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/269776 | 6 | Let $X\_0$ be a locally noetherian scheme and $\mathcal{F}\_0$ a coherent $\mathcal{O}\_{X\_0}$-module. Let $C$ be an artin ring with residue field $k$ and let $X \to Spec C$ be a (flat) deformation of $X\_0$ over $C$ (meaning the closed fiber isomorphic to $X\_0$).
>
> **Definition 1**: A **deformation of $\mathc... | https://mathoverflow.net/users/22810 | Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?) | You ask many questions. I will answer the question that you labelled "question". The way that you phrase the question is ambiguous. When you write "**finite homological dimension $n<\infty$**", you do not specify whether you are assuming that $\mathcal{F}$ has finite homological dimension, or whether you are assuming t... | 8 | https://mathoverflow.net/users/13265 | 269797 | 120,860 |
https://mathoverflow.net/questions/269789 | 4 | Suppose $M$ is a countable transitive model for ZFC. Suppose $X\subset M\cap \textbf{ON}$ is bounded in $M\cap \textbf{ON}$. Why does $M[X]$ (the least ctm $N$ such that $M \subset N$ and $X \in N$) exists? Furthermore, is it true that there exists a forcing notion $P \in M$ and a $P$-generic filter over $M$, $G$, such... | https://mathoverflow.net/users/nan | If $X$ a subset of ordinals, is $M[X]$ a generic extension? | Concerning your modified question, as it is commented by Andres Caicedo, the answer is yes; in fact we have the following theorem:
**Theorem 1.** Assume $V[G]$ is a set forcing extension of $V$ by a complete Boolean algebra $\mathbb{B}$ and suppose $N$ is a model of $ZFC$ with $V \subseteq N \subseteq V[G]$. Then for... | 3 | https://mathoverflow.net/users/11115 | 269798 | 120,861 |
https://mathoverflow.net/questions/269804 | 3 | Let $(X,\tau)$ be a topological space. For $x,y\in X$ we set $$\Delta(x,y) = \{(z,z):z\in X\}\cup \{(x,y),(y,x)\}.$$
Is there a connected Hausdorff space $(X,\tau)$ with more than $1$ point such that for all $x,y\in X$ we have $X\cong X/\Delta(x,y)$ with respect to the quotient topology?
| https://mathoverflow.net/users/8628 | Connected Hausdorff space that is stable under collapsing two points | Start with the space $X\_0=S^1$ (any connected manifold will do), and choose a basepoint in $X\_0$. Construct a sequence of pointed spaces recursively as follows: $$X\_{n+1}=\bigvee\_{x,y\in X\_n, x\ne y} X\_n/
\Delta(x,y).$$ Let $X=\bigvee\_{n=1}^\infty X\_n$.
Then $X$ is connected and Hausdorff. Moreover, $X$ is na... | 7 | https://mathoverflow.net/users/6668 | 269807 | 120,863 |
https://mathoverflow.net/questions/269799 | 6 | Suppose $a^2+b^2-c^2=0$ are formed by a (integral) Pythagorean triple. Then, there are $3\times3$ integer matrices to generate infinitely many more triples. [For example, take](https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples#Pythagorean_triples_by_use_of_matrices_and_linear_transformations)
$$... | https://mathoverflow.net/users/66131 | "Nearly" Fermat triples: case cubic | Any such matrix $M$ would give rise to an automorphism of the cubic surfaces
$$a^3 + b^3 = c^3 \pm d^3 \quad \subset \mathbb{P}^3.$$
These are both just different ways of writing the Fermat cubic surface
$$x\_0^3 + x\_1^3 + x\_2^3 + x\_3^3 = 0 \quad \subset \mathbb{P}^3.$$
It is well-known that the automorphism group o... | 19 | https://mathoverflow.net/users/5101 | 269809 | 120,864 |
https://mathoverflow.net/questions/269746 | 2 | Let $G$ be a connected reductive algebraic group and $X$ be a $G$-variety. Let $Y$ be a $G$-invariant irreducible sub variety of $X$ which has non-trivial intersection with the semi stable locus $X^{ss}$ (for a fixed line bundle $L$ on $X$). Let $Z$ be an irreducible codimension one sub variety of $Y$ which is **not** ... | https://mathoverflow.net/users/104316 | G-sweep of irreducible sub variety | First, if $Z$ is contained in the unstable locus $X^{us}=X\setminus X^{ss}$ then the ``$G$-sweep'' $G\cdot Z$ is still in $X^{us}$ since the latter is $G$-invariant. In particular, it can't be all of $Y$.
Secondly, if $Z$ is of codimension one in $Y$ then it is an irreducible component of $Y\cap X^{us}$. The latter b... | 2 | https://mathoverflow.net/users/89948 | 269811 | 120,865 |
https://mathoverflow.net/questions/269492 | 11 | Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$, which is just the set of finite functions from $I$ to $2$. Then, ${\rm HOD}^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.
I assume that the proof is just a modification of the proof that ${\rm HOD}$ in $M[G]$ i... | https://mathoverflow.net/users/17968 | Kunen (2011) exercise IV.8.17: $M[G]\vDash HOD^\mathbb{R}\subseteq L[\mathbb{R}]$ | This question seems to be a little more subtle than the usual homogeneity arguments. Here is one way that I found to do it.$\newcommand\R{{\mathbb{R}}}\newcommand\HOD{\text{HOD}}\newcommand\Ord{\text{Ord}}$
Allow me to describe the problem like this. We start in $L$ and force with
$\newcommand\P{\mathbb{P}}\P=\text{A... | 8 | https://mathoverflow.net/users/1946 | 269827 | 120,871 |
https://mathoverflow.net/questions/269803 | 3 | I have Von Neumann algebra $\mathfrak{U}$ and a weakly dense \*-subalgebra $A$. I have another Von Neumann Algebra $\mathfrak{V}$ and an injective \*-homorphism $$\phi: A\longrightarrow \mathfrak{V}$$ such that $\phi(A)$ generates $\mathfrak{V}$. Is it possible to say that one can extend $\phi$ to an \*-isomorphism fro... | https://mathoverflow.net/users/40832 | Von Neumann Algebra isomorphism extension | No. This will almost never be true (subalgebras of the compacts are the only cases I can think of where it could work). The easiest example is probably $C[0,1].$ It has an injective homomorphism to $\ell^\infty(\mathbb{N})$--by point evaluation at the rationals--that generates $\ell^\infty(\mathbb{N})$. It also sits in... | 10 | https://mathoverflow.net/users/34640 | 269828 | 120,872 |
https://mathoverflow.net/questions/269808 | 3 | The original question is asked [here](https://math.stackexchange.com/questions/2271312/inequality-of-quadratic-form-and-generalized-inverse) by myself. But seems like nobody give an answer. So I decide to ask this question in mathoverflow. Sorry for duplication.
I want to prove the following results,
Given two non-... | https://mathoverflow.net/users/81944 | Inequality of quadratic form and generalized inverse | Yes, this is true. I write $(x,y)=x'y=y'x$ for an inner product of (column) vectors $x,y$. We have $a=Ax=By$ for certain $x,y$ and also $a=(A+B)z$ (since the image of the operator $A+B$ is an orthogonal complement to the kernel of $A+B$ and thus contains the image of $A$.) We have $(Ax,x)(Az,z)\geqslant (Ax,z)^2=(a,z)^... | 2 | https://mathoverflow.net/users/4312 | 269833 | 120,873 |
https://mathoverflow.net/questions/269801 | 5 | We say that $\Omega$ is a strongly star shaped domain (with respect to $0$ for example) in $\mathbb R ^n$ if:
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x\right \|})\} $$ and
$$\partial \Omega = \{x\in \mathbb R ^n : \left \| x \right \| = g(\frac{x}{\left \| x \right \|})\} $$
with... | https://mathoverflow.net/users/102228 | Decompostition of a Lipschitz domain | A Lipschitz domain $\Omega$ is an open set and any open set is a union of balls, which are strongly star-shaped. So I assume you meant $\overline\Omega$. By definition, any point of $\partial\Omega$ has a nbd in $\overline\Omega$ which is isometric to a sub-graph of a positive $k$-Lipschitz function $f:B(0,r)\to(0,+\in... | 4 | https://mathoverflow.net/users/6101 | 269834 | 120,874 |
https://mathoverflow.net/questions/269740 | 10 | Let $a\in (0,1), \;\;\psi\_a(x):=\prod\_{j=0}^\infty (1-a^{2j+1}x).$
**Question**. Is it true that, for all $x\in [0,1]$ and all $k\in\mathbb{N},$ the following inequality holds:
$$\frac{x^k}{(1-a)(1-a^3)\dots (1-a^{2k-1})}\leq\frac{1}{\psi\_a(x)}\quad ?$$
For $a\leq 0.5,$ this is obvious. When $a$ approaches 1, it... | https://mathoverflow.net/users/78726 | Inequality for functions on [0,1] | It looks like the calculus part is doable (though I still don't like the way I approach it), so let me post the simple (analysis) part.
Consider the symmetric curve $e^{-s}+e^{-t}=1, s,t>0$ (the graph of $f(t)=-\log(1-e^{-t})$
The first observation is that for every $S,T>0$, we have
$$
\int\_S^\infty f(s)\,ds\ge \int... | 11 | https://mathoverflow.net/users/1131 | 269846 | 120,878 |
https://mathoverflow.net/questions/269815 | 3 | Let $R$ be a commutative ring with identity, and let $T = T\_2(R)$ be the ring of $2\times 2$ upper triangular matrices over $R$. Is it true that the following identity holds?
$$GL\_2(T)=U(T)E\_2(T)$$
Here, $U(T)$ represent the units of $T$ and we abuse notation by identifying $U(T)$ with its embedding $\begin{pmatr... | https://mathoverflow.net/users/102861 | For $T$ the $2\times 2$ triangular matrices over $R$, can we write $GL_2(T)=U(T)E_2(T)$? | The answer is **no** in general because $T$ cannot be a $GE\_2$-ring in the sense of P. M. Cohn if $R$ isn't.
Let us recall some definitions. Let $n \ge 2$. A ring $\Lambda$ (not necessarily commutative) is said to be a $GE\_n$-ring if the identity $GL\_n(\Lambda) = D\_n(\Lambda) E\_n(\Lambda)$ holds. Here $D\_n(\La... | 5 | https://mathoverflow.net/users/84349 | 269875 | 120,887 |
https://mathoverflow.net/questions/269880 | 10 | Let $G$ be a connected, reductive group over a field $k$. Let $\Gamma = \textrm{Gal}(k\_s/k)$. I think my question is better suited using the classical language: think of $G$ as an affine $\overline{k}$-variety with $k$-structure. This gives a continuous action of $\Gamma$ on $G$. A *$1$-cocyle* is a continuous map $c:... | https://mathoverflow.net/users/38145 | A reductive group has a quasi-split inner form | Nothing is "better-suited to using the classical language"; if you cannot express things clearly via schemes then think harder about it until you can. Also, any connected reductive group over a field has a *unique* quasi-split inner form. (See Proposition 7.2.12 in the article *Reductive Group Schemes* in the Proceedin... | 14 | https://mathoverflow.net/users/81332 | 269885 | 120,892 |
https://mathoverflow.net/questions/269884 | 37 | I have been in situations when I submitted a paper to a math journal, but at the end of the refereeing process the final report was not sent to me. It has happened both in case when my paper was rejected or accepted.
>
> Do I have a right to see the report(s) on my paper? Does it depend on a journal?
>
>
>
If ... | https://mathoverflow.net/users/16183 | Requesting a referee's report on my paper from a math journal | No, you do not have this right. When this first happened to me (more precisely, to my student), I also was outraged and demanded a report. They replied that this is a journal policy: they decide when to send a report to an author and when not to.
Gradually I understood that this is a right of a journal, to establish an... | 39 | https://mathoverflow.net/users/25510 | 269888 | 120,893 |
https://mathoverflow.net/questions/269542 | 9 | The title has it all. I'm looking for a reference to the following:
>
> **Q.** Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the category of topological spaces)?
>
>
>
This is, at the end of the day, a special instance of a famous problem... | https://mathoverflow.net/users/16537 | When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces) | For simplicity of presentation, I follow Lovasz in his paper Operations with Structures and work with structures with one finitary relation which I will choose to be binary. The extension to multiple finitary relations and algebraic structures is straightforward, and the paper of Lovasz and the book *Algebras, Lattices... | 1 | https://mathoverflow.net/users/3402 | 269892 | 120,894 |
https://mathoverflow.net/questions/269893 | 26 | It is a classic result that the irrational rotations of the circle are ergodic. Formally, let $T:\mathbb{T}\to \mathbb{T}$ be defined by $Tz=ze^{2\pi i\alpha}$. If $\alpha$ is irrational, then $T$ is ergodic. This result appears in many textbooks (e.g., Walters, *An Introduction to Ergodic Theory*), and even [in Wikipe... | https://mathoverflow.net/users/42864 | Who first proved ergodicity of irrational rotations of the circle? | the proof goes back to [Nicole Oresme](https://en.wikipedia.org/wiki/Nicole_Oresme) in his paper *De commensurabilitate vel incommensurabilitate motuum celi [On the Commensurability or Incommensurability of the Motions of the Heavens]*, dated around 1360, see
* [Nicole Oresme and the commensurability or incommensurab... | 28 | https://mathoverflow.net/users/11260 | 269895 | 120,896 |
https://mathoverflow.net/questions/269890 | 3 | Let $P = \{n^k: n,k\in\mathbb{N}\setminus\{0,1\}\}$ denote the set of powers. For any $n,r\in\mathbb{N}$ we set $B\_r(n)=\{m\in\mathbb{N}: |m-n| \leq r\}$.
Is there a "global" constant $K\in\mathbb{N}$ such that the set $$\{q\in P: (B\_K(q)\setminus \{q\})\cap P \neq \emptyset\}$$ is infinite?
| https://mathoverflow.net/users/8628 | Bounded gaps between powers | According to the Wikipedia page on [Catalan's conjecture](https://en.wikipedia.org/wiki/Catalan%27s_conjecture), the problem you pose is open. (Look under the "generalization" heading.) A more general problem is Pillai's conjecture.
| 3 | https://mathoverflow.net/users/3199 | 269896 | 120,897 |
https://mathoverflow.net/questions/269842 | 4 | Let $G$ be the group given by the presentation
$$\langle x,y,z,w \ | \ xy = yx, yz = zy, zw = wz\rangle.$$
This is a right-angled Artin group (RAAG) whose graph is a path on $4$ vertices.
We can also write $G$ as $(\mathbb{Z} \times F\_2) \*\_{\mathbb{Z}} \mathbb{Z}^2$ where $F\_2$ is a free group of rank $2$ gene... | https://mathoverflow.net/users/38889 | Finite index subgroups of a RAAG | This is not a satisfactory answer, but in the absence of anything better I will put it on record. It would be nice if someone could come up with a more illuminating answer.
Computer experiments strongly indicate that the answer is yes, and I could write down a proof by hand if forced, but it would be technical and te... | 5 | https://mathoverflow.net/users/35840 | 269912 | 120,901 |
https://mathoverflow.net/questions/269908 | 3 | Let $G$ be a bounded degree graph and fix a vertex $v\_0$. Suppose that the simple random walk on $G$ is transient, and let $g:G\to\mathbb{R}$ be defined by $$g(v)=\mathbb{P}\_v[T\_{v\_0}<\infty].$$
That is, $g(v)$ is the probability that a simple random walk starting at $v$ will ever hit $v\_0$.
Now consider $g(G)$... | https://mathoverflow.net/users/74799 | Accumulation points of Green function on a transient graph | The answer is yes. To see this, take a transient graph $G$ (say $\mathbb{Z}^3$) and glue a copy of $\mathbb{N}$ to some vertex (say $v\_0$). The vertices of $\mathbb{N}$ now all have $g(v)=1$.
If you want infinitely many vertices with distinct values of $g$, all greater then some $a>0$, take a ladder instead of $\mat... | 3 | https://mathoverflow.net/users/1061 | 269914 | 120,902 |
https://mathoverflow.net/questions/269920 | 8 | I am trying to understand exactly which role the Euler characteristic plays in (smooth) cobordism theory, and especially why the answer seems to depend on the dimensions of the manifolds in question. Suppose $M$ and $N$ are two closed smooth $n$-manifolds and that we have a smooth cobordism $(W;M,N)$ (i.e. $W$ is a smo... | https://mathoverflow.net/users/47228 | Cobordisms and Euler characteristics | René Thom proved that two closed $n$-manifolds $M$ and $N$ are (unoriented) cobordant iff their [Stiefel-Whitney numbers](https://en.wikipedia.org/wiki/Stiefel%E2%80%93Whitney_class#Stiefel.E2.80.93Whitney_numbers) agree: for any partition $i\_1 + \dotsb + i\_k = n$,
$$[M]\frown w\_{i-1}(M)w\_{i\_2}(M)\dotsm w\_{i\_k... | 16 | https://mathoverflow.net/users/97265 | 269922 | 120,906 |
https://mathoverflow.net/questions/269654 | 5 | I'm constructing [a Coq library for Big-O notation](https://github.com/siddharthist/coq-big-o). Naturally, I'd like it to be as general as possible. The Wikipedia page on Big-O notation [says](https://en.wikipedia.org/wiki/Big_O_notation#Generalizations_and_related_usages)
>
> The generalization to functions takin... | https://mathoverflow.net/users/101048 | Generalizing Big O notation to arbitrary vector spaces | Here is a straightforward generalization of the asymptotic notation to functions taking values in normed vector spaces:
Let $X$ be a topological space, $\overline X\supseteq X$ a superspace in which $X$ is dense, and $a\in\overline X$. Let $Y$ be a normed vector space, and $f,g\colon X\to Y$. Then we say
* $f(x)=O(... | 4 | https://mathoverflow.net/users/12705 | 269934 | 120,910 |
https://mathoverflow.net/questions/269904 | 9 | I've seen various statements that the Boolean-valued models of ZFC occurring in model-theoretic forcing are "really" the topos of sheaves on an appropriate site, but never a fully precise statement. What *exactly* is the relationship, and where can I find this written down in a clear, simple way?
More precisely, let ... | https://mathoverflow.net/users/56878 | What is the precise relationship between forcing on a poset and the topos of double-negation sheaves on this poset? | One good way of seeing the connection is given in Section 4 of Michael Fourman’s paper [*Sheaf models for set theory*](http://ac.els-cdn.com/0022404980900961/1-s2.0-0022404980900961-main.pdf?_tid=6cf55fba-3a47-11e7-b3e9-00000aab0f01&acdnat=1494946590_56a4d39faa131b99b4959c2ba7fad74c) (JPAA, Vol 19, 1980).
For any Gro... | 10 | https://mathoverflow.net/users/2273 | 269937 | 120,912 |
https://mathoverflow.net/questions/269940 | 12 | Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.**Question** Show the flow of $JX$ is by biholomorphisms.
I know one reference to show the answer is yes (where it is stated that: flow of $X$ is by biholomorphisms $\iff$ $X-iJX$ is a holomorphi... | https://mathoverflow.net/users/99732 | Vector field with holomorphic flow | A vector field $X$ has flow preserving $J$ just when $L\_X J=0$, i.e. just when $[X,JY]=J[X,Y]$ for any vector field $Y$. Take $Y$ to also have such flow. Many such $Y$ exist locally, spanning the tangent bundle, because $(M,J)$ is a complex manifold. Vanishing of the Nijenhuis tensor gives $[JX,JY]=[X,Y]+J([JX,Y]+[X,J... | 10 | https://mathoverflow.net/users/13268 | 269958 | 120,919 |
https://mathoverflow.net/questions/269966 | 11 | Is there some simple proof that $\mathbb{Z}$ is not isomorphic to the fundamental group of any compact Kähler manifold? This follows from the main result of <https://arxiv.org/abs/0709.4350> which states that any $3$-manifold group which is Kähler must be finite. The simplest non-finite 3-manifold group is $\mathbb{Z} ... | https://mathoverflow.net/users/99732 | Why is $\mathbb{Z}$ not a Kähler group? | If $X$ is a compact Kähler manifold, then $h^{p,q}(X) = h^{q,p}(X)$ and $b\_k(X) = \sum\_{p+q=k}h^{p,q}(X)$, so in particular, $b\_1(X) = h^{1,0}(X) + h^{0,1}(X) = 2h^{1,0}(X)$ is even. Now,
$$b\_1(X) = \operatorname{rank} H^1(X; \mathbb{Z}) = \operatorname{rank} \operatorname{Hom}(\pi\_1(X), \mathbb{Z}).$$
If $\... | 28 | https://mathoverflow.net/users/21564 | 269968 | 120,925 |
https://mathoverflow.net/questions/269964 | 24 | Let $k$ be a field, $\mathsf{Vect}$ the category of finite dimensional vector spaces, and $\mathsf{C} = Fun(\mathsf{Vect},\mathsf{Vect})$ the abelian category of pointed endofunctors (sending $0$ to $0$). The question has 2 parts, one for characteristic $0$ fields and the other for characteristic $p>0$.
>
> **Assum... | https://mathoverflow.net/users/22810 | About the abelian category of endofunctors of $\mathsf{Vect}$ | There are a couple of equivalent ways to characterise polynomial functors. One is to say that $F$ is a polynomial functor of degree $n$ if the function $$\hom(U, V)\to \hom(F(U), F(V))$$ is polynomial of degree n. A second, equivalent formulation goes via cross-effects: A functor is polynomial of degree $n$ if its $n+1... | 26 | https://mathoverflow.net/users/6668 | 269975 | 120,928 |
https://mathoverflow.net/questions/269932 | 4 | Let $d(n)$ be the divisor function defined by $d(n) = \sum\_{m|n} 1$. I am in need of estimate of the following type:
$$
\sum\_{Q \leq n \leq Q + H} d^2(n) \ll H (\log (Q + H))^T
$$
where $T$ can be any positive number, and the implicit constant in $\ll$ is independent of $Q$ and $H$. I would appreciate any references ... | https://mathoverflow.net/users/84272 | An upperbound for divisor function squared on a short interval | Let $d\_5(n)$ be the number of solutions of the equation $x\_1x\_2x\_3x\_4x\_5=n$ in positive integers. Observe that $d\_5(p^\alpha)=\frac{(\alpha+1)(\alpha+2)(\alpha+3)(\alpha+4)}{24}$ for any prime $p$ and any $\alpha \geq 0$. For $\alpha=1$ we have $d\_5(p^\alpha)=5>d(p^\alpha)^2=4$ and for $\alpha>1$ the inequaliti... | 4 | https://mathoverflow.net/users/101078 | 269982 | 120,930 |
https://mathoverflow.net/questions/265832 | 3 | We define the Laplace transform of a non-negative function $f : \mathbb{R\_+} \to \mathbb{R\_+}$ by
$$\mathcal{L}f(q) \triangleq \int\_0^{+\infty}f(t)e^{-qt}dt,$$
where $q$ is in the domain of convergence of the integral.
Suppose that $\mathcal{L}f(q) \leq C(q)$ for some $C(q) \in \mathbb{R}\_+$ that can depend on $q... | https://mathoverflow.net/users/91121 | Upper bounds on the inverse Laplace transform | Without any assumptions (like monotonicity of your function) you cannot get any **nice** bound. That is, there exists the sequence of functions $f\_n$ such that $0\leq f\_n(t)\leq 1$ for all $t \in \mathbb R\_{\geq 0}$ and $C(q)(n)$ tends to 0 uniformly in every finite interval $(\varepsilon,M)$, but $\sup\limits\_{t \... | 2 | https://mathoverflow.net/users/101078 | 269986 | 120,933 |
https://mathoverflow.net/questions/269984 | 1 | 1. Do the symmetry group generators of a regular convex polytope and a marked $\{0,1\}^n$ vertex point suffice to embed the polytope uniquely with $\{0,1\}^n$ vertex set?
2. If so can we find the John's ellipsoid of this polytope in polynomial time?
| https://mathoverflow.net/users/10035 | Uniqueness of polytope embedding from symmetry group | I am not sure I understand the question: if you know a vertex, and the symmetry group generators, and the symmetry group is transitive on the vertex set, don't you know all the vertices? And then your polytope is just the convex hull?
As for the second question, in sufficiently high dimensions the only regular polyto... | 1 | https://mathoverflow.net/users/11142 | 269989 | 120,935 |
https://mathoverflow.net/questions/213084 | 6 | Consider the following setup: $(X, Y)$ is a doubly symmetric binary source with parameter $0 < p < 1/2$, i.e., $X \sim \text{Bernoulli}(1/2)$, $Z \sim \text{Bernoulli}(p)$ and $Y = X \oplus Z$. Let $(U, V)$ be another pair of binary variables, where $U$ is derived from $X$ and $V$ is derived from $Y$, i.e., the Markov ... | https://mathoverflow.net/users/76321 | Do binary symmetric channels maximize mutual information? | **No. You can get a higher $I(U;V)$ using asymmetric channels. Below I construct a counterexample, but first a more succinct restatement of the question.**
---
Restatement
-----------
To summarize, there is an input $U$ distorted by three independent binary hops, each described by $2\times 2$ stochastic matrice... | 3 | https://mathoverflow.net/users/10668 | 269996 | 120,937 |
https://mathoverflow.net/questions/238445 | 3 | Which countable discrete groups (apart from the infinite amenable ones) admit uncountably many mutually non-conjugate free ergodic probability measure preserving actions that are all mutually orbit equivalent?
| https://mathoverflow.net/users/46541 | Groups with many non-conjugate but orbit equivalent actions | We consider infinite countable groups and ask about the *this property*: admitting uncountably many mutually non-isomorphis free ergodic probability measure preserving (pmp) actions that are all mutually orbit equivalent.
You ask which groups satisfy *this property*. I do not know exactly. I think no one knows. But i... | 3 | https://mathoverflow.net/users/89334 | 269999 | 120,938 |
https://mathoverflow.net/questions/269942 | 3 | I am looking for a reference on certain modules of invariants. I think that the question is quite natural so that I believe there should be some results already, but I am not able to find anything.
The situation is the following: let $\mathfrak{S}\_n$ be the symmetric group that acts on $\mathbb{C}[x\_1,y\_1,x\_2,y\... | https://mathoverflow.net/users/45285 | Alternating multisymmetric functions | If I am not mistaken, the space $Alt$ has a basis given by $\{ \Delta\_M(\mathbf{x},\mathbf{y}) \mid M \subset \mathbb{N}\times\mathbb{N} \text{ finite}\}$.
Here, $\Delta\_M(\mathbf{x},\mathbf{y}) = \det[x\_i^{p\_j}y\_i^{q\_j}]\_{1\leq i,j\leq n}$ for $M = \{ (p\_1,q\_1),\ldots,(p\_n,q\_n) \}$.
You find this for exam... | 3 | https://mathoverflow.net/users/21291 | 270011 | 120,945 |
https://mathoverflow.net/questions/269990 | 5 | Let $A$ and $B$ be two matrices of eigenvalues $\lambda\_i$ and $\mu\_i$, respectively.
The *spectral variation* of $B$ w.r.t. $A$ and the *eigenvalue variation* of $B$ and $A$ are, respectively,
\begin{align} s\_B(A)&=\max\_i\min\_j\vert\lambda\_i-\mu\_j\vert, \\
v(A,B)&=\min\_{\pi}\max\_i\vert\lambda\_i-\mu\_{\pi(... | https://mathoverflow.net/users/66131 | Norm bounds on spectral variation and eigenvalue variation | The Hermitian case is more like a state-of-art answer. A good review of results in given in [Holbrook].
$\nu(A,B)\leq\|A-B\|$ for the operator norm. This is a direct consequence from Weyl's inequality.
This problem about spectral variation bound is fully discussed in [Bhatia] Chap 3&4(with a supplement in Chap7&8 ... | 2 | https://mathoverflow.net/users/25437 | 270022 | 120,951 |
https://mathoverflow.net/questions/259031 | 2 | I'm looking for the simplest possible example (one that's easy to remember) for the situation described in the title. More precisely I'm looking for the following example:
A (probably has to be singular) algebraic surface $X$ (2-dimensional, reduced, integral, finite type over $k$ algebraically closed) and a complex ... | https://mathoverflow.net/users/22810 | Simple example of a perfect complex not isomorphic to a strictly perfect complex? | Perhaps it is silly to post this as an answer. My answer to your other question also answers this question.
[Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?)](https://mathoverflow.net/questions/269776/can-the-homol... | 1 | https://mathoverflow.net/users/13265 | 270024 | 120,952 |
https://mathoverflow.net/questions/270020 | 10 | Let $\ell^\infty(\mathbb{N})$ denote the set of bounded real sequences $(a\_n)\_{n\in\mathbb{N}}$. The $\lim$ operator is a *partial* linear operator from $\ell^\infty(\mathbb{N})$ to $\mathbb{R}$. With the [Hahn-Banach theorem](https://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem), $\lim$ can be extended to a *ge... | https://mathoverflow.net/users/8628 | Generalized limits on $\ell^\infty(\mathbb{N})$ | Yes. In fact, if you work in $\mathsf{ZF}+\mathsf{DC}+$ "all sets of reals have the property of Baire" ($\mathsf{BP}$), say the [Solovay](https://en.wikipedia.org/wiki/Solovay_model) or Shelah models, you can prove that $(\ell^\infty)^\* = \ell^1$, so that the only continuous linear functionals on $\ell^\infty$ are tho... | 19 | https://mathoverflow.net/users/4832 | 270026 | 120,953 |
https://mathoverflow.net/questions/269973 | 8 | **Question:** Let's assume we have a pair $(X,\check{X})$ that are mirror dual to each other in the sense of Homological mirror symmetry (EDIT: this does not have to be CY n-folds, but can also be a Fano variety and a Landau-Ginzburg partner etc). Now take a subvariety $Y \subset X$ and consider $Z = Bl\_Y X$.
* Are ... | https://mathoverflow.net/users/105615 | How to construct the mirror partner of a blowup? | You can find a lot of such examples in the paper of Abouzaid-Auroux-Katzarkov: <https://link.springer.com/article/10.1007/s10240-016-0081-9>.
Basically, they studied the case when $X$ is $(\mathbb{C}^\times)^{n-1}\times\mathbb{C}$, and $Y\subset X$ is codimension 2 and is a hypersurface in $(\mathbb{C}^\times)^{n-1}$... | 5 | https://mathoverflow.net/users/43423 | 270043 | 120,961 |
https://mathoverflow.net/questions/269844 | 5 | Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let
$$
\mathcal{E}\_{I}
\stackrel{\text{df}}{=}
\{ x \in \mathcal{E} \mid \langle x,x \rangle\_{\mathcal{E}} \in I \}.
$$
Using the Cauchy-Schwarz Inequality for Hilbert $ C^{\ast} $-modules, it is ... | https://mathoverflow.net/users/50614 | On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilbert $ C^{\ast} $-module by a certain linear subspace | Yes, I think this is true. Here is an argument to show why. First for $E=A$, then this is just the standard theorem saying that the quotient norm on $A/I$ is the $C^\*$-algebra norm. (any ref on $C^\*$-algebras has a proof)
Then you argue directly that the case $E=A$ implies that it is also true for $E=l^2(\mathbb{N}... | 3 | https://mathoverflow.net/users/89956 | 270051 | 120,963 |
https://mathoverflow.net/questions/269899 | 0 | Please refer to [this link](http://snap.stanford.edu/class/cs224w-readings/erdos59random.pdf). It is Erdos and Renyi's first paper on Random Graphs (1959). I am trying to work through it.
I'm struggling with equations (16), (17) and (21).
(16)
====
* I'm not sure why they are using those two bounds in the summat... | https://mathoverflow.net/users/109984 | Random graphs- Erdos and Renyi 1959 paper | **(16)** They split the sum in (13) for $M<s\leq \frac n 2$ and $\frac n 2 < s < n- \frac {2 N\_c} n$. In the first case they use (14) and, since the terms are positive, they enlarge the sum to $M<s<\infty$. In the second range they use (15) to estimate the sum, then they substitute $s'=n-s$, and then they extend to $\... | 7 | https://mathoverflow.net/users/58242 | 270054 | 120,965 |
https://mathoverflow.net/questions/269907 | 6 | (This is a cross-post from [MSE](https://math.stackexchange.com/questions/2276367/universal-differential-identities)).
Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f\_{xy}=f\_{yx}$. This identity is "*universal*" in the sense that it holds for *any* smooth map.
**Question:**
Are ... | https://mathoverflow.net/users/46290 | "Universal" differential identities | Let's work in two dimensions for notational simplicity. We claim that there is no non-trivial polynomial identity of the form
$$ P( f, f\_x, f\_y, f\_{xx}, f\_{xy}, \dots ) = 0$$
relating some finite number of derivatives of $f$, for any smooth $f$. Suppose for contradiction that this is the case. Taking a functional d... | 7 | https://mathoverflow.net/users/766 | 270062 | 120,968 |
https://mathoverflow.net/questions/270049 | 5 | Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions:
$$
B(B^\intercal AB)^\star B^\intercal = A^\star,
$$
and such that $A^\star=0$ if and only if $A=0$?
Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) \to \operatorname{M}(n,m,\mathbb R)$.
Edit:... | https://mathoverflow.net/users/84393 | Existence of generalized inverse-like operator | No. We suppose it is defined for $2\times 2$ matrices and we get a contradiction. Unless you drop the requirement on $\operatorname{rank} A^\star$ in which case $A^\star=0$ trivially works.
Let $A=\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}$ and let $B=\begin{pmatrix} a & b\\ c & d\end{pmatrix}$ with arbitrary $a,b,c... | 7 | https://mathoverflow.net/users/58242 | 270066 | 120,969 |
https://mathoverflow.net/questions/269933 | 5 | If $R$ is commutative ring, $n\in\mathbb{N}$, $\mathsf{M}$ the category of $R[x\_1,\dotsc,x\_n]$-modules,and $F\colon\mathsf{M}\to\mathsf{M}$ an endofunctor of $\mathsf{M}$ which preserves all finite limits, does it follow in general that there exists an $M\in\mathsf{M}$ and a natural transformation $\tau\colon F\right... | https://mathoverflow.net/users/108556 | On some sense of representing an endofunctor of the category of modules over polynomial rings | Replacing $R$ by $R[x\_1,\ldots,x\_n]$ if necessary, we may assume $n = 0$. Note that preserving all finite limits is equivalent to being (additive and) left exact (see e.g. [Tag 010N](http://stacks.math.columbia.edu/tag/010N)).
**Lemma.** Let $R$ be a commutative ring, let $F \colon \operatorname{Mod}\_R \to \operat... | 5 | https://mathoverflow.net/users/82179 | 270076 | 120,972 |
https://mathoverflow.net/questions/269431 | 8 | The study of matrix quantum group coactions on the noncommutative disk algebra turns up the following series, which is a $q$-deformation of the negative binomial series, for integer $t\ge 0$, complex $z$ and $q\in[0,1]$:
$$\sum\_{n\ge 0}(-1)^n q^{n(n-1)/2}\frac{[n+t]\_q!}{[t]\_q![n]\_q!}\,z^n =
\sum\_{n\ge 0} (-1)^n... | https://mathoverflow.net/users/29625 | A not quite theta not quite basic hypergeometric function | "Basic hypergeometric series are series $\sum c\_n$,
with $c\_{n+1}/c\_n$ a rational function of $q^n$, for a fixed parameter $q$,
which is usually taken to satisfy $|q|<1$, but at other times is a power of a prime."
This quote is from the Forward (by Richard Askey) to "Basic Hypergeometric Series", by Gaspar and Ra... | 2 | https://mathoverflow.net/users/7222 | 270077 | 120,973 |
https://mathoverflow.net/questions/270079 | 2 | I'm looking for a proof or a reference for the following statement:
Let $R$ be a DVR (Discrete Valuation Ring) and $p$ a prime element, and let $\mathfrak a$, $\mathfrak b$ and $\mathfrak c$ be finitely generated $p$-modules. A $p$-module is a right $R$-module $V$ such that $Vp^e = 0$ for some $e \in \mathbb N$. Let ... | https://mathoverflow.net/users/110074 | $C=A \cdot B$ matrices and exact sequence of $DVR$-modules | **Remark.** Note that there is no need to mention $\alpha$, $\beta$, and $\gamma$. Indeed, the relation between $A$ and $\mathfrak a$ is that there is a short exact sequence
$$0 \to R^n \stackrel A\to R^n \to \mathfrak a \to 0.$$
Since $A$ has nonzero determinant (i.e. $A$ is injective), the quotient $\mathfrak a$ is t... | 4 | https://mathoverflow.net/users/82179 | 270083 | 120,975 |
https://mathoverflow.net/questions/270047 | 9 | I am reading Abouzaid's paper "A geometric criterion for generating the Fukaya category" (<https://arxiv.org/abs/1001.4593>), and it is claimed there, without proof, in section C.4 in the appendix (pp.34) that the compactificaiton of the moduli space of holomorphic annuli is given by two discs meeting in an interior no... | https://mathoverflow.net/users/14105 | Deligne Mumford Compactification of Moduli Space Of Annuli | The study of the Deligne-Mumford compactification of Riemann surfaces with boundary can be reduced to the study of the Deligne-Mumford compactification of closed Riemann surfaces together with the study of separating antiholomorphic involutions on them.
More precisely, the data of a compact Riemann surface A with (no... | 4 | https://mathoverflow.net/users/66777 | 270085 | 120,976 |
https://mathoverflow.net/questions/270089 | 0 | In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as shadows of 4-D objects. How does this helps us visualize 4-D objects?
I searched that we can atleast see their 3-D cross-sections. A tesseract hypercube would be a good example. Can we conclude that a 3-D cube is a shadow of a... | https://mathoverflow.net/users/109672 | Visualizing the 4th dimension | From a mathematical viewpoint, the dimension is just a number. You can of course define something like a "shadow" as the intersection of an object in 4-D and a subspace of dimension three, but this is but a definition. It might help to not think too much about how it would look like (because, as you noticed, you can't ... | 2 | https://mathoverflow.net/users/109932 | 270090 | 120,977 |
https://mathoverflow.net/questions/270072 | 10 | I don't know if this is already answered somewhere in MO. The dynkin involution of $SL\_{2n}$ that is $\alpha\_i \mapsto \alpha\_{2n-i}$ gives an outer automorphism of $SL\_{2n}$ and then the maximal parabolic $P\_n$ corresponding to the simple root $\alpha\_n$ is invariant under this automorphism. So it induces an aut... | https://mathoverflow.net/users/109746 | Lagrangian Grassmannian from an Involution | I think the confusion comes from the term "gives an outer automorphism" which has to be made precise. The automorphism, say $\phi$, is only well-defined up to multipliciation by conjugation with an element of the torus. To make it unique one usually requires that a "pinning" $e\_\alpha$ is fixed. For the $SL(n)$ this m... | 3 | https://mathoverflow.net/users/89948 | 270099 | 120,981 |
https://mathoverflow.net/questions/270088 | 34 | I'm conscious that this isn't necessarily a research level question, but I've asked this question on mathstackexchange, and received no answer. So I'm trying it here.
A usual mantra in field theories is the assertion that *only massless theories can be conformally invariant*. By a *theory* I mean an action $$ S = \in... | https://mathoverflow.net/users/104213 | Why is conformal invariance only possible for massless theories? | A physicist would answer this question as follows. (Everything I'll say can be expressed in a way that the purest of mathematicians would understand, but that translation would take a lot of work, so I'll only do it on demand.)
In physics we have units of mass ($M$), length ($L$) and time ($T$).
In special relativ... | 39 | https://mathoverflow.net/users/2893 | 270100 | 120,982 |
https://mathoverflow.net/questions/270094 | 10 | Let $S$ be a scheme and let $X$ be a quasi-separated algebraic space over $S$. Does there exist an open subspace of $X$ which is a scheme and which is dense in each fiber $X\_s$, $s \in S$?
I am happy to make the following additional assumptions:
(1) $S$ is the spectrum of a complete dvr with algebraically closed r... | https://mathoverflow.net/users/13302 | Schematic locus of algebraic spaces in fibers | There are quasi-separated counterexamples. If you assume that $X$ is separated, then this is true. The basic reference is Donald Knutson's book.
MR0302647 (46 #1791)
Knutson, Donald
Algebraic spaces.
Lecture Notes in Mathematics, Vol. 203.
Springer-Verlag, Berlin-New York, 1971. vi+261 pp.
If $... | 5 | https://mathoverflow.net/users/13265 | 270101 | 120,983 |
https://mathoverflow.net/questions/269840 | 3 | I have an optimization problem of the following form
$$\text{minimize} \,\|Qa-b\|\_2 \quad \text{ subject to } Q \succeq 0$$
where $a,b \in \mathbb{R}^n$ are given and the $n \times n$ square matrix $Q$ is the variable.
* It is most probably a semidefinite programming problem. Is there a standard answer to this... | https://mathoverflow.net/users/109953 | Standard solution to semidefinite program | I will assume throughout that the definition of positive semidefiniteness includes symmetry. The problem is to find the Euclidean projection of $b$ onto the convex set $R = \{Qa \mid Q\succeq 0\}$. Let $S = \{c \mid c^Ta \geq 0\}$, which is closed. We first show that $S$ is the closure of $R$. If $c\in R$ then $c=Qa$ f... | 2 | https://mathoverflow.net/users/5963 | 270114 | 120,989 |
https://mathoverflow.net/questions/270073 | 4 | Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and let $G=Aut(\Omega)$ be the **full** group of self-biholomorphisms of $\Omega$. Assume that there is $z\in \Omega$, such that the orbit of $z$ is somewhere dense, i. e. $\mathrm{int}~ \overline{G\cdot z}\ne\varnothing$. Does it follow that $\Omega$ is homogeneous,... | https://mathoverflow.net/users/53155 | Criterion for homogeneity | **Edit: (21 May 2017)** I have modified my answer to cover the case that the OP meant to ask, i.e., the assumption is that the *closure* of an orbit has nonempty interior.
Now that you have added the assumption of boundedness (and, I assume, connectedness, since, without it, the answer would clearly be 'no'), we can ... | 4 | https://mathoverflow.net/users/13972 | 270119 | 120,991 |
https://mathoverflow.net/questions/270107 | 2 | Let $L/K$ be a finite extension of algebraic number fields of degree prime $p$. Is it true that the index $(U\_K:\text{Norm}(U\_L))$ divides $[L:K]$,
where $U\_K$ denotes the unit group and Norm denotes the ideal norm map of the relative extension of $L/K$?
| https://mathoverflow.net/users/98582 | Does the unit index divide the degree of an extension of number fields? | No, not necessarily. For example, you can take $K\neq \mathbb{Q}$ to be totally real, and take $L$ to be a quadratic CM extension of $K$ whose unit group is equal to that of $K$ (e.g. make sure that $L/K$ ramifies at some place that is not above $2\infty$). Then ${\rm Norm}(U\_L) = U\_K^2$, and the unit index is $2^{{\... | 7 | https://mathoverflow.net/users/35416 | 270121 | 120,992 |
https://mathoverflow.net/questions/270115 | 12 | I like the following binomial identity in that the RHS extracts the indeterminate $w$ from the LHS.
>
> **Question.** Can you show that
> $$\sum\_{k=0}^n\binom{x+kw}k\binom{y-kw}{n-k}=\sum\_{k=0}^n\binom{x+y-k}{n-k}w^k\,\,\,?$$
>
>
>
It would be great if we can see alternative proofs? I've a bias for combinat... | https://mathoverflow.net/users/66131 | An interesting identity: in search of a proof -Part I | This is known as Jensen's identity and dates back to 1902. See here an overview of this identity and related ones, and a proof: <https://arxiv.org/abs/1005.2745>, a paper by Victor Guo.
| 19 | https://mathoverflow.net/users/31469 | 270124 | 120,995 |
https://mathoverflow.net/questions/270125 | 6 | For any imaginary quadratic field $F$, the Hilbert class field $H$ is generated by the $j$-invariant of any elliptic curve with complex multiplication (CM) by $\mathcal O$, the ring of algebraic integers of $F$.
What is the simplest generalization of this well known and useful fact? Since an imaginary quadratic fiel... | https://mathoverflow.net/users/56594 | Analogue of j-invariant for CM fields | The simplest generalisation to abelian surfaces is, I believe, the statement (which is a theorem, not a conjecture) that the [Igusa invariants](http://www.lmfdb.org/knowledge/show/g2c.igusa_invariants) of an abelian surface with CM by $K$ generate an abelian unramified extension of the reflex field of $K$. Note that th... | 7 | https://mathoverflow.net/users/35416 | 270128 | 120,997 |
https://mathoverflow.net/questions/270131 | 2 | Let $[n]\_q=1+q+\cdots+q^{n-1}$. Is there a nice closed form of $\sum\_{s=1}^i[s]\_{q}$? One would expect that the answer will be some q-analog of $\frac{i(i+1)}{2}$, since $\sum\_{s=1}^i s=\frac{i(i+1)}{2}$.
Also I'm quite unfamiliar with q-theory, so if my terminology/notation is imprecise please let me know!
| https://mathoverflow.net/users/106377 | Does this q-analogue have a nice closed form? | $[n]\_q=\frac{q^n-1}{q-1}$
$\Sigma\_{s=1}^i\frac{q^s-1}{q-1}=\frac{(q-1)+(q^2-1)+...+(q^i-1)}{q-1}=\frac{1+q+q^2+...+q^i-(i+1)}{q-1}=\frac{q^{i+1}-1}{(q-1)^2}-\frac{i+1}{q-1}=\frac{[i+1]\_q-(i+1)}{q-1}$
| 4 | https://mathoverflow.net/users/38448 | 270136 | 121,000 |
https://mathoverflow.net/questions/270108 | 1 | I saw the following question at mathstackexchang < <https://math.stackexchange.com/questions/2282194/when-an-ideal-is-locally-comaximal-with-idempotents>>. It seems to be a nice question and I need it in my research. I want to know is it true? It is clearly true for indecomposable rings.
>
> Let $R$ be a commutativ... | https://mathoverflow.net/users/108198 | When an ideal is locally comaximal with idempotents(restated) | No,this is false. Take $R=\mathbb{C}\times\mathbb{C}$.
The primes ideals of $R$ are $P\_1=\mathbb{C}\times (0)$ and $P\_2=(0)\times \mathbb{C}$.
The idempotents in $P\_1$ are $(0,0)$ and $(1,0)$, and those in $P\_2$ are $(0,0)$ and $(0,1)$. Hence $I(P\_1)=P\_1$ and $I(P\_2)=P\_2.$
Now take $I=P\_1=\mathbb{C}\times (0... | 1 | https://mathoverflow.net/users/36683 | 270137 | 121,001 |
https://mathoverflow.net/questions/270129 | 12 | As usual, denote $[n]\_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q}$ and $[n]\_q!=[1]\_q[2]\_q\cdots[n]\_q$. Furthermore, we write
$$\binom{n}k\_q=\frac{[n]\_q!}{[k]\_q!\cdot[n-k]\_q!}.$$
As a follow up on [this MO question](https://mathoverflow.net/questions/270115/an-interesting-identity-in-search-of-a-proof-part-i), I... | https://mathoverflow.net/users/66131 | A "quantum" identity: in search of a proof -Part II | Both sides are equal to $\binom{x+y+1}{n}\_q$. This enumerates lattice paths in an $n\times (x+y-n+1)$ rectangle, according to the area statistic. We will assume that these paths start at $(0,0)$ and end at $(x+y-n+1,n)$, and they are only directed East or North. Here are two ways to enumerate it:
**First count**: Fo... | 20 | https://mathoverflow.net/users/2384 | 270141 | 121,003 |
https://mathoverflow.net/questions/270146 | 15 | Here's the output of the GAP command "SmallGroupsInformation(512)"
>
> There are 10494213 groups of order 512.
>
>
>
> ```
> 1 is cyclic.
> 2 - 10 have rank 2 and p-class 3.
> 11 - 386 have rank 2 and p-class 4.
> 387 - 1698 have rank 2 and p-class 5.
> 1699 - 2008 have rank 2 and p-class 6.
> 2009 - 2039 have ... | https://mathoverflow.net/users/1048 | Is there anything significant about GAP's SmallGroup(512,2045)? | It is the "free" group of nilpotency class at most $2$, exponent dividing $4$, on three generators. (In other words, every group in that class is a quotient of it.)
According to
<https://etd.ohiolink.edu/rws_etd/document/get/osu1086112148/inline>
(ON SYLOW $2$-SUBGROUPS OF FINITE SIMPLE GROUPS OF ORDER UP TO $2^... | 19 | https://mathoverflow.net/users/22377 | 270148 | 121,006 |
https://mathoverflow.net/questions/270160 | 3 | If I'm slightly misusing definitions forgive me I'm not an algebraist.
I have $N$ polynomials $f\_n(x)$, $n=1,\ldots,N$ where $x\in \mathbb{R}^N$ and the set $\{x:f\_n(x)=0\text{ for all }n\}$ is finite. If I slightly change the coordinates in the polynomials are there some conditions and a result that lets me ensur... | https://mathoverflow.net/users/110114 | Continuity in the roots of an algebraic variety with respect to the coordinates | Of course, (real) solutions can just disappear under small perturbations, so you'd definitely have to assume that the number of solutions, counted appropriately, stays constant. But that's not enough either because you can move zeros around between factors. Consider for example
$$
x=a, \quad (x^2+y^2)(xy-1)=0 .
$$
For ... | 3 | https://mathoverflow.net/users/48839 | 270162 | 121,011 |
https://mathoverflow.net/questions/270173 | 2 | Is there for every positive integer $n\in\mathbb{N}$ a finite, simple, undirected graph $G=(V,E)$ with the following property?
>
> $G$ does not have a complete minor with more than $\frac{\delta(G)}{n}$ points (where $\delta(G)$ denotes the minimal degree).
>
>
>
| https://mathoverflow.net/users/8628 | Complete minors and minimal degree | Not an answer but an observation: for any *fixed* order $r$ of the complete minor, your question can be answered by a *finite* search over the set of all graphs of order at most $c\_{\mathrm{Kostochka}}\sqrt{\log r}$: let any $n\in\mathbb{N}$ be given. Let any graph $G=(V,E)$ be given. Let $\delta:=\delta(G)$. Let $c=c... | 4 | https://mathoverflow.net/users/108556 | 270176 | 121,014 |
https://mathoverflow.net/questions/270130 | 2 | Let $X$ be a random variable with infinitely divisible and symmetric distribution $F$ distributed on $\mathbb{R}$.
It is well known that the characteristic function of $X$ has a canonical representation of the form:
\begin{align}
\phi(t)=e^{ -\frac{\sigma^2}{2}t^2-\int\_{-\infty}^\infty (1-cos(tx) ) dV(x)}
\end{alig... | https://mathoverflow.net/users/69661 | Questions about Levy measure in the canonical representation of infinitely divisible distributions | I would discuss your questions partially under the following condition:
$$ \int\_{-\infty}^\infty x^2\,dF(x)<\infty, \tag{1}$$
or equivalently $\varphi^{\prime\prime}(0)$ exists and finite or $\int\_{-\infty}^\infty y^2\,dV(y)<\infty$. Let $K(x)\triangleq\int\_{-\infty}^x y^2\,dV(y)<\infty$, it is easy to obtain the fo... | 2 | https://mathoverflow.net/users/103256 | 270178 | 121,015 |
https://mathoverflow.net/questions/270180 | 3 | If $B$ is a $k$-algebra, let $T^1(B/k;M)$ denote the first cotangent functor. It classifies first order deformations of the scheme $\mathrm{Spec} B$.
Now, if $X \subset \mathbb P^n$ is a smooth projective scheme, we can look at its homogeneous coordinate ring $B$. Then $T^1(B/k;B)$ is a graded module, an under certa... | https://mathoverflow.net/users/11173 | Reference for a folklore result about $T^1(B/k;M)$ | This is almost true. Indeed $T^1(B )$ classifies first order deformations of the affine cone over $X$. If the depth at the vertex $v$ of the affine cone satisfies $depth\_v B \geq 3$ there is a long exact sequence in cohomology (coming from the relative tangent)
$$ \ldots H^1(X, \mathcal{O}\_X) \to (T^1\_B)\_0 \to H^1... | 2 | https://mathoverflow.net/users/52811 | 270186 | 121,019 |
https://mathoverflow.net/questions/270181 | 3 | Let $k$ be a field of characteristic zero and $X,Y$ be integral schemes of finite type. Assume we have a dominant morphism $\pi\colon X\to Y$.
Then we know that $\pi$ is generically smooth (i.e. on an open subset of the source $X$)
Question: can we say that there exists an open subset $U\subset Y$ such that the dif... | https://mathoverflow.net/users/40038 | Differential map of a dominant morphism in char zero | I am just posting my comment as an answer. The question has a positive answer.
**Generic Smoothness**. For every dominant morphism $\pi:X\to Y$ of finite type, integral $k$-schemes, denote by $V$ the maximal open subscheme of $X$ on which $\pi$ is smooth. Then $V$ contains the generic point of $X$ (and hence is a den... | 1 | https://mathoverflow.net/users/13265 | 270195 | 121,022 |
https://mathoverflow.net/questions/270197 | 8 | Let $X$ be a 4-manifold and $\Sigma\_g\subset X$ be an embedded closed orientable genus = $g$ surface. Suppose $\Sigma\_g\subset X$ has a trivial closed normal bundle $N(\Sigma\_g) = \Sigma\_g\times D^2$. For a given self-diffeomorphism $h$ on $\Sigma\_g\times S^1$, we can define a new 4-manifold:
$$X(\Sigma\_g,h):=(... | https://mathoverflow.net/users/85988 | Surgery along an embedded surface in a 4-manifold | For $g\geq 2$, this construction doesn't yield anything new. In this situation, any self-diffeomorphism of $\Sigma\_g \times S^1$ extends to a self-diffeomorphism of $\Sigma\_g \times D^2$, and hence your $X$ is unchanged.
| 9 | https://mathoverflow.net/users/3460 | 270203 | 121,025 |
https://mathoverflow.net/questions/270157 | 1 | First, I'm no expert in symmetry analysis of evolution equations and so I apologize if this post is a bit of a cobble. The question I have is about the evolution of $\psi: \mathbb{R}^{1+1}\to \mathbb{C}$ given by
$i \psi\_{t} + \psi\_{xx} + \frac{1}{2}|\psi|^{2} \psi + \lambda \left(\left[|\psi|^{2} \psi\right]\_{xx... | https://mathoverflow.net/users/110090 | Infinitesimal generators and conserved quantities (Schrodinger type evolution) | Perhaps part of your question could be phrased as follows. Does there exists a (formally defined) functional $S\_1[\psi,\bar{\psi}] = \int dt\,dx\, L\_1[\psi,\bar{\psi}]$ such that its Euler-Lagrange derivative gives $E[\psi,\bar\psi] = \frac{\delta S\_1}{\delta\bar{\psi}} = \left[ |\psi|^2 \psi \right]\_{xx}$, and by ... | 2 | https://mathoverflow.net/users/2622 | 270206 | 121,026 |
https://mathoverflow.net/questions/270190 | 5 | Suppose that $G=S^1$ acts on a smooth, connected, compact manifold with discrete fixed points, additionally assume that there is at least one fixed point.
Let $\alpha \in H^{2}\_{S^1}(M)$ be such that $\alpha|\_{p} = 0$ for any fixed point $p$. The space of such $\alpha$ forms an additive sub-group $A \subset H^{2}\... | https://mathoverflow.net/users/99732 | Equivariant cohomology defined by restrictions? | This subgroup $A$ is precisely the torsion subgroup in $H^2\_{S^1}(M)$ (since $H^2\_{S^1}(M^{S^1})$ is obviously free, and the relative $H^2\_{S^1}(M,M^{S^1})$ is torsion). So, it will be trivial if and only if the equivariant cohomology in degree 2 is free. This is true for compact $M$ with Hamiltonian $S^1$-action by... | 2 | https://mathoverflow.net/users/66 | 270209 | 121,027 |
https://mathoverflow.net/questions/270207 | 3 | Let $p(n)$ denote the number of partitions of a positive integer $n$. It is known that $\{p(n)\}\_{n>25}$ is log-concave.
Dietrich Burde said [in this MathOF post](https://mathoverflow.net/questions/138321/is-the-sequence-of-partition-numbers-log-concave) that property $PF\_3$ for partition function $p(n)$ semms to hol... | https://mathoverflow.net/users/110052 | What is the relationship between Partition function and Betti numbers for nilpotent Lie algebras? | The relationship is as follows. Denote by $\mathfrak{f}\_n$ the standard graded filiform nilpotent Lie algebra of dimension $n$. Then the following result is known.
*Theorem (Armstrong, Sigg 1996):* Let $L$ be a complex nilpotent Lie algebra having an abelian ideal of codimension $1$. Then the sequence of Betti numbe... | 2 | https://mathoverflow.net/users/32332 | 270210 | 121,028 |
https://mathoverflow.net/questions/270140 | 2 | A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions.
Motivated by conversations on [this questions](https://mathoverflow.net/questions/269940/vector-field-with-holomorphic-flow) we ask:
>
> Is the flow of a Harmonic vector field on $\ma... | https://mathoverflow.net/users/36688 | The flow of Harmonic vector fields | There is a "known true version", but it is basically what you have already as the converse.
**Defn** Let $(M,g)$ and $(N,h)$ be Riemannian manifolds, and let $\phi: M\to N$ be a smooth mapping. The *tension field* of $\phi$ is defined to be
$$ \tau(\phi) = \mathrm{trace}\_g D \mathrm{d}\phi $$
where $D$ is the pull-... | 5 | https://mathoverflow.net/users/3948 | 270212 | 121,029 |
https://mathoverflow.net/questions/269554 | 7 | Let $X$ be a Calabi-Yau threefold. Ordinary Donaldson-Thomas theory is formulated as a virtual count of ideal sheaves $\mathcal{I}$ with discrete invariants $\text{ch}(\mathcal{I}) = (1,0, -\beta, -n)$, which is equivalent to counting one-dimensional subschemes $Y \subseteq X$ with $[Y] = \beta \in H\_{2}(X, \mathbb{Z}... | https://mathoverflow.net/users/105661 | Incorporating Divisors (D4-branes) into Donaldson-Thomas Theory? | The original definition of DT invariants ( <https://arxiv.org/abs/math/9806111> ) works for any ch such that there is no strictly semistable objects. Later, this was generalized by Joyce and Song to arbitrary ch (giving rational invariants satisfying a conjectural multicovering formula). In general these invariants dep... | 6 | https://mathoverflow.net/users/25309 | 270213 | 121,030 |
https://mathoverflow.net/questions/270204 | 2 | The [Cauchy Identity](https://en.wikipedia.org/wiki/Schur_polynomial#The_Cauchy_identity)
$$ \sum\_{\nu}s\_{\nu}(x)s\_{\nu}(y) = \prod\_{j,k=1}^{\infty}\frac{1}{1-x\_{j}y\_{k}} $$
expresses the sum over all integer partitions of the product of pairs of [Schur polynomials](https://en.wikipedia.org/wiki/Schur_polynomial... | https://mathoverflow.net/users/110135 | Cauchy identity, with sum restricted over partitions with first part $\leq n$ | The sum (in any number of variables) is equal to the determinant
$$\det(B\_{j-i})\_{1\le i,j\le n},$$
where
$$B\_i=\sum\_{l=0}^\infty e\_{l+i}(x)e\_l(y),$$
and $e\_l$ is the elementary symmetric function, with $e\_l=0$ for $l<0$.
This follows by applying the involution $\omega$ in $x$ and $y$ to Theorem 16 of my paper ... | 6 | https://mathoverflow.net/users/10744 | 270215 | 121,031 |
https://mathoverflow.net/questions/270214 | 4 | Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix and $D\in\mathbb{R}^{n\times n}$ be a diagonal positive definite matrix. Let $\mathrm{diag}(X)\in\mathbb{R}^{n\times n}$ denote the diagonal part of a matrix $X\in\mathbb{R}^{n\times n}$.
>
> **My question:** Does the following inequality hold true
... | https://mathoverflow.net/users/62673 | $\mathrm{diag}\left[(A+D)^{-1}\right] \ge \left[\mathrm{diag}(A)+D\right]^{-1}$? | This is true. As Darij suggests, denote $B=A+D$, let $e=e\_i$ be a basic unit vector. We have to prove $(B^{-1}e,e)\geqslant (Be,e)^{-1}$. Denote further $C=B^{1/2}$, this rewrites $\|Ce\|\cdot \|C^{-1}e\|\geqslant 1$. This follows from Cauchy-Bunyakovsky-Schwarz, as $(Ce,C^{-1}e)=(e,e)=1$.
| 9 | https://mathoverflow.net/users/4312 | 270219 | 121,032 |
https://mathoverflow.net/questions/270175 | 0 | Assume that $(M,g)$ is a Riemannian manifold. A vector field $X$ on $M$ is called a harmonic vector field if the corresponding $1$-form $\alpha$ with $\alpha(Y)= \langle X,Y \rangle\_g$ is a harmonic $1$-form.
Motivated by [this conversations](https://mathoverflow.net/questions/269940/vector-field-with-holomorphic-fl... | https://mathoverflow.net/users/36688 | Vector field with Harmonic flow | Well, right away, you can see that the answer is 'no', in general. Consider the round $n$-sphere $S^n$ with its standard metric. When $n>1$, it has no nonzero harmonic $1$-forms, but it has nontrivial Killing vector fields since it is a homogeneous Riemannian manifold. Since the flow $\phi\_t$ of such a Killing field $... | 12 | https://mathoverflow.net/users/13972 | 270220 | 121,033 |
https://mathoverflow.net/questions/270061 | 3 | A positive integer $n$ is called a *Hilbert number* if $\exists a,b,d \in \mathbb{N}$ such that $ 4ab-a-b = d n$ and $d|a b$.
I ran an algorithm checking divisors for all $0\lt a,b\le500$, and the only numbers $n\le500$ for which I did not find a solution are
$\{1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,2... | https://mathoverflow.net/users/106239 | Hilbert Numbers | Your equation $$4ab-a-b=dn$$ gives
$$(1)\quad \frac{4}{n} =\frac{1}{ab/d}+ \frac{1}{an}+\frac{1}{bn}.$$
This is (almost) the Erdos-Straus equation.
See here for the well known [Erdos-Straus conjecture](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus_conjecture), which asks
if the equation
$$\frac{4}{n} =\frac... | 20 | https://mathoverflow.net/users/36707 | 270222 | 121,034 |
https://mathoverflow.net/questions/269941 | 3 | Do you know a basic reference to introduce an undergraduate student with more physical rather than mathematical background to De Rham cohomology?
The Student (from a Bachelors program in material science) should work on a summer project aiming to understand some of the mathematics behind the quantum hall efect, expl... | https://mathoverflow.net/users/21985 | Reference for de Rham cohomology for physicists | If you are familiar with some topological field theory, then the de Rham cohomology of $M$ can be viewed as the vector space of supersymmetric vacua in the $\sigma$-model associated to $M$. I think this was first noted by Prof. Witten in his work on sypersymmetry and Morse theory. You can look it up in the paragraph 4 ... | 2 | https://mathoverflow.net/users/nan | 270224 | 121,036 |
https://mathoverflow.net/questions/225115 | 8 | I wonder whether there is a notation for such thing, which I denote $[a;b]\_q^n$ for a moment:
$$
[a;b]\_q^n:=(a-b)(a-qb)\dots (a-q^{n-1}b)=a^n(b/a;q)\_n,
$$
this last equation uses [$q$-Pochhammer symbol](https://en.wikipedia.org/wiki/Q-Pochhammer_symbol) notation.
My motivation is that it is a natural analogue of $(a... | https://mathoverflow.net/users/4312 | notation for $(a-b)(a-qb)\dots (a-q^{n-1}b)$ | There are some different notations in the literature:
[B.A. Kupershmidt](http://de.arxiv.org/pdf/math/0004187.pdf) used $ {(a\dot - b)^n} $,
Victor Kac and Pokman Cheung in “Quantum Calculus” used $(a-b)\_q^n$.
In my [lecture notes](https://homepage.univie.ac.at/johann.cigler/skripten.html) I used ${(a\underset{... | 3 | https://mathoverflow.net/users/5585 | 270233 | 121,040 |
https://mathoverflow.net/questions/270228 | 11 | Let $G$ be a finitely generated group. Let $S(G)$ be the quotient of $\text{Hom}(G,\mathbb{R}) \setminus \{0\}$ by the equivalence relation that identifies two homomorphisms if they differ by scaling by a positive constant. For a nonzero $\phi \in \text{Hom}(G,\mathbb{R})$, write $[\phi] \in S(G)$ for the associated eq... | https://mathoverflow.net/users/317 | Subtle point in definition of BNS invariant | Let $S$ and $T$ be generating sets for $G$, and suppose that $X\_\phi$ is $T$-connected (i.e., spans a connected subgraph in the Cayley graph of $G$ with respect to $T$.) Let $[X\_\phi]\_S$ be the subgraph of $\text{Cay}(G,S)$ spanned by $X\_\phi$. We must show that $[X\_\phi]\_S$ is connected.
Claim 1: For any $n\in... | 16 | https://mathoverflow.net/users/36155 | 270234 | 121,041 |
https://mathoverflow.net/questions/270171 | 9 | Let $D\in\mathbb{R}^{n\times n}$ be a diagonal positive definite matrix s.t. $D\leq I$ ($I$ denotes the $n$-dim. identity matrix) and let $\alpha$ be a strictly positive real number. Consider the optimization problem over the set of positive semidefinite matrices with trace less or equal than one
$$
\max\_{A\in\mathbb{... | https://mathoverflow.net/users/62673 | Maximizing a ratio of determinants | Yes, of course (provided that you mean that the maximum is attained on a diagonal matrix, not that *every* matrix on which it is attained is diagonal).
First notice that it is a bit more convenient to have the denominator in the form $\det(A+Q)$ where $Q$ is $\alpha D^{-1}$. Now $A=R^\*BR$ where $B$ is diagonal and ... | 6 | https://mathoverflow.net/users/1131 | 270240 | 121,043 |
https://mathoverflow.net/questions/270241 | 6 | In [this paper](http://ir.library.osaka-u.ac.jp/dspace/bitstream/11094/3071/1/PRIMS41_3_683.pdf), the following result is proved.
For any prime $p$, all the Fourier coefficients of
$$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod\_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$
are non-negative.
As seeing [t... | https://mathoverflow.net/users/69834 | Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative? | Yes this is true and these Fourier coefficients actually enumerate some combinatorial objects called $m$-cores. These are partitions with no hooklength divisible by $m$. In fact for $m\geq 4$ these coefficients are positive. This last result was proved in ["Defect zero $p-$blocks for finite simple groups"](http://www.a... | 15 | https://mathoverflow.net/users/2384 | 270243 | 121,044 |
https://mathoverflow.net/questions/270238 | 3 | Hi I'm looking for an Ito-type lemma for rough paths but am having difficulty finding something. Could someone kindly point me in the right direction?
| https://mathoverflow.net/users/36886 | Reference: Ito lemma for rough paths | In addition to what Nate Eldredge mentioned, here is a recent paper that you might like to consider.
Keller, Christian; Zhang, Jianfeng *Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients*. Stochastic Process. Appl. 126 (2016), no. 3, 735–766.
[This copy](https://arxiv.org/abs/14... | 1 | https://mathoverflow.net/users/66131 | 270244 | 121,045 |
https://mathoverflow.net/questions/270225 | 5 | Let $X$ be a connected reduced stable curve of genus $g$ over an algebraically closed field $k$ of characteristic not dividing $|G|$. Let $G$ be a finite group acting faithfully on $X$.
Let $A := Aut(X\rightarrow X/G)$ be the group of automorphisms $\alpha : X\rightarrow X$ which induce the identity on $X/G$ and comm... | https://mathoverflow.net/users/88840 | Can a "stable" $G$-torsor have automorphisms which do not lie in the center $Z(G)$? | Assume $k$ has characterstic not $2$. Take two curves $Y$ and $Z$ of genus $> 1$ over $k$ each with an automorphism of order $2$ having the same number of fixed points. Say the fixed points are $y\_1, \ldots, y\_n$ on $Y$ and $z\_1, \ldots, z\_n$ on $Z$. Let $X$ be obtained by glueing $y\_i$ to $z\_i$ in $Y \amalg Z$ a... | 3 | https://mathoverflow.net/users/110156 | 270247 | 121,046 |
https://mathoverflow.net/questions/269972 | 6 | By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an arbitrary abelian subcategory $\mathcal{C}$ of the category $R\text{-mod}$ of unitary left $R$-modules.
>
> We say that a... | https://mathoverflow.net/users/33026 | Direct sum of injective modules is injective | **EDIT** Thanks to Jeremy Rickard for several corrective insights in the comments!
In the direction of positive conditions, I'm not sure whether the following conditions are reasonable for your purposes:
**Proposition:** Let $\mathcal{C}$ be an abelian category which is
* [locally finitely presentable](https://n... | 3 | https://mathoverflow.net/users/2362 | 270254 | 121,049 |
https://mathoverflow.net/questions/270201 | 4 | If one can define such a thing as a discrete dynamical system, can a Lagrangian and/or Hamiltonian for this also be defined and if so how, i.e. what do these correspond to?
Apologies if this is "dynamical systems 101", or little more. If so, or perhaps in any event, I guess the most helpful answer might just be to ci... | https://mathoverflow.net/users/10454 | Can a Lagrangian and/or Hamiltonian be defined for a discrete dynamic system? | [Geometry and Hamiltonian mechanics on discrete spaces](http://iopscience.iop.org/article/10.1088/0305-4470/37/41/008/meta) (2004). This paper$^\ast$ also contains an overview of earlier publications on this topic:
>
> The goal of this paper is to provide a discrete analogue of
> differential geometry, and to defi... | 2 | https://mathoverflow.net/users/11260 | 270262 | 121,054 |
https://mathoverflow.net/questions/270205 | 8 | Let $G$ be a split, almost-simple connected reductive group over a field $F$ with split maximal torus $T$. I am trying to understand precisely the groups $[G\_{\alpha}, G\_{\alpha}]$, where $\alpha$ is a root, $G\_{\alpha} = \mathcal{Z}\_G(T\_{\alpha})$, and $T\_{\alpha} = (\mathrm{ker} \ \alpha)^{\circ}$. Sometimes th... | https://mathoverflow.net/users/61816 | Centralizers of subtori in reductive groups, derived subgroups | We can ask this question for an arbitrary nilpotent element of the Lie algebra ${\mathfrak g}$ of $G$ (not just root elements as in the question). I'm sure this must already be written down somewhere, but it's also (mostly) quite easy to work out.
In characteristic zero or large enough, any nilpotent element $e$ of $... | 6 | https://mathoverflow.net/users/26635 | 270272 | 121,057 |
https://mathoverflow.net/questions/270271 | 3 | I read in Milnor's article "Link groups", where he defines invariants to classify links up to link homotopy, that the linking number is a complete invariant which can tell almost trivial two components links apart up to link homotopy.
My question is how do I prove that the linking number is a complete invariant for lin... | https://mathoverflow.net/users/110173 | Linking number a complete invariant of link homotopy | The linking number is the same as the homology class that one component represents in the complement of the other. You can reduce any $2$-component link to a normal form by first homotoping one component to be an unknot. Then the complement of this component is homeomorphic to a solid torus, which has $\pi\_1\cong\math... | 7 | https://mathoverflow.net/users/9417 | 270282 | 121,061 |
https://mathoverflow.net/questions/270277 | 5 | Let $K\_n = \Bbb Q(\mu\_{p^{n+1}})$ and let $A\_n$ be it's class group. Iwasawa theory tells us a lot about the $p$-part of $A\_n$. For instance, we know quite a lot about how it varies with $n$.
I am interested in the $\ell$-part of $A\_n$ where $\ell \neq p$ is a prime. What is known about these groups? For instanc... | https://mathoverflow.net/users/58001 | The $\ell$- part of the class groups of the $p$-cyclotomic fields | As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in *Invent. Math*. He proved that for each *abelian* number field $K/\mathbb{Q}$ and every pair $\ell\neq p$ of odd primes (I am not sure at what ... | 6 | https://mathoverflow.net/users/18238 | 270288 | 121,064 |
https://mathoverflow.net/questions/270292 | 1 | Assume that $M\_1, M\_2, \ldots , M\_k$ are $k$ disjoint compact submanifolds of $\mathbb{R}^n$ of the same dimension $m$. Assume that $\lambda\_{ij}, \; 1\leq i,j\leq k$ are $k^2$ arbitrary real numbers.
>
> Are there $k$ differential $m$- forms $\alpha\_1, \alpha\_2,\ldots, \alpha\_k$ with $\int\_{M\_{i}} \alpha\... | https://mathoverflow.net/users/36688 | Integral Separation of disjoint submanifolds of $\mathbb{R}^n$ | In order for integration of differential forms to make sense, we need $M\_1,\dotsc,M\_k$ to be oriented. Let $\omega\_i\in\Omega^m(M\_i)$ be such that $\int\_{M\_i}\omega\_i = 1$.
Since $M\_1,\dotsc,M\_k$ are pairwise disjoint, compact subsets of $\mathbb R^n$, then they have open neighborhoods $U\_1\supset M\_1,\dot... | 2 | https://mathoverflow.net/users/97265 | 270299 | 121,070 |
https://mathoverflow.net/questions/268470 | 51 | Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7\*. In a [recent preprint](https://arxiv.org/abs/1704.05490), Schoen and Yau show how the usual techniques can be generalised to arbit... | https://mathoverflow.net/users/109141 | What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint? | I know two applications:
a) The convergence of the Yamabe flow in dimension 6 and higher
Simon Brendle, Invent. math. 170, 541–576 (2007)
DOI: 10.1007/s00222-007-0074-x
b) Solution of the equivariant Yamabe problem
Farid Madani, Hebey-Vaugon conjecture II. (English, French summary)
C. R. Math. Acad. S... | 17 | https://mathoverflow.net/users/110127 | 270301 | 121,072 |
https://mathoverflow.net/questions/233144 | 61 | So far I made several attempts to really learn Atiyah-Singer theorem. In order
to really understand this result a rather broad background is required: you need
to know analysis (pseudodifferential operators), algebra (Clifford algebras,
spin groups) and algebraic topology (characteristic classes, K-theory, Chern
cha... | https://mathoverflow.net/users/24078 | Atiyah-Singer theorem-a big picture | I agree with @coudy's answer that the best approach is to first understand the theorem's special cases / applications / generalizations. That can help highlight some of the key pain points in the various proofs, and motivate some of the ideas involved. Still, I'll take a crack at the main thrust of the question: how do... | 18 | https://mathoverflow.net/users/4362 | 270305 | 121,073 |
https://mathoverflow.net/questions/267330 | 6 | There's a (fairly basic) fact I want to use in a paper I'm writing; it's not entirely trivial, so I don't feel comfortable just stating the result and moving on, but I don't have a citation for it. Can anyone provide a reference? (I'd be willing to include it if I thought it was new, but it's the sort of thing that at ... | https://mathoverflow.net/users/8133 | Reference request: a version of $\Sigma^1_1$ bounding for structures | After asking around for a while, it appears that there is no prior reference for this fact. I'm posting and accepting this answer to move this question off the unanswered list. That said, if anyone does later find a reference for this, please post it as an answer - I'll accept it and delete this one.
*(I've made this... | 1 | https://mathoverflow.net/users/8133 | 270309 | 121,076 |
https://mathoverflow.net/questions/270117 | 1 | Let $f(x,y) = a^2 x^2 - (b^2 - 2ac)xy + c^2 y^2$ be a positive definite binary quadratic form with co-prime integer coefficients such that $b \ne 0$. For a given pair of integers $(u,v)$ and a prime $p | f(u,v)$, we see that $u$ is a quadratic residue modulo $p$ if and only if $v$ is because
$$\displaystyle b^2 uv \e... | https://mathoverflow.net/users/10898 | Averaging the Jacobi symbol over an ellipse | I will work with $f=u^2+v^2$ and the region $|u|,|v|\leq X$
but the argument easily generalises. The numbers $m$ go up to $X^2$ but by Dirichlet's divisor trick we can assume that they go up to $\sqrt{X^2}$.
Then the congruence $u^2+v^2\equiv 0 \mod{m}$ can be written equivalently as $u\equiv \alpha v \mod{m}$ for some... | 3 | https://mathoverflow.net/users/9232 | 270328 | 121,083 |
https://mathoverflow.net/questions/270313 | 6 | I'm trying to learn about Freudenthal triple systems. Here is the definition given by Helenius [1], start of Section 5:
>
> A Freudenthal triple system is a finite-dimensional vector space $V$
> over a field $F$ (with characteristic not 2 or 3) such that
>
>
> • There is a nonzero quartic form $q$ defined on $V$... | https://mathoverflow.net/users/41139 | Is the triple product in a Freudenthal triple system fully symmetric? | You say that the 4-linear form $Q$ is obtained by 'linearization' of the quartic form $q$. (I can't stand giving them both the same name.) There will be many 4-linear forms $Q$ related to $q$ by
$$ Q(x,x,x,x) = q(x) $$
but, except perhaps in characteristic 2 or 3, there is just one that is totally symmetric, meanin... | 6 | https://mathoverflow.net/users/2893 | 270334 | 121,084 |
https://mathoverflow.net/questions/270333 | 2 | Let $G$ be the base change of a connected reductive group over ${\mathbb{F}}\_q$ to $\overline{\mathbb{F}}\_q$, and let $F$ be the associated geometric Frobenius on $G$. Let $W$ be the Weyl group of a quasisplit $F$-stable maximal torus of $G$.
By definition, $F$ acts on $W$. We know that, the $G^F$-conjugacy classe... | https://mathoverflow.net/users/56217 | On cuspidal maximal tori of a connected reductive group | No. For example, in (split, simply connected) type $\mathrm C\_2$, there are the tori whose rational points are isomorphic to $\ker \mathrm N\_{\mathbb F\_{q^4}/\mathbb F\_{q^2}}$, and those whose rational points are isomorphic to the product of $\ker \mathrm N\_{\mathbb F\_{q^2}/\mathbb F\_q}$ with itself.
| 2 | https://mathoverflow.net/users/2383 | 270339 | 121,085 |
https://mathoverflow.net/questions/270331 | 3 | Maybe this question is off topic or duplicate, but maybe there is some information which I am not aware of.
Let $G\_n$ denote the set of non-isomorphic simple graphs and $|G\_n|=g\_n$. Also, let $U\_{n,m}$ denote the set of non-isomorphic simple graphs with $n$ vertices and $m$ edges, and $|U\_{n,m}|=u\_{n,m}$. I am ... | https://mathoverflow.net/users/19885 | The probability that two graphs have same number of edges | Since the vast majority graphs has trivial groups, and this is even more true (to exponential precision) for graphs with likely numbers of edges, the answer will be the same if you consider labelled graphs. This makes it a simple exercise if you know the identity
$$2^{-2N} \sum\_{i=0}^N \binom Ni^2 = \frac{\Gamma(N+\fr... | 7 | https://mathoverflow.net/users/9025 | 270341 | 121,086 |
https://mathoverflow.net/questions/270329 | 8 | Let $k$ be a field or a commutative ring with unit and let $F:M\_n(k)\to M\_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A\_1XB\_1 + \cdots + A\_m X B\_m$ for some $A\_i,B\_i\in M\_n(k)$ (note that any $k$-linear $F$ can be written in this form, though not uniquely).
>
> Is there a form... | https://mathoverflow.net/users/1849 | Determinants (and traces) of linear maps of matrices | We have $F(X)=\sum\_i A\_i X B\_i = \sum\_i (B\_i^T \otimes A\_i) vec(X)$ (see [here](https://en.wikipedia.org/wiki/Vectorization_(mathematics))), i.e. $F \sim \sum\_i (B\_i^T \otimes A\_i)$. Because of some formulas [here](https://en.wikipedia.org/wiki/Kronecker_product) we have $tr(F)=\sum\_i tr(A\_i)tr(B\_i)$. For t... | 5 | https://mathoverflow.net/users/100908 | 270342 | 121,087 |
https://mathoverflow.net/questions/270337 | 5 | Let $F(x,y), A(x,y), B(x,y)$ be homogeneous polynomials with integer coefficients (a.k.a. binary forms) such that the degrees of $A(x,y)$ and $B(x,y)$ match. Define
$$R(x,y) := F\left(A(x,y), B(x,y)\right).$$
Then $R(x,y)$ is a homogeneous polynomial with integer coefficients of degree $dr$, where $d := \deg(F)$ an... | https://mathoverflow.net/users/22733 | Discriminant of a composition of binary forms | If I did not mess up the adaptation to your special case, Theorem 3.31 for $n=2$ in ["A computational approach to the discriminant of homogeneous polynomials"](https://arxiv.org/abs/1210.4697) by Busé and Jouanolou
says that
$$
{\rm Disc}(R)=({\rm Disc}(F))^r\times
{\rm Res}(A,B)^{d(d-1)}\times K(F,A,B)
$$
where $K$ is... | 3 | https://mathoverflow.net/users/7410 | 270351 | 121,088 |
https://mathoverflow.net/questions/270347 | 2 | I been reading Nagy and Foias' book "Harmonic analysis of operators on Hilbert space". They prove the existence of a isometric (also unitary) dilation.
However Douglas-Muhly-Pearcy in <http://projecteuclid.org/download/pdf_1/euclid.mmj/1029000093> seem to call this a co-isometry. Moreover they refer to Sarason's resu... | https://mathoverflow.net/users/83682 | Why do Douglas-Muhly-Pearcy consider the following operator a co-isometry? | An operator $A$ is a co-isometry if its adjoint $A^\*$ is an isometry, that is $AA^\* = I$. It really is a matter of taste whether you do dilation theory with isometries or co-isometries, in the end you really need a mix of the two to get to the unitary dilation. One consideration is that the minimal co-isometric exten... | 4 | https://mathoverflow.net/users/76593 | 270367 | 121,095 |
https://mathoverflow.net/questions/270336 | 23 | Let's call a sequence of nonnegative integers $x\_1,x\_2,\ldots$ *matrix-realizable*, if there exists a $k\times k$ nonnegative integer matrix $A$ (for some $k$), as well as nonnegative integer vectors $u,v$, such that
$x\_n = u^T A^n v$
for all $n$. (So in particular, all matrix-realizable sequences are [linearly ... | https://mathoverflow.net/users/2575 | Possible behaviors of integer sequences that arise from powering nonnegative integer matrices | The answer is "no" for both questions.
The rational functions $a\_0+a\_1x+a\_2x^2+\cdots$ with non-negative integer entries which can be obtained by $a\_n=u^{T}A^nv$ for some nonnegative vectors $u,v$ and matrix $A$ are called $\mathbb N$-rational in the literature. [Here](http://people.brandeis.edu/~gessel/homepage... | 16 | https://mathoverflow.net/users/2384 | 270372 | 121,096 |
https://mathoverflow.net/questions/270276 | 3 | **Definition:** A subgroup $H$ of $G$ is said to be *pronormal* in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.
**Definition:** A subgroup $H$ of $G$ is said to be *weakly pronormal* in $G$ if for $g \in G$, there exists $x \in H^{\langle g \rangle} := \langle g^nHg^{-n} ... | https://mathoverflow.net/users/92488 | Weak Pronormality of a finite group | Here is a suggestion on a possible way to look for an example, though I have not pursued it myself: suppose that $H$ is a non-maximal subgroup of maximal order of a finite simple group $G$ and that $H$ is self-normalizing( given the first assumption, the second one is equivalent to requiring that $H$ is not a (proper n... | 3 | https://mathoverflow.net/users/14450 | 270377 | 121,099 |
https://mathoverflow.net/questions/270378 | 7 | Are there interesting examples of linear binary codes that are closed under the operation of removing the first bit of the codeword and appending the complement of that bit to the end of the codeword?
| https://mathoverflow.net/users/3621 | Codes with a twisted cyclic action | *This is an extended comment that may be too long for the comment section.*
If I understand your question correctly, the linear codes you described are automatically trivial.
Define $\pi\_{c}$ to be the map that takes any $n$-dimensional vector $\boldsymbol{v} = (v\_0, v\_1, \dots, v\_{n-1}) \in \mathbb{F}\_2^n$ ov... | 7 | https://mathoverflow.net/users/27829 | 270384 | 121,101 |
https://mathoverflow.net/questions/270318 | 6 | Let $Y\_1(n)$ (for $n\ge 4$) be the fine moduli scheme over $\mathbb{Q}$ parametrizing elliptic curves with a rational point of order $n$. Let $\mathbb{A}^1\_j$ be the $j$-line over $\mathbb{Q}$, the coarse moduli scheme of the moduli stack of elliptic curves $\mathcal{M}\_{1,1}$ (over $\mathbb{Q}$). Let $\mathcal{M}\_... | https://mathoverflow.net/users/15242 | Torsion points on twists of elliptic curves and products of fine modular curves over $\mathcal{M}_{1,1}$ vs over the $j$-line | I think your degree calculation is bogus. You write:
>
> $Y\_1(7)^\circ$ is finite etale over $\mathcal{M}\_{1,1}^\circ$ of degree $[SL\_2(\mathbb{Z}):\Gamma\_1(7)] = 48$, in the sense that the fiber category above a geometric point $\text{Spec }k\rightarrow\mathcal{M}\_{1,1}^\circ$ is a groupoid with 48 objects, b... | 3 | https://mathoverflow.net/users/1310 | 270391 | 121,105 |
https://mathoverflow.net/questions/270387 | 3 | At the end of [*Fontaine’s rings and p-adic L-functions*](http://staff.ustc.edu.cn/~yiouyang/colmez.pdf), P. Colmez states a Theorem 8.4.8 ([click here](https://i.stack.imgur.com/JEz0b.png)) of Faltings-Tusji-Saito without references.
So I am wondering is there any references for this theorem?
| https://mathoverflow.net/users/108486 | Reference on a result on local Galois representation associated to classic modular form in p-adic Hodge theory | The three articles referenced presented in logical order of exposition are respectively
Faltings, Gerd *Hodge-Tate structures and modular forms* Math. Ann. 278 (1987)
Tsuji, Takeshi
*$p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case.* Invent. Math. 137 (1999)
and
Saito, Take... | 4 | https://mathoverflow.net/users/2284 | 270404 | 121,112 |
https://mathoverflow.net/questions/270405 | 2 | Consider the heat flow $e^{t\Delta}$, $t>0$, on an Euclidean domain (say $\mathbb{R}^3$). I expect, in analogy with the Strichartz estimates for the Schrodinger equation, that the following estimates hold:
$$\Vert e^{t\Delta}f\Vert\_{L^s(\mathbb{R}^+,L^p(\mathbb{R}^3))}\lesssim \Vert f\Vert\_{L^2(\mathbb{R}^3)},$$
wher... | https://mathoverflow.net/users/54552 | Strichartz estimates for the heat equation | Sure. Actually you have in addition the maximum principle (i.e. an $L^\infty$--$L^\infty$ estimate) which combined with the others gives a full triangle of indices for the Strichartz type estimates in your question. Even more, the strong decay of the symbol $e^{-t|\xi|^2}$ can be exploited for some gains in regularity.... | 4 | https://mathoverflow.net/users/7294 | 270406 | 121,113 |
https://mathoverflow.net/questions/270418 | 3 | I know that generally the geodesic distance $d\_x$ from a point $x$ in a Finsler space is not smooth ($C^\infty$). According to Shen, the square of it is just $C^1$ at $x$. Now I am wondering if there exists any number $n$ that makes $(d\_x)^n$ smooth in $x$?
| https://mathoverflow.net/users/110243 | Smoothness of some power of the geodesic distance in a Finsler geometry | Without more restrictions on the Finsler structure, the answer is 'no, there need not be any finite $n$ such that the $n$-th power of the distance function is smooth'.
For example, consider the Finsler structure on the $xy$-plane whose norm is
$$
F(x,y;\dot x, \dot y) = \sqrt{\dot x^2+\dot y^2}+\sqrt{\dot x^2 + 2\do... | 9 | https://mathoverflow.net/users/13972 | 270420 | 121,118 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.