group index formula in Lang-Algebraic number theory | If there is a real embedding then $n=2$, because all the solutions to the equation $x^n-1$ in
$\mathbb R$ are $\pm 1$ and the field contains the $n$-th roots of unity. Hence, there are 2 possibilities $r\_1=0$ and $n$ arbitrary and the formula is ok, or $r\_1>0$, $n=2$ and the formula is also ok.
| 2 | https://mathoverflow.net/users/19229 | 102611 | 59,524 |
https://mathoverflow.net/questions/102575 | 4 | What are modular forms or cusps forms, resp. ?
==============================================
We start with defining their common domains $\mathbb{H}\_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The symplectic group $Sp(g,\mathbb{Z})$ is the subgroup of $SL(2g,\mathbb{Z})$... | https://mathoverflow.net/users/24392 | Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard) | Though I haven't seen it, I've heard that Tomoya Kiyuna, a student at Kyushu
University, has done this for the case $j=4$ as part of his master's thesis. In particular, he finds eighteen explicit generators of the module of vector-valued Siegel modular forms of the symmetric tensor 8.
| 4 | https://mathoverflow.net/users/16419 | 102613 | 59,525 |
https://mathoverflow.net/questions/102627 | 0 | Let $k$ be an integer. The following inequality is standard.
$$
(a+b)^{k+1} - b^{k+1} \leq (k+1)a(a+b)^k
$$
for $a,b > 0$.
However, does the following inequality still hold
$$
(a+b)^{k+1} - b^{k+1} \leq (k+1)a\left(a+ \frac{b}{(k+1)^{1/(k+1)}} \right)^k
$$
for $a,b > 0$? While $k \rightarrow \infty$, the term $(k+1)... | https://mathoverflow.net/users/12981 | Strengthening an inequality | Let's denote $x = \frac{b}{a}$, then the second inequality may be rewritten in following form
$$ (1+x)^{k+1} - x^{k+1} \leqslant (k+1) \left(1+ \frac{x}{(k+1)^{\frac{1}{k+1}}}\right)^{k}, $$
but the leading coefficient of RHS is equal to $(k+1)^{\frac{1}{k+1}}$, while in the LHS it is $k+1$, which is strictly greater u... | 2 | https://mathoverflow.net/users/24953 | 102630 | 59,531 |
https://mathoverflow.net/questions/102279 | 2 | Let us consider a map from a $\Sigma\_g \longrightarrow N$, where $N$ is a symplectic manifold.
Then we define the moduli space as
$M= \{ f | f \mbox{ is a pseudoholomorphic map } \Sigma\_g \to N, f\_\* ([\Sigma]) = [A]\}$, where $A \in N$.
We assume the dimension of $N$ is $2n$. We can cauculate the dimension... | https://mathoverflow.net/users/25054 | Problem:Gromov-Witten;Moduli space | I am trying to assemble the answers to the question(s) that were hashed out in the comments (and also in a separate discussion with Jonny Evans). This answer is community wiki since it is the outcome of collaborative discussion. Please feel free to edit this.
Rephrasing of question: does there exist a non-constant ho... | 5 | https://mathoverflow.net/users/477 | 102638 | 59,536 |
https://mathoverflow.net/questions/102631 | 2 | A Morse function $f(x)\colon \mathbb{R}^n\to \mathbb{R}$ is a smooth function s.t. all singular points are non-degenerate. A theorem of Sard implies that for any smooth $f(x)$ and almost all $a\in \mathbb{R}^n$ the function $f(x) + ax$ is Morse.
A Morse function $f$ is called perfect Morse if in addition the singula... | https://mathoverflow.net/users/2042 | Looking for the perfect morse | The answer to your question is negative. A counterexample is given by the Morse function
$f:\mathbb{R}\to \mathbb{R}, \;\; f(x)=\sin(x). $
Using this example it is easy to produce higher dimensional counterexamples.
(The Morse functions with at most one critical point per level set were called *excellent* by R. T... | 7 | https://mathoverflow.net/users/20302 | 102646 | 59,540 |
https://mathoverflow.net/questions/102642 | 6 | Before I start, let me make a note on terminology. Curves are always smooth projective connected curves over an algebraically closed field of characteristic zero.
Let $\mathcal C$ be a class of curves. We say that $\mathcal C$ is dominant if, for all curves $X$, there exists a curve $Y$ in $\mathcal C$ and a finite m... | https://mathoverflow.net/users/4333 | Is the class of $k$-gonal curves dominant | The points of $X^{(2)}$ correspond to divisors of degree 2; two points of $X^{(2)}$ are in the image of a morphism $\mathbb P^1 \to X^{(2)}$ if and only if the two corresponding divisors are linearly equivalent. Hence there is a non-trivial morphism $\mathbb P^1 \to X^{(2)}$ if and only if $X$ is hyperelliptic.
More ... | 6 | https://mathoverflow.net/users/4790 | 102657 | 59,546 |
https://mathoverflow.net/questions/102654 | 2 | This question was originally posted to Math.StackExchange, but having got no response there, I'm reposting it here. I apologise if it is too elementary for this site.
(Original post: <https://math.stackexchange.com/questions/170310/calculating-hng-mathbbzg-as-co-homology-with-compact-support-of-a-prope> )
I came acro... | https://mathoverflow.net/users/25223 | Calculating $H^n(G, \mathbb{Z}G)$ as co-homology with compact support of a proper co-compact $G$-CW-complex $X$ | Perhaps the "spectral sequence argument" Brown is referring to is the cohomology analogue of Proposition 7.3 in Chapter VII. It seems that for any $G$-CW-complex $X$ and $G$-module $M$, there should be a spectral sequence
$$
H^p(G;H^q(X;M))\Rightarrow H^{p+q}\_G(X;M).
$$
It then should follow (using Proposition 5.3 in ... | 2 | https://mathoverflow.net/users/8103 | 102665 | 59,548 |
https://mathoverflow.net/questions/102663 | 0 | Hello everyone, could someone gives the conditions for $\lambda$ that the following inequality is correct for any $0\leq x\leq\alpha$
$$\lambda x-1+e^{-\lambda x}\leq\lambda^2x\sqrt{\alpha x}$$
Here we know $2\leq\alpha\leq 4$ and $\lambda\alpha\geq 1$.
Thanks for help!
| https://mathoverflow.net/users/25005 | inequality of a function | It is true for any positive $\lambda$.
If $x\le\alpha$, the right hand side is greater or equal to $\lambda^2x^2$. Hence it suffices to show that $u-1+\exp(-u)-u^2\le 0$ for positive $u$. It is easy to show that the function
$f(u)=u-1+\exp(-u)-u^2$ is monotone decreasing and concave for $u\ge 0$.
| 2 | https://mathoverflow.net/users/12120 | 102669 | 59,552 |
https://mathoverflow.net/questions/102670 | 3 | Question. Are there smooth complex surfaces of general type with an irregularity $q = 1$ and
Euler characteristic $3$.
If the answer is yes, what is known about the geometry of such surfaces? Are there some explicit constructions?
| https://mathoverflow.net/users/25225 | Algebraic surfaces of general type | If you mean *holomorphic* Euler characteristic equal to $3$, i.e. $p\_g=3$, then the answer is *yes*.
For some explicit constructions, look at the paper by Takahashi
[Certain algebraic surfaces of general type with irregularity one and their canonical mappings](https://projecteuclid.org/journals/tohoku-mathematical... | 6 | https://mathoverflow.net/users/7460 | 102672 | 59,554 |
https://mathoverflow.net/questions/102434 | 1 | Fix polynoms g1(x), g2(x) over F\_2[x].
**Question:** How to find minimum over polynoms p(x) of the:
*HammingWeight(p(x) g1(x) ) + HammingWeight(p(x) g2(x) )* ?
By HammingWeight of polynom I mean number of non-zero monoms, let me denote by || \*||
Trivial estimate: minimum <= || g1(x)|| + || g2(x)||. Proof jus... | https://mathoverflow.net/users/10446 | Given g1(x), g2(x) minimize over p(x) Hamming weight of [p(x)g1; p(x)g2(x) ] ? (Or how to find minimal distance of convolutional code?) | Presumably you forgot to add the condition $\gcd(g\_1(x),g\_2(x))=1$ for otherwise you would allow catastrophic encoders (a finite number of channel errors may cause an infinite number of
errorneously interpreted input bits).
The minimum distance of a convolutional code can be calculated by a slightly modified Viterb... | 1 | https://mathoverflow.net/users/15503 | 102673 | 59,555 |
https://mathoverflow.net/questions/102656 | 17 | Does there exists a symplectic formulation of statistical physics?
I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated symplectic, but I do not know anything which tries to relate them 'symplectilly'. Partial results are also welcome!
| https://mathoverflow.net/users/17047 | Symplectic formulation of statistical physics | You want to read Chapter IV "Statistical Mechanics" in *[Structure of Dynamical Systems](http://www.ams.org/mathscinet-getitem?mr=1461545)* (1970 French original available [here](http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm)) by J.-M. Souriau, one of the pioneers of symplectic mechanics.
Given a sym... | 19 | https://mathoverflow.net/users/19276 | 102674 | 59,556 |
https://mathoverflow.net/questions/101471 | 2 | Everything over F\_2.
For any matrix $A$ define the number $N(A) = min\_{x}$ [HammingWeight](http://en.wikipedia.org/wiki/Hamming_weight) $( [x , Ax])$.
Where $x$ is vector and [a,b] is just concatenation of vectors: (a\_1,...a\_n, b\_1,...,b\_m).
**Question** What is $max\_{A \in Mat(n,m) } (N(A))$ ?
Particula... | https://mathoverflow.net/users/10446 | What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?) | This is, indeed, an open question for most values of $n$. A.D. Brouwer maintains a database of the best known lower
and upper bounds, and everything that is known for small $n$ can be found there. A minute of googling points me at
<http://mint.sbg.ac.at/desc_CBrouwerTable-Bound.html>
Asymptotically the best bound ... | 5 | https://mathoverflow.net/users/15503 | 102675 | 59,557 |
https://mathoverflow.net/questions/84231 | 2 | I stumbled upon a problem that seems simple but I cannot tackle it. Let $X\_n$ be a discrete process defined by the following algorithm.
Choose $X\_0\in[0,1]$, set $\kappa>0$ small enough and
$X\_{n+1}=X\_n+\kappa(I\_n-X\_n)$
with $I\_n=1$ with probability $X\_n$ and $I\_n=0$ with probability $1-X\_n$.
In oth... | https://mathoverflow.net/users/18812 | Finding a stochastic differential equation as limit of a discrete stochastic process | Here is a sketch of how you might approach this. Let
$$
\xi\_j = \frac{\kappa(I\_{j-1} - X\_{j-1})}
{\sqrt{X\_{j-1}(1 - X\_{j-1})}}.
$$
The two "hard" results that must be proven are: (1) For each $t$,
$$
\lim\_{\kappa\to0} E\big[\max\{|\xi\_j|: 1 \le j
\le \lfloor\kappa^{-2}t\rfloor\}\big] = 0,
$$
and (2) for ea... | 3 | https://mathoverflow.net/users/15575 | 102690 | 59,568 |
https://mathoverflow.net/questions/102621 | 3 | Let $P$ be a poset and denote by $\operatorname{Hom}(P, \mathbb N)$ the set of all monotone functions from $P$ to natural numbers $\mathbb N$. Under what conditions on $P$ Is it possible to recover the order on $P$ from the knowledge of $\operatorname{Hom}(P, \mathbb N)$?
I should mention here that the only example I... | https://mathoverflow.net/users/7041 | Characterizing posets by functions into natural numbers | Sitting inside $Hom(P,{\bf N})$ is the set $P^\*$ of monotone functions from $P$ into $\{0,1\}$. This set carries the structure of a "Stone lattice", and the normal lattice homomorphisms from any Stone lattice into $\{0,1\}$ will be a poset, in the case of $P^\*$ it recovers $P$. In fact I have just described a dual eq... | 7 | https://mathoverflow.net/users/23141 | 102695 | 59,571 |
https://mathoverflow.net/questions/102697 | 2 | In a comment for this old [question](https://mathoverflow.net/questions/92081/hopf-comodule-decompositions-and-coinvariant-elements), it was said that
>
There is a theorem that Hopf modules are (up to isomorphism) the tensor products of their coinvariants with H. (This theorem is usually worded in terms of a categor... | https://mathoverflow.net/users/1867 | $H$-Hopf modules equal the tensor products of their coinvariants with H | The result you mention is a classical result on Hopf modules, first proved by Larson and Sweedler. My favorite reference is
Pareigis: When Hopf Algebras are Frobenius Algebras. J. of Alg. 18(1971), 588-596. Lemma 2.
There you can also find the maps you describe in your question (so the answer is yes, they are inv... | 3 | https://mathoverflow.net/users/10194 | 102699 | 59,572 |
https://mathoverflow.net/questions/102645 | 3 | I am posting the following question from [Math.Stackexchange](https://math.stackexchange.com/questions/172528/limit-connected-with-a-periodic-function):
>
> Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula
> $$
> f(x)=2x-1.
> $$
> For a real number $x$ conside... | https://mathoverflow.net/users/18483 | Limit connected with a periodic function | No, convergence does not hold for all irrational $x$. I posted a full answer to the question at [math.stackexchange](https://math.stackexchange.com/a/173029/1321) and I'll summarize the result here.
There are uncountably many values of $x$ for which the partial sums $\sum\_{n=1}^N\frac{f(nx)}n$ are unbounded. In fact... | 3 | https://mathoverflow.net/users/1004 | 102702 | 59,573 |
https://mathoverflow.net/questions/102492 | 10 | Recall the notion of an $n$-cateogry $C$ enriched in a symmetric monoidal category. Instead of a *set* of $n$-morphisms $mor(a, b)$ (where $a$ and $b$ are compatible $(n{-}1)$-morphisms), we have an *object* $mor(a, b)$ in some symmetric monoidal category $S$. Composition of $n$-morphisms in $C$ takes the form of an $S... | https://mathoverflow.net/users/284 | n-categories enriched in (n+1)-categories | The case $n=1$ is well-known in the category-theoretic literature and is called (of course) "categories enriched in a bicategory". It was introduced by Betti and Carboni (*Cauchy-completion and the associated sheaf*) and Walters (*Sheaves and Cauchy-complete categories*), and studied by Street (*Enriched categories and... | 8 | https://mathoverflow.net/users/49 | 102721 | 59,580 |
https://mathoverflow.net/questions/102720 | 6 | One of Mertens' theorems gives that
$$\sum\_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$
where $B$ is a constant near $0.26$ in value and $E(k)$ is an error
term whose size is dominated by something close to $4/\log{k}$, when $k$
is large enough to make the sum meaningful.
I want to work with pa... | https://mathoverflow.net/users/3402 | Better error bounds for partial sums of reciprocals of primes? | In case you also want computationally explicit bounds look at Theorem 6.10 of:
Pierre Dusart: Estimates of Some Functions Over Primes without R.H.
<http://arxiv.org/abs/1002.0442>
Probably the proof is for you more interesting than the Theorem itself.
On the one hand side it shows (as Eric did) the connection with ... | 3 | https://mathoverflow.net/users/7673 | 102724 | 59,583 |
https://mathoverflow.net/questions/79819 | 3 | In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:
>
> **Proposition.** Let $R$ be a regular local ring and $I=(x\_1,\ldots,x\_r)\subset R$ an ideal generated by a regular sequence. Let $A:=R/I$. Then,
... | https://mathoverflow.net/users/9947 | A particular Isomorphism of graded algebras over a regular local ring | One reference is Bruns-Herzog, Theorem 1.1.8. You don't really need any assumption on $R$.
| 2 | https://mathoverflow.net/users/25237 | 102728 | 59,585 |
https://mathoverflow.net/questions/102731 | 0 | Has anyone read G. DaPrato and P. Grisvard *Equations d'evolution abstraites nonlineaires de type parabolique*?
It's not available in my library. I am wondering if it's worth me acquiring it: is it all in French? I only know English. Are there articles/papers that cover the same material and that are more easily ava... | https://mathoverflow.net/users/23013 | Reference: DaPrato and Grisvard parabolic PDEs. |
>
> Has anyone read the paper?
>
>
>
I haven't; [but someone did](http://www.ams.org/mathscinet-getitem?mr=551075). `:-)`
>
> Is it all in French?
>
>
>
Yes
>
> Are there articles/papers that cover the same material...?
>
>
>
It depends on which results you are looking for. It maybe that the r... | 2 | https://mathoverflow.net/users/3948 | 102735 | 59,586 |
https://mathoverflow.net/questions/102725 | 9 | Let $F$ be a finite field of odd size $q$, and $\phi\_0 : F \mapsto F$ be any map from $F$ to itself. For each $a \in F$, set $\phi\_a : x \in F \mapsto \phi\_0 (x) + ax $.
When $\phi\_0 : x \mapsto x^2 $ , each image $\phi\_a (F)$ has size $\frac{q+1}{2}$. It turns out that for *any* $\phi\_0$, there's always some ... | https://mathoverflow.net/users/25235 | A mixing property of linear map over finite fields | Consider the $q(q-1)$ quotients $\frac{\phi\_0(x)-\phi\_0(y)}{x-y}$ for all pairs of distinct elements $x,y\in F$. There is a value $a\in F$ which appears at most $\frac{q(q-1)}{q}=q-1$ times. Fix this $a$. So $\phi\_a(x)=\phi\_a(y)$ has at most $q-1$ solutions with distinct $x,y$.
Let $r$ be the size of $\phi\_a(F)$... | 13 | https://mathoverflow.net/users/18739 | 102736 | 59,587 |
https://mathoverflow.net/questions/102733 | 5 | I am trying to prove a small proposition that got me completely stumped, and I cannot find a single counterexample.
>
> (**ZF**) Suppose that $E$ is such that for every $A\subseteq\mathcal P(E)$ either $|E|\leq|A|$ or $|A|\leq|E|$, then $E$ can be well-ordered.
>
>
>
It is not a biconditional statement since... | https://mathoverflow.net/users/7206 | Comparability implies well-orderability? | It is open whether the continuum hypothesis for an infinite set $E$ implies the well-orderability of $E$. Of course, if $CH(E)$ holds, then the assumption in your (first) statement holds.
($CH(E)$ is the statement that any subset $A$ of $\mathcal P(E)$, either $A$ injects into $E$, or else $A$ is in bijection with $... | 8 | https://mathoverflow.net/users/6085 | 102743 | 59,590 |
https://mathoverflow.net/questions/102746 | 5 | Let $G$ be an abelian connected complex linear algebraic group.
Is it true that $G$ is isomorphic to $(\mathbb{G}\_m)^k\times (\mathbb{G}\_a)^\ell$, where the nonnegative exponents denote repeated direct products of linear algebraic groups, $\mathbb{G}\_m=\mathbb{C}^\*$, and $\mathbb{G}\_a=(\mathbb{C},+)$?
This see... | https://mathoverflow.net/users/24957 | Structure of abelian connected complex linear algebraic groups? | A reference:
Lie Groups and Algebraic Groups (Springer Series in Soviet Mathematics) by Arkadij L. Onishchik, Ernest B. Vinberg and Dimitry A. Leites (new printing will appear on Amazon.com on Jul 31, 2012). See Ch. 3, Section 2.5, page 116, Corollary of Theorem 8: Any irreducible commutative (linear) algebraic group i... | 3 | https://mathoverflow.net/users/4149 | 102752 | 59,594 |
https://mathoverflow.net/questions/102628 | 4 | I would like to ask a simple question. Let $A=\mathbb{C}\langle x\_{1},\dots,x\_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x\_{i}x\_{j}=a\_{ij}x\_{j}x\_{i}$ for $1\le i,j\le n$. We say a $\mathbb{C}$-algebra $B$ is n Calabi-Yau if the category of left $B$-modules is a Calabi-Yau n category, i.e. $Ex... | https://mathoverflow.net/users/25101 | When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? | Note as in my notes mentioned in B. Bischof's response, there is a nontrivial condition required to be Calabi-Yau instead of merely twisted Calabi-Yau.
Finite global dimension would follow from your condition since in particular it would imply that any $\operatorname{Ext}$ vanishes above degree $n$.
Your definition... | 9 | https://mathoverflow.net/users/25244 | 102760 | 59,597 |
https://mathoverflow.net/questions/102092 | 20 | I was on Math.Stack Exchange the other day and I found a question that said:
>
> How many zeroes are there in 100! ?
>
>
>
I quickly factored it out and said that there were 24 zeroes. However, that are only the trailing zeroes, as the person who asked the question quickly pointed out. As the days passed no o... | https://mathoverflow.net/users/24478 | number of zeroes in 100 factorial | Using well known approximations for the length and number of trailing zeroes of n!, and making the reasonable assumption that the inside zeros appear with frequency $\frac{1}{10}$, we get the following approximation of the total number of zeros, t, in n!:
$t = \lfloor \frac{1}{10}(\frac{\log (2 \Pi n)}{2}+n\log (\fra... | 7 | https://mathoverflow.net/users/1320 | 102761 | 59,598 |
https://mathoverflow.net/questions/102753 | 5 | A [coverage](http://www.ncatlab.org/nlab/show/coverage) $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for every arrow $f: V \to U$ in $C$, there is a function $J(U) \to J(V)$ which ... | https://mathoverflow.net/users/18355 | Coverage, itself considered as a presheaf | First of all, Andreas' comment is right: a coverage gives no specified way to "pull back" a covering family of $U$ to a covering family of $V$. However, if you consider what *Sketches of an Elephant* calls "sifted" coverages, meaning that all covering families are sieves, then there is a canonical choice: the pullback ... | 7 | https://mathoverflow.net/users/49 | 102762 | 59,599 |
https://mathoverflow.net/questions/102763 | 15 | A reader of one of my drafts found fault with my use of parentheses; I put the word "bounded" in parentheses in a statement of a certain theorem, and he replied "But the statement isn't true if the assumption of boundedness is dropped!"
That reader seemed to be thinking that parentheses mark things that are in some w... | https://mathoverflow.net/users/3621 | the use of parentheses to mean "I won't tell you this again" | Re: "Does this use of parentheses have a name?",
[preterition](http://en.wikipedia.org/wiki/Apophasis#Paralipsis) |ˌpretəˈri sh ən|
noun (...) the rhetorical technique of making summary mention of something by professing to omit it.
ORIGIN late 16th cent.: from late Latin praeteritio(n-), from praeterire ‘pass, g... | 14 | https://mathoverflow.net/users/19276 | 102768 | 59,601 |
https://mathoverflow.net/questions/102714 | 3 | To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta\_g$ be the Laplace-Beltrami operator w.r.t. $g$.
**Prove**: Suppose $f\in C^\alpha(M)$ satisfies $\int\_M f\, dVol\_g=0$, then there exists a function $u\in C^{2,\alpha}(M)$ such that $\Delta\_g u=f$ in $M$, an... | https://mathoverflow.net/users/15289 | How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds? | Careful. C^\infty is not dense in C^{\alpha}. Nonetheless you can first produce an L^2 solution then use Schauder estimates to show that the solution is C^{2,\alpha}. This last step is localizable -- you can multiply by a smooth cutoff function (and use some preliminary
estimates to say that the solution is C^2, for ex... | 7 | https://mathoverflow.net/users/17969 | 102769 | 59,602 |
https://mathoverflow.net/questions/102767 | 6 | Let $\mathcal{A}$ be an abelian category and let
$$
0 \rightarrow E \rightarrow F \rightarrow G \rightarrow 0
$$
be a short exact sequence. Then in $D(\mathcal{A})$, the derived category of $\mathcal{A}$ we have a distinguished triangle
$$
E \rightarrow F \rightarrow G \rightarrow E[1].
$$
Moreover, the above short... | https://mathoverflow.net/users/6330 | Cone of a morphism in an abelian category when considered as a morphism in derived category. Connection between 4-term exact sequences and distinguished triangles. | I'll expand on Steve's comment, but I'm sure you must know this stuff already.
if $f: A \to B$ you take the cone $D$ and cohomology long exact sequence tells you that the cohomology of $D$ is $K = \text{ker } f$ in degree minus one and $C = \text{coker } f$ in degree zero.
Explicitly a cone is given by the mapping co... | 2 | https://mathoverflow.net/users/16857 | 102775 | 59,606 |
https://mathoverflow.net/questions/102068 | 5 | Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q\_{X}$ is equal to a fixed integer $k \neq 0, \pm 1$. Is it true that the number of possible intersection forms for such $X$ is finite? Any... | https://mathoverflow.net/users/25085 | Intersection forms of 4-manifolds with boundary | The classification of integral quadratic forms is discussed in [Chapter 15 of Conway-Sloane](http://books.google.com/books?id=upYwZ6cQumoC&lpg=PR5&ots=_L-R4Qi9DS&dq=conway%2520sloane&lr&pg=PA352#v=onepage&q&f=false). In particular, the discussion there implies that there are only finitely many integral quadratic forms ... | 4 | https://mathoverflow.net/users/1345 | 102776 | 59,607 |
https://mathoverflow.net/questions/33016 | 8 | **Definition**
Consider a domain $\Omega \subseteq \mathbb{R}^3$ and three differentiable maps $f\_i:\Omega \rightarrow \mathbb{R}$, $i = 1, 2, 3$. If at every point $x \in \Omega$, $\nabla f\_i(x) \cdot \nabla f\_j(x) = 0$ whenever $i \ne j$, then the $f\_i$ are *triply orthogonal coordinates* on $\Omega$.
**Conte... | https://mathoverflow.net/users/1557 | Existence of triply orthogonal coordinates | I give an explicit counterexample below, so I have modified my answer: No, these sorts of orthogonal coordinates on 'shells' of parallel compact surfaces don't always (or even usually) exist. The reason has to do with whether there are nonconstant functions on the given surface $\mathcal{M}$ that are constant on one of... | 13 | https://mathoverflow.net/users/13972 | 102783 | 59,612 |
https://mathoverflow.net/questions/102777 | 2 | Suppose I have a r.v. $Z = X + \alpha Y$ and that $F\_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F\_Z(q) = \mathbb{P}(X+\alpha Y < q)$ as a function $p = p(q, \alpha)$, how can we write the derivative $\partial\_{\alpha} p(q, \alpha)$ supposing as much regularilty on the dis... | https://mathoverflow.net/users/8916 | Derivative of the CDF of a family of random variables | If $X$ and $Y$ have joint density $f(x,y)$, we have
$p(q,\alpha) = \int\_{-\infty}^\infty dy\ \int\_{-\infty}^{q - \alpha y} dx\ f(x,y)$
and thus (assuming sufficient regularity) $\partial\_\alpha p(q,\alpha) = - \int\_{-\infty}^\infty dy \ y f(q-\alpha y,y)$.
| 4 | https://mathoverflow.net/users/13650 | 102788 | 59,616 |
https://mathoverflow.net/questions/102795 | 7 | It is known that there exist infinitely many non-trivial zeros of the Riemann zeta function in the critical strip. Also, we know that infinitely many zeros are on the critical line - more than 1/3 among all, asymptotically - as well as most of the zeros live near the critical line.
In 1984, the theorem of Rademacher ... | https://mathoverflow.net/users/14467 | If a non-trivial zero of the zeta function existed off the critical line, would infinitely many zeros exist with the same real part? | 1) Yes, this is still possible
2) No. We know that they have to distributed with low density, i.e. the number of zeros $z$ with $\Im z < T$ with $\Re z > \sigma >1/2$ is bounded by $T^{4\sigma(1-\sigma) + \epsilon}$ for $\epsilon >0$. So no uniform distribution is possible, since the gaps between consecutive zeros h... | 9 | https://mathoverflow.net/users/10400 | 102796 | 59,620 |
https://mathoverflow.net/questions/102794 | 5 | I apologize for the vague title. Let $M$ be a compact smooth manifold, then we have $T^\*M$ and hence $\wedge^pT^\*M$ as vector bundles on $M$. There for we have
$$
\sum (-1)^p[\wedge^pT^\*M] \in K(M).
$$
It seems to have close relation with the Euler class of $M$.
My question is: Is there any study on $\sum (-1)^p[... | https://mathoverflow.net/users/24965 | What is the role of $\sum (-1)^p[\wedge^pT^*M]$ in the K-theory $K(M)$ | I think the main topological significance of the element you identify is its close relationship with the Thom isomorphism $K(M) \cong K(T^\*M)$. I would imagine that there is also some connection with the Euler characteristic as well, perhaps by taking Chern characters, but this is probably secondary to its role in the... | 5 | https://mathoverflow.net/users/4362 | 102808 | 59,623 |
https://mathoverflow.net/questions/102798 | 0 | I would like to know examples of $G/K$ where $G$ is a locally compact group and $K$ is a compact subgroup of $G$.
I know about Riemannian symmetric spaces of Euclidean, compact and non-compact type. They can be realized as $G/K$. I am wondering what are other prominent spaces (discrete and continuous) which can be wr... | https://mathoverflow.net/users/25252 | Examples of locally compact group quotiented by compact subgroup, but not Riemannian symmetric space | There are lots of Riemannian homogeneous spaces that are not symmetric. For example, consider the chain $\mathrm{SO}(2)\subset \mathrm{SO}(3)\subset \mathrm{SO}(4)$. The quotient $\mathrm{SO}(4)/\mathrm{SO}(2)$ is not a symmetric space.
| 3 | https://mathoverflow.net/users/13972 | 102814 | 59,628 |
https://mathoverflow.net/questions/102764 | 1 | It is well known that the gravitational forces due to a spherical shell of uniform density cancels in the interior of the shell (in three dimensions). Another way to state this is that the gravitational potential is uniform in the interior of the sphere.
Suppose you are given a sphere of fixed radius and you can desi... | https://mathoverflow.net/users/18931 | A constant potential due to a designed radial force inside a spherical shell | I'm answering my own question after a night of sleep.
---
Without loss of generality we consider the unit sphere centered at the origin. Define the potential between two point masses separated by a distance $\ell$ as $Q(\ell)$. The potential at a point at radius $r$ inside the unit sphere can be found by taking a... | 1 | https://mathoverflow.net/users/18931 | 102818 | 59,631 |
https://mathoverflow.net/questions/102810 | 6 | I don't known the correct credit for the following: every non-empty $\Pi^1\_1$ set of reals contains some $X \in L\_{\alpha}$ for some $X$-recursive $\alpha$. (Addison-Kondo?)
So, my question is: what is the least $\beta$, s.t. $\mathcal{P}(\omega) \cap L\_\beta$ is a basis for all non-empty $\Pi^1\_1$ sets?
| https://mathoverflow.net/users/15424 | To find an element of a $\Pi^1_1$ set | I can't access the full paper at the moment, but I'm pretty sure this is exactly the question addressed in the paper "A Note on the Kondo-Addison Theorem" by D. Guaspari
| 5 | https://mathoverflow.net/users/1682 | 102820 | 59,632 |
https://mathoverflow.net/questions/102829 | 9 | The Mostowski collapse lemma (see [here](http://en.wikipedia.org/wiki/Mostowski_collapse_lemma) for a quick ref) is one of the key basic tools in the set-theory arsenal. I wonder if the collapse is natural, in the functorial sense.
>
> More precisely, is this a
> reflection from the large category of
> well-foun... | https://mathoverflow.net/users/15293 | Is the Mostowski collapse natural? | You didn't say what the morphisms in your categories are supposed to be; Trevor assumed you meant elementary embeddings, but you could also have meant mere embeddings, or something else. Nevertheless, unless you make a very strange choice of morphisms, the answer to your question is yes. Not only is the Mostowski colla... | 13 | https://mathoverflow.net/users/6794 | 102833 | 59,638 |
https://mathoverflow.net/questions/102823 | 5 | The question mark complex is a finite spectrum whose cohomology looks like a "question mark" (when drawn as a module over the Steenrod algebra): that is, there is an element in dimension zero $a\_0$, an element $a\_2 = \mathrm{Sq}^2 a\_0$, and an element $a\_6 = \mathrm{Sq}^4 a\_2$. It can be constructed by starting wi... | https://mathoverflow.net/users/344 | Reference on the question mark cell complex | This is hardly a "canonical example", but one place I've seen the question mark complex $Q$ in action is in Hovey and Sadofsky's paper *Invertible spectra in the $E(n)$-local stable homotopy category*. There they compute that the even part of the Picard group of the $E(1)$-local stable category at $p = 2$ is generated ... | 5 | https://mathoverflow.net/users/1094 | 102841 | 59,643 |
https://mathoverflow.net/questions/102839 | 65 | I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, <http://www.jmilne.org/math/xnotes/MOT102.pdf>), the category of motives is defined to be a category such that every Weil cohomology (viewed as a functor) factors through it. This does not define the ... | https://mathoverflow.net/users/5309 | What is the relationship between motivic cohomology and the theory of motives? | Classically, Grothendieck's motives are only the pure motives, meaning abelian-ish things which capture the (Weil-cohomology-style) $H^i$ of smooth, projective varieties. To see the relationship with motivic cohomology, one should extend the notion of motive so that non-pure (i.e. "mixed") motives are allowed, these mi... | 77 | https://mathoverflow.net/users/3931 | 102842 | 59,644 |
https://mathoverflow.net/questions/102802 | 7 | This must be a naive question, but I'm wondering about the definition of quasi-unipotent monodromy for general families, not only 1-parameter families. The problem is that usually, in the books of algebraic geometry, quasi-unipotent monodromy is only discussed over a disc $\Delta^{\ast}$, i.e. for a 1-parameter family.... | https://mathoverflow.net/users/18000 | Quasi-unipotent monodromy for general families | Quasi-unipotency is a well defined notion at any point of the discriminant. If we have a proper family $f : X \to S$ of varieties with a smooth total space and a smooth base, and if $p \in D \subset S$ is a point of the discriminant, then we say that the local monodromy of the family near $p$ is quasi-unipotent if we c... | 15 | https://mathoverflow.net/users/439 | 102847 | 59,646 |
https://mathoverflow.net/questions/102846 | 20 | Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in [this way](http://en.wikipedia.org/wiki/Grothendieck_universe)). Also, say that "Ca" is the axiom that "For each cardinal κ, t... | https://mathoverflow.net/users/24611 | What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory? | Yes. Assume ZFC. If there is a proper class of inaccessible cardinals, then Tarski's Axiom A holds because whenever $\kappa$ is inaccessible, the rank initial segment $V\_\kappa$ of $V$ is a Tarski set. Conversely, if Tarski's Axiom A holds then for every set $x$ there is a Tarski set $y$ with $x \in y$. We will show t... | 19 | https://mathoverflow.net/users/1682 | 102849 | 59,647 |
https://mathoverflow.net/questions/102858 | 1 | Hello!
I want to prove that $x = 0.5$ is the global maximum of the function
$f(x) = \frac{(1-a)^2e^{(2x\cdot(x-1)a^2)}}{(1-a)(e^{(2x\cdot(2x-1)a^2)}+e^{((2x-1)\cdot(2x-2)a^2)})-2(1-2a)e^{(4x\cdot(x-1)a^2)}}$
where $a\in(0,1)$ and $x\in[0,1]$. I tried to show this using "conventional" ways, but unfortunately the ... | https://mathoverflow.net/users/25265 | Global maximization of a particular function | I have to run out now, so will write only the hint as of now. Later I might complete the details. Let me do one case here, the other one probably follows similarly.
Let $a \in (1/2,1)$, and write $a=1/2 + \epsilon$, for an appropriate $\epsilon$.
Now consider the function $1/f(x)$, which is (after some cleanup) se... | 0 | https://mathoverflow.net/users/8430 | 102859 | 59,650 |
https://mathoverflow.net/questions/102803 | 3 | Let $m$ be an even integer greater than $8$. Is there any software I can use to determine for some small $m$ whether the following constraints on $t\_0,\ldots,t\_{m-1}$ and $w$ have solutions?
$-\pi\leq t\_r\leq\pi,\ r=0,\ldots,m-1,$
$0\leq w\leq\frac{1}{m},$
$\sum\limits\_{j=1}^{2r-1}\cos(t\_j+t\_{2r-j})+\sum\l... | https://mathoverflow.net/users/16475 | software for numerical constraint satisfaction problems | It seems to me that methods based on interval analysis are very efficient for solving hard constraints satisfaction problems. In particular, SIVIA (Set Inversion Via Interval Analysis) is an algorithm which can approximate using small "boxes" a subset of $\mathbb{R}^n$ satisfying a given set of constraints. It's a bran... | 1 | https://mathoverflow.net/users/13552 | 102867 | 59,653 |
https://mathoverflow.net/questions/102622 | 0 | Consider a finitely presented group $G$ with presentation $P$ given by $\left\langle g\_1,\ldots,g\_n|\, r\_1,\ldots,r\_m\right\rangle$, equipped with a homomorphism $\rho\colon\, G\to H$ to a finitely generated group $H$ (generators in the presentation are ordered). Perform a sequence of [Tietze transformations](http:... | https://mathoverflow.net/users/2051 | Monodromy in presentations of one group over another | The question is effectively answered by HW and by Misha in the comments. Namely, a sequence of Tietze transformations starting and ending at "the same" presentation defines an automorphism of $G$. But the action of $\mathrm{Out}(G)$ on $\mathrm{Hom}(G,H)/\mathrm{Aut}(H)$ is non-trivial in general.
| 0 | https://mathoverflow.net/users/2051 | 102871 | 59,656 |
https://mathoverflow.net/questions/102862 | 4 | (I asked this question on MSE but I did not receive an answer so I hope I can post here.)
Let $S$ be a **compact** set in $\mathbb{R}^2$ and let $C^{k, \alpha}(S)$ denote the usual Holder space with $k$ continuous derivatives and finite $k$-th order seminorms with exponent $\alpha$.
1) Is it true that if $f \in C^... | https://mathoverflow.net/users/25266 | Composition of $C^{k, \alpha}$ function with $C^\infty$ function on a compact domain | From the mean value theorem, we have
$$
|f(u(x))-f(u(y))| \leq \max\_{\xi\in U}|f'(\xi)| \cdot |u(x)-u(y)|,
$$
where $U=[\min u, \max u]$. This argument applied to $f^{(k)}$ should give $f\circ u \in C^{k,\alpha}$.
Something similar can be done also for the second question.
| 2 | https://mathoverflow.net/users/824 | 102873 | 59,657 |
https://mathoverflow.net/questions/102855 | 4 | Let $\mathcal{lie}\_n$ be the free Lie algebra generated by $n$ elements $x\_1,\ldots, x\_n$. A derivation $u\in \text{Der}(\mathcal{lie}\_n)$ is called tangential if there exist $a\_i\in \mathcal{lie}\_n, i=1\ldots n$ such that $u(x\_i)=[x\_i,a\_i]$. The definition can be found in section 3 of Alekseev and Torossian's... | https://mathoverflow.net/users/24965 | About the term "tangential derivation" on a free Lie algebra. | You may be interested in these [Bar Natan's lecture](http://katlas.math.toronto.edu/drorbn/dbnvp/wClips-120530.php).
The geometric intuition is that the $n$th Lie algebra of tangential derivations can be realized into the algebra of tangential differential operators (not derivations !) on $\mathfrak g^n$ for any (say... | 4 | https://mathoverflow.net/users/13552 | 102885 | 59,665 |
https://mathoverflow.net/questions/102887 | 2 | Dear all,
I have recently found the following discussion:
[Different Measures On R2](https://mathoverflow.net/questions/102061/hausdorff-measure-and-minkowskis-content-boundary-measures)
regarding different boundary measures on $\mathbb{R}^n $ .
The discussion made me wondering:
Is there any example in the oppo... | https://mathoverflow.net/users/25272 | Geometric Measure Theory On Euclidean Spaces | Take $[0,1]\times \{0\} \subset \mathbb{R}^2$.
| 3 | https://mathoverflow.net/users/11716 | 102891 | 59,670 |
https://mathoverflow.net/questions/102835 | 6 | This is my first question on mathoverflow! It relates to a project I'm undertaking with a student.
Work by [Tamura](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-33/issue-3/The-average-edge-order-of-triangulations-of-3-manifolds/ojm/1200787102.full) (extending results by [Luo and Stong](https... | https://mathoverflow.net/users/25075 | What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant) | Below is the data from Ben Burton's census of triangulations of $S^3$ with up to 9 tetrahedra. An Euler characteristic argument shows that the average edge degree is equal to $6\left(\frac{T}{V+T}\right)$, where $T$ is the number of tetrahedra and $V$ is the number of vertices, so the numbers below are all of that form... | 6 | https://mathoverflow.net/users/6015 | 102893 | 59,672 |
https://mathoverflow.net/questions/102879 | 37 | Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of $S\_n$)?
| https://mathoverflow.net/users/8381 | Bijection between irreducible representations and conjugacy classes of finite groups | This is a different take on Steven Landsburg's answer. The short version is that conjugacy classes and irreducible representations should be thought of as being dual to each other.
Fix an algebraically closed field $k$ of characteristic not dividing the order of our finite group $G$. The group algebra $k[G]$ is a fi... | 44 | https://mathoverflow.net/users/290 | 102894 | 59,673 |
https://mathoverflow.net/questions/102902 | 3 | I am thinking about homomorphisms $\mathrm{Hom}(\Gamma,G)$, where $G$ is a Lie group and $\Gamma$ is a discrete, finitely generated subgroup.
[This question](https://mathoverflow.net/questions/34640/local-vs-infinitesimal-rigidity) talked about the difference of infinitesimal rigidity vs. local rigidity and [the seco... | https://mathoverflow.net/users/11084 | Infinitesimal rigidity vs. local rigidity | First, there are many examples of representations $\rho: \Gamma \to G$ which are locally but not infinitesimally rigid. The earliest example is due to Lubotzky and Magid, it is a reducible representation $\rho\_0$ of a Euclidean Coxeter group of rank 3 (namely, $T(3,3,3)$, the affine Weyl group of type $A\_2$) to $SO(3... | 5 | https://mathoverflow.net/users/21684 | 102912 | 59,683 |
https://mathoverflow.net/questions/102880 | 1 | Consider a Clebsch-Gordan expansion $R\_i\bigotimes{R\_j}=\bigoplus\_p{R\_p}$. Assume the irrep $R\_k$ does NOT appear in the sum on the right side. Does it now follow that the "triangle" ${R\_i,R\_j,R\_k}$ is "inaccessible" and consequently the 6j symbol
$
\begin{Bmatrix}
R\_i & R\_j & R\_k\\\
R\_l & R\_m & R\_n
\end{... | https://mathoverflow.net/users/11504 | Simple 6j symbol question | Doc, not sure what "inaccessible" but the second question is a tautology. Your $6j$ symbol is a linear map $\hom(R\_k,R\_i\otimes R\_j)\otimes\hom(R\_m,R\_k\otimes R\_l)\rightarrow \hom(R\_n,R\_j\otimes R\_l)\otimes \hom(R\_m,R\_i\otimes R\_n)$. It is zero because the left vector space is zero.
| 3 | https://mathoverflow.net/users/5301 | 102918 | 59,685 |
https://mathoverflow.net/questions/102920 | 6 | Let $\sigma(n,m)$ be the number of trees with $n$ vertices $\{ v\_1, \dots, v\_n \}$ such that the matching number (the size of a maximum matching) is $m$.
I have been trying to compute the value of $\sigma(n,m)$ but I have been unsuccessful. It looks hard. Is it known? Any idea or suggestion about how can $\sigma$ b... | https://mathoverflow.net/users/22236 | Number of trees with the same matching number | If $n=2m$, the answer is $2^{m-2} m! \binom{2m}{m} m^{m-2}$. Let $T\_m = \sigma(n,m)$. We go through several transformations. We write $[n]$ for the set $\{ 1,2,\ldots, n \}$ and $[a,b]$ for $\{ a, a+1, \ldots, b \}$.
$T\_m$ number of fully matched trees on $[2m]$.
$2 T\_m$ number of fully matched trees on $[2m]$ e... | 4 | https://mathoverflow.net/users/297 | 102929 | 59,688 |
https://mathoverflow.net/questions/102931 | 5 | Let $\omega$ be an $n$-form on $S^n$, nowhere vanishing.
**Is there a Riemannian metric $g$ on $S^n$, so that its volume form is $\omega$, and $(S^n,g)$ is homogeneous? Is it unique, and if not, what transformations of $S^n$ relates such metrics?**
For example, for $n=1$, the unique metric with these properties is ... | https://mathoverflow.net/users/10095 | Homogeneous metrics of given volume element, on the n-sphere. | There is. By Moser's Theorem, there exists a diffeomorphism $f:S^n\to S^n$ such that $f^\*\omega = c\omega\_{std}$ where $c>0$ and $\omega\_{std}$ is the standard volume form homogeneous under $\mathrm{SO}(n{+}1)$. Now use $f$ to transfer the right multiple of the standard metric (under which $S^n$ is homogeneous) back... | 18 | https://mathoverflow.net/users/13972 | 102938 | 59,690 |
https://mathoverflow.net/questions/97819 | 2 | Fix natural numbers $n,m\in\mathbb{N}$. Given a partition $\lambda\vdash d$ with at most $n$ rows (and at most $m$ columns), we can define a partition $\lambda^\ast=(\lambda^\ast\_1,\ldots,\lambda^\ast\_n)$ by setting $\lambda^\ast\_i:=m-\lambda\_{n+1-i}$. Graphically, it is obtained by taking the complement of the You... | https://mathoverflow.net/users/9947 | Box-dual of a partition - what is it called? | The "right" answer was given by Gjergji Zaimi in his comment above: Both "reverse partition" and "complementary partition" are terms that seem to be in use.
| 1 | https://mathoverflow.net/users/9947 | 102941 | 59,691 |
https://mathoverflow.net/questions/97440 | 1 | For $G,H$ Hopf algebras, and $\pi:G \to H$ a Hopf algebra map, can some-one give me an example of a (right) $H$-comodule $(V,\Delta\_R)$, such that
$$
(G \otimes V)^{\text{co}H} = \lbrace g\_{(1)} \otimes v\_{(0)} \otimes \pi(g\_{(2)})v\_{(1)} = g \otimes v \otimes 1 ~ | ~ g \otimes v \in G \otimes V) \rbrace = \lbrac... | https://mathoverflow.net/users/2612 | Trivial Hopf Coinvariant Subspace Example | Zero comodule!!
If you want something more halloweeny, take $G$ to be a field, $H$ - a Hopf algebra, not a field, $V$ - any simple nontrivial $H$-comodule...
**EDIT:** as asked, $G$ is any Hopf algebra with a simple non-trivial comodoule, $H=G\otimes G$, the map is $x\mapsto x \otimes 1$, $V$ is a simple nontrivia... | 2 | https://mathoverflow.net/users/5301 | 102942 | 59,692 |
https://mathoverflow.net/questions/102932 | 17 | This is a naive question but I hope that the answers will be educational. When is it the case that a finitely presented group $G$ admits a faithful $2$-dimensional complex representation, e.g. an embedding into $\text{GL}\_2(\mathbb{C})$? (I am mostly interested in sufficient conditions.)
I think I can figure out th... | https://mathoverflow.net/users/290 | What finitely presented groups embed into $\operatorname{GL}_2$? | If the group $G$ does not have [property FA](http://en.wikipedia.org/wiki/Property_FA), then a necessary and sufficient
condition is that the group embeds in $\operatorname{GL}\_2(\mathcal{O}\_K)$, for some number
field $K$ (although there are such subgroups which do not have property FA).
This follows from [Bass-Serre... | 17 | https://mathoverflow.net/users/1345 | 102944 | 59,694 |
https://mathoverflow.net/questions/102939 | 1 | Let $\frak G$ be a Lie algebra and let $M$ be a $\frak G$-module generated by a vector $v$ satisfying some set of defining relations denoted by $R$. I mean, $M = U(\frak G)/\langle R \rangle$, where $\langle R \rangle$ is the $U(\frak G)$-submodule generated by $R$ and $U(\frak G)$ denotes the universal enveloping alge... | https://mathoverflow.net/users/25284 | Module given by generators and relations | If I understand you correctly, you want to write $M$ as $U(\frak G/\frak B)/\langle R'\rangle$ where $R'$ is somehow determined by $S$.
This can indeed be done. There is a natural map $U(\frak G) \to U (\frak G/\frak B)$, the unique ring homomorphism that sends elements of $G$ to the corresponding elements of $G/B$. ... | 1 | https://mathoverflow.net/users/18060 | 102951 | 59,699 |
https://mathoverflow.net/questions/102936 | 2 | Let $R$ be a commutative ring and let $M$ and $N$ be $R$-modules. Let $\sigma:R\rightarrow R$ be a ring automorphism.
Let $f: M\rightarrow N$ be a $\sigma$-semilinear map, i.e. a map of abelian groups satisfying
$f(rm)=\sigma(r)f(m)$. Let $Hom\_{R,\sigma}(M,N)$ denote the set of all $\sigma$-semilinear maps from $M$... | https://mathoverflow.net/users/3969 | Dual of a semilinear morphism | No.
Take $R=k[x,y]/(x^2,xy,y^2)$ and let $\sigma=id$ and $\sigma'$ swap $x$ and $y$.
Let $M=N=R/(x,y)$, then the identity map $M\to N$ is $\sigma$ and $\sigma'$ semilinear.
One element of $N^\*$ sends the generator to $x$. But $\sigma^{-1}$ sends $x$ to itself and $\sigma'^{-1}$ sends $x$ to $y$.
| 3 | https://mathoverflow.net/users/18060 | 102955 | 59,700 |
https://mathoverflow.net/questions/102947 | 1 | In the paper by GUIDO DE PHILIPPIS AND ALESSIO FIGALLI:
<http://arxiv.org/abs/1111.7207>
They proved the $W^{2,1}$ estimate for standard monge-ampere equation
$detD^{2}u=f$
with $f$ bounded from below and above. But there is one part I can not understand very well.
In page 5, at the beginning of section 3, they s... | https://mathoverflow.net/users/25293 | W^{2,1} REGULARITY FOR SOLUTIONS of Monge-Ampere equation | One possible approximation argument is spelt out in detail in section 5 of [this paper of Schmidt](http://cvgmt.sns.it/media/doc/paper/1779/eps.pdf).
There the author proves the stronger result that under your same assumption $0<\lambda\leq f\leq \Lambda$ then $u\in W^{2,1+\varepsilon}\_{\rm loc}$ for some $\varepsi... | 1 | https://mathoverflow.net/users/13168 | 102958 | 59,702 |
https://mathoverflow.net/questions/102917 | 7 | Hello everyone,
I wanted to lecture on Grassmann and his works , and I have been reading the collected works of Grassmann " Die Lineale Ausdehnungslehre ". There Grassmann introduced something called " Interior product " ( Left and Right interior products ) . So I was completely stuck up there, the Bourbaki papers d... | https://mathoverflow.net/users/24713 | Urge/reason for inventing interior product ( Grassmann algebra ) | Here is an elementary motivation for interior products .
Suppose $V$ is an $n$-dimensional vector space .
To every non-zero vector $ v\in V$ you can associate the complex $$ 0\to V\to...\to \Lambda ^kV\to \Lambda ^{k+1}V\to...\to \Lambda ^nV\to 0 \\quad (\star)$$
where the linear map $ext\_k(v):\Lambda ^kV\to \L... | 11 | https://mathoverflow.net/users/450 | 102959 | 59,703 |
https://mathoverflow.net/questions/102898 | 1 | Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J\_\epsilon$ on a compact manifold $M$, defined by the following three properties:
1) $J\_\epsilon \in \Psi^{-\infty}(M)$ for each $\epsilon \in (0,1... | https://mathoverflow.net/users/20264 | Generalized Friedrichs Lemma | The topology on $\Psi^m$ is a ''bornology''. A sequence of symbols is converging if it is bounded in the Fréchet space $\Psi^m$ and if it converges in $C^\infty$. That bornology implies the strong convergence of operators. Then $J\_\epsilon$ is bounded in $\Psi^0$ implies that the commutator $[A,J\_\epsilon]$ is bounde... | 2 | https://mathoverflow.net/users/21907 | 102960 | 59,704 |
https://mathoverflow.net/questions/102956 | 4 | Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$.
>
> What conditions on $\iota$ -- if any -- guarantee that $\mu\circ\iota:N \to \mathbb{R}$ is a Morse function on $N$?
>
>
>
Of cou... | https://mathoverflow.net/users/18263 | Classifying smooth embeddings which yield Morse functions | Saying a function $f : M \to \mathbb R$ is Morse amounts to saying that $D^\* f : M \to T^\* M$ is transverse to the zero section, where $D^\*f$ is the derivative of $f$, though of as a section of the cotangent bundle, alternatively that the Hessian $Hf\_p$ is non-degenerate at the critical points $p$ of $f$.
A crit... | 7 | https://mathoverflow.net/users/1465 | 102961 | 59,705 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.