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https://mathoverflow.net/questions/258958 | 3 | A result of Borel-Remmert in 1961/1962 published in Math. Ann. states that a compact homogeneous Kähler manifold must be the product of a complex torus and a projective-rational manifold. This implies that a simply-connected compact homogeneous Kähler manifold has nonzero Euler characteristic.
My question is, if a si... | https://mathoverflow.net/users/36974 | A question about simply-connected homogeneous compact complex manifold | Yes, it appears to be true. An argument is written up in a recent preprint by Ping Li:
Nonnegative Hermitian holomorphic vector bundles and Chern numbers <https://arxiv.org/pdf/1702.01701.pdf>
>
> Theorem 4.6.
> Suppose $M^n$ is a compact connected homogeneous complex manifold. Then
> we have the following impl... | 1 | https://mathoverflow.net/users/40950 | 264237 | 118,791 |
https://mathoverflow.net/questions/264240 | 3 | Is there a cardinal $\kappa>2^\omega$ and a connected space $X$ such that
(1) $|X|=\kappa$, and
(2) every connected subset of $X$ (with at least 2 points) has cardinal $\kappa$?
Let's assume every space is regular, so the answer would be YES if we allowed $\kappa=2^\omega$.
I seem to recall a paper with an e... | https://mathoverflow.net/users/95718 | Cardinality of connected subspaces | The answer is yes. Consider any limit ordinal $\gamma$ and let
$X$ be the space of all binary $\gamma$-sequences that are not
eventually ones, ordered under the lexical ordering. We may put the
order topology on $X$. This space has size $\kappa=2^\gamma$.
I claim that this space, in the order topology, is
connected. ... | 3 | https://mathoverflow.net/users/1946 | 264245 | 118,797 |
https://mathoverflow.net/questions/264218 | 2 | Suppose that $\beta$ is a $2n$-strand braid with plat closure $L$. We can multiply $\beta$ on either side by a member of the Hilden subgroup to get a new braid whose plat closure is still $L$. Or we could replace $\beta$ with $\beta\sigma\_{2n-1} \in B\_{2n+2}$ to get another braid whose plat closure is $L$. (Here $\si... | https://mathoverflow.net/users/14626 | Plat representations of unlinks | [Otal showed](http://www.ams.org/mathscinet-getitem?mr=679942) that stabilization is not needed for the unknot: any bridge representative of the unknot is equivalent to a stabilization
of the trivial 1-bridge representative. This theorem can be regarded a variant of Waldhausen's theorem that every Heegaard splitting of... | 3 | https://mathoverflow.net/users/1345 | 264249 | 118,799 |
https://mathoverflow.net/questions/264228 | 3 | By famous KPT theorem, if $\mathsf{T^i\_2}\vdash \mathsf{T^j\_2}$ for some $0<i<j$, then the polynomial hierarchy collapses. So it seems this bounded arithmetic hierarchy does not collapse. Also, conservative results for some classes of formulas lead to collapsing complexity classes which is not expected.
My question ... | https://mathoverflow.net/users/83598 | On subtheories of $\mathsf{T_2+EXP}$ | All these theories coincide: if exponentiation is total, a bounded formula is equivalent on any bounded domain to a sharply bounded formula (with an exponentially large parameter), hence bounded induction follows from sharply bounded induction, i.e., $T^0\_2+\mathrm{EXP}=T\_2+\mathrm{EXP}$.
| 4 | https://mathoverflow.net/users/12705 | 264258 | 118,803 |
https://mathoverflow.net/questions/264150 | 8 | I have found the determinant of the following matrix of order $n\ge3$ using some elemetary operations
$$\begin{bmatrix}0& 1 & 1& \dots & 1 \\ 0 & 0 & 1& \ddots & 1\\ 1 & 0 & 0 & \ddots & 1 \\ \vdots & \ddots & \ddots & \ddots & 1 \\ 1 & 1 & 1 & \dots & 0 \end{bmatrix}\_{n\times n},$$ that is, a matrix which have diago... | https://mathoverflow.net/users/91089 | Determinant of a matrix having diagonal and subdiagonal entries zero | I will try to give a different explicit solution. From the problem formulation of answer 1, $$\det(A+uu^T)=(1+u^TA^{-1}u).\det(A).$$ From [here](https://en.wikipedia.org/wiki/Tridiagonal_matrix), we need to solve some recursive expressions, in order to calculate the determinant and inverse of $A$. I solve these recursi... | 3 | https://mathoverflow.net/users/91089 | 264264 | 118,805 |
https://mathoverflow.net/questions/264259 | 7 | *Question: Let $E$ be a Hilbert space. Can there exist a strictly finer bornological topology on $E$?*
The background to my question is as follows. I am looking at locally complete, locally convex spaces $E$ with bounded inner products $\gamma$. Then the topology induced by $\gamma$ is weaker than the bornological to... | https://mathoverflow.net/users/13970 | Strictly finer bornological topology on Hilbert space | Every (real or complex) vector space $E$ can be endowed with its *finest locally convex topology* $\tau\_{flc}$ where every seminorm is continuous and (equivalently) every absolutely convex absorbing set is a $0$-neighborhood.
This topology can be described as the locally convex inductive limit of all finite dimensio... | 9 | https://mathoverflow.net/users/21051 | 264266 | 118,806 |
https://mathoverflow.net/questions/264272 | 1 | Dirchlet's Approximation Theorem states that for $k$ real numbers $b\_1, b\_2, \ldots, b\_n$, and $N\in \mathbb{N}$ there exists a $q\in\mathbb{N}$ such that for all $1\le i\le k$, there exists integers $p\_i$ such that $|b\_iq-p\_i|\le \frac{1}{N^{\frac{1}{k}}}$.
Note that in some sense these approximations can be ... | https://mathoverflow.net/users/105971 | "Less than or Equal To" Rational Approximations | This is false in general, consider any irrational $b\_1\in (0,1)$ and $b\_2=1-b\_1$. Then $qb\_1-\lfloor qb\_1 \rfloor+qb\_2-\lfloor qb\_2 \rfloor=1$ for any positive integer $q$, thus if $\epsilon<1/2$, your $q$ does not exist.
On the other hand, if $b\_i$'s and 1 are rationally independent, we may say even more: fo... | 4 | https://mathoverflow.net/users/4312 | 264274 | 118,807 |
https://mathoverflow.net/questions/263894 | 5 | The [Quillen-Suslin](https://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem) theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$.
Let's work over the complex numbers. What can be said about vector bundles on the *punctured* affine space $X\_n=\mathbb{A}^... | https://mathoverflow.net/users/4721 | Classification of (complex algebraic) vector bundles on punctured affine space | I spoke to my colleague Song Sun, and he reminded me of a discussion that he and I had about Question 2 some time ago. For $n\geq 2$, there are many examples of locally free sheaves on $X\_{n} = \mathbb{A}^{n+1}\setminus\{0\}$ that admit no equivariant structure. Denote by $S$ the polynomial ring $\mathbb{C}[x\_0,\dots... | 4 | https://mathoverflow.net/users/13265 | 264293 | 118,810 |
https://mathoverflow.net/questions/178764 | 13 | One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi\_1(X\_\overline K)$, where $X$ is a (anabelian) variety and $X\_\overline K=X\times\_K \overline K$. This yields a representation by outer automorphims of the fundamental group.
... | https://mathoverflow.net/users/48554 | Applications of anabelian geometry to Galois representations? | The standard theory of Galois representations is concerned with the actions of absolute Galois groups of number fields over abelian groups (in particular, vector spaces). Anabelian geometry is about the actions (by outer automorphisms) of the same absolute Galois group over étale fundamental group of varieties which ar... | 9 | https://mathoverflow.net/users/9317 | 264300 | 118,811 |
https://mathoverflow.net/questions/264295 | 2 | Let $f: A \rightarrow B$ be a morphism of abelian varieties defined over a finite field $k$. Let $G$ be a finite group of $A$ and $\pi:A\rightarrow A/G$ the quotient morphism.
Looking at just the group structure, it is enough to have that $G\subset \ker f$ to ensure the existence of a group morphism $g$ such that $f... | https://mathoverflow.net/users/103121 | morphism of abelian variety | This is Theorem 4 on page 73 of Mumford's *Abelian Varieties*, where you will also find a proof. Here's the statement: Let $X$ be an abeian variety. There is a 1-1 correspondence between the two sets of objects:
(a) finite subgroups $K\subset X$
(b) separable isogenies $f:X\to Y$, where two isogenies $f\_1:X\to Y\_... | 5 | https://mathoverflow.net/users/11926 | 264306 | 118,814 |
https://mathoverflow.net/questions/264230 | 0 | I'm trying to read a paper called "Graph Embedding Discriminant Analysis on Grassmannian Manifolds for Improved Image Set Matching" and I came across a sentence that confused me (the last one):
>
> A manifold is a topological space that is locally similar to Euclidean space. At an intuitive level, manifolds can be... | https://mathoverflow.net/users/92027 | How can Kernel functions make a Grassmann manifold into an Euclidean vector space? | If you have you have a set $X$ and a kernel function $k: X \times X \to \mathbb R$, then you can associate to $k$ a Hilbert space $H$ of functions on $X$ in the following manner: set
$$H\_0 = \operatorname{span} \{ k(\cdot,x) \,:\, x \in X \}\,,$$
and define on $H\_0$ the inner product
$$ \langle \sum\_i \alpha\_i k(\... | 4 | https://mathoverflow.net/users/13970 | 264318 | 118,817 |
https://mathoverflow.net/questions/263650 | 13 | As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation problem" a differential graded Lie algebra or $L\_{\infty}$-algebra that controls it.
I've seen this idea realized i... | https://mathoverflow.net/users/66688 | DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons | The Quillen-Drinfeld-Deligne-etc. philosopy should not be looked at as something too mysterious.
Namely, it reduces to the fact that if the set of objects one is interesting in the infinitesimal deformations of is not too wild, then it can be described in the form $f(v)+Q(v)=0$, where $f:V\to W$ is a linear function ... | 18 | https://mathoverflow.net/users/8320 | 264322 | 118,818 |
https://mathoverflow.net/questions/264325 | 2 | Let $G$ be a finite group, and let $P$ be a finitely generated group.
Consider the number $$n=\#Hom\_{Grp}(P,G).$$
It is known (see [Number of solutions to equations in finite groups](https://mathoverflow.net/questions/198306/number-of-solutions-to-equations-in-finite-groups)) that under relative mild assumptions on $P... | https://mathoverflow.net/users/41644 | Number of homomorphism, or number of solution to equations, in finite groups | There are $70$ homomorphisms from $\mathbb{Z}\times\mathbb{Z}$ to the dihedral group of order $14$.
| 6 | https://mathoverflow.net/users/22989 | 264327 | 118,820 |
https://mathoverflow.net/questions/264330 | 37 | I know that Ciprian Manolescu has settled the triangulation conjecture in the negative: Not all manifolds can be triangulated. I've only read secondary literature on this result, which did not detail in which dimensions it is known
that there exist untriangulable manifolds. My understanding is that Freedman and Casson ... | https://mathoverflow.net/users/6094 | Not all manifolds can be triangulated: In which dimensions? | In dimensions up to three, every manifold is triangulable (this is classical). In dimension 4, there are simply connected non-triangulable manifolds (such as the E8 manifold); in fact, a closed 4-manifold is triangulable if and only if it's smoothable. (Pick a triangulation; the links of vertices are always both homolo... | 67 | https://mathoverflow.net/users/40804 | 264333 | 118,824 |
https://mathoverflow.net/questions/264335 | 18 | Given $a,b,c\in \Bbb{N}$ such that $\{a,b,c\}$ are coprime natural numbers and $a,b,c>1$. When
$$\frac{a^2}{b+c}＋\frac{b^2}{c+a}＋\frac{c^2}{a+b}\in\mathbb Z\,?$$
I know the solution $\{183,77,13\}$. Is there any other solution?
| https://mathoverflow.net/users/105999 | When $\frac{a^2}{b+c}＋\frac{b^2}{a+c}＋\frac{c^2}{a+b}$ is integer and $a,b,c$ are coprime natural numbers, is there a solution except (183,77,13)? | Yes, there is another solution. The next one I found is a bit big, namely
$$ a = 15349474555424019, b = 35633837601183731, c = 105699057106239769. $$
This solution also satisfies the property that
$$ \frac{a^{2}}{b+c} + \frac{b^{2}}{a+c} + \frac{c^{2}}{a+b} = \frac{31}{21} (a+b+c),
$$
which was true of $a = 13$, $b =... | 40 | https://mathoverflow.net/users/48142 | 264342 | 118,829 |
https://mathoverflow.net/questions/264355 | 1 | Let $X$ be a locally Noetherian scheme defined over an algebraically closed field $k$ and let $Y\subset X$ be a closed subscheme. Suppose there is an algebraic group $G$ acting on $X$ and a subgroup $H\subset G$ acting on $Y$, in such a way that the closed immersion $Y\to X$ descends to a morphism
$$
f:[Y/H]\to [X/G].
... | https://mathoverflow.net/users/97902 | Condition for a morphism of stacks to be locally of finite type | Yes.
Since the question is local in $X$ we can assume that it is a finite type affine scheme. We always have the induced map of stacks $$f:[Y/H]\rightarrow [X/H]\rightarrow [X/G]$$ given by tensoring a locally trivial $H$ torsor with $G$ over $H$. Now we need to show that the fibre product $[Y/H]\times\_{[X/G]} X$ i... | 1 | https://mathoverflow.net/users/21637 | 264362 | 118,833 |
https://mathoverflow.net/questions/264210 | 2 | Here is a slightly anecdotical notational question.
Let $S$ be a scheme and let $X$ be a scheme over $S$, with structural morphism $s\colon X\to S$. Is there a good suggestive notation for the group $\lbrace (f,g)\in \mathrm{Aut}(X)\times \mathrm{Aut}(S)~\vert~sf=gs\rbrace $ ?
After chatting with a categorical frie... | https://mathoverflow.net/users/47722 | Notation for the automorphisms of a $S$-scheme over automorphisms of $S$ | Indeed, this already exists in the literature. The automorphisms of $G\to\mathrm{Spec}(k)$ are called *semilinear automorphisms* of $G$, or just *semi-automorphisms* of $G$, and the corresponding group is denoted by $\mathrm{SAut}(G)$. See Subsection 3.2 of [this paper](https://arxiv.org/abs/1207.1329) and references t... | 5 | https://mathoverflow.net/users/4149 | 264365 | 118,834 |
https://mathoverflow.net/questions/264142 | 6 | The following paper gives a classification of the character tables of irreducible representations of $SL(3,GF(q))$ where $q$ is a power of a prime number, and $ GF(q)$ a finite field of $q$ elements.
WILLIAM A. SIMPSON AND J. SUTHERLAND FRAME
Can. J. Math., Vol. XXV, No. 3,1973, pp. 486-494
THE CHARACTER TABLES FOR
S... | https://mathoverflow.net/users/17787 | the character tables of irreducible representations of $SL(3,Z_q)$ | As was already mentioned, the answer to the question asked is "no, we currently do not have a good classification". Here I wish to describe a successful and interesting recent line of research which does not aim at giving such a classification, rather merely at counting how many representations we do have.
To put thi... | 8 | https://mathoverflow.net/users/89334 | 264366 | 118,835 |
https://mathoverflow.net/questions/264280 | 16 | Let $G$ be a finite group, $L(G)$ its subgroup lattice and $\mu$ the Möbius function.
Consider the Euler totient of $G$ defined as follows:
$$ \varphi(G) = \sum\_{H \le G}\mu(H,G) |H| $$
Let $X=\{M\_1, \dots, M\_n \}$ be the set of maximal subgroups of $G$. By applying the *Crosscut Theorem* with $X$ (see [this c... | https://mathoverflow.net/users/34538 | Existence of a faithful irreducible representation using Möbius function | No, the modular maximal-cyclic group [$M\_4(2)$](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/M4(2).html), of order $16$, has a **f**aithful **i**rreducible **c**omplex **r**epresentation (f.i.c.r.) of dimension $2$, whereas $\hat{\varphi}(M\_4(2)) = 0$.
Let $B$ be the subgroup generated by the minimal subgrou... | 6 | https://mathoverflow.net/users/34538 | 264368 | 118,836 |
https://mathoverflow.net/questions/204722 | 7 | The space $P1$ of my earlier question [203755](https://mathoverflow.net/questions/203755/two-spaces-attached-to-mod-2-level-9-modular-forms-a-conjectural-hecke-isomorph) "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear insi... | https://mathoverflow.net/users/6214 | Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism? | Like the corresponding level 9 question, this is best understood using the Fricke involution. Let M(odd) be the space of odd mod 2 modular forms of level Gamma\_0 (25), and J (I call it P above) be the kernel of U\_5 acting on M(odd); it is a Z/2[G^2] module of rank 24 stabilized by the T\_p. (Throughout p is an odd pr... | 2 | https://mathoverflow.net/users/6214 | 264375 | 118,839 |
https://mathoverflow.net/questions/264363 | 3 | Is this a theorem?
>
> Given any integer $m$, there exists $N(m)$, such that for all $n>N(m)$, we have
> $$n = a\_1+a\_2+\cdots+a\_m$$
> where $\gcd(a\_i,a\_j) = 1$ with $a\_i>1$ for all $i,j>1$.
>
>
>
For instance, it is a textbook exercise to show for small values of $m$; Any integer exceeding $6$ is a sum... | https://mathoverflow.net/users/106014 | Is every positive integer (eventually) a sum of $m$ relatively prime integers for every $m$? | We may induct on $m$. Assume that for $m-1$ this is proved. Choose $a\_1$ as a prime number between $n/2$ and $n-N(m-1)$ and apply induction proposition for $n-a\_1$.
| 9 | https://mathoverflow.net/users/4312 | 264376 | 118,840 |
https://mathoverflow.net/questions/264373 | 0 | I don't fully understand this other question, but there's a clear relationship between logic and number theory
[the strength of saying "each sentence of true arithmetic has a recursive proof"](https://mathoverflow.net/questions/264201/the-strength-of-saying-each-sentence-of-true-arithmetic-has-a-recursive-proof)
He... | https://mathoverflow.net/users/1358 | what ordinals were used in proving unique factorization over $\mathbb{Z}$ | Let me first address the specific issue of how statements in number theory correspond to recursive "infinitary tree proofs."
Well, first of all, the point is that *every* true statement about natural numbers has a recursive proof. So asking *whether* something like the (statement of the effectiveness of the) Euclidea... | 2 | https://mathoverflow.net/users/8133 | 264384 | 118,841 |
https://mathoverflow.net/questions/264171 | 4 | Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p > 0$. Is there an affine Lefschetz theorem and Poincaré duality for sheaves represented by finite flat commutative group schemes of $p$-power order for flat cohomology of $X$?
For a surface and certain sheaves, I have f... | https://mathoverflow.net/users/nan | Poincaré duality and affine Lefschetz for flat cohomology | No. Roughly speaking, finite group schemes are objects with slope between 0 and 1 and so there is no Poincare duality without twisting except on curves (see the article of Artin and Milne for that). There is a flat analogue of the etale $\mathbb{Z}\_l(r)$ duality --- see the article of Milne you mention and later artic... | 1 | https://mathoverflow.net/users/106022 | 264392 | 118,843 |
https://mathoverflow.net/questions/264396 | 3 | I have heard mentioned the following theorem:
If a Riemannian metric on a product of two Riemann surfaces has non-positive sectional curvature then the metric is a product metric.
And am trying to find a reference for it but have not found anything, any suggestions?
| https://mathoverflow.net/users/85369 | Non-positive metric on a product is a product metric | This is a very special case of Theorem 10.3.9 in Eberlein's book "Geometry of nonpositively curved manifolds".
| 7 | https://mathoverflow.net/users/21684 | 264397 | 118,845 |
https://mathoverflow.net/questions/264379 | 2 | I asked a similar [question](https://math.stackexchange.com/q/2139530/133781) a few weeks ago in M.SE but it didn't receive any answers, so I decided to post it here with some modifications.
My motivation comes from a theorem given in Pete L. Clark's [notes](http://alpha.math.uga.edu/%7Epete/factorization2010.pdf) on... | https://mathoverflow.net/users/97665 | About Euclidean domains | At the request of the OP, I'm turning my comments above into an answer, though different answers are possible and the question sounds a bit soft to me. Let it be as it may, here are my two cents.
From the point of view of factorization theory, Euclidean domains can be understood as a rather special subclass of the cl... | 6 | https://mathoverflow.net/users/16537 | 264400 | 118,847 |
https://mathoverflow.net/questions/262792 | 15 | Let $\mathcal{B}$ be the Borel $\sigma$-algebra of $[0,1]$, and let $\mathcal{M}$ be the set of probability measures on $([0,1],\mathcal{B})$, equipped with the evaluation $\sigma$-algebra $\ \sigma(\rho \mapsto \rho(A):A \in \mathcal{B})$.
Let $\mathcal{M}\_2$ be the set of probability measures on $([0,1] \times [0,... | https://mathoverflow.net/users/15570 | Is "conditioning to a sub-$\sigma$-algebra" a measurable operation? | Okay, I think I've worked out that the answer is **no**, i.e. there exists a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{B}$ such that $\mathbb{E}\_\mathcal{G}$ is not universally measurable.
(We will write $\lambda$ for the Lebesgue measure on $([0,1],\mathcal{B})$.)
It is well-known that the evaluation $\sigm... | 4 | https://mathoverflow.net/users/15570 | 264407 | 118,850 |
https://mathoverflow.net/questions/264405 | 15 | This question is out of curiosity and came to me thinking about another MO question which is linked below.
>
> **Question:** Do there exist positive integers $a,b,c$ such that $\gcd(a,b,c) =1 $ and each of $\frac{a^2}{b+c},\frac{b^2}{a+c},$ and $\frac{c^2}{a+b}$
> are also integers?
>
>
>
My question was insp... | https://mathoverflow.net/users/51668 | Dividing squares by sums | First, notice that such $a$, $b$, $c$ must be pairwise coprime (e.g., if prime $p\mid \gcd(a,b)$, then $(a+b)\mid c^2$ implies $p\mid c$, a contradiction to $\gcd(a,b,c)=1$).
As divisors of pairwise coprime numbers, $a+b$, $a+c$, $b+c$ are also pairwise coprime.
Now, since $(a+b)\mid c^2$, $(a+c)\mid b^2$, $(b+c)\mi... | 20 | https://mathoverflow.net/users/7076 | 264408 | 118,851 |
https://mathoverflow.net/questions/264235 | 1 | I found some absurd observation which I could not fix by myself. For an elliptic curve $E$ over $\mathbb{Q}$, let $\overline{E}=E\otimes\overline{\mathbb{Q}}$. Every multiplication-by-$n$ map $\overline{E}\to \overline{E}:P\mapsto nP$ defines an étale covering of $E$. It means that for any $n$ and $n$-torsion point $P$... | https://mathoverflow.net/users/105934 | Local parameters and etale coverings of of elliptic curves | I can only speak about the case that $p=\Bbb O$, when there’s one uniformizer involved. I’ll follow you and call $x/y=\varpi$ — now we need no subscripts. *If* you’re satisfied with a formal response to your question, then all is answered by the formal group of the elliptic curve in question. Since you’re working over ... | 2 | https://mathoverflow.net/users/11417 | 264410 | 118,852 |
https://mathoverflow.net/questions/263093 | 12 | Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z\_1,\dots,z\_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum\_{|\alpha| \leq \mu} c\_{\alpha} z^{\alpha},
\end{equation}
where as usual for every $\alpha=(\alpha\_1,\dots,\alpha\_\nu) \in \mathbb{N}^{\nu}$ (here and in the following $\mathbb{N}$ denote... | https://mathoverflow.net/users/99197 | On an Inequality of Lars Hörmander | Finally, I realized how to modify Hörmander's proof in order to prove (I).
I will describe here the necessary changes we have to do to Hörmander's proof. Clearly, all the notation is that of Hörmander's paper.
Let $C$ be a closed, convex set of $\mathbb{R}^\nu$. Then make the following changes in Section (4) of Hörma... | 3 | https://mathoverflow.net/users/99197 | 264422 | 118,857 |
https://mathoverflow.net/questions/264130 | 4 | I have been trying to find an analytical expression for the following:
$\frac{\partial {X^{+}}}{\partial {X}}$
In my case, $X$ has a constant rank.
I've found the formula for differentiating a pseudoinverse in [Golub's paper](http://i.stanford.edu/pub/cstr/reports/cs/tr/72/261/CS-TR-72-261.pdf) (equation 4.12):
... | https://mathoverflow.net/users/105750 | Derivative of pseudoinverse with respect to original matrix |
>
> but I can't see how to input the original matrix.
>
>
>
I think when people write $\tfrac {\partial f} { \partial X}$ or $\tfrac {df}{dX}$ they really mean to find the Fréchet derivative, denoted by $df$, or $Df$. Your formula becomes (appears also in [this answer](https://math.stackexchange.com/questions/16... | 2 | https://mathoverflow.net/users/45979 | 264426 | 118,859 |
https://mathoverflow.net/questions/264423 | 11 | **EDIT.** Fix $n,d,k\in\mathbb{N}$. Let us consider the set $\mathcal{P}\_{n,d,k}$ of polynomials $P$ in one variable for which there exits a closed irreducible subvariety $X\_P\subset \mathbb{C}\mathbb{P}^n$
(which may be assumed to be smooth if necessary) of dimension $k$ and degree $d$ whose [Hilbert polynomial](ht... | https://mathoverflow.net/users/16183 | Counting Hilbert polynomials of projective varieties | **Edit.** I edited the answer below so that it also applies to geometrically reduced schemes of degree $d$ and pure dimension $k$. Also, the argument shows that there is a single finite set $\mathcal{P}\_{n,d,k}$ for all fields simultaneously, i.e., $\mathcal{P}\_{n,d,k}$ is independent of the characteristic. I also ed... | 13 | https://mathoverflow.net/users/13265 | 264428 | 118,860 |
https://mathoverflow.net/questions/264380 | 11 | This problem is some variation of [another MO question](https://mathoverflow.net/questions/264150/determinant-of-a-matrix-having-diagonal-and-subdiagonal-entries-zero/264167#264167). Consider the matrix
$$M\_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c & \ddots & a \\ \vdots & \ddots & \... | https://mathoverflow.net/users/66131 | yet another determinant and inverse of a matrix | Based on the answer by მამუკა ჯიბლაძე, I can see a generalized claim for which we need a proof.
Define the polynomial $P\_n(x)=\prod\_{k=1}^n(c\_k-x)$ and consider the matrices
$$A\_n:=\begin{bmatrix}c\_1& a & a& \dots & a \\ b & c\_2 & a& \ddots & a\\ b & b & c\_3& \ddots & a \\ \vdots & \ddots & \ddots & \ddots & a... | 10 | https://mathoverflow.net/users/66131 | 264429 | 118,861 |
https://mathoverflow.net/questions/264438 | 4 | Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of cardinality $\mathfrak{c}$? This can be proved with AC, but I suspect a much weaker form of Choice, or maybe none at all,... | https://mathoverflow.net/users/19444 | Number of torsion-free abelian groups | You can construct antichains of size $\frak c$ without using choice, you can even have them to be antichains in a stronger sense of the word: i.e. every two have a finite intersection.
The one thing you'd want choice for this is to make sure your antichains are maximal. So we can't do that. But do we need that? No. O... | 7 | https://mathoverflow.net/users/7206 | 264439 | 118,863 |
https://mathoverflow.net/questions/264389 | 2 | *EDIT: There are serious problems with the definition below; see the comment thread below for those problems and some thoughts on addressing them. I'm leaving the question up for now since I think the **idea** is still interesting, but it is definitely broken at the moment.*
---
Suppose $\mathbb{P}$ is a forcing ... | https://mathoverflow.net/users/8133 | Invariant names and submodels of forcing extensions | The original definition has a problem with the hereditary requirement, unless your name is something particularly nice, e.g. a name for a subset of the ground model. But in the latter case, you just get a model of $\sf ZFC$, since you essentially get $V[A]$ for some $A\subseteq V$.
In the comments you suggest instead... | 1 | https://mathoverflow.net/users/7206 | 264441 | 118,865 |
https://mathoverflow.net/questions/264436 | 7 | Who first published a proof that
$$\sum\_{n\leq x}d\_{k}(n)d\_k(n+h)=O(x(\log x)^{2k-2})$$
for fixed $k$ and $h$ please? I am struggling to find a reference. Thank you.
| https://mathoverflow.net/users/10980 | Who proved the upper bound for the autocorrelation of higher order divisor functions? | I'm not sure if it was the first, but the bounds would follow from the main theorem in
*Nair, Mohan; Tenenbaum, G\'erald*, [**Short sums of certain arithmetic functions**](http://dx.doi.org/10.1007/BF02392880), Acta Math. 180, No.1, 119-144 (1998). [ZBL0917.11048](https://zbmath.org/?q=an:0917.11048).
See also the ... | 9 | https://mathoverflow.net/users/766 | 264442 | 118,866 |
https://mathoverflow.net/questions/264433 | 13 | Let $f(x)$ be a polynomial in the ring $\mathbb{R}[x]$, the roots are all real and $f(0)=1$. Write the Taylor series of $1/f(x)$ around the origin as
$$\frac1{f(x)}=\sum\_{k=0}^{\infty}a\_kx^k,$$
and denote $P\_n(x)=\sum\_{k=0}^na\_kx^k$.
>
> **Question.** Is it true that the polynomial $P\_{2n}(x)$ has no real roo... | https://mathoverflow.net/users/66131 | $f$ real-rooted forbid truncated $\frac1f$ to be so? | We use a standard notation $[x^n]h(x)$ for a coefficient of $x^n$ in the series of $h(x)$.
Assume that $t$ is a real root of $P\_n$. First of all, note that $$0=a\_0+a\_1t+\dots+a\_nt^n=t^n[x^n]\frac1{f(x)(1-x/t)},$$
thus for $g(x)=f(x)(1-x/t)$ we just need to prove that all coefficients $b\_{2n}=[x^{2n}]1/g(x)$ are... | 11 | https://mathoverflow.net/users/4312 | 264447 | 118,868 |
https://mathoverflow.net/questions/264445 | 6 | In *Hyperbolic groups* (page 82), Gromov claims that, by a standard application of Thurston's method of *geodesic (hyperbolic) simplices*, it can be prove that a hyperbolic group contains finitely many pairwise non conjugate subgroups isomorphic to a fixed one-ended finitely presented group.
I took a look on the usu... | https://mathoverflow.net/users/43559 | One-ended finitely presented subgroups of hyperbolic groups | There is a proof of this theorem due to Thomas Delzant:
T. Delzant, L’image d’un groupe dans un groupe hyperbolique, Comment. Math. Helv.
70 (1995), no. 2, 267–284.
There is also a version for relatively hyperbolic groups du to Francois Dahmani:
Accidental Parabolics and Relatively Hyperbolic Groups, Israel Journ... | 9 | https://mathoverflow.net/users/69797 | 264458 | 118,871 |
https://mathoverflow.net/questions/264231 | 7 | I am looking for the largest Wasserstein distance to the uniform distribution among all probability distributions with uniform marginals.
More specifically, let $\Xi=\{1,2,\ldots,N\}^2$, and let $\nu$ be a uniform distribution on $\Xi$, namely, $\nu\_{ij}:=\nu(\{(i,j)\})=\frac{1}{N^2}$, for all $1\leq i,j\leq N$.
C... | https://mathoverflow.net/users/42644 | The largest Wasserstein distance to uniform distribution among all probability distributions with uniform marginals | I think I have an answer for the case p = 1, K = 2. I write "I think" because my computation does not coincide with the example values for $N=4$ posted earlier by OP in a comment, but I really cannot find any error in my proof, so I wanted to share it.
As already mentioned, we only need to consider permutation measur... | 2 | https://mathoverflow.net/users/106046 | 264477 | 118,882 |
https://mathoverflow.net/questions/264377 | 1 | Let $A$ be a $C^\*$-algebra and let $\phi:A\to B(H)$ be a completely positive map. The Stinespring representation theorem constructs a representation of $A$ on a Hilbert space $K$, which is constructed as follows. Define $K\_0=A\otimes H$ with the inner product
$$\langle a\otimes v, b\otimes w\rangle\_{\phi}= \langle \... | https://mathoverflow.net/users/104535 | Bounded operators on the Stinespring representation space | Rather than use the proof of Stinespring, I prefer to work with the uniqueness part of the result. Given $\phi$ we can find a Hilbert space $K$, a linear map $V:H\rightarrow K$ and $\pi:A\rightarrow B(K)$ a $\*$-representation with $\phi(a) = V^\*\pi(a)V$. Under the assumption that $\{ \pi(a)V\xi : a\in A, \xi\in H \}$... | 3 | https://mathoverflow.net/users/406 | 264479 | 118,883 |
https://mathoverflow.net/questions/264474 | 1 | The following problem is related to work on topological dynamics, but I feel like the question is interesting on its own. I think the answer to the question below is likely to be well-known and I hope someone here will be able to help.
Here goes: a theorem of Lyapunov implies that, given atomless Borel probability me... | https://mathoverflow.net/users/8923 | An infinite-dimensional counterexample to a theorem of Lyapunov? | Sure: Let $(q\_i)\_{i\in\mathbb N}$ be an enumeration of the rationals in $(0,1)$. Let $\mu\_i$ be the product measure on $\{0,1\}^{\mathbb N}$ such that each coordinate is 1 with probability $q\_i$ and 0 with probability $1-q\_i$. Now let $C$ be any clopen set. It is a union of finitely many cylinder sets: that is - t... | 1 | https://mathoverflow.net/users/11054 | 264482 | 118,884 |
https://mathoverflow.net/questions/264448 | 5 | Let $M$ be a smooth manifold (with boundary). Suppose I have a smooth vector field $T$ defined on the complement of a compact subset $K$ of $M$ and I wish to extend $T$ to the whole of $M$. What are the obstructions to doing so? Does $M$ need to be compact? Does $K$ need to be a submanifold?
I have a feeling that an ... | https://mathoverflow.net/users/104213 | Extensions of local vector fields to whole manifold | The sheaf of smooth functions on a manifold is [fine](https://ncatlab.org/nlab/show/fine+sheaf) and hence [soft](https://ncatlab.org/nlab/show/soft+sheaf), so we can extend sections on closed subsets to global sections. However, it is not generally [flabby (flasque)](https://ncatlab.org/nlab/show/flabby+sheaf): local s... | 7 | https://mathoverflow.net/users/56938 | 264484 | 118,885 |
https://mathoverflow.net/questions/261860 | 8 | Consider a function $f:\mathbb{CP}^1\times\mathbb{CP}^1\to \mathbb{CP}^1 $ defined by $f([x\_1,x\_2],[y\_1,y\_2])=[x\_1y\_1,x\_2y\_2]$. This function is well defined except at $([0,1],[1,0])$ or vice versa (in therms of the Riemann sphere $\mathbb{C}\_\infty$ we do not have a well defined zero times infinity). Now cons... | https://mathoverflow.net/users/29625 | Higher dimensional residues in complex analysis | While it's hard to guess exactly what you are after, I believe such residues were first studied by Poincaré ([Sur les résidus des intégrales doubles](http://projecteuclid.org/euclid.acta/1485888747), *Acta Math.* **9** (1887) 331–380), with your sought invariance encoded in the statement that "residue" takes a *closed ... | 6 | https://mathoverflow.net/users/19276 | 264490 | 118,886 |
https://mathoverflow.net/questions/264491 | 5 | Let $\|\cdot\|$ be the spectral norm, i.e., largest singular value. The condition number of an invertible complex matrix $X$ is defined as $\kappa(X):=\|X\|\|X^{-1}\|$.
I am able to prove
**Proposition** Let $A, B$ be $n\times n$ positive definite matrices. If $X$ is an $n\times n$ invertible matrix such that $AX... | https://mathoverflow.net/users/54458 | A matrix norm inequality II | No the norm of the left side can be very large.
For example $\left\| X^{-1}AXB \right\| $ is an unbounded function in $(x,y)$ where $A,X,B$ are the following matrices:
Put $A=B= \begin{pmatrix} 1&0\\0&2 \end{pmatrix}$ and $X^{-1}=\begin{pmatrix} -x^3&y\\y&x \end{pmatrix}$
where $(x,y) \in \mathbb{R}^{2} \setminus... | 7 | https://mathoverflow.net/users/36688 | 264493 | 118,887 |
https://mathoverflow.net/questions/264495 | 7 | For any cardinal $\kappa$ set $$\log(\kappa) = \min\{\mu\in \kappa\cup\{\kappa\}: 2^\mu \geq \kappa\}.$$
Clearly $\log(\omega) = \omega$ and in $\textsf{GCH}$ we have $\log(\aleph\_\omega) = \aleph\_\omega$.
Is it consistent that $\log(\kappa)<\kappa$ for all uncountable cardinals $\kappa$?
| https://mathoverflow.net/users/8628 | Fixed points of cardinal logarithm | No. Because strong limits cardinals exist in $\sf ZFC$. These are the $\beth\_\alpha$ for a non-successor $\alpha$.
| 12 | https://mathoverflow.net/users/7206 | 264496 | 118,888 |
https://mathoverflow.net/questions/264497 | 2 | Let $p$ be a prime number and $n$ a positive integer. I want to know what is the (right) socle of the group ring $A=\mathbb Z\_{(p)}C\_n$, where $\mathbb Z\_{(p)}$ is the localization of integers at the prime ideal $(p)$, and $C\_n$ is the cyclic group of order $n$. Thanks for any help!
| https://mathoverflow.net/users/48889 | Right socle of a group ring | It's zero. If $M$ is any non-zero right ideal, then $pM$ is a strictly smaller right ideal, so there are no minimal right ideals.
| 4 | https://mathoverflow.net/users/22989 | 264499 | 118,889 |
https://mathoverflow.net/questions/263403 | 2 | Is there a Hausdorff topological group $(G,\cal T)$ such that $G$ is a solvable group with a cardinality strictly greater than $\frak c$ and such that there is not any nontrivial (not necessarily Hausdorff) group topology $\cal S$ on $G$ with $\cal S\subsetneqq T$?
There not such topological group if "solvable" is re... | https://mathoverflow.net/users/47958 | An atomic solvable Hausdorff topological group with a cardinality greater than that of real line | **Theorem** (suggested by I.V.Protasov). Every solvable Hausdorff topological group $G$ is topologically solvable in the sense that $G$ contains an increasing sequence of *closed* subgroups $\{1\}=G\_0\subset G\_1\subset\dots\subset G\_n=G$ such that for every $i\le n$ the subgroup $G\_{i-1}$ is normal in $G\_i$ and th... | 2 | https://mathoverflow.net/users/61536 | 264508 | 118,892 |
https://mathoverflow.net/questions/264525 | 5 | Consider a discrete-time Markov chain on $n$ states $S = \{1,2,\ldots,n\}$, with transition dynamics
$$ \mathbf{x}^{(t+1)} = \mathbf{P} \mathbf{x}^{(t)},$$
where $\mathbf{P}$ is the *transition matrix*, a stochastic matrix (nonnegative entries and unit column sums). We are interested in the possibility of decomposi... | https://mathoverflow.net/users/78539 | Factorization of a Markov chain as the product of smaller chains | In dynamical systems, there is a concept of "almost-invariance", which generalizes invariance of a set, under the action of dynamics. The analogy is roughly the following:
If you create a markov chain on $S$ from the given dynamics $F$, then if the chain is reducible (into $A$ and $B$), it just implies that sets $A$ ... | 3 | https://mathoverflow.net/users/30684 | 264527 | 118,895 |
https://mathoverflow.net/questions/264521 | 13 | Recently, working in some calculations I needed to use the Prokhorov's theorem
about compactness for probability measures. However, a friend warned me that
I had not the hypotesis of separability required by the theorem.
After some searching, over the books which I have reach, this is the version of the theorem that... | https://mathoverflow.net/users/98969 | Prokhorov's theorem in non separable metric spaces | It is correct, see Theorem 8.6.7 in volume 2 of Bogachev's "Measure Theory" monograph. See also his Theorem 8.6.8 for a version of the second statement which covers the case of a non-separable space.
| 13 | https://mathoverflow.net/users/38566 | 264530 | 118,897 |
https://mathoverflow.net/questions/264546 | 7 | I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations like Schrödinger and others.
| https://mathoverflow.net/users/106086 | Reference for Feynman-Kac | Please see p.282 of the following ref, it does not call it Feynman-Kac Formula but the whole section 5 is discussing it. It formalized the diffusion using a tensor field over a manifold which is the most general and in-depth treatment from stochastic equation perspective that I know of.
>
> Ikeda, Nobuyuki, and Shi... | 5 | https://mathoverflow.net/users/25437 | 264562 | 118,909 |
https://mathoverflow.net/questions/264555 | 8 | Assume $s, j \in\mathbb{N}$. Define the set
$$\mathcal{A}\_{j,s}:=\{(n\_1,n\_2,\dots,n\_j)\in\mathbb{Z}\_{\geq0}^j\vert \,
n\_1+2n\_2+\cdots+jn\_j=j, \, n\_1+n\_2+\cdots+n\_j=s\}.$$
>
> **Question.** Is there a combinatorial argument for this?
> $$\sum\_{s=0}^j(-1)^{j-s}\sum\_{\mathcal{A}\_{j,s}}\binom{s}{n\_1,\d... | https://mathoverflow.net/users/66131 | In search of a combinatorial reasoning for a vanishing sum | For fixed $s,j>0$, $\sum\_{\mathcal{A}\_{j,s}}\binom{s}{n\_1,\dots,n\_j}$ enumerates the compositions of $j$ into $s$ positive parts by ordering the corresponding partitions. That is, we have
$$\sum\_{\mathcal{A}\_{j,s}}\binom{s}{n\_1,\dots,n\_j}=\binom{j-1}{s-1}$$
and the identity in question follows trivially.
| 16 | https://mathoverflow.net/users/7076 | 264566 | 118,911 |
https://mathoverflow.net/questions/264518 | 7 | The following theorem appears without proof in :
Helmke, Uwe, and John B. Moore. Optimization and dynamical systems. Springer Science & Business Media, 2012.
Let $A$ be a symmetric $n\times n$ real matrix. Define the Stiefel manifold as
$St(k,n)=\{X\in \mathbb{R}^{n\times k}|X^TX=I\}$.
Then, we consider the follo... | https://mathoverflow.net/users/106076 | Generalized Rayleigh-quotient gradient flow on Grassmannian | On the Stiefel manifold, consider the function $f(X)=\operatorname{tr}(X^TAX)$. It evolves under the flow of the given vector field as
$$\frac d{dt}f=2\operatorname{tr}(X^TA(1-XX^T)AX)\;.$$
Because $1-XX^T$ describes the projection onto the orthogonal complement of the span $V\_X$ of the columns of $X$, we have
\begin{... | 5 | https://mathoverflow.net/users/70808 | 264577 | 118,915 |
https://mathoverflow.net/questions/264554 | 7 | I wonder how to characterize the characters of a (say, finite) group $G$ as special class functions, in particular for the case $G=S\_n$ (symmetric group). The answer to this is presumably well known to people working in group theory, so it is more of a reference request.
Any character $\chi$ is "positive" in the sen... | https://mathoverflow.net/users/23753 | Characterization of group characters | The best answer I can think of is **[Brauer's characterization of (generalized) characters](https://en.wikipedia.org/wiki/Brauer%27s_theorem_on_induced_characters):** Recall that a *generalized character* is a difference of two characters. A *Brauer elementary group* is a group that is the direct product of a $p$-group... | 10 | https://mathoverflow.net/users/10266 | 264607 | 118,924 |
https://mathoverflow.net/questions/264592 | 2 | Let $A = (a\_{ij})\_{i,j=1,\dots,n}$ be a matrix such that
* $a\_{ij} \ge 0$ for all $i,j = 1, \dots, n,$ and
* $A$ is positive definite.
Let $I$ be the identity matrix, and $\pmb{1}$ the vector containing only ones.
Suppose that the solution $x = (x\_i)\_{i=1,\dots,n}$ to the system of linear equations
$$
(A+I)x=... | https://mathoverflow.net/users/75070 | Perturbation of linear system of equations: Is the solution still non-negative? | I found a proof under the additional assumption that $(A+f(t)I)^{-1}$ is a [$Z$-matrix](https://en.wikipedia.org/wiki/Z-matrix_(mathematics)).
It works for arbitary vectors $x$, not just the vector containing only ones.
Let me write $M \ge 0$ ($M > 0$) if every entry of the matrix $M$ is non-negative (positive).
Pi... | 1 | https://mathoverflow.net/users/75070 | 264613 | 118,927 |
https://mathoverflow.net/questions/264631 | 12 | Let $K$ be a number field, and let $P(t,X)$ be a monic polynomial in $X$ with coefficients in $K(t)$.
I would like to understand the set $T$ consisting of those $t\_0 \in K$ such that the polynomial $P(t\_0,X)$ is (totally) split over $K$.
**Q1.** Is there an algorithm to decide whether $T$ is empty, finite or infi... | https://mathoverflow.net/users/6506 | Splitting of polynomials over rational function fields | Your question 1 is open and is equivalent to the problem of determining the rank of an elliptic curve over a number field. (In most "practical cases" it should be solvable.) In particular, we don't have an algorithm to determine if an elliptic curve $E/K$ has positive rank, if $E : y^2 = x^{3} + Ax + B$, this is asking... | 15 | https://mathoverflow.net/users/48142 | 264633 | 118,934 |
https://mathoverflow.net/questions/264637 | 7 | Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ such that $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the absolute Galois group of $\mathrm{F}$, denoted by $G\_F$, and its **open** subgroup $G\_{F[\sqrt[p]{a}]}$ of index $p$ (w... | https://mathoverflow.net/users/106123 | Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism? | The corestriction map on cohomology is indeed the norm in degree zero (see [Tate's notes on Galois cohomology](http://wstein.org/edu/2010/582e/refs/tate-galois_cohomology.pdf) for example). By a dimension shifting argument, it then easily follows that the corestriction in degree one also corresponds to taking norms (un... | 7 | https://mathoverflow.net/users/17907 | 264641 | 118,935 |
https://mathoverflow.net/questions/264646 | 8 | Ramanujan graphs are the best spectral expanders: $\lambda\_2 \le 2\sqrt{d-1}$. I'm looking for some intuition for this value $2\sqrt{d-1}$.
Friedman showed that every random $d$-regular graph achieves $\lambda\_2 = 2\sqrt{d-1}+\epsilon$ and Alon-Boppana shows that every sufficiently large $d$-regular graph must hav... | https://mathoverflow.net/users/70060 | Where does $2\sqrt{d-1}$ come from in Ramanujan graphs? | Yes, see [this paper by Ram Murty.](http://www.mast.queensu.ca/~murty/ramanujan.pdf) The basic point is that the sum of squares of the eigenvalues is the trace of the square of the adjacency matrix, which is equal to $d n.$
| 8 | https://mathoverflow.net/users/11142 | 264650 | 118,937 |
https://mathoverflow.net/questions/264652 | 13 | In *Introduction to Higher-Order Categorical Logic*, Lambek & Scott remark that Brouwer's Theorem (all functions $\mathbb{R}\to\mathbb{R}$ are continuous) holds in the free topos $\mathcal{T}$.
***EDIT***: L&S do not say that it holds in the free topos, they are incorrectly quoted as saying this in the [nLab](https:/... | https://mathoverflow.net/users/51336 | Brouwer's Theorem in the free topos? | To summarize, the Lambek and Scott book actually says that functions on the reals in the free topos *represent continuous functions*. The nLab previously made the stronger claim that Brouwer's Theorem holds in the free topos (which is not the case for the reasons I described), but I have edited the page appropriately.
... | 10 | https://mathoverflow.net/users/51336 | 264656 | 118,938 |
https://mathoverflow.net/questions/264606 | 6 | Given $A=BC$ where $A\in\mathbb{R}^{m\times n}$ and for some $B\in\mathbb{R}^{m\times k}, C\in\mathbb{R}^{k\times n}$. We assume that $k>=\min(m,n)$ so that this decomposition always exists for any matrix $A\in\mathbb{R}^{m\times n}.$
Can we prove that any perturbation $\bar{A}$ of $A$ can be represented as the produ... | https://mathoverflow.net/users/90183 | Can a perturbation of a matrix product always be represented as product of perturbations of its factor matrices? | The condition you want is exactly that the matrix multiplication map be locally open at the pair $(B,C)$. This is the topic of the recent paper [Where is matrix multiplication locally open?](http://www.sciencedirect.com/science/article/pii/S0024379516306036) by Behrends. The paper contains a complete characterization i... | 9 | https://mathoverflow.net/users/5963 | 264658 | 118,940 |
https://mathoverflow.net/questions/264489 | 12 | Let $A\subset B$ be an integral extension of commutative unital rings.
Let $\mathfrak{p}\_0\subset\mathfrak{p}\_1\subset\mathfrak{p}\_2$ be a saturated chain of primes in $A$ of length $2$.
Suppose $\mathfrak{q}\_0,\mathfrak{q}\_2$ lie over $\mathfrak{p}\_0,\mathfrak{p}\_2$, and $\mathfrak{q}\_0\subset\mathfrak{q}\... | https://mathoverflow.net/users/12419 | How bad does a ring have to be for a failure of "going-in-between"? | This is a partial answer, giving a two dimensional Noetherian counterexample. It starts from your observation that one has to consider the case where $A/\mathfrak{p}\_0$ is not integrally closed. Geometrically the construction is as follows: take a normal affine surface $Y$, a curve $C$ on $Y$, a point $P\_1 \in C$ and... | 2 | https://mathoverflow.net/users/1508 | 264663 | 118,943 |
https://mathoverflow.net/questions/264549 | 5 | I am trying to understand "de Cataldo, Migliorini. The perverse filtration and the Lefschetz hyperplane theorem. *Annals of Mathematics*, 171(2010), 2089-2113." My question is about one detail in the paper. Here is my reproduction of the situation.
Let $\Sigma$ be a stratification of $\mathbb{P}^N$ adapted to a boun... | https://mathoverflow.net/users/104949 | Base change and the octahedron axiom | I'm not sure what's going on, but here's a guess:
Firstly, the base change map goes the other way: $i^\*J\_\*\mathcal{F}\rightarrow J\_\*i^\*\mathcal{F}$. Say this is an isomorphism for the Verdier dual of $K$ (they end up saying that the condition they impose makes the base change morphism an isomorphism independent... | 2 | https://mathoverflow.net/users/21637 | 264676 | 118,952 |
https://mathoverflow.net/questions/264539 | 12 | We are interested in the following ´relative´ version of residual finiteness for fundamental groups of surfaces. Similar discussions where given in this question: [injectivity radius of hyperbolic surface](https://mathoverflow.net/questions/126352/injectivity-radius-of-hyperbolic-surface) (in particular with [this answ... | https://mathoverflow.net/users/5753 | Finite covers of hyperbolic surfaces and the `second systole´ | [17/4/17: edited to correct proof.]
This is true. First, we need a lemma which builds a related cover. Throughout, $\alpha$ is a simple closed geodesic of length $\ell$, and $\beta\_1,\ldots,\beta\_n$ are the finitely many (not necessarily simple) closed geodesics on $S$ of length at most $K$ which are not equal to $... | 5 | https://mathoverflow.net/users/1463 | 264684 | 118,955 |
https://mathoverflow.net/questions/264651 | 3 | Do there exist three non-constant entire functions $f,g,h:\mathbb{C}\to\mathbb{C}$ such that for any $z\in\mathbb{C}$, at least two of $f(z)$, $g(z)$ and $h(z)$ belong to the closed unit disk?
| https://mathoverflow.net/users/51203 | A generalization of Liouvilles Theorem for entire functions | To elaborate on my comment, choose three pairwise disjoint Jofrdan $U\_1$, $U\_2$, $U\_3$ whose boundary passes through infinity, and let $\gamma\_i\subset U\_i$ be a curve to infinity for each $i$. Fix $\newcommand{\eps}{\varepsilon}\eps>0$ (e.g. $\eps=1$).
By Arakelyan's approximation theorem (see, e.g. [this expo... | 5 | https://mathoverflow.net/users/3651 | 264687 | 118,957 |
https://mathoverflow.net/questions/261105 | 16 | Hilbert's lecture at the ICM in Paris in 1900 presented 10 of the famous 23 open problems. It is well known that the idea of the lecture came from Hermann Minkowski. Hilbert was at Gottingen at the time where he was hired through untiring efforts of Felix Klein. As detailed by historian David Rowe and others, both Hilb... | https://mathoverflow.net/users/28128 | Did Hilbert discuss his 23 problems with Felix Klein? | As [Constance Reid](https://books.google.de/books?id=RtnvBgAAQBAJ&pg=PA88&dq=scarcely+%22devoted+himself+to%22+encyclopedia+%22schools+commission%22+%22bring+together%22+%22technical+and+mathematical+training%22+%22a+number+of+projects%22scarcely&hl=de&sa=X&ved=0ahUKEwi-idjXs9jSAhXlJsAKHWqDAW0Q6AEIGjAA#v=onepage&q=scar... | 2 | https://mathoverflow.net/users/96921 | 264690 | 118,959 |
https://mathoverflow.net/questions/264693 | 1 | In [the paper](http://www.ams.org/journals/proc/1994-120-03/S0002-9939-1994-1169886-7/S0002-9939-1994-1169886-7.pdf), a set $L$ associated to an element $w$ in a Coxeter group $W$ is defined as follows. Let $w=s\_{i\_1} \cdots s\_{i\_m}$ be a reduced expression. Define $L=\{\beta\_1, \ldots, \beta\_m\}$, where
\begin{a... | https://mathoverflow.net/users/11877 | Positive roots and elements in a Coxeter group. | This is a standard observation in Lie theory: if you let the Weyl group act on the root system on the left, $\beta\_k$ is the unique root such that $s\_{i\_k}\cdots s\_{i\_1}\beta\_k$ is negative and $s\_{i\_{k-1}}\cdots s\_{i\_1}\beta\_k$ is positive. That is, for a positive root $\gamma$, the number of times it appea... | 5 | https://mathoverflow.net/users/66 | 264695 | 118,961 |
https://mathoverflow.net/questions/264678 | 8 | For the moment we work over the complex numbers.
Suppose that $G = \mathrm{SL}(V)$, or $G = \mathrm{Sp}(V)$, or $G = \mathrm{SO}(V)$.
Weyl gave explicit constructions of irreducible representations of a given heighest weight in terms of the faithful representation $V$. This is done using Schur functors and possibly int... | https://mathoverflow.net/users/105976 | Generalisations of Weyl's construction of irreducible representations | The short answers to your questions are all No.
The reason is that Weyl's construction works when the centraliser algebra
of a tensor power of $V$ is a quotient of a Brauer algebra. This is similar
to Schur's construction of representations of $SL(n)$ which assumes that the centraliser algebra is a quotient of the gr... | 6 | https://mathoverflow.net/users/50658 | 264696 | 118,962 |
https://mathoverflow.net/questions/264698 | 10 | By using the LLL algorithm, I tried to find the best simultaneous Diophantine approximation of the three numbers $\sqrt{2} $ and $ \sqrt{2 \pm \sqrt{3}} $. I was expecting that to get a precision of $\epsilon$, the common denominator $q$ should be on the order of $\epsilon^{-3}$, because I have three irrational numbers... | https://mathoverflow.net/users/91370 | Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$ | $$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$
| 27 | https://mathoverflow.net/users/4312 | 264699 | 118,963 |
https://mathoverflow.net/questions/264692 | 3 | Let $X$ be a smooth complex variety. Is it always possible to find an embedding $\varphi: X\to \mathbb CP^n$ for some $n$, such that the blow up of $\mathbb CP^n$ at $\varphi(X)$ is a Fano variety?
Let us call the above class of varieties $F$-embeddable. How large is this class? How large is its complement?
| https://mathoverflow.net/users/13441 | Fano blow ups of $\mathbb CP^n$ | **Note.** My comment above was wrong; I had the wrong denominators. When you correct the denominators, the formula gives an asymptotic result.
In my comment I wrote the wrong formula for the denominator of that fraction. The correct statement is that the blowing up of $\mathbb{P}^n$ along a smooth subvariety $X$ of ... | 2 | https://mathoverflow.net/users/13265 | 264700 | 118,964 |
https://mathoverflow.net/questions/264701 | 8 |
>
> I am looking for some non-trivial examples of (smooth) 4-mflds $M,N$ such that $M$ and $N$ are STABLY diffeomorphic. I.e. $$M\sharp\_n (S^2\times S^2) \cong N \sharp\_r (S^2\times S^2)$$ for $r,n$ not necessarily the same
>
>
>
By non trivial I mean that $M$ and $N$ are not diffeomorphic or that $M \cong N\s... | https://mathoverflow.net/users/93538 | Non-trivial examples of Stably diffeomorphic 4-manifolds | The main examples of this go back to work of Moishezon and Mandelbaum in the late 1970s. For instance, if a simply-connected elliptic surface E(2n+1) is homotopy equivalent to the connected sum of $4n+1$ copies of $\mathbb{C}P^2$ blown up $20n +9$ times, but those manifolds are not diffeomorphic for $n>1$. This is a th... | 13 | https://mathoverflow.net/users/3460 | 264702 | 118,965 |
https://mathoverflow.net/questions/264704 | 7 | Suppose I'm given a finite set of possibly unbounded commuting self-adjoint operators $T\_i : \mathfrak H \supset \mathscr D(T\_i)\to \mathfrak H, i = 1 , \dots , N$ on a Hilbert space (in the sense of commuting resolvents $\forall i, j: \exists z\_i \in \rho (T\_i) , z\_j \in \rho (T\_j):[R\_{T\_i} (z\_i ) , R\_{T\_j}... | https://mathoverflow.net/users/106160 | Simultaneous diagonalization of self-adjoint operators on Hilbert space | Apply the SNAG (Stone-Naimark-Godement-Ambrose) theorem to the unitary group generated by these operators.
| 6 | https://mathoverflow.net/users/13650 | 264706 | 118,966 |
https://mathoverflow.net/questions/264655 | 6 | By a $d$-permutation hypergraph, I mean, for some fixed integer $k$, a $d$-uniform hypergraph on $[dk]$ with $k$ disjoint edges such that every edge has exactly one vertex from each of $\{1,\ldots,k\}$, $\{k+1,\ldots,2k\}$, $\ldots$, and $\{(d-1)k+1,\ldots,dk\}$. (So each vertex is contained in exactly one edge.)
By ... | https://mathoverflow.net/users/100631 | Is an $O(n^{d-1})$ bound known for the maximum number of edges in an ordered $n$-vertex hypergraph avoiding a fixed $d$-permutation hypergraph? | Yes, the $O(n^{d-1})$ bound holds in this case as well, with a proof that is similar to the proof of the $d$-uniform case.
I would guess that this also follows from the proof of Klazar and Marcus with some easy modification, but I don't know it as well as our own proof, which you can find in Section 3 here:
<https://ar... | 3 | https://mathoverflow.net/users/955 | 264712 | 118,968 |
https://mathoverflow.net/questions/264715 | 6 | I'm trying to pin down a notion of inductive definability in category-theoretic terms.
The sorts of inductively defined sets (and classes) I'm most interested in are those that admit of induction and recursion. So far, I've noticed that ([weakly?](https://mathoverflow.net/questions/127540/does-induction-for-a-functor... | https://mathoverflow.net/users/90691 | Inductive Definitions in Category Theory | It is indeed the case that initial algebras for functors are the category-theoretic manifestation of inductive definitions. There's a whole industry surrounding this idea.
You are more specifically asking about (pieces of) the cumulative hierarchy. On that topic you should look at [algebraic set theory](http://www.ph... | 6 | https://mathoverflow.net/users/1176 | 264732 | 118,976 |
https://mathoverflow.net/questions/264340 | 4 | Let $D$ be a bounded domain in $\mathbb{C}^n$, and let $\gamma$ be a biholomorphism of $D$ such that for all $\epsilon>0$ there is a point $z\in D$ such that the Kobayashi distance from $z$ to $\gamma(z)$ is at most $\epsilon$ (and $\gamma$ is not the identity).
Is it ever possible to have a non-constant holomorphic... | https://mathoverflow.net/users/101909 | When do quotients of bounded domains contain closed Riemann surfaces? | **Proposition.** If $D$ is a bounded connected domain in ${\mathbb C}^n$ and $\Gamma$ is a nilpotent group of biholomorphic transformations of $D$ acting freely and properly discontinuously on $D$, then every map from a compact Riemann surface to $D/\Gamma$ is constant.
*Proof.* I will use the following theorem, whi... | 4 | https://mathoverflow.net/users/21684 | 264747 | 118,982 |
https://mathoverflow.net/questions/264740 | 6 | On Hilbert spaces, the following is true:
Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.
The obvious proof I see to show this uses explicitly the Hilbert space structure which is why I would like to ask:
Is the same result true for operators on Banach spaces?
| https://mathoverflow.net/users/105722 | Non-empty resolvent set, then operator closed? | What I would consider the obvious proof uses only the Banach space structure.
If $\lambda$ is in the resolvent set, the graph $G(T)$ of $T$ maps in an obvious way to the graph of $(T-\lambda I)^{-1}$:
$G(T) = f^{-1}(G((T-\lambda I)^{-1}))$ where
$$f:\;(x, y) \mapsto (y-\lambda x, x)$$
Since $f$ is continuous from $X... | 6 | https://mathoverflow.net/users/13650 | 264749 | 118,984 |
https://mathoverflow.net/questions/264744 | 3 | Suppose $A$, $B$ are finite sets of positive integers.
Let $$\mathcal{S}\_n = \{C \subset [1,n] \, : \, A+C = B+C \}, $$ and denote $a\_n = |\mathcal{S}\_n|$.
Note that for any $X \in \mathcal{S}\_n$ and $Y \in \mathcal{S}\_m$, the set $X \cup (Y+n) \subset [1,m+n]$ is in $\mathcal{S}\_{m+n}$, hence $a\_m \cdot a\... | https://mathoverflow.net/users/83085 | Limit measuring failure of sum-set cancellability | $m(A,B)$ may be computed as follows. By translation we may assume that $A,B \subset [1,k]$ for some natural number $k$.
Let $V = 2^{[1,k]}$ denote the power set of $[1,k]$. Observe that a subset $C$ of $[1,n]$ can be identified with a sequence $v\_1, v\_2, \dots, v\_{n+k+1}$ in $V$ by setting $v\_i := \{ j \in [1,k]:... | 6 | https://mathoverflow.net/users/766 | 264750 | 118,985 |
https://mathoverflow.net/questions/264753 | 5 | Introduce the $2^{n-1}\times 2^{n-1}$ matrix $B\_n$ recursively as follows: $B\_1(b\_1)=\begin{pmatrix} b\_1\end{pmatrix}$ and
$$B\_n(b\_1,\dots,b\_n)=\begin{pmatrix} B\_{n-1}(b\_1,\dots,b\_{n-1})& b\_nJ\_{n-1}\\ b\_nJ\_{n-1}&B\_{n-1}(b\_1,\dots,b\_{n-1})
\end{pmatrix}.$$
Here $J\_n$ is a $2^{n-1}\times 2^{n-1}$ matrix... | https://mathoverflow.net/users/66131 | dyadically recursive matrices: Part I | For a vector $v$ of length $2^{n-2}$, denote by $v'$ the "mirrored" vector $J\_{n-1}v$.
If $v$ is an eigenvector of $B\_{n-1}$ for the eigenvalue $\lambda$, then $B\_n\binom v{\pm v'}=(\lambda\pm b\_n)\binom v{\pm v'}$. So this gives you all the eigenvalues of $B\_n$ as the sums $ b\_1\pm\cdots\pm b\_n$ (note that $ b... | 5 | https://mathoverflow.net/users/29783 | 264769 | 118,990 |
https://mathoverflow.net/questions/264764 | 3 | It is well known that a (smooth complete) fan $\Delta$ corresponds to a (smooth proper) toric variety $X= X\_\Delta$.
My question is whether there is a relationship between the number of maximal cones in $\Delta$ and a geometric invariant of $X$.
(as the number of 1-dimensional cones is the number of torus invariant ... | https://mathoverflow.net/users/106190 | Relation between the number of maximal cones in a fan and the geometry of corresponding toric variety | Yes, the number of maximal cones is the same as the topological Euler characteristic.
This is the result of the natural generalization of the thing you already pointed out: the number of codimension $r$ irreducible $T$-invariant subvarieties is the number of $r$-dimensional cones in $\Delta$.
So more precisely, max... | 4 | https://mathoverflow.net/users/21637 | 264771 | 118,992 |
https://mathoverflow.net/questions/264751 | 4 | A follow up on [an earlier MO question](https://mathoverflow.net/questions/258386/kasteleyns-formula-for-domino-tilings-generalized).
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod\_{j=1}^n\prod\_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\right)$ is known to be ... | https://mathoverflow.net/users/66131 | Asymptotics for 'generalized" Kasteleyn's formula | Take the logarithm and note that for large $n$ we may replace the sum by an integral:
$$\lim\_{n\rightarrow\infty}\frac{1}{n^r}\log K\_r(n)=2\log 2+C\_r$$
with an $r$-dependent numerical coefficient $C\_r$ given by an integral over the $r$-dimensional unit cube:
$$C\_r=\int\_0^1 dx\_1 \cdots\int\_0^1 dx\_r \log\left[... | 5 | https://mathoverflow.net/users/11260 | 264773 | 118,993 |
https://mathoverflow.net/questions/261349 | 9 | Given $\large q=e^{2\pi i \tau}$. Define,
$$\alpha(\tau) = \sqrt2\,q^{1/8}\prod\_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$
$$\beta(\tau) = q^{1/5}\prod\_{n=1}^\infty\frac{ (1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})}$$
$$\gamma(\tau) = q^{1/4}\prod\_{n=1}^\infty\frac{ (1-q^{6n-1})(... | https://mathoverflow.net/users/12905 | Some nice functional equations for $q$-continued fractions | Throughout this answer, I'll be referencing Kubert-Lang, "Modular Units" Chapter 2, sections 1 and 2, and Chapter 3 section 4.
Each of your functions is a ratio of the Siegel functions of the form
$$A\_N(\tau)=\left(\frac{g\_{\frac1N,0}}{g\_{\frac aN,0}}(N\tau)\right)^m.$$
If $r=(r\_1,r\_2)\in \mathbb Q^2$, $B\_2(... | 4 | https://mathoverflow.net/users/61910 | 264778 | 118,995 |
https://mathoverflow.net/questions/264779 | 0 | Consider the following number field $K = \frac{\mathbb{Q}[x]}{(x^{(p^2-1)/2}-p)}$ where $p$ is a prime. What is the factorization of the prime ideal $p$ in the ring of integers of $K$?
| https://mathoverflow.net/users/11392 | Ramification of a prime in a number field | If you are interested only in the ramification above a single prime number $p$, it is best to forget the field $\mathbf{Q}$ and work instead over the field $\mathbf{Q}\_p$. For every $n>0$, the polynomial $f=x^n-p$ is an Eisenstein polynomial over $\mathbf{Q}\_p$, so it is irreducible, and adjoining a root of $f$ gives... | 4 | https://mathoverflow.net/users/2821 | 264782 | 118,997 |
https://mathoverflow.net/questions/264788 | 6 | Suppose $E \to B$ is a symplectic vector bundle, i.e. it possesses a fibrewise linear symplectic form $\omega\_F$. Further, suppose $\omega\_B$ is a symplectic form on $B$.
Question: is there a symplectic form $\omega$ on $E$, such that (a) on any fibre of $E$, it restricts to $\omega\_F$, and (b) on the zero sectio... | https://mathoverflow.net/users/15197 | A symplectic form on a symplectic vector bundle | Such a symplectic form does not exist in general. In my paper (theorem 8.2). Gerbes, 2-gerbes and symplectic fibrations, I have shown that the obstruction of the existence of such a symplectic form on $E$ can geometrically be represented by a $2$-gerbes. My paper is available at
<http://lanl.arxiv.org/pdf/math/05042... | 9 | https://mathoverflow.net/users/80891 | 264792 | 119,003 |
https://mathoverflow.net/questions/264801 | 4 | Assume $X$ is a smooth projective variety over $\mathbb{C}$ of dimension $n$, here $n\geq 3$, with a reduced normal crossing divisor $D\subset X$, such that $D=\sum\limits\_{i=1}^r D\_i$ where the $D\_i$ are the irreducible and nonsingular components of $D$ and $sing(D)=\bigcup\limits\_{i\neq j}(D\_i\cap D\_j)$.
* Ca... | https://mathoverflow.net/users/70593 | Existence of regular conic bundles with a given discriminant divisor | No. There is a reciprocity law at play here which places additional restrictions on the discriminant locus.
The precise relations are a bit complicated and are easier to phrase in terms of the corresponding Brauer group element. See e.g. Theorem 1 from Section 3 of "Artin, Mumford - Some elementary examples of unirat... | 5 | https://mathoverflow.net/users/5101 | 264806 | 119,007 |
https://mathoverflow.net/questions/264803 | -1 | Is the line graph of a bipartite $2K\_2$-free graph a $2K\_2$-free graph?
Definition of line graph :click <https://fr.wikipedia.org/wiki/Line_graph>
| https://mathoverflow.net/users/106205 | the line graph of bipartite 2k2-free graph | No, it is not the case.
Complete bipartite graphs $K\_{n,m}$ are $2K\_2$-free. But their line graphs $L(K\_{n,m})$ are not, for $m$ and $n$ big enough. Specifically, the vertices of $L(K\_{n,n})$ may be indexed by pairs $(i,j)$, with $1\leq i,j\leq n$; two vertices $(i,j)$ and $(p,q)$ are adjacent if they have a common... | 1 | https://mathoverflow.net/users/11100 | 264816 | 119,009 |
https://mathoverflow.net/questions/264780 | 3 | For a prime power $p^a$ define $\gamma(p^a) := (p^a-1)(p^{a-1}-1)\cdots(p^{2}-1)(p-1)$. Moreover, for a natural number $n = \prod {p\_{i}}^{\alpha\_{i}}$ define $\gamma(n) := \prod \gamma({p\_{i}}^{\alpha\_{i}}) $ where $p\_1,\dots,p\_k$ are
distinct prime numbers and $\alpha \_{i}$ are natural numbers.
The question... | https://mathoverflow.net/users/58321 | primitive prime divisor of $2^{8n+4} - 1 $ | No. We have that $p = 709$ is a primitive prime divisor of $2^{708} - 1$. However, $\frac{2^{708} - 1}{2^{177} + 1}$ is a multiple of the prime $q = 5521693$ and therefore $q-1 | \gamma\left(\frac{2^{708} - 1}{2^{177}+1}\right)$. Since $q \equiv 1 \pmod{709}$, we have that $p | \gamma\left(\frac{2^{708} - 1}{2^{177}+1}... | 7 | https://mathoverflow.net/users/48142 | 264817 | 119,010 |
https://mathoverflow.net/questions/264805 | 38 | Is it true, that a path connecting two opposite points (i.e. such that the segment joining them passes through the centre of mass of the cube) on the surface of the $d$-dimensional unit cube (with $d>1$) is not shorter than $2$?
| https://mathoverflow.net/users/2158 | Shortest path connecting two opposite points on a cube | Consider the sphere with equator 4.
Divide it into *spherical cubes*, the central projections from an inscribed cube.
Note that the exponential map from tangent plane to the sphere is short.
Note also that if one maps a unit cube centered at the origin by the exponential map it will cover the *spherical cube*.
It is ... | 23 | https://mathoverflow.net/users/1441 | 264819 | 119,011 |
https://mathoverflow.net/questions/264772 | 2 | Let $G=(V,E)$ be a connected graph. Here $V$ is the set of all vertices of $G$, and $E$ is the set of all edges of $G$. Suppose that $G$ is locally finite, i.e., $\sharp\{y\sim x:y \in V \}$ is finite for any $x\in V$. We denote by $m(x)=\sharp\{y\sim x:y \in V \}$. We denote by $d(\cdot,\cdot):V\times V\to [0,\infty)$... | https://mathoverflow.net/users/84068 | Volume doubling implies that the degree is uniformly bounded above? | Yes. More precisely for every $x \in V$, $m(x) \leq (C-1)^2$.
This follows from the volume doubling inequality for $r=1$, which says that for all $x \in V$, $m(x) + \sum\_{y \sim x} m(y) \leq C m(x)$, or equivalently $\frac{1}{m(x)} \sum\_{y \sim x} m(y) \leq C-1$. In particular, there is $y \sim x$ such that $m(y) \... | 1 | https://mathoverflow.net/users/10265 | 264834 | 119,018 |
https://mathoverflow.net/questions/264716 | 2 | For $m > n$, I want to calculate the number of binary matrices with $m$ rows and $n$ columns for which two conditions hold:
* the rank of a matrix is $n$ (i.e. it is full-rank, as $m > n$);
* there are no rows with a single 1 - in other words, there are no rows of Hamming weight 1.
I need either closed formula or s... | https://mathoverflow.net/users/101533 | Number of full-rank binary matrices with no rows of weight 1 | First let us count the number of matrices $m\times n$ over $\mathbb F\_2$ of full rank (without the second restriction). Let $r\_i$ ($i=1,2,\dots,m$) be the $i$-th row of such a matrix, $V\_i=\langle r\_1,\dots,r\_i\rangle$ be the vector space spanned by the first $i$ rows, and $d\_i = \dim V\_i$. Then $|V\_i|=2^{d\_i}... | 3 | https://mathoverflow.net/users/7076 | 264835 | 119,019 |
https://mathoverflow.net/questions/264837 | 8 | Let $H$ be a handlebody with $\Sigma = \partial H$. Given an automorphism $f : \Sigma \to \Sigma$ we can glue to obtain a closed 3-manifold $M = H \cup\_f H$ and in fact all such 3-manifolds are obtained this way. Since only the isotopy class of $f$ is necessary in order to specify the homeomorphism type of the resulti... | https://mathoverflow.net/users/99414 | What does the matrix of a mapping class tell you about the 3-manifold? | In addition to the first homology group, it also determines the Seifert pairing on the torsion in the first homology group. What is more, in an appropriate sense this is all it determines. This is all contained in my paper "Symplectic Heegaard splittings and linked abelian groups" with Joan Birman and Dennis Johnson, w... | 10 | https://mathoverflow.net/users/317 | 264838 | 119,020 |
https://mathoverflow.net/questions/264754 | 7 | Find all the non-trivial integer solutions to the equation
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4.$$
| https://mathoverflow.net/users/37096 | Solution to a Diophantine equation | FWIW, using Michael Stoll' answer to [Estimating the size of solutions of a diophantine equation](https://mathoverflow.net/questions/227713/estimating-the-size-of-solutions-of-a-diophantine-equation), we use Magma to do:
```
F := EllipticCurve(x^3 + 109* x^2 + 224 * x);
IntegralPoints(F);
[ (-100 : -260 : 1), (-56 :... | 5 | https://mathoverflow.net/users/11142 | 264842 | 119,022 |
https://mathoverflow.net/questions/264832 | 11 | Two vectors from Leech lattice - as defined on wikipedia - have scalar product $\pm 32,\pm 16, \pm 8$ or $0$. Do there exist 24 vectors having scalar product 8 pairwise ? When we consider unit vectors in Leech lattice then scalar product is $\frac{1}{4}$ pairwise.
Motivation for this question is the fact that for 2b ... | https://mathoverflow.net/users/nan | 24 vectors in Leech lattice having scalar product $\frac{1}{4}$ pairwise | Yes, such a configuration exists, even with all inner products positive
(as the title of the question requires, even though the text allows
either sign).
We shall use the standard normalization of the Leech lattice $\Lambda$
that makes it unimodular, so the $196560$ minimal vectors $(v,v) = 4$,
not $32$. Thus we clai... | 16 | https://mathoverflow.net/users/14830 | 264866 | 119,030 |
https://mathoverflow.net/questions/258643 | 12 | EDIT: as pointed-out below, this has been posted on math.stackexchange. I'll leave it up to the community whether or not to delete this question, but I do think there is room for a more technical answer than the one posted on math.stackexchange.
---
A famous question related to [Hilbert's third problem](https://e... | https://mathoverflow.net/users/1231 | Intuition behind the Dehn Invariant | The Dehn functional of a polyhedron $P$ with edge lengths $\ell\_i$ and exterior dihedral angles $\theta\_i$ is $D(P) = \sum\_i \ell\_i \otimes\_{\mathbb Q} (\theta\_i\, \mathrm{mod}\, \mathbb{Q}\pi)$. It looks similar to the (twice) the total mean curvature $S(P) = \sum\_i \ell\_i \theta\_i$. Both of them satisfy the ... | 13 | https://mathoverflow.net/users/98590 | 264880 | 119,034 |
https://mathoverflow.net/questions/264668 | 5 | If ${\cal C}$ is a collection of subsets of a set $X$, we associate to ${\cal C}$ a graph $G\_{\cal C} = (V,E)$ where $V = {\cal C}$ and $$E = \big\{\{A,B\}: A\neq B\in {\cal C} \land A\cap B \neq \emptyset\big\}.$$
If $G$ is a simple, undirected graph, we define its *intersection number* $\iota(G)$ to be the smalles... | https://mathoverflow.net/users/8628 | Intersection number of infinite graphs | I think the answer is no, at least not without some additional assumptions. Suppose that $|\omega| < |\omega\_1| < \mathfrak c$ and let $G$ be the union of the complete graph on $\mathfrak c$ and an independent set of size $|\omega\_1|$. As you mention in the comments, the clique can be represented as an intersection g... | 3 | https://mathoverflow.net/users/25485 | 264887 | 119,036 |
https://mathoverflow.net/questions/264730 | 2 | Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C\_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t) + Z u(t) \varphi(t)$$
with $\varphi(0)=\varphi\_0 \in D(A)$ has a unique solution.
I would like to know if the foll... | https://mathoverflow.net/users/105722 | Evolution equation invariance of sets | If $\varphi'(t)$ is orthogonal to $V^\perp$ then it belongs to $V^{\perp\perp}=V$, hence $$\varphi(s)=\varphi(0)+\int\_0^s\varphi'(t)dt \in V.$$
| 3 | https://mathoverflow.net/users/21051 | 264890 | 119,038 |
https://mathoverflow.net/questions/227651 | 5 | Can one describe the set $\{e^f+e^g: f, g\in H(C)\}$ in some way?
For example, in unital Banach algebras, every element has this form.
I am in particular interested in the problem whether the function $u=(1-e^L)^2$ with
$L=(1-z)^{-3}$ is a sum of two exponentials in H(D).
Here $L$ is chosen so that every value is taken... | https://mathoverflow.net/users/61993 | sums of zero-free entire functions and its siblings on the disk | Describing the set $\{ e^f+e^g:f,g\in H(C)\}$ is difficult, and probably it cannot be described in any reasonable form. However many conditions can be given for
a function $h$ not to be of this form. For example, $h$ cannot be a square (or any power) of an entire function. Proof: Suppose that there are entire functions... | 2 | https://mathoverflow.net/users/25510 | 264891 | 119,039 |
https://mathoverflow.net/questions/264896 | 20 | The curvature operator on $\Lambda^2(TM)$ is defined on decomposable bivectors by $$g(\mathfrak{R}(X \wedge Y), V \wedge W) = R(X,Y,W,V)$$ and then extended by linearity to all elements of $\Lambda^2(TM)$. It is self-adjoint, so defines a symmetric bilinear form on $\Lambda^2(TM)$. If this form is positive definite, th... | https://mathoverflow.net/users/98590 | Positive sectional curvature does not imply positive definite curvature operator? | For the first question: positive curvature operator on a compact manifold implies that the manifold is diffeomorphic to a space form, i.e., a manifold of sectional curvature one. This is due to C. Boehm and B. Wilking [Manifolds with positive curvature operators are space forms](http://annals.math.princeton.edu/wp-cont... | 21 | https://mathoverflow.net/users/1573 | 264899 | 119,041 |
https://mathoverflow.net/questions/264898 | 11 | Riemann famously introduced the notion of what we now call a Riemannian manifold and introduced the Riemann curvature tensor $R\_{ijk}{}^l$, showing that it is an obstruction the local existence of Euclidean coordinates. It is now well known that $R\_{ijk}{}^l = 0$ is also a **sufficient** condition for the local exist... | https://mathoverflow.net/users/2622 | Who first proved that a vanishing Riemann tensor is sufficient for the existence of Euclidean coordinates? | Presumably it was Riemann himself who proved the sufficiency of $R=0$ for the flatness. Spivak's Chapter 4 in Volume II of "A comprehensive introduction to differential geometry" is a good source on Riemann's work. It gives a translation of Riemann's inaugural lecture and of a part of his prize essay. At the end of the... | 10 | https://mathoverflow.net/users/98590 | 264909 | 119,046 |
https://mathoverflow.net/questions/262890 | 2 | Let $X\_{(1)}\leq X\_{(2)}\leq \cdots X\_{(n)}$ be the order statistics for a random sample from a continuous distribution with c.d.f. $F(x)$ and density $f(x)$. Define $U\_{i}$, $i=1,2\ldots,n,$ by
$$U\_{i}=\frac{F(X\_{(i)})}{F(X\_{(i+1)})}, \quad i=1,\ldots,n-1, \: \mbox{and }\: U\_{n}=F(X\_{(n)}).$$
Find the join... | https://mathoverflow.net/users/nan | Find the joint distribution of $U_{1},\ldots,U_{n}$ where $U_{i}=\frac{F(X_{(i)})}{F(X_{(i+1)})}$ and $X_{(i)}$ are order statistics | See the discussion after Theorem 1.2.9 in the book you refer to (Introduction to the Theory of Nonparametric Statistics, Randles & Wolfe).
If we let $V\_i=F(X\_{(i)})$ then it is explained that, because (by Theorem 1.2.9) $F(X)\sim U(0,1)$, we have that $V\_1\leq \cdots \leq V\_n$ are distributed as the order statist... | 2 | https://mathoverflow.net/users/83613 | 264913 | 119,048 |
https://mathoverflow.net/questions/264914 | 2 | Let Y be a smooth projective curve over **C** and prescribe a branch divisor B on Y. I want to know if the number of coverings of Y of fixed degree and branched along B is finite. If so, why? Or, where is the reference for this? I was told it is claimed in an article by Manin ("On branched coverings of algebraic curves... | https://mathoverflow.net/users/106244 | Is the number of ramified coverings of given degree of a curve with prescribed branch divisor finite? | This can be understood by looking at fundamental groups.
Covers of $Y$ branched only along $B$ of degree $n$ correspond to homomorphisms $\pi\_1(Y-B)\rightarrow S\_n$ ($S\_n$ is the symmetric group on $n$ letters). (The cover is connected if and only if the image is a transitive subgroup, and the cover is Galois iff ... | 6 | https://mathoverflow.net/users/15242 | 264917 | 119,050 |
https://mathoverflow.net/questions/264903 | 3 | For each $ n\ge 1$ Define the vectors $e\_n = (e\_{nk})$ where $ k\ge 1$ and $ e\_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
| https://mathoverflow.net/users/50438 | Determining if a set is a Basis for l^2 | You want to know if $\sum\_{k=1}^\infty a\_k k^{-n}=0$ for $(a\_k)\in l^2$ and every positive integer $n$ implies $a\_k=0$. This is true. First we note that if $(a\_k)\in l^2$, then $(a\_k/k)\in l^1$. So w.l.o.g. we may consider the case where $(a\_k)\in l^1$ to begin with. Now note that the condition implies that $\su... | 7 | https://mathoverflow.net/users/12120 | 264919 | 119,052 |
https://mathoverflow.net/questions/264904 | 6 | Let $G$ be a connected reductive group over an algebraically closed field and consider a semisimple element $s \in G$ and let $L$ be a Levi subgroup containing $s$.
My question is about the two ways we can look at $C\_L(s)^\circ$. On the one hand, it is the connected centralizer of $s$ in the Levi subgroup $L$, on t... | https://mathoverflow.net/users/68519 | Dynkin diagram of the centralizer of a semisimple element in a Levi subgroup | In the Bourbaki numbering ($\alpha\_i = \epsilon\_i - \epsilon\_{i + 1}$ for $i < 10$ and $\alpha\_{10} = \epsilon\_{10}$), I believe that you can take $s = \alpha\_1^\vee(-1)\alpha\_3^\vee(-1)$ (with connected centraliser of type $D\_4 \times B\_6$, where the base for the $D\_4$ piece is $\{\alpha\_3, \alpha\_2, \alph... | 3 | https://mathoverflow.net/users/2383 | 264924 | 119,054 |
https://mathoverflow.net/questions/264931 | 2 | I'm reading Hartshorne's *Deformation Theory* book. In exercise 5.9(b), he claims:
Let $C$ be a reduced projective locally complete intersection curve embedded inside a smooth surface $X$ (over some alg. closed field $k$), as:
$$i : C\hookrightarrow X$$
Tensoring the exact sequence
$$0\rightarrow T\_{C/k}\rightarrow ... | https://mathoverflow.net/users/88840 | For a LCI curve $C$ embedded inside a smooth surface $i : C\rightarrow X/k$, is $i^*T_X\otimes\omega_X = i^*\Omega_X$? | Let $\mathscr F$ be a rank $n$ locally free sheaf on a ringed space $X$. Then the wedge product map
$$\bigwedge^r \mathscr F \otimes \bigwedge^{n-r} \mathscr F \to \bigwedge^n \mathscr F$$
is a perfect pairing. Indeed, it suffices to check this locally, where one does this by choosing a basis $x\_i$ for $\mathscr F$. I... | 3 | https://mathoverflow.net/users/82179 | 264934 | 119,057 |
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