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/301193 | 2 | I wanted to know if it is possible to construct an indecomposable self-injective finite-dimensional algebra $\Lambda$ whose Auslander-Reiten quiver $\Gamma\_\Lambda$ is not connected. I'd love to see examples as well as construction methods, if they exist, since I may later want to obtain examples satisfying further pr... | https://mathoverflow.net/users/12166 | In search of disconnected indecomposable self-injective finite-dimensional algebras | Probably any example you try (that is not of finite representation type) will work. It is conjectured that the Auslander-Reiten quiver of a finite dimensional algebra $A$ is never connected, and in fact has infinitely many connected components, unless $A$ has finite representation type. See (2) and (3) in the list of c... | 5 | https://mathoverflow.net/users/22989 | 301198 | 132,015 |
https://mathoverflow.net/questions/301202 | 3 | In [this](https://mathoverflow.net/q/288105/118366) question I had asked about proof of the property of selective ultrafilter. As was answered, the proof is trivial if we know that ultrafilter is selective iff it is Ramsey ultrafilter. The proof of latter fact can be found in book "Theory of ultrafilters" by Comfort an... | https://mathoverflow.net/users/118366 | Selective ultrafilter on $\omega$ is normal. Clear proof | I don't have my copy of Comfort & Negrepontis handy, so the following might be essentially the same as their proof, but I think it's clear enough.
The very rough idea is that, because $\mathcal F$ is a Q-point, the proof would be easy if each $A\_i$ were a final segment of $\omega$ (Steps 3 and 4 below), and, because... | 5 | https://mathoverflow.net/users/6794 | 301207 | 132,018 |
https://mathoverflow.net/questions/301209 | 3 | It is well known how to find a solution for the following linear difference equation
$$h\_{m} = h\_{m-1} + a \cdot h\_{m-2}$$
Finding the roots $r\_1$ and $r\_2$ of $r^2 - r - a$, we have that the solutions are of the type $\lambda r\_1^m + \mu r\_2^m$, and then we can solve for $m = 0$ and $m = 1$ to find $\lambda... | https://mathoverflow.net/users/124913 | Linear difference inequality | You can find the sharpest possible bound for any $m$ and numerical value of $a$ by solving a Linear Program.
Make use of $h\_0 = 0$ and $h\_1 = 1$ in the below.
**Minimize $h\_m$** with respect to $h\_2,..,h\_m$
subject to
$$h\_i \ge 0, i= 2,., m$$
$$h\_i \le 1, i = 2,.., m$$
$$h\_i \ge h\_{i-1}+ ah\_{i-2}, i=2... | 1 | https://mathoverflow.net/users/75420 | 301214 | 132,020 |
https://mathoverflow.net/questions/301223 | 6 | I got very lost in checking that simplicial sets with Quillen model structure are indeed a simplicial model category.
Recall that a model category $\mathcal{M}$ is simplicial if it is enriched in $\mathbf{SSet}$, powered by $\mathbf{SSet}$ and tensored by $\mathbf{SSet}$ so that the natural adjunctions exist and and... | https://mathoverflow.net/users/123731 | How are simplicial sets with Quillen model structure a simplicial model category? | The trick is to check that the corner map $$\lambda^n\_k\bar{\times}\delta^m:\Lambda^n\_k \times \Delta^m \coprod\_{\Lambda^n\_k\times \partial \Delta^m} \Delta^n \times \partial \Delta^m \hookrightarrow \Delta^n\times \Delta^m$$ is anodyne for all $k, m, n$ appropriate. This is proven in Higher Topos Theory chapter 2,... | 7 | https://mathoverflow.net/users/1353 | 301227 | 132,022 |
https://mathoverflow.net/questions/301213 | 0 | Most probably this question should be well studied in the theory of stochastic processes, but I am not educated in that area. Sorry if this question is too elementary.
Let $V\colon \mathbb{R}\to \mathbb{R}$ be a function to be specified later. Fix two points $a,b\in \mathbb{R}$. I am interested in convergence of the ... | https://mathoverflow.net/users/16183 | Convergence of an integral with respect to the Wiener measure | The conditional Wiener measure is concentrated on the space $C(L,a,b)$ of continuous curves $x : [0,L] \to \mathbb R$ such that $x(0) = a$ and $x(L) = b$, endowed with the topology given by the distance $d(x,y)= \sup \_{t \in [0,L]} |x(t) - y(t)|$. Since it is a Borel, regular measure, all continuous and bounded functi... | 1 | https://mathoverflow.net/users/54780 | 301229 | 132,023 |
https://mathoverflow.net/questions/301212 | 4 | Let say we have a symmetric matrix $A(\omega)$ depending smoothly on some variables $\omega \in \Omega$ with $\Omega \subset \mathbb{R}^d$ a $d$-dimensional parameterspace (this means the eigenvalues are real). For calculating the eigenvalues we can make use of the characteristic polynomial $\rho(A( \omega))$ and by se... | https://mathoverflow.net/users/114495 | when is an eigenvalue differentiable with respect to a parameter? | When the roots are simple, they can be chosen as smooth functions of $\omega$ if the matrix $A$ is smooth of $\omega$ ; both "smooth" above can be replaced by "analytic". This is a consequence of the implicit function theorem, since the characteristic polynomial is $P\_{A(\omega)}(\lambda)$ and the simplicity of a give... | 5 | https://mathoverflow.net/users/21907 | 301232 | 132,026 |
https://mathoverflow.net/questions/301222 | 8 | Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?
| https://mathoverflow.net/users/69275 | Subspaces isomorphic with quotients | Every separable Banach space is a quotient of $\ell\_1$, so in particular every subspace of $\ell\_1$ is a quotient of $\ell\_1$.
| 11 | https://mathoverflow.net/users/15129 | 301233 | 132,027 |
https://mathoverflow.net/questions/300891 | 6 | Let $\Gamma$ be a group with a probability measure preserving action on $(X,\mu)$, and $H$ another group. Recall that a *cocycle* is a map $c:\Gamma\times X\to H$ such that $c(gg',x)=c(g,g'x)c(g',x)$. Two cocycles $c,c'$ are cohomologous if there is $f:X\to H$ such that $c(g,x)=f(gx)^{-1}c'(g,x)f(x)$.
Let $H$ be a di... | https://mathoverflow.net/users/81562 | Cocycle superrigidity | Yes.
Instead of writing down explicitly such a cocycle (which is not hard), let me take this opportunity to explain how to think of such objects in a "cooridnate free" manner.
Given a cocycle $c:\Gamma\times X\to H$ you can consider the space $Y=X\times H$ and endow it with the product measure and with the $\Gamma\t... | 5 | https://mathoverflow.net/users/89334 | 301246 | 132,032 |
https://mathoverflow.net/questions/297461 | 2 | Are there any Euclidean spaces, in which the maximal vertex degree of MSTs (Minimum Spanning Trees) of a finite set of points and edge weights equal to Euclidean distance, isn't equal to the [kissing number](https://en.wikipedia.org/wiki/Kissing_number_problem)?
**Remark:**
The fact, that the kissing number is an... | https://mathoverflow.net/users/31310 | Maximal Vertex Degree of MSTs in Euclidean Spaces | The answer (maybe)1 is no. In the following paper,
*Gabriel Robins and Jeffrey S. Salowe*, [**On the maximum degree of minimum spanning trees.**](http://dx.doi.org/10.1145/177424.177978) Proceedings of the tenth annual symposium on Computational Geometry (SCG '94), 250-258 (1994). [PDF](https://www.cs.virginia.edu/~r... | 1 | https://mathoverflow.net/users/53059 | 301248 | 132,034 |
https://mathoverflow.net/questions/301244 | 3 | Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X\_s$ satisfies $h^2(\mathcal{O}\_{X\_s})=0$. This means in particular that given an invertible sheaf $\mathcal{L}\_0$ on $X\_{s\_0}$, for some $s\_0 \in S$, there exists no obstruction to infinitesimal deforma... | https://mathoverflow.net/users/43198 | Local to global deformation of invertible sheaves | The answer is no, even for locally constant families. Let $F$ be a smooth projective variety with an automorphism $\sigma$, and let $L$ be a line bundle on $F$ such that $\sigma^\* L$ is not isomorphic to $L$. For example, take $F=\mathbf{P}^1\times \mathbf{P}^1$ with $\sigma$ the coordinate-switching involution, and $... | 7 | https://mathoverflow.net/users/3847 | 301250 | 132,035 |
https://mathoverflow.net/questions/301220 | 3 | I am going through James Maynard's paper, *Small Gaps between Primes*, and have a number of questions regarding his approach. First, I am wondering why uses weights in his approach. While I generally understand the meaning of (2.1) on pg. 3:
$$
S(N,\rho)=\sum\_{N\le n<2N}\Bigl(\sum\_{i=1}^k\chi\_{\mathbb{P}}(n+h\_i)-\r... | https://mathoverflow.net/users/124907 | Use of weights in the GPY's and Tao-Maynard's work on the twin prime conjecture | Very briefly, the role of the weights is to pick $n$'s for which the shifted admissible $k$-tuple $\{n+h\_1,\dots,n+h\_k\}$ has a better chance to contain at least two primes. Without any weighting the average number of primes lying in this tuple would be zero, so the weights are really there to
counterbalance the fac... | 4 | https://mathoverflow.net/users/11919 | 301261 | 132,040 |
https://mathoverflow.net/questions/301216 | 5 | Let $\mathcal{K}$ be a $2$-category. Keeping in mind the Cat-intuition on $\mathcal{K}$ I say that:
>
> **Def** : A $1$-cell $A \stackrel{f}{\to} B$ is a left split subobject if $f$ is a right adjoint and the counit is invertible.
>
>
>
In Cat this condition is equivalent to be fully faithful (and a right adjo... | https://mathoverflow.net/users/104432 | Left split subobject in a $2$-category | This is true if by "monomorphism" you mean "representably fully faithful" as in the result of Lack: just argue representably. If $f:A\to B$ has a left adjoint $\ell:B\to A$ in $\mathcal K$, then $\mathcal K(X,f): \mathcal K(X,A) \to \mathcal K(X,B)$ has a left adjoint $\mathcal K(X,\ell)$ in $\mathrm{Cat}$ for any $X\i... | 5 | https://mathoverflow.net/users/49 | 301263 | 132,041 |
https://mathoverflow.net/questions/301206 | 8 | Let $k=\Bbb F\_q$ be a finite field, $G$ be a connected reductive group over $k$, $\mathcal{N}$ be the nilpotent cone of $G$ which consists of nilpotent elements in the lie algebra of $G$.
Motivation & example: If $G=GL\_n$, then $\#\mathcal{N}(k)=q^{n^2-n}$ ([here](https://projecteuclid.org/euclid.ijm/1255629831) i... | https://mathoverflow.net/users/102104 | Number of points of the nilpotent cone over a finite field and its cohomology | When formulating a question about the cohomology of a variety it's important to determine which cohomology group you want to ask about. One rule of thumb is:
>
> If you want to understand the number of $\mathbb F\_q$-rational points, or you already understand the number of $\mathbb F\_q$-rational points and want to... | 11 | https://mathoverflow.net/users/18060 | 301269 | 132,045 |
https://mathoverflow.net/questions/301271 | 8 | Let $f: E\rightarrow B$ be a Kan fibration between pointed connected Kan complexes with fibre the Eilenberg-MacLane space $\mathrm{K}(M, n), n\geq 2, M$ an abelian group. Assume $f$ induces an isomorphism on $\pi\_1$ with common value $G$, and so $G$ acts naturally on $\pi\_n\mathrm{K}(M, n)=M$ hence on (the simplicial... | https://mathoverflow.net/users/42571 | About fibrations with fibre Eilenberg-MacLane spaces | No. If this were the case then there would be a section $s: B \to E$ to $f$ induced by the $G$-equivariant map $\widetilde{s}:\widetilde{B} \to \widetilde{B} \times {\rm K}(M,n)$ sending $x$ to $(x,0)$ (where $0 \in {\rm K}(M,n)$ is the neutral element, which is fixed by the action of $G$). However, there exists fibrat... | 7 | https://mathoverflow.net/users/51164 | 301274 | 132,047 |
https://mathoverflow.net/questions/301288 | 3 | I am trying to solve the following exponential Diophantine equation:
$$ 9^{k\_1} -2^{j\_1} = 9^{k\_2}-2^{j\_2}$$
My conjecture is that this implies $k\_1=k\_2$ and $j\_1=j\_2$, apart from eventually some small few exceptions. Is this true?
| https://mathoverflow.net/users/66594 | Solution to an exponential Diophantine equation | Yes, this follows from a conjecture of Pillai (1945), which was proved by Stroeker and Tijdeman (1982). For references and generalizations see:
M. A. Bennett, Pillai’s conjecture revisited, J. Number Theory 98 (2003), 228-235.
R. Scott, R. Styer, On $p^x-q^y=c$ and related three term exponential Diophantine equatio... | 6 | https://mathoverflow.net/users/11919 | 301295 | 132,051 |
https://mathoverflow.net/questions/300893 | 6 | I would like to illustrate my question with an example:
It is well-known that $\Delta$ is the generator of a strongly continuous semigroup $(T(t))$ on $L^2(\mathbb R^n),$ i.e. the heat-semigroup.
It is also known that if one has a strongly continuous semigroup and the domain of the generator is the entire Banach sp... | https://mathoverflow.net/users/119875 | Uniform continuity of heat semigroup | **Setting.** Throughout, let $E$ be a complex Banach space and denote the space of bounded linear operators on $E$ by $\mathcal{L}(E)$. Let $X \subseteq E$ be a closed subspace and let $\mathcal{T} = (T(t))\_{t \ge 0}$ be a $C\_0$-semigroup on $E$ with generator $A: E \supseteq D(A) \to E$.
As already mentioned in th... | 2 | https://mathoverflow.net/users/102946 | 301304 | 132,055 |
https://mathoverflow.net/questions/301302 | 13 | Let $\mu$ be a finite positive measure on a set $M$:
$$
\mu(M)<\infty.
$$
As is known, the Banach dual space $L\_\infty(\mu)^\*$ to the space $L\_\infty(\mu)$ contains $L\_1(\mu)$, but (excluding some trivial situations) does not coincide with $L\_1(\mu)$:
$$
L\_1(\mu)\subseteq L\_\infty(\mu)^\*, \qquad L\_1(\mu)\ne L\... | https://mathoverflow.net/users/18943 | A characterization of $L_1(\mu)$ in $L_\infty(\mu)^*$ | It is shown in "Linear Operators, Part I" 1988 by Dunford and Schwarz as IV.8.16 on page 296 that the dual of $L\_\infty(\mu)$ can be identified with the space of finitely additive bounded signed measures that are absolutely continuous with respect to $\mu$ (with the variation norm).
Every such finitely additive mea... | 13 | https://mathoverflow.net/users/35357 | 301318 | 132,062 |
https://mathoverflow.net/questions/301317 | 3 | I have an operator $C$ that I wish to diagonalize on a Riemmanian manifold $M$ with *constant curvature* $\Lambda$
$$C = A + B$$
Now I know that these operators $A$ and $B$ commute in flat space, but on a curved space, they give
$$[A,B] = \Lambda B$$
Here's an example on $M = H^2$, where we will consider $A$ to be th... | https://mathoverflow.net/users/60770 | Simultaneous diagonalization on spaces with constant curvature | You can try the following trick. Using the identity
$$
e^{-B/\mu} A e^{B/\mu} = A + \mu^{-1} [A,B] + \frac{\mu^{-2}}{2!} [[A,B],B] + \cdots = A + (\Lambda/\mu) B
$$
and setting $\mu = -\Lambda$ we have
$$
e^{B/\Lambda} C e^{-B/\lambda} = (A - B) + B = A .
$$
So if we can diagonalize $A = UDU^{-1}$, with $D$ diagonal,... | 2 | https://mathoverflow.net/users/2622 | 301324 | 132,063 |
https://mathoverflow.net/questions/301316 | 1 | I am looking for a suitable reference to put in a bibliography for the following fact:
Let $f: X \rightarrow Y$ be a surjective morphism between normal projective varieties. Let $D$ be a $\mathbb{Q}$-divisor on $Y$. If $f^\*D$ is semi-ample, so is $D$.
It is part of the content of Lemma 3.6 in these notes:
<http:... | https://mathoverflow.net/users/89459 | Reference request: $f^*D$ semi-ample, then $D$ semi-ample | This is proved in (1.20) THEOREM of Fujita, T. Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 353–378.
You can download this paper from <https://repository.dl.itc.u-tokyo.ac.jp/index.php?action=repository_view_main_item_detail&item_id=39567&item_no=1&page_id=28&block_id=31>
| 3 | https://mathoverflow.net/users/42636 | 301326 | 132,064 |
https://mathoverflow.net/questions/299143 | 4 | I'm studying complexifications of compact Lie groups on "Representation of compact Lie groups- Dieck Brocker".
I found on internet that there is a bijection between complexifications of compact Lie groups and complex algebraic linear reductive groups.
In my book they show that given a representation $\ r:G \rightar... | https://mathoverflow.net/users/123935 | Complexification of compact Lie groups and complex algebraic linear reductive groups | Your first question appears answered in the comments.
With regard to your second question, Maxime Bergeron has a well-written, well-referenced exposition of the characterization of complex reductive affine algebraic groups as complexifications of compact Lie groups titled *Complex reductive algebraic groups* (one can... | 4 | https://mathoverflow.net/users/12218 | 301331 | 132,066 |
https://mathoverflow.net/questions/301332 | 1 | Let $ (\xi\_i)\_{i \ge 1} $ be independent identically distributed random variables, taking values in $ (1,3]$.
Can we show:
$P( \exists N \in \mathbb{N}, \text{ s.t. } \forall k \ge 0, \prod\_{i=1}^{N}\xi\_{i+kN} > 6N ) =1 ?$
i.e, almost surely, for each consecutive block with length $ N$, its product is great... | https://mathoverflow.net/users/124254 | property of iid random variable | $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newco... | 3 | https://mathoverflow.net/users/36721 | 301337 | 132,067 |
https://mathoverflow.net/questions/301188 | 1 | Let $(H, \|\cdot\|)$ be a Hilbert space, $A \colon D(A)\subset H \longrightarrow H$ generates an analytic semigroup $T(t)$ on $H$. We define the following Banach space with the respect norm
$$F=\{x\in H : \int\_0^{+\infty} \|AT(t)x\|^2 dt <\infty\},$$
$$\|x\|\_F =\|x\|+ \left(\int\_0^{+\infty} \|AT(t)x\|^2 dt\right)^{1... | https://mathoverflow.net/users/124904 | Relation between a norm and norm of Besov spaces | Yes, your identities are correct. Theorem 1.14.5 in Triebel's book [T] says that $$F = (H,D(A))\_{1/2,2},$$ and $(1/2,2)$-real interpolation spaces between Hilbert spaces are in fact exactly the $1/2$-complex interpolation spaces, so $$(H,D(A))\_{1/2,2} = [H,D(A)]\_{1/2}$$ as proven for example in [P]. If the operator ... | 1 | https://mathoverflow.net/users/85906 | 301352 | 132,070 |
https://mathoverflow.net/questions/301347 | 2 | The concept of neighborhood maps was looked at in [a previous question](https://mathoverflow.net/questions/300356/neighborhood-maps-for-graphs-g-with-deltag-geq-2).
Let $G= (V,E)$ be a simple, undirected graph. For $v\in V$ we set $N(v) = \{w\in V: \{v,w\} \in E\}$. Note that we always have $v\notin N(v)$. A function... | https://mathoverflow.net/users/8628 | Injective, but no bijective neighborhood map | The answer is no.
If $f:V \to V$ is an injective map, it is a disjoint union of finite cycles, copies of the successor function in $\mathbb{Z}$ and copies of the successor function in $\mathbb{N}$. Leave the first two kinds alone and replace all of the third kind by (the appropiate copy of) the function that swaps each... | 5 | https://mathoverflow.net/users/17836 | 301363 | 132,076 |
https://mathoverflow.net/questions/301365 | 7 | Suppose that $X$ is a smooth projective variety $/\mathbb{C}$ with a $\mathbb{C}^{\*}$-action with isolated fixed points. Must $X$ be rational?
| https://mathoverflow.net/users/99732 | Does a torus action with isolated fixed points imply rational? | Yes. This follows from the Białynicki-Birula decomposition (see Theorem 4.4 in [the original paper](https://www.jstor.org/stable/1970915)).
| 9 | https://mathoverflow.net/users/3847 | 301368 | 132,077 |
https://mathoverflow.net/questions/300943 | 9 | I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me.
Suppose $W$ is a Brownian motion, and we have regularly-sampled times $\{t\_k = T / k, 1 \leq k \leq n \}$. Then the covariance of the vector $(W\_{t\_1}, \dots, ... | https://mathoverflow.net/users/9564 | Covariance function of Brownian motion and the second derivative operator | I'm not sure what "deeper reason" you are aiming at, but for one thing, there is no surprise here. The restriction of $W\_n$ of $W$ to $(t\_1,\dots,t\_n)$ is a Gaussian vector whose density form is given by
$$
(\Sigma^{-1}\_n v;v)=v^2\_1\cdot t\_1+\sum\_{i=2}^n (v\_i-v\_{i-1})^2(t\_i-t\_{i-1})=(\nabla\_n v;\nabla\_n... | 4 | https://mathoverflow.net/users/56624 | 301370 | 132,078 |
https://mathoverflow.net/questions/301335 | 4 | Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that
$\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that
$$\mathrm{trace}\,B \le \sum\_{i=1}^n \sqrt{\lambda\_i(A)\lambda\_i(C)},$$
where for any symmetric $M\in\mathbb{R}^{n \times n}$, $\lambda\_1(M) \le \lambda\_2(M) \le \cdots... | https://mathoverflow.net/users/27261 | Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix | If $\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$ then there exists a contraction $K$ (i.e. $\lambda\_n(K)=\|K\| \leq 1$) such that $B = A^{1/2} K C^{1/2}$ (e.g. see Theorem IX.5.9 of Bhatia's book: Matrix Analysis). By [Von-Neumann's trace inequality](https://en.wikipedia.org/wiki/Trace_inequalit... | 7 | https://mathoverflow.net/users/53059 | 301371 | 132,079 |
https://mathoverflow.net/questions/301272 | 8 | I am trying to calculate
$$Y = A^{\frac 12} X$$
where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any method that can compute or approximate $A^{\frac{1}{2}}$ efficiently?
I noticed that matrix $A$ can be written as $D... | https://mathoverflow.net/users/117734 | Square root of a large sparse symmetric positive definite matrix | I completely agree with fedja: there is a nice method here (which, unfortunately, does not always work well). If you know bounds for the spectrum of $A$, say $0<a<\lambda<b$, then you (sometimes) can compute the square root very efficiently by approximating the square root by a polynomial on this interval
$$\sqrt{x}\ap... | 4 | https://mathoverflow.net/users/9833 | 301376 | 132,080 |
https://mathoverflow.net/questions/301366 | 7 | It is well known, as well as absolutely intuitive, that the Riemannian holonomy of a generic Riemannian manifold is $O(n)$, the Riemannian holonomy of a generic orientable Riemannian manifold is $SO(n)$, and the Riemannian holonomy of a generic Kähler manifold is $U(n/2)$.
I guess that in this context by "generic" on... | https://mathoverflow.net/users/9871 | Riemannian holonomy of generic manifolds | Here are proofs for the Riemannian and Kähler case which rely on the fact that the curvature can be seen as parallel transport around infinitesimal loops. It uses explicit deformations which are hard to build with the holonomy is further restricted, that's why this proof doesn't work for smaller homotopy groups.
Riem... | 7 | https://mathoverflow.net/users/8887 | 301380 | 132,081 |
https://mathoverflow.net/questions/301375 | 6 | Let $\mathcal{H}(f)$ be the Hopf invariant of a map $f:\mathbb{S}^{4n-1}\to \mathbb{S}^{2n}$.
>
> When is the suspension map $$ \sigma:\{f\in
> \pi\_{4n-1}(\mathbb{S}^{2n}):\, \mathcal{H}(f)\neq 0\}\to
> \pi\_{4n}(\mathbb{S}^{2n+1}) $$ a non-zero map? When is it a
> surjection?
>
>
>
Clearly it is a surjec... | https://mathoverflow.net/users/121665 | Hopf invariant and the Freudenthal theorem | I'm on shaky ground here, this is what I think happens. There are the Hopf invariant one maps $\nu:S^7\to S^4$ and $\sigma:S^{15}\to S^8$, and these suspend to generators of the stable stems $\pi^S\_3\cong \mathbb{Z}/24$ and $\pi^S\_7\cong\mathbb{Z}/240$. So you also have surjectivity when $n=2, 4$, also.
The Hopf in... | 6 | https://mathoverflow.net/users/8103 | 301382 | 132,082 |
https://mathoverflow.net/questions/301339 | 6 | I am reading [this paper](http://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.pdf) on Homotopy for functors by Ming-Jung
Lee.
The author gives a definition (at the beginning of section $3$) as follows :
>
> Let $\varphi,\varphi':\Lambda\rightarrow \Gamma$ be covaria... | https://mathoverflow.net/users/118688 | Homotopy for functors | The author means there is a zigzag of natural transformations. That is, "a natural transformation between $\varphi\_i$ and $\varphi\_{i+1}$" is intended to be nonspecific as to the direction of the transformation: it could go from $\varphi\_i$ to $\varphi\_{i+1}$ or from $\varphi\_{i+1}$ to $\varphi\_{i}$.
This is a ... | 8 | https://mathoverflow.net/users/118688 | 301383 | 132,083 |
https://mathoverflow.net/questions/301215 | 7 | *Notations*: Recall that $\omega\_1$ is the first uncountable ordinal.
Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$
A function $\rho:F(X)\to \omega\_1$ is called an *ordinal rank*.
In other words, ordinal rank assigns an ordinal to ... | https://mathoverflow.net/users/42411 | Reference Request: Existence of Ordinal Rank Theory? | This is a topic I briefly looked at in 1992-1993 (I was mostly interested in ranks for differentiable functions and for nowhere differentiable continuous functions), when I was working on my dissertation, and at that time I collected a few papers on the topic (and some more throughout the 1990s). However, even though I... | 4 | https://mathoverflow.net/users/15780 | 301401 | 132,088 |
https://mathoverflow.net/questions/301287 | 3 | (I have asked the question
[The commutativity of minimal extension $\cdots$](https://mathoverflow.net/questions/301134/the-commutativity-of-minimal-extension-and-direct-image-by-blowing-down) and I simplify this question to the next simple question:)
Let $X$ be a rational variety over $\mathbb{C}$, $\phi : \hat{X} \r... | https://mathoverflow.net/users/124883 | Is direct image of simple $D$-module is also simple? | No. I’ll provide an example for what WillSawin suggested. Let $X$ be $\Bbb C^2$. Let $M$ be the irreducible holonomic module on $\hat X$ supported on the special fiber whose restriction to the special fiber is the structure sheaf. Then the direct image of $M$ is just the de Rham complex of $\Bbb P^1$, or rather the dir... | 3 | https://mathoverflow.net/users/36720 | 301408 | 132,092 |
https://mathoverflow.net/questions/296056 | 27 | It is well known that for $n>0$
$$d(n)=\det\left(\binom{2i+2j+1}{i+j}\right)\_{i,j=0}^{n-1}=1.$$
Computer experiments suggest that more generally
$$d(n,k)=\det\left(\binom{2i+2j+2k+1}{i+j}\right)\_{i,j=0}^{(2k+1)n-1}=(2n+1)^{k}.$$
Has anyone an idea how to prove this for general $k$?
| https://mathoverflow.net/users/5585 | Some binomial coefficient determinants | Johann Cigler and I have posted a solution on arXiv:
["An interesting class of Hankel determinants"](https://arxiv.org/abs/1807.08330), arXiv:1807.08330.
Let $d\_r(N)=\det\left({2i+2j+r\choose i+j}\right)\_{i,j=0}^{N-1}$. We show that for $k,n\ge 1$,
\begin{align}
&d\_{2k+1}((2k+1)n)=d\_{2k+1}((2k+1)n+1)=(2n+1)^k,\... | 5 | https://mathoverflow.net/users/112641 | 301413 | 132,093 |
https://mathoverflow.net/questions/301385 | 45 | I have now some problems about my research Career, I would like to tell my stories. I am a Chinese guy, but now a Ph.D. candidate in Germany, in the field of so called 'Geometric Analysis', but I do not feel happy when I work in such a field. Somebody said that a wrong choice of study field or supervisor means several ... | https://mathoverflow.net/users/124934 | A second Ph.D. in mathematics? | Finish this degree, then switch to whatever interests you. Many (most?) mathematicians change fields at some point in their research careers.
Bob Solovay told me once that the most important research was the first new thing you did after you got your degree. His thesis was *A Functorial Form of the Differentiable Rie... | 34 | https://mathoverflow.net/users/45581 | 301417 | 132,094 |
https://mathoverflow.net/questions/286332 | 8 | Let $(X,\tau)$ be a topological space.
Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming form $\mathcal{E}$ and $\tau$ are the same. Can we conclude that $X$ is **second countable** ?!
This question is also asked when $X$ is a locally convex space. Ple... | https://mathoverflow.net/users/84390 | A criterion for second countability | A counterexample to this question (and its locally convex version) is any non-metrizable (locally convex) space $X$, which is hereditarily Lindelof. The hereditary Lindelofness of $X$ implies that any open set is a countable union of basic open sets and this implies that the $\sigma$-algebra generated by any base of th... | 4 | https://mathoverflow.net/users/61536 | 301427 | 132,098 |
https://mathoverflow.net/questions/301333 | 2 | I have asked this question in Mathematics StackExchange, but there is no response yet. I've just realized that here is the right forum for asking research level questions... :'(
In game theory, in the attempt to define sequential equilibrium, for every tuple $b$ of behavioral strategies for each player, the correspon... | https://mathoverflow.net/users/124960 | On the limit of assessments that are not sequentially rational | Consider a situation in which some player has a strictly dominated strategy. Any strategy profile that assigns a strictly positive probability to all actions at all information sets will involve a player not playing something sequentially rational given any beliefs.
If we would not allow this, even the prisoner's dil... | 1 | https://mathoverflow.net/users/35357 | 301431 | 132,100 |
https://mathoverflow.net/questions/301373 | 7 | (The following is crossposted from [Math.SE](https://math.stackexchange.com/questions/2759503/surjectivity-of-a-map-on-inverse-limits), where the question did not receive any answers.)
I am looking for a proof of the following lemma from P. Gabriel's *Des catégories abéliennes* (Chap. IV, §3, Lemme 1):
>
> **Lemm... | https://mathoverflow.net/users/60903 | Surjectivity of a map on inverse limits | Your question can be interpreted as the vanishing of the first derived projective limit functor ${\lim\limits\_{\leftarrow}}^{(1)} \mathcal M$ for the projective spectrum of the kernels. If these kernels are Artenian even all derivatives vanish. This is shown in Corollary 7.2 of C.U. Jensen's *Les Foncteurs Dérivés de ... | 4 | https://mathoverflow.net/users/21051 | 301433 | 132,101 |
https://mathoverflow.net/questions/301422 | 2 | consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T\_i)\_{i \ge 1} $, where operator space is endowed with operator norm $ || T\_i||= \sup\_{v \in \mathcal{B},||v||=1}||T\_iv||$.
assume $ \sup\_i||T\_i|| < \infty $, spectr... | https://mathoverflow.net/users/124254 | iid random operator and its spectrum | Let $\mathbb{P}$ be a Borel probability measure on the space of bounded operators on $\mathcal{B}$, equipped with the operator norm topology. By the subadditive ergodic theorem, the limit
$$\lim\_{n \to \infty} \frac{1}{n}\log \|T\_{\omega\_n}\cdots T\_{\omega\_1}\|$$
exists a.s., where the operators $T\_{\omega\_i}$ a... | 3 | https://mathoverflow.net/users/1840 | 301440 | 132,105 |
https://mathoverflow.net/questions/301419 | 3 | I am currently doing some research on surfaces of general type and I need some results from Bogomolov's paper:
**Bogomolov, F. A.**
*Families of curves on a surface of general type.*
Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041–1044.
14J25 (14J05). **([MR0457450](https://mathscinet.ams.org/mathscinet-getitem?mr=045... | https://mathoverflow.net/users/44192 | Where to find "Families of curves on a surface of general type" (MR0457450)? | My local library has the paper version, here is a [scan.](https://www.dropbox.com/s/tnqqaf8royx7i6k/bogomolov.pdf?dl=0)
| 11 | https://mathoverflow.net/users/13168 | 301445 | 132,107 |
https://mathoverflow.net/questions/301369 | 2 | Let $f:X \to Y$ be an affine morphism; is it true that the counit map
\begin{equation\*} f^\* f\_\* \mathcal{F} \to \mathcal{F}
\end{equation\*}
is surjective for every (coherent) sheaf $\mathcal{F}$?
| https://mathoverflow.net/users/112415 | Surjectivity counit of pushforward-pullback adjunction of an affine morphism | Yes. Since $f$ is affine, it is enough to check surjectivity after applying $f\_\*$. But the composition
$$ f\_\* F \to f\_\* f^\* f\_\* F \to f\_\* F $$
of $f\_\*({\rm counit})$ with the unit of $f\_\* F$ is the identity (triangle identities).
| 3 | https://mathoverflow.net/users/3847 | 301446 | 132,108 |
https://mathoverflow.net/questions/301444 | 2 | Does $L^1(R)\cap L^2(R)$ have finite or infinite corank in $L^2(R)$?
I guess the latter is the case but I have never seen this discussed, and would like to see a simple proof.
| https://mathoverflow.net/users/56920 | A question regarding $L^1(R)\cap L^2(R)$ | It is a truth universally acknowledged, that a Banach space in possession of a continuous and dense inclusion in a strictly larger Banach space must have infinite co-dimension in it. This is also the case of
$$L^1(\mathbb{R})\cap L^2(\mathbb{R}),\ \|\cdot\|\_1+\|\cdot\|\_2 \longrightarrow L^1(\mathbb{R}) ,\ \|\cdot\|... | 8 | https://mathoverflow.net/users/124646 | 301453 | 132,110 |
https://mathoverflow.net/questions/301460 | 5 | Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as *something*.$x\_1x\_2x\_3\dots$ in the binary system, then set $$f(x)=\lim\_{n\to\infty}\frac{x\_1+\cdots+x\_n}{n}$$ if the limit exists and belongs to $(... | https://mathoverflow.net/users/81488 | An example of an open discontinuous function | I saw it as an exercise (E1.4.2) in the book "Analysis Now" by "Gert. K. Pedersen" with slight difference; $f$ is defined by
$$f:\mathbb R\rightarrow\mathbb R,\quad x\mapsto\limsup\frac1n{\sum\_{k=1}^nx\_n}$$
where $x-\lfloor x\rfloor=0.x\_1x\_2\dots$ is the binary expansion of the
fractional part of $x$.
| 9 | https://mathoverflow.net/users/84700 | 301464 | 132,111 |
https://mathoverflow.net/questions/301022 | 28 | This question was first asked by Mehtaab Sawhney in Alex Postnikov's combinatorics class.
Given a rational function $F=P(x\_1,...,x\_n)/Q(x\_1,...,x\_n)$ with (say) integer coefficients, it is often of interest in combinatorics whether $F$ has a *subtraction-free expression*, that is, whether $F$ is formally equal t... | https://mathoverflow.net/users/33089 | Determining if a rational function has a subtraction-free expression | $\def\RR{\mathbb{R}}$I'm pretty sure I know what the answer is, but I found it surprisingly hard to find references for the facts I needed. So, here is what I think the truth is: Let $f(x\_1, \ldots, x\_n)$ be a polynomial with real coefficients; we write
$$f(x\_1, \ldots, x\_n) = \sum\_{a \in A} f\_a x^a$$
for some f... | 16 | https://mathoverflow.net/users/297 | 301468 | 132,113 |
https://mathoverflow.net/questions/301458 | 9 | I am trying to understand basic notions from Hill-Hopkins-Ravenel paper: <https://arxiv.org/abs/0908.3724>
In the Example 3.10 we are considering equviariant cellular chain complex for $n$-dimensional representation $V$ of a group $G$
$$
\ldots\to C^{cell}\_n(S^V;\underline{\mathbb{Z}})\to C^{cell}\_{n-1}(S^V;\underl... | https://mathoverflow.net/users/123432 | "Oriented representation" sphere | First of all, note that right before example 3.9 they prove that
$$H^G\_\*(S^V;\underline{\mathbb{Z}})=H\_\*(C^{cell}\_\*(S^V)^G)\,,$$
where $C^{cell}\_\*(S^V)$ is the cellular complex for some $G$-CW-structure on $S^V$ (and so levelwise is just a sum of permutation modules). In particular, since $S^V$ is $n$-dimension... | 9 | https://mathoverflow.net/users/43054 | 301479 | 132,117 |
https://mathoverflow.net/questions/301471 | 0 | Given integers $m,n\geq 1$, let $W\_{n,m}$ denote the family of all sequences $S\_1,S\_2,\cdots,S\_m$ satisfying
(1) every $S\_i$ is a subset of $\{1,2,\cdots,n\}$;
(2) $\mid S\_i\cap S\_j\mid\geq 3$ for all $1\leq i<j\leq m$.
How to calculate $\mid W\_{n,m}\mid$? Is there any known formula for $\mid W\_{n,m}\mi... | https://mathoverflow.net/users/58096 | The number of a family of sequences of subsets of $\{1,2,\cdots,n\}$ | One can represent such a sequence as an $m$ by $n$ (binary) Incidence matrix. The problem is then one of counting such matrices, and the exact enumeration is (as far as my ignorance permits) not expressible by a nice formula. However, some bounds can be nicely expressed.
The trivial upper bound $2^{nm}$ comes from co... | 1 | https://mathoverflow.net/users/3402 | 301480 | 132,118 |
https://mathoverflow.net/questions/296227 | 3 | Is it known that local connectivity of the Mandelbrot set (MLC) is sufficient prove the density of hyperbolic conjecture of qudratic family.
I wondered is it known that the MLC is not enough (or enough) to prove the density of hyperbolic conjecture for the family of unicritical polynomial family with degree d ($d\ge... | https://mathoverflow.net/users/11966 | Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational maps (or not) | I guess "MLC implies HD" holds for any unicritical polynomial family (probably the same proof applies but I haven't checked). On the other hand, Lavaurs proved in his thesis that the connectedness locus of cubic polynomial family is not locally connected, so your question does make sense only for (essentially) unicriti... | 1 | https://mathoverflow.net/users/73630 | 301481 | 132,119 |
https://mathoverflow.net/questions/301467 | 1 | Let $(X,\mathcal{X},\mu)$ be a standard probability space. A measure $\mu$ is atomic if $\mu$ is supported on at most countable many atoms, and is atomless if $\mu$ has no atom (a point x is an atom if $\mu(\{x\})>0$).
Now let $(X,\mathcal{X},\mu)$ be an extension of $(Y,\mathcal{Y},\nu)$ and $\mu=\int\_{Y}\mu\_{y}d... | https://mathoverflow.net/users/125014 | decomposing a measure into relative atom and atomless parts | It is generally true that if $(X,\mathcal{X})$ is a measurable space, $(Y,\mathcal{Y})$ a standard Borel space, and $\kappa:X\to\mathcal{P}(X)$ a transition probability, then the measure-valued function that maps $x$ to the atomic part of $\kappa(x)$ is measurable. The rest is a red herring.
With the usual $\sigma$-a... | 1 | https://mathoverflow.net/users/35357 | 301482 | 132,120 |
https://mathoverflow.net/questions/301483 | 3 | Given a convex function $f : \mathbb{R}^n \to [0,\infty)$, the objective is to find the farthest point in the level set $\left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1\right\rbrace$ (Assuming that such set is non empty, and closed and compact), i.e.
$$
\begin{aligned}
& \underset{x \in \mathbb{R}^n}{\text{maximize}... | https://mathoverflow.net/users/88894 | Maximizing a convex function with a convex constraint | Under your assumptions, this is a concave programming problem (i.e., minimization of a concave function subject to convex constraints) with compact constraint set, and therefore has a global minimum at an extreme of the feasible set, i.e., satisfying $f(x) = 1$. (Although there may be other globally optimal points not ... | 4 | https://mathoverflow.net/users/75420 | 301485 | 132,121 |
https://mathoverflow.net/questions/301470 | 16 | What kind of ‘category’ is Cubical type theory the internal language of?
Its known that Martin-Löf type theories are the internal language of Locally cartesian closed categories, adding higher inductive types you get the internal language of locally cartesian closed $(\infty , 1)$-categories, a.k.a HoTT without Univa... | https://mathoverflow.net/users/54401 | What kind of category is generated by Cubical type theory? | There are two kinds of answers as to what kind of category a "homotopy type theory" is the internal language of. On the one hand there is a kind of $(\infty,1)$-category that is the semantic object of real interest; but on the other hand there is a 1-categorical presentation of the latter that corresponds more closely ... | 18 | https://mathoverflow.net/users/49 | 301489 | 132,123 |
https://mathoverflow.net/questions/301488 | 4 | Let $\mathcal{B}$ be the (unique up to isomorphism) countable atomless boolean algebra, and $\mathrm{Aut}(\mathcal{B})$ its automorphism group, with pointwise convergence topology.
My question: Does $\mathrm{Aut}(\mathcal{B})$ contain a **closed** (non-abelian) free subgroup?
This question arises from reading about... | https://mathoverflow.net/users/16107 | Closed free subgroups of the automorphism group of the countable atomless boolean algebra | Yes.
This group is widely documented as "(self-)homeomorphism group of the Cantor set" (by Stone duality).
It admits, for every prime $p$ and $p$-adic field $K$, and $d\ge 2$, the group $\mathrm{PGL}\_d(K)$ as a closed subgroup (viewed as acting on the projective space $\mathbb{P}^{d-1}(K)$ which is homeomorphic t... | 6 | https://mathoverflow.net/users/14094 | 301491 | 132,125 |
https://mathoverflow.net/questions/301495 | 4 | *I asked this question on [math.SE](https://math.stackexchange.com/q/2789364/), but even with a bounty, there were no answers/comments. I hope this is not too low-level for this site.*
Suppose I have a covering map $\pi:E\rightarrow B$, and a path in $B$, which is just a map $f$ from $I=[0,1]$ to $B$; then I know I c... | https://mathoverflow.net/users/104963 | Path-lifting property: function space interpretation | Yes, $p$ is continuous. In proposition 3.7 of [this paper](https://projecteuclid.org/download/pdf_1/euclid.hha/1355321064) it is shown that the lifting map $Map((I,0),(B,b))\to Map((I,0),(E,e))$, $f\mapsto \tilde{f}$ is continuous (and therefore a homeomorphism) when you fix $p(e)=b$. Essentially the same proof will te... | 5 | https://mathoverflow.net/users/5801 | 301499 | 132,127 |
https://mathoverflow.net/questions/301496 | 10 | **This problem is a restatement of [this question](https://math.stackexchange.com/questions/2788189/is-there-a-triangle-which-makes-dense-set-of-angles/2788526#2788526), first announced in MathStackExchange.**
We start with a triangle $T$ in the Euclidean plane and we define $A\_n$ as the set of angles of the $6^n$ t... | https://mathoverflow.net/users/123652 | Is there a triangle which makes dense set of angles by drawing medians? | The answer to the second question is **yes**, for any non-flat triangle $T$, the set of angles $A$ is dense in $(0, \pi)$. This follows from a stronger result of Barany et al. [Theorem 1, 1]:
>
>
> >
> > **Theorem**. Successive barycentric subdivisions of a non-flat triangle contain
> > triangles which, to withi... | 15 | https://mathoverflow.net/users/6101 | 301505 | 132,129 |
https://mathoverflow.net/questions/301507 | 0 | This is a cross-post of [this](https://math.stackexchange.com/q/2801027/64809) and [this](https://math.stackexchange.com/q/2800921/64809) questions from math.stackexchange.com since I have not received any response there. I would like to seek help here.
Suppose $x(t,\omega): [0,T]\times\Omega\rightarrow \mathbf R$ i... | https://mathoverflow.net/users/32660 | Does sequence almost sure convergence imply almost sure convergence? | No. A counterexample for all of your questions is as follows. Let $\Omega$ be $[0,1)$, with probability measure $\mathbb P$ being Lebesgue measure. Set $x(t,\omega)=1$ if the fractional part of $1/t$ is $\omega$ and 0 otherwise.
This is a version of the standard example satisfying convergence in probability, but not... | 3 | https://mathoverflow.net/users/11054 | 301511 | 132,131 |
https://mathoverflow.net/questions/301514 | 9 | **Context:** In formulating problems for secondary school mathematics teachers (and students) about **absolute value functions**, which we define as functions $\mathbb{R} \rightarrow \mathbb{R}$ that send $x \mapsto a|x-h|+k$ for fixed parameters $a, h, k \in \mathbb{R}$, I was able to rewrite the **nested** absolute v... | https://mathoverflow.net/users/22971 | De-Nesting Absolute Value Function into Linear Combination of Absolute Value Functions | $\newcommand{\al}{\alpha}
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\newco... | 7 | https://mathoverflow.net/users/36721 | 301515 | 132,132 |
https://mathoverflow.net/questions/301463 | 4 | Let $X$ be a set. For $B\subseteq X$ and ${\cal H}\subseteq {\cal P}(X)$ we set $$\text{ST}^1(B,{\cal H}) = \{H\in {\cal H}: H\cap B\neq \emptyset\},$$ and $\text{st}^1(B,{\cal H}) = \bigcup \text{ST}^1(B,{\cal H})$. For any integer $n>1$ we inductively set $$\text{ST}^{n+1}(B,{\cal H}) = \{H\in {\cal H}: H\cap\text{st... | https://mathoverflow.net/users/8628 | $n$-star-compactness | As I wrote in my comments, D.N. Sarkhel constructs such examples in Section 4 of "Some generalizations of countable compactness" (Indian J. pure appl. math 17 (6), 1986). It works as follows. Fix some $n\ge 2$. Take a partition of $[0,1]$ into pairwise disjoint dense subsets $A\_i$, $i = 1,\dots, 2n$. If $x\in[0,1]$ th... | 4 | https://mathoverflow.net/users/29491 | 301537 | 132,141 |
https://mathoverflow.net/questions/301533 | -1 | For any undirected simple graph $G=(V,E)$ we define for $v\in V$ the set $N(v) = \{w\in V: \{v,w\}\in E\}$.
Suppose $A, B$ are finite, disjoint sets, and $G = (A\cup B, E)$ is a bipartite graph with bipartition $(A,B)$ -- that is, for every $e\in E$ we have $e\cap A \neq \emptyset \neq e\cap B$. Informally speaking, ... | https://mathoverflow.net/users/8628 | On a condition concerning the number of neighbors in bipartite graphs | This is wrong. Consider, for instance, the situation where the partite sets $A$ and $B$ are two copies of $\mathbb F\_p$, with $p\equiv 1\pmod 4$ prime, and $a\in A$ is adjacent to $b\in B$ whenever $a-b$ is a square in $\mathbb F\_p$; that is, $a=b$ or $a-b$ is a quadratic residue. In this case each $b\in B$ has $(p+1... | 2 | https://mathoverflow.net/users/9924 | 301538 | 132,142 |
https://mathoverflow.net/questions/301396 | 2 | Let $\lambda = (P,\pi,M;G)$ be a smooth principal $G$-bundle (projection $\pi : P \to M)$, $V$ a finite dimensional vector space, and $\rho : G \to GL(V)$ a smooth representation of $G$ in $V$.
We can associate to $\lambda$ a vector bundle on $M$ with total space $P \times\_{\rho} V$, wich I note $\lambda \times\_{\r... | https://mathoverflow.net/users/74372 | Connection 1-form of the frame bundle associated to a vector bundle with a connection | I think I have the answer, and if I am not wrong it's so simple that I even regret to have asked this question :
$$ \omega\_i = \Gamma\_i$$
This comes form the fact that the space of linear connection on $\xi$ and the space of principal connection on $\lambda\_F(\xi)$ are affine isomorphic, and the correspondances
$$H ... | 3 | https://mathoverflow.net/users/74372 | 301544 | 132,144 |
https://mathoverflow.net/questions/301512 | 16 | Lately, I have been constructing finite involution monoids that generate varieties with $2^{\aleph\_0}$ subvarieties. One construction requires groups that violate the identity ${ [x,y]^2 \approx 1 }$, where ${ [x,y] = x^{-1} y^{-1} xy }$.
Is there a name for groups satisfying the identity ${ [x,y]^2 \approx 1 }$? H... | https://mathoverflow.net/users/57297 | Groups that satisfy ${ [x,y]^2 \approx 1 }$ | This variety of groups has indeed been considered in the literature. It is known that the following conditions hold for every group $G$ satisfying the identity $[x,y]^2=1$:
1. $[[x,y\_1,\ldots,y\_m],[x,z\_1,\ldots,z\_n]]=1$ for all
$x,y\_1,\ldots,y\_m,z\_1,\ldots,z\_n\in G$ (see [1]).
2. $[[x\_1,x\_2],[x\_3,x\_4]]=[[... | 24 | https://mathoverflow.net/users/40723 | 301558 | 132,149 |
https://mathoverflow.net/questions/301561 | 6 |
>
> Let $\mathscr F$ be the collection of smooth functions $f \colon
> \mathbb R \to \mathbb R$ such that
>
>
> 1. $f \in C^\infty\_c(\mathbb R)$, with $\text{supp } f \subset [-1,1]$;
> 2. $\int\_0^1 x f(x) dx = \frac{1}{2\pi}$.
>
>
> I would like to compute $$ \inf\_{\mathscr F} \int\_0^1 \vert f^\prime
> (x... | https://mathoverflow.net/users/119793 | A one-dimensional integral minimization problem | $\newcommand{\al}{\alpha}
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\newco... | 8 | https://mathoverflow.net/users/36721 | 301568 | 132,151 |
https://mathoverflow.net/questions/301535 | 7 | The following question on simultaneous resolutions is a follow-up to earlier questions posed here (e.g. [Resolution of singularities for flat families.](https://mathoverflow.net/questions/87998/resolution-of-singularities-for-flat-families/)).
What I'm interested in is an "obstruction theory" for simultaneous resolutio... | https://mathoverflow.net/users/109326 | Obstructions to simultaneous resolution | I am writing my comments as a solution. Assume that $k$ has characteristic $0$, for simplicity. First I will make precise one interpretation of "vanishing monodromy around the discriminant". Denote by $\Delta$ the complement of $B^{\text{sm}}$ in $B$. For simplicity, assume that $\Delta$ has pure codimension $1$ in $B$... | 5 | https://mathoverflow.net/users/13265 | 301571 | 132,152 |
https://mathoverflow.net/questions/301349 | 8 | Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
>
> For every $p \in M$ there exist a coordinate system around $p$, such that the co-frame associated with it satisfy:
>
>
> $$ \delta(dx^{i\_1} \we... | https://mathoverflow.net/users/46290 | Obstructions for the wedge of coordinate differentials to be harmonic | **Update (1 June 2018):** I have now figured out the 'little linear algebra lemma' in all dimensions $d = 2n$ and can give a complete answer to the OP's question: A metric $(M^{2n},g)$ possesses coordinate charts of the kind that the OP desires if and only if it is 'locally conformally unimodular Hessian', i.e., every ... | 6 | https://mathoverflow.net/users/13972 | 301584 | 132,156 |
https://mathoverflow.net/questions/300147 | 2 | Consider a 1D zero-energy Schrödinger equation on the half-line,
$(-\partial\_x^2 + V(x))\psi(x)=0, \quad x \in (0, \infty)$
with a zero boundary condition $\psi(0) = 0$.
Is it true that if the zero-energy solution $\psi(x)$ has $N$ nodes in $(0, \infty)$, then $V(x)$ has exactly $N$ negative energy bound states?... | https://mathoverflow.net/users/74032 | A nodal theorem in 1D | Yes, this is true. This is essentially the Sturm oscillation theorem.
A pedagogical reference is
B. Simon, in *Sturm-Liouville Theory* (Birkhäuser Basel, 2005), pp. 29—43.
Many thanks to Christian Remling for the right keyword.
| 2 | https://mathoverflow.net/users/74032 | 301589 | 132,158 |
https://mathoverflow.net/questions/301593 | 4 | I asked [this question at MSE](https://math.stackexchange.com/questions/2792892/a-precise-definition-of-contractible-banach-algebras) but I did not received any answer. So I ask it here at MO
I am sorry if this question is elementary:
What is a precise definition of a contractible Banach algebra?
What is my mista... | https://mathoverflow.net/users/36688 | A precise definition of contractible Banach algebras | $A$ is contractible if $H^1(A,X)=0$ for all Banach $A$-bimodules $X$ (here $H^1$ denotes continuous Hochschild cohomology for Banach algebras, as defined in the works of Johnson or Helemskii). It is an implicit conjecture, going back to the 1970s, that there are no "interesting" examples of contractible Banach algebras... | 8 | https://mathoverflow.net/users/763 | 301598 | 132,162 |
https://mathoverflow.net/questions/301607 | 5 | Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a matrix ring over $D$?
(Note that the answer with general base field is no, as there are countably many finite groups and... | https://mathoverflow.net/users/117693 | Do all finite-dimensional division algebras appear as Wedderburn factors of rational group rings? | The subgroup of the Brauer group generated by (and in fact, consisting of) such division algebras is called "the Schur group". Brauer-Witt theorem asserts that it is given by cyclotomic algebras, so the answer is negative.
I learned this [here](https://ac.els-cdn.com/S0021869385713646/1-s2.0-S0021869385713646-main.pdf?... | 10 | https://mathoverflow.net/users/89334 | 301622 | 132,166 |
https://mathoverflow.net/questions/301630 | 29 | Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the existing ones in a fairly *controlled* and *comprehensible* way and connect them to one another through forcing extensions.
... | https://mathoverflow.net/users/82843 | What is the dimension of the mathematical universe? | My co-authors and I introduced a notion of dimension for forcing
extensions in the following paper:
* *Hamkins, Joel David; Leibman, George; Löwe, Benedikt*, [**Structural connections between a forcing class and its modal logic**](http://dx.doi.org/10.1007/s11856-015-1185-5), Isr. J. Math. 207, Part 2, 617-651 (2015)... | 35 | https://mathoverflow.net/users/1946 | 301637 | 132,168 |
https://mathoverflow.net/questions/301603 | 6 | What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?
Only 2 results I have found so far are,
* That the $\infty$-norm covering number for $L$-Lipschitz functions constrained to map $[0,1]^d \rightarrow [0,1]$ is $\exp\left(\Theta\le... | https://mathoverflow.net/users/89451 | Covering number of Lipschitz functions | Here is a reference to a more general result: Lipschitz functions over a doubling metric space (rather than $[0,1]^d$). The $||\cdot||\_\infty$ $\epsilon$-metric entropy of such functions is, disregarding log factors, of order $(D/\epsilon)^{ddim}$, where $D$ and $ddim$ are the diameter and doubling dimension of the me... | 5 | https://mathoverflow.net/users/12518 | 301647 | 132,171 |
https://mathoverflow.net/questions/301409 | 5 | Let $M$ be a compact manifold and $E$ a complex vector bundle. We will consider differential operators $P$ acting between $\Gamma^{\infty}(M,E)$. Let $\mathcal{P}$ be the algebra of all differential operators: then $\mathcal{P}$ is *filtered*. Once we have filtered algebra, we can associate the *graded* algebra $\mathc... | https://mathoverflow.net/users/24078 | Elements of graded algebra associated with the algebra of differential operators as smooth sections | The vector space $U\_x$ will be infinite dimensional, so it's not immediately clear what $\Gamma^\infty(M,U)$ denotes. I assume you mean $\Gamma^\infty(M,U):=\bigoplus\_k \Gamma^\infty(M,U^k)$ where $U^k\_x:=\mathcal{S}^k/I\_x\mathcal{S}^k$. Then the question might be: Why is $\mathcal{S}^k$ isomorphic to $\Gamma^\inft... | 4 | https://mathoverflow.net/users/745 | 301651 | 132,173 |
https://mathoverflow.net/questions/301612 | 1 | Let $\{Y\_i\}\_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y\_i)=m$ for all $i$) and $\{P\_i\}\_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of *positive definite* matrices. Let $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{m\times n}$ and consider the following equation
$$\ta... | https://mathoverflow.net/users/62673 | On a condition for a matrix sum to be zero | EDIT: Now with an embarrassingly simple counterexample.
Same as my other integer counterexample, but now with very small magnitude integer entries in all matrices. $A$ and $B$ full rank, all $Y\_i'$ s are identity matrices, $m = n = N = 2$.
```
>> disp(P1)
1 0
0 1
>> disp(P2)
1 0
0... | 1 | https://mathoverflow.net/users/75420 | 301652 | 132,174 |
https://mathoverflow.net/questions/301628 | 4 | Let $X$ be a ***separable topological vector space*** with size (cardinal number) no larger than $\mathfrak{c}$. Does there exist any sequence of finite rank linear maps $\phi\_n:X\to X$ pointwise converging to the identity mapping $id:X\to X$?
| https://mathoverflow.net/users/84390 | pointwise convergence to the identity | Just to chat, an easier counterexample is $X:=L^p(\mathbb{R})$ for $0\le p<1$, a complete metric separable TVS. The identity map can't be approximated by finite rank continuous linear operators, for the simple reason that there aren't any. The only convex open set of $X$ is $X$ itself. As a consequence, there aren't an... | 9 | https://mathoverflow.net/users/6101 | 301671 | 132,181 |
https://mathoverflow.net/questions/301605 | 2 | I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability.
The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation:
$$\int\_{-T}^T\varphi\_n(x)... | https://mathoverflow.net/users/18560 | Are the Prolate Spheroidal Wave Functions absolutely integrable? | They are not in $L^1$. The principal term of the asymptotics is
$$\frac{e^{\pm iTx}}{x}.$$
This asymptotics is written for example here:
Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F.
New characterizations for the eigenvalues of the prolate spheroidal wave equation. (English summary)
Stud. Appl. Math. 138 (... | 2 | https://mathoverflow.net/users/25510 | 301679 | 132,187 |
https://mathoverflow.net/questions/301645 | 27 | Let $a\_1,\dots,a\_n$ and $b\_1,\dots,b\_n$ be two sequences of non negative numbers such that for every positive integer $k$,
$$ a\_1^k+\cdots+a\_n^k \leq b\_1^k+\cdots+b\_n^k,$$
and
$$a\_1+\cdots+a\_n = b\_1+\cdots+b\_n.$$
Can we conclude
$$\sqrt{a\_1}+\cdots+\sqrt{a\_n}\geq \sqrt{b\_1}+\cdots+\sqrt{b\_n}$$
| https://mathoverflow.net/users/51663 | Is this inequality on sums of powers of two sequences correct? | As a counter-example with $n=3$
$$a\_1=6, a\_2=42,a\_3=52$$
$$b\_1=12, b\_2=22, b\_3=66$$
Then
* $a\_1+a\_2+a\_3 = 100 = b\_1+b\_2+b\_3$
* $a\_1^p+a\_2^p+a\_3^p \lt b\_1^p+b\_2^p+b\_3^p$ for $p \gt 1$
* $\sqrt{a\_1}+\sqrt{a\_2}+\sqrt{a\_3} \lt 16.2 \lt \sqrt{b\_1}+\sqrt{b\_2}+\sqrt{b\_3}$
though notice that $... | 22 | https://mathoverflow.net/users/12565 | 301681 | 132,188 |
https://mathoverflow.net/questions/301627 | 9 | We say that $f:\mathbb{R}\to\mathbb{R}$ is of **Baire Class $1$** if it is a pointwise limit of a sequence of continuous functions.
One can generalize the definition above by taking pointwise limit of each 'previous' level(s) to obtain 'next' level.
More precisely,
>
> For any countable ordinal $\xi\geq 1,$ we s... | https://mathoverflow.net/users/42411 | Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$ | A somewhat concrete example of function which is Baire class $\zeta$ but not Baire class $\gamma$ for any $\gamma < \zeta$ is the $\zeta$-th Turing jump. This is essentially the iterated version of Shoenfield's limit lemma. The (iterated) Turing jump naturally gives us a function $J : \{0,1\}^\omega \to \{0,1\}^\omega$... | 5 | https://mathoverflow.net/users/15002 | 301689 | 132,190 |
https://mathoverflow.net/questions/301692 | 0 | Let $K>0$ be a constant. Suppose $\{z\_n\}\_{n=1}^\infty$ is a non-decreasing positive sequence. Then the series
$$\sum\_{n=1}^\infty\frac{z\_n}{(K+z\_1)(K+z\_2)\cdots(K+z\_n)}K^n=K$$
This is a quite interesting result as the series is convergent and the limit doesn't depend on the choice of $\{z\_n\}\_{n=1}^\inft... | https://mathoverflow.net/users/112346 | An interesting series converging to a constant | The point is that the partial sum
$$ \sum\_{n=1}^N \frac{z\_n}{(K+z\_1)\ldots(K+z\_n)} K^n = K - \frac{K^{N+1}}{(K+z\_1)\ldots(K+z\_N)} $$
as is easy to prove by induction.
| 3 | https://mathoverflow.net/users/13650 | 301693 | 132,191 |
https://mathoverflow.net/questions/301653 | 3 | Let $X$ be a locally compact, second countable Hausdorff topological space and let $Y$ be a Hausdorff quotient of $X$. Let $q:X\to Y$ denote the quotient map. Then for $y\in Y$, $q^{-1}(y)$ is a closed subset of $X$ but not necessarily compact.
Is it possible to find another locally compact, second countable Hausdor... | https://mathoverflow.net/users/121269 | Hausdorff quotient space with compact or finite inverse images | I think the answer is no. I'll basically point to two exercises in Engelking.
A map $q:X\to Y$ is called *hereditary quotient*, if for any $B\subset Y$, the restriction of $q$ on $q^{-1}(B)$ is a quotient map (see Exercise 2.4.F for some characterizations). In particular, it is said there that any quotient map onto a... | 2 | https://mathoverflow.net/users/53155 | 301694 | 132,192 |
https://mathoverflow.net/questions/301654 | 3 | Let $M$ and $N$ be smooth manifolds. Consider an isotopy of $M$ inside $N$. This means that we have a level preserving embedding $J\colon M\times [0,1] \to N \times [0,1]$. Put $J(x,t)=(\phi\_t(x),t)$. Hence $\phi\_t\colon M \to N$ is an embedding for each $t\in [0,1]$.
If $M$ is compact and $N$ has no boundary, then... | https://mathoverflow.net/users/99088 | Isotopy extension theorem: how non-unique is ambient isotopy | If I interpret the question correctly then the answer is "yes". You seem to be asking whether, if $H'$ is an isotopy satisfying the same conditions as $H$, there must be a one-parameter family of such isotopies joining $H$ to $H'$. I claim that the space of all such isotopies $H'$ is not only path-connected but contrac... | 2 | https://mathoverflow.net/users/6666 | 301698 | 132,193 |
https://mathoverflow.net/questions/301700 | 3 | I know the name of the [heptadecagon](https://en.wikipedia.org/wiki/Heptadecagon) (17 sides) and the [diacosipentacontaheptagon](https://en.wikipedia.org/wiki/257-gon) (257 sides). But what is the name of the [polygon](https://en.wikipedia.org/wiki/65537-gon) with 65537 sides? I am unable to figure it.
| https://mathoverflow.net/users/6129 | What is the name of the 65537-gon? | Following the portuguese nomenclature (I am from Brazil) and translating to english its results:
hexacontakaipentachiliakaipentahectakaitriacontakaiheptagon.
Best regards!!
| 5 | https://mathoverflow.net/users/125121 | 301713 | 132,199 |
https://mathoverflow.net/questions/299842 | 4 | Let $f=f(x,y),g=g(x,y) \in \mathbb{C}[x,y]$, each of degree $\geq 1$, and $f,g$ are algebraically independent over $\mathbb{C}$ (= their Jacobian $\in \mathbb{C}[x,y]-\{0\}$).
>
> **(1)** Is there a sufficient condition that will guarantee that $\mathbb{C}(f,g)=\mathbb{C}(x,y)$?
>
>
>
Perhaps it would help if ... | https://mathoverflow.net/users/72288 | Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$ | The two polynomials that you are giving provide a morphism $\tau\colon\mathbb{C}^2\to \mathbb{C}^2$, given by $(x,y)\mapsto (f(x,y),g(x,y))$.
This map $\tau$ is dominant if and only if $f$ and $g$ are algebraically independent (which seems what you already ask).
This map $\tau$ is birational if and only if $\mathb... | 4 | https://mathoverflow.net/users/23758 | 301714 | 132,200 |
https://mathoverflow.net/questions/301696 | 1 | This question is related to my question [here](https://math.stackexchange.com/questions/2803363/behavior-of-int-ll-x2k-operatornameerfxk-dx) such that i want to find a closed form of $\int\_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ , for $k$ is even integer because for odd integer is $0$ as we have integrand of od... | https://mathoverflow.net/users/51189 | Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx $ for $ k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$ | I understand from the OP that the motivation for this question is to find a series expansion in powers of $t$ of
$$I(t)=\int \_{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx=\sum\_{p=1}^\infty c\_p t^p.$$
The coefficients $c\_p=p^{-1}d\_{p-1}$ follow from the series expansion
$e^{-x^2\,{\rm erf}\,x}=\sum\_{p=0}^\infty d\_p... | 4 | https://mathoverflow.net/users/11260 | 301719 | 132,201 |
https://mathoverflow.net/questions/301727 | 2 | Let $A\_1$ and $A\_2$ be two commuting self-adjoint (or normal) operators on an infinite-dimensional complex Hilbert space $E$, then there exists a measure space $(X,\mathcal{E},\mu)$,
two functions $\varphi\_1,\varphi\_2\in L^\infty(\mu)$ and a unitary operator $U:E\longrightarrow L^2(\mu)$, such that each $A\_k$ is u... | https://mathoverflow.net/users/113054 | References for the Spectral Theorem ( Multiplication Operator Form) | You can see for example section 1.4 - spectral theorem II(1.47) - in the book "A course in abstract harmonic analysis" by "Gerald B. Folland".
| 2 | https://mathoverflow.net/users/84700 | 301732 | 132,204 |
https://mathoverflow.net/questions/301733 | 1 | I am interested in a quantum algorithm that has the following characteristics:
1. output = 2n bits OR 2 sets of n bits (e.g. 2 x 3 bits)
2. the number of 1-bits in the first set of n-bits must be equal to the number of 1-bits in the second set. E.g. correct output = `0,0,0, 0,0,0` (both 3-bit sets have zero 1-bits); ... | https://mathoverflow.net/users/125134 | How to create a quantum algorithm that produces 2 n-bit sequences with equal number of 1-bits? | Here is one way to achieve this, for concreteness described for $n=2$: Start with two registers of $2$ qubits, initialised as $|00\rangle|00\rangle$; apply a Hadamard transformation to each of the qubits in the first register, resulting in
$$(|00\rangle+|10\rangle+|01\rangle+|11\rangle)|00\rangle$$
(I leave out the nor... | 1 | https://mathoverflow.net/users/11260 | 301737 | 132,205 |
https://mathoverflow.net/questions/301769 | 1 | For any positive integer $n$, let $X\_n$ be the family of all subsets of $\{1,2,\cdots,n\}$.
Let $(X\_n,d)$ be the metric space such that
$$d(A,B)=|\,A\triangle B\,|,\ \forall A,B\in X\_n$$
where $A\triangle B$ is the symmetric difference of $A$ and $B$.
Let real $k>0$ (be fixed), and let $\ S\_n\subseteq X\_n\ $ ... | https://mathoverflow.net/users/58096 | A question about a $2^n$-point metric space | No: let $S\_n$ be the collection of subsets of $\{1,\ldots,n\}$ whose size is a multiple of $\lfloor k\sqrt n\rfloor$. Then every subset of $\{1,\ldots,n\}$ may be approximated by an element of $S\_n$ with error at most $\frac 12\lfloor k\sqrt n\rfloor$. The ratio $|S\_n|/|X\_n|$ is approximately $1/(k\sqrt n)$.
| 3 | https://mathoverflow.net/users/11054 | 301770 | 132,216 |
https://mathoverflow.net/questions/301768 | 0 | If $C$ is a characteristic $0$ algebraically closed field over $\mathbb{Q}\_p$ which is complete with respect to a non-trivial non-archimedean valuation. Now let $x\_1, \cdots,x\_n$ be any $n$ elements of $C$. Is $x\_1, \cdots,x\_n$ contained in a sub-field of $C$ with discrete valuations?
| https://mathoverflow.net/users/111816 | Discrete valuation sub-fields | Not necessarily.
Assuming $x\in C$ is an element with valuation $\sqrt{2}$, $x$ is not contained in any subfield of $C$ which has discrete valuation.
| 1 | https://mathoverflow.net/users/89334 | 301771 | 132,217 |
https://mathoverflow.net/questions/301684 | 2 | Let M be a symmetric non-negative definite $n\times n$ matrix. Let $K\_n$ denote the complete graph on $n$ vertices. Under what conditions is it possible to assign edge weights to $K\_n$ in such a way that $M$ is the corresponding graph Laplacian? Obviously the nullspace of M must contain the constant vector $(1,1,...)... | https://mathoverflow.net/users/105314 | Conditions for a matrix to be a Graph Laplacian | If $M$ is the Laplacian of an undirected graph, then $M$ has to satisfy $a\_{ii} = -\sum\_{j: i \not = j} a\_{ij}$ where $a\_{ij} = a\_{ji}$ is the $ij$-the entry of $M$, and $a\_{ij}=a\_{ji}$ (symmetric)
These conditions are sufficient though: Give edge $\{i,j\}$ in $K\_n$ the weight $a\_{ij}$ and then $M$ will be t... | 2 | https://mathoverflow.net/users/122188 | 301772 | 132,218 |
https://mathoverflow.net/questions/33062 | 14 | It is well-known that the space of $S$-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 Riemann surface $M$ is $CP^3$ (more concretely $PH^0(Jac(M),L(2\theta)$). Especially, the points corresponding to semistable (and not stable) bundles are smooth points. On the... | https://mathoverflow.net/users/4572 | Moduli space of semistable bundles | Disclaimer: This answer is rewritten in response to Chris Woodward's insightful comments.
The moduli space of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 surface $\Sigma$ is homeomorphic to the character variety $$\mathfrak{X}\_\Sigma(SU(2)):=\mathrm{Hom}(\pi\_1(\Sigma),SU(2))/S... | 5 | https://mathoverflow.net/users/12218 | 301775 | 132,219 |
https://mathoverflow.net/questions/301673 | 5 | Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$.
For any positive function $v$, I set
$$
J(v) = \int\_M \left|\nabla(v^a)\right|^p d\mu^g.
$$
Assume now that $(v\_t)\_{t \geq 0}$ is a solution of the heat equation
$$
\frac{d}{dt} v\_t = \Delta\_g v\_t.
$$
Is i... | https://mathoverflow.net/users/24271 | Functional decaying under the heat flow (?) | OK, here is the story. Consider the case $a=2$. Then we want to figure out what happens with $\int |ff'|^p$. I want to create the situation when $ff'$ is the largest and positive at $0$ and goes up at that point. Then for large enough $p$ we are in trouble because once we went down from the maximum of $(f^2)'$, we can ... | 2 | https://mathoverflow.net/users/1131 | 301776 | 132,220 |
https://mathoverflow.net/questions/301501 | 3 | Let $S$ be a Noetherian scheme, let $Y$ be a scheme of finite type over $S$, and let $X$ be an algebraic space of finite type over $S$. Suppose that there is a morphism $f:Y \rightarrow X$ which is proper and birational. Must $X$ be a scheme?
See also
[When is an algebraic space a scheme?](https://mathoverflow.net/q... | https://mathoverflow.net/users/4690 | Algebraic space birational to a scheme | I am just posting my comment as an answer. Chow's Lemma for algebraic spaces is Theorem IV.3.1, p. 192 of the following.
MR0302647 (46 #1791)
Knutson, Donald
Algebraic spaces.
Lecture Notes in Mathematics, Vol. 203.
Springer-Verlag, Berlin-New York, 1971. vi+261 pp.
For every separated, Noetheri... | 2 | https://mathoverflow.net/users/13265 | 301778 | 132,222 |
https://mathoverflow.net/questions/301780 | 14 | I am looking for a guidance in $K$-theory. My master thesis was in the field of Algebraic K-theory and its relation
and interaction with the field of Algebraic Topology. I mainly had
concentrated on the study of the third K-group of an infinite field. I studied the Anderie Suslin paper, which was titled as the "$K\_3$ ... | https://mathoverflow.net/users/117508 | Entering to the K-theory realm | I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks in the subject. Let me mention two of them:
* *Stable homotopy and generalized cohomology* by J. Frank Adams is an ol... | 19 | https://mathoverflow.net/users/43054 | 301797 | 132,230 |
https://mathoverflow.net/questions/301784 | 7 | Apart from the direct products, what are some "interesting" or "naturally occurring" examples of extensions
$$
1 \to N \to G \to Q \to 1
$$
of finite groups such that *neither* $N$ *nor* $Q$ is solvable?
I feel a bit stupid for asking the question, but I don't think I know a single example that isn't split.
Just to... | https://mathoverflow.net/users/17064 | Examples of extensions of non-solvable groups by one another | Here is a way you can construct nonsplit examples in which all composition factors are nonabelian.
Many finite nonabelian simple groups do not split over their automorphism groups. The smallest such example is the group often known as $M\_{10}$, which is the point stabilizer in the Mathieu group $M\_{11}$. It has ord... | 10 | https://mathoverflow.net/users/35840 | 301805 | 132,231 |
https://mathoverflow.net/questions/286965 | 5 | Let $n$ be a positive integer, and $p$ a prime number. Let $K\_i$ be the cyclotomic field containing exactly the $np^i$th roots of unity. Let $H$ be the inverse limit of $p$-power torsion of the class groups of the $K\_i$. Let $V$ be the $\mathbb{Q}\_p$ - vector space $H \otimes\_{\mathbb{Z}\_p} \mathbb{Q}\_p$. It is a... | https://mathoverflow.net/users/7935 | Rationality of trace of endomorphism of Iwasawa-thing | This is not really an answer, just a (very!) long comment. Everything I write is obvious for people working in Iwasawa theory, and I apologize for the trivialities.
Let me start by your final paragraph, where you discuss the analogy with curves over finite fields and the action of Frobenius: I guess you are aware tha... | 2 | https://mathoverflow.net/users/18238 | 301807 | 132,232 |
https://mathoverflow.net/questions/301641 | 2 | Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step filtration:
$$V\_0 \subset V\_1 \subset V\_2$$
Lets denote $A=V\_0$, $B=V\_1/V\_0$, $C= V\_2/V\_1$, $D=V\_1$, $E=V\_2/V\_... | https://mathoverflow.net/users/22810 | Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data | Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V\_0 \subseteq V\_1 \subseteq V\_2$. However, a variant on your construction does have a positive answer, and this is closely related to Massey products. Rec... | 6 | https://mathoverflow.net/users/51164 | 301808 | 132,233 |
https://mathoverflow.net/questions/301811 | 4 | I am wondering whether there exists an algebraic structure(group or modules etc) possess some kind of self-similarity, (sub-group or sub-module have an identical structure with itself) and "irregularity"(undefined) at the same time? if it exists can we find a functor between the category of fractals and this category o... | https://mathoverflow.net/users/83349 | algebraic structure of fractals | It is a theorem of Douady and Hubbard that the hyperbolic points in the Mandelbrot set form a free noncommutative monoid in a natural way, and that this monoid has a natural action on the whole Mandelbrot set. This is part of a large body of results about the Mandelbrot set that deserve to be better known. I learned ab... | 6 | https://mathoverflow.net/users/10366 | 301817 | 132,237 |
https://mathoverflow.net/questions/301835 | 0 | Let $n>1$ be an integer. We say that two points $(x\_1,\ldots,x\_n),(y\_1,\ldots,y\_n)\in\mathbb{Z}^n$ are a member of the edge set $E\_n$ if and only if $$\sum\_{i=1}^n|x\_i-y\_i| = 1.$$
**Question.** Given an integer $n>1$, is there a maximal integer $m(n)$ such that the complete graph $K\_{m(n)}$ is a [minor](http... | https://mathoverflow.net/users/8628 | Complete minors of the grid graphs $\mathbb{Z}^n$ | Either I am missing something or for $n>2$ you have $m(n)=\infty$. It is enough to show this for $n=3$. Choose any $m$ and a set of edges in $E\_3$ which is the union of the following three sets:
* $\{(i,j,0),(i,j+1,0)\}\mbox{ such that } 1\leq i\leq m\mbox{ and } 1\leq j < m$
* $\{(i,j,1),(i+1,j,1)\}\mbox{ such that... | 3 | https://mathoverflow.net/users/16678 | 301842 | 132,244 |
https://mathoverflow.net/questions/301752 | 4 | Let $(M, g)$ be a compact Riemannian manifold.
Assume that $u\_0$ is a positive smooth function on $M$ and let $u\_t = e^{t \Delta} u\_0$ be the solution to the heat equation on $(M, g)$ with initial data $u\_0$.
Given $2a > 1$, is it true that the function
$$
f: t \mapsto \int\_M (u\_t)^{2a}
$$
is a convex functio... | https://mathoverflow.net/users/24271 | $L^p$-norm under the heat flow | If I didn't do any miscalculations I believe I have proven the case $1\leq p\leq 2$. I will write $u$ instead of $u\_t$. Let $(p)\_k=p(p-1)\ldots(p-k+1)$ and
$$
w=\begin{pmatrix}
pu^{p-1}\Delta^2 u\\ (p)\_2u^{p-2}\nabla\Delta u\cdot \nabla u\\ (p)\_2u^{p-2}(\Delta u)^2\\ (p)\_3u^{p-3}(\Delta u)|\nabla u|^2\\ (p)\_4u^{p... | 9 | https://mathoverflow.net/users/100908 | 301845 | 132,245 |
https://mathoverflow.net/questions/301844 | 49 | Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the squares of the entries of $\mathbf{v}$ and $\mathbf{w}$.
My question is how to prove that
\begin{align\*}
\|\mathbf{v}^2\... | https://mathoverflow.net/users/11236 | A strengthening of the Cauchy-Schwarz inequality | Here is a proof for every $n$. Using the notation $\mathbf{v}=(v\_1,\dots,v\_n)$ and $\mathbf{w}=(w\_1,\dots,w\_n)$, the inequality reads
$$\left(\sum\_i v\_i^4\right)^{1/2}\left(\sum\_i w\_i^4\right)^{1/2}-\sum\_i v\_i^2 w\_i^2\leq
\left(\sum\_i v\_i^2\right)\left(\sum\_i w\_i^2\right)-\left(\sum\_i v\_i w\_i\right)^... | 41 | https://mathoverflow.net/users/11919 | 301855 | 132,248 |
https://mathoverflow.net/questions/301854 | 1 | For any $k \ge 3$ construct a non hamiltonian, connected k-regular bipartite graph. I have tried to find such graphs for small $k$-s but i got nothing. Can anybody help?
| https://mathoverflow.net/users/125148 | Non hamiltonian k-regular bipartite graphs | Denote by $G$ a complete bipartite graph $K\_{k,k}$ without an edge. Take $k$ copies of $G$, call them $G\_1,\dots,G\_k$, vertices of degree $k-1$ in $G\_i$ are $v\_i,u\_i$. Add two vertices $a,b$ and join $a$ with all $v\_i$ and $b$ with all $u\_i$. The new graph is connected and bipartite, but it is not Hamiltonian s... | 3 | https://mathoverflow.net/users/4312 | 301856 | 132,249 |
https://mathoverflow.net/questions/301840 | 2 | Let $K$ be an extension field of $\mathbb{Q}\_p$, let $O$ be the ring of integers of $K$, and let $P$ be the maximal ideal of $O$.
If $K$ is a finite extension of $\mathbb{Q}\_p$, there is the well-known algebraic isomorphism
$$O[[X]]\cong\varprojlim O[X]/((1+X)^{p^n}-1).$$
A proof of this fact can be found, for ... | https://mathoverflow.net/users/109085 | On an isomorphism between $p$-adic power series and an inverse limit | There is unique division with remainder by a monic polynomial in $O[X]$, where $O$ is any commutative ring. When $O$ is a $p$-adically complete ring, the Weierstrass division theorem tells us there is unique division with remainder by a polynomial in $O[[X]]$ that is *distinguished*: monic with lower degree coefficient... | 6 | https://mathoverflow.net/users/3272 | 301870 | 132,253 |
https://mathoverflow.net/questions/301871 | 4 | Let $\text{Sym}(\omega)$ denote the set of all bijections $f:\omega\to\omega$ together with composition as group operation. Does $\text{Sym}(\omega)$ have $2^{\aleph\_0}$ pairwise non-isomorphic subgroups?
| https://mathoverflow.net/users/8628 | Does $\text{Sym}(\omega)$ have $2^{\aleph_0}$ pairwise non-isomorphic subgroups? | To get a continuum of a selection of different subgroups, take that many proper countable infinite subsets of primes. For each such subset S consider the abelian subgroup where an element is composed of one or more disjoint cycles each cycle of length a prime p belonging to S. As an isomorphism must map an element of f... | 11 | https://mathoverflow.net/users/3402 | 301872 | 132,254 |
https://mathoverflow.net/questions/301865 | 4 | Is it possible for a vector field on a smooth manifold $M$ to be a gradient with respect to a Riemannian metric $g$, but not a gradient with respect to a different Riemannian metric $h$?
For completeness: the gradient of a smooth function $f:M\to \mathbb{R}$ with respect to the metric $g$ is the unique smooth vector ... | https://mathoverflow.net/users/89166 | Are there vector fields which are gradients with respect to one metric but not another? | Consider the vector field $$X=(y-10x)\partial\_x-x\partial\_y$$ it is not a gradient vector field with respect to the standard Riemannian metric of $\mathbb{R}^2$ but it is a gradient vector field with respect to the Riemannian metric $$g=5dx\otimes dx-dx\otimes dy -dy\otimes dx +5dy\otimes dy$$
For this metric we ha... | 11 | https://mathoverflow.net/users/36688 | 301874 | 132,255 |
https://mathoverflow.net/questions/301827 | 2 | Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?
| https://mathoverflow.net/users/122182 | Partial derivatives of spherical harmonics | The following formula for derivatives of associated Legendre functions is given in <https://www.sciencedirect.com/science/article/pii/S0377042709004385>
(New formulae for higher order derivatives and applications, by R.M. Slevinsky and H. Safouhi):
$$\frac{d^k}{dx^k}P\_l^m(x)=\frac{(-1)^m}{2^ll!}\sum\limits\_{n=0}^k\bi... | 0 | https://mathoverflow.net/users/32389 | 301878 | 132,258 |
https://mathoverflow.net/questions/301801 | 3 | What is the smallest subfield $F\subset N\_0$ such that $$(F,+,\times,\leq)\ncong(N\_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N\_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is not sufficient to delineate what we mean by 'smallest', so we use the following notion. For two proper class sized o... | https://mathoverflow.net/users/92164 | 'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group | $\DeclareMathOperator{\Noo}{\mathbf{No}}$This might actually be a dead end.
This is because if $F$ is isomorphic as an ordered group to $\Noo$, then their value classes under natural ordered group valuation, that is, the underlying orders of their value groups under natural ordered field valuation, must be isomorphic... | 3 | https://mathoverflow.net/users/45005 | 301879 | 132,259 |
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