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 |
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https://mathoverflow.net/questions/247238 | 3 | Is the real KO-theory of real Stiefel $V\_k(\mathbb{R}^n)$ manifold known? Or have the real vector bundles over the real Steifel manifolds $V\_k(\mathbb{R}^n)$ been studied?
| https://mathoverflow.net/users/33699 | KO-theory of real Stiefel manifold | The case of $V\_1(\mathbb{R}^n) = \mathbb{R}P^{n-1}$ is in
*Michikazu Fujii*, MR 219060 [**$K\_{O}$-groups of projective spaces**](http://www.ams.org/mathscinet-getitem?mr=219060), *Osaka J. Math.* **4** (1967), 141--149.
Perhaps some of the methods there will generalise to $k>1$? The complex case is also known, a... | 3 | https://mathoverflow.net/users/8103 | 247271 | 112,530 |
https://mathoverflow.net/questions/247136 | 0 | Many authors used the **Tanaka connection** in their papers such as
[1]
to define new Tanaka connection so-called *Generalized Tanaka connection* $^\*\nabla$ on a contact Riemannian manifold $(M,\eta,\xi,\phi,g)$ as follows:
$$^\*\Gamma\_{ik}^j = \Gamma\_{ik}^j + \eta\_j\phi\_k^i-\nabla\_j\xi^i\eta\_k+\xi^i\nabla\_j... | https://mathoverflow.net/users/90655 | On generalized Tanaka connection | The short answer to your question is that the Levi-Civita connection is perfectly adapted to the metric, but not compatible with the additional structure around. In particular, it does not preserve the contact distribution (in your notation, this should be the kernel of $\phi$). Indeed, a torsion-free connection can ne... | 4 | https://mathoverflow.net/users/64141 | 247272 | 112,531 |
https://mathoverflow.net/questions/244955 | 2 | Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.
Now let $\Lambda\_{\xi, q}$, with $\xi \in C$, be a flat family of linear spaces parametrized by $C$. Assume that for a general $\xi\in C$ the intertsection mu... | https://mathoverflow.net/users/nan | Intersection multiplicity of limit linear spaces | The answer is no. Take $X \subset \mathbb{P}^4$ be a hyperplane section of $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$. So $X$ is a cubic ruled surface, I denote by $L$ the ruling.
Let $H$ be a generic hyperplane containing $L$. Then $H \cap X = L \cup D\_1 \cup D\_2$, where $D\_1$ and $D\_2$ are two disj... | 0 | https://mathoverflow.net/users/37214 | 247295 | 112,536 |
https://mathoverflow.net/questions/247264 | 14 | For some work I'm doing, I need a version of the Hodge to de Rham spectral sequence for stacks. I am not at all an expert on stacks, so please excuse me if I make minor technical mistakes in stating it.
I only need to deal with quotient stacks. Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$ and let $G$... | https://mathoverflow.net/users/97118 | Hodge to de Rham spectral sequence for stacks | Short answer: yes.
As I recall, Teleman constructs such a spectral sequence for fairly general stacks. You can look at his paper *The quantization conjecture revisited* Annals 2000. But this may fall into the category of "more generality than you need".
The case you want can probably done by hand. The point is if ... | 15 | https://mathoverflow.net/users/4144 | 247296 | 112,537 |
https://mathoverflow.net/questions/247304 | 4 | In an answer [here](https://mathoverflow.net/questions/190837/entire-function-bounded-at-every-line), several users reference entire functions which are bounded along many or every line. I have two questions tending in the opposite direction:
1) Is there an entire function (not a polynomial) which diverges along ever... | https://mathoverflow.net/users/35158 | Entire function which diverges along every path | For (1), just take $f(z) = z$. (Or do you want to call that "converging to $\infty$"?)
For (2), the problem is what happens on the imaginary axis. Whether $f(it)$ diverges or converges to $0$ as $t \to +\infty$, you can take a continuous function $g: [0,\infty) \to (0,\infty)$ that goes to $0$ so rapidly that $f(it)$... | 11 | https://mathoverflow.net/users/13650 | 247307 | 112,542 |
https://mathoverflow.net/questions/247300 | 5 | It is well known that the Large Cardinal Axiom which asserts the existence of a measurable cardinal number is inconsistent with V=L. Do any of the well-known Large Cardinal Axioms contradict V=OD?
| https://mathoverflow.net/users/4423 | A question about ordinal definable sets and large cardinal axioms | Almost all of the usual large cardinal axioms can be preserved by the reverse-Easton support class forcing to make every set of ordinals coded into the GCH pattern, or the $\Diamond^\*$ pattern, and these axioms imply V=HOD.
For example, this is true of the supercompact cardinals by the usual Laver argument, and ana... | 9 | https://mathoverflow.net/users/1946 | 247310 | 112,543 |
https://mathoverflow.net/questions/247280 | 5 | Let the cocomma-square $\require{AMScd}$
\begin{CD}
A @>T>> Y\\
@V S V V \Uparrow\_\beta @VV i\_1 V\\
X @>>i\_0> (S\star T)
\end{CD}
where :
* $Ob\,(S\star T) :=Ob\,X\sqcup Ob\,Y$
and $\hom\_{(S\star T)}(r,r') $ is given by :
* $\hom\_{(S\star T)}(r,r'):=\hom\_{X}(x,x')$ if $r=x, r'=x' \in X$,
* $\hom\_{(S... | https://mathoverflow.net/users/97124 | An "explicit" description of cocomma-categories ? | Your construction is basically correct. The only slight mistake (probably you had the correct thing in mind and this was just a mistake in phrasing) is that the formal arrows $\underline{a}$ are not sufficient to give the hom-sets $\hom\_{S\star T}(r,r')$ when $r\in X$ and $r'\in Y$; you also need the "formal composite... | 6 | https://mathoverflow.net/users/49 | 247311 | 112,544 |
https://mathoverflow.net/questions/247318 | 1 | What is the number of solutions of $(a\_i)\_{i=1}^n$ such that
$$\sum\_{i=1}^nia\_i\le b,\quad a\_i\in\{-1,1\},\quad \sum\_{i=1}^n{a\_i}=c$$
given $b,c\in\mathbf Z$?
Is there a generating function solution?
| https://mathoverflow.net/users/32660 | Sign Enumeration | The g.f. equals
$$\frac1{1-y}\prod\_{i=1}^n \left(xy^i + (xy^i)^{-1}\right).$$
That is, the number of solutions is given by the coefficient of $x^cy^b$.
| 4 | https://mathoverflow.net/users/7076 | 247319 | 112,547 |
https://mathoverflow.net/questions/247332 | 5 | On the first page of [this preprint](https://math.berkeley.edu/~gbergman/papers/faithful_OX.pdf), I give a proof (a simplification
of an argument suggested by Ken Goodearl) that if *k* is a field,
*A* and *B* are *k*-algebras (not necessarily commutative), *M* is a faithful
left *A*-module, and *N* is a faithful left *... | https://mathoverflow.net/users/39608 | Is this result on tensor products of faithful modules known? | See Lemma 1.1 of <http://www.math.wisc.edu/~passman/balgebra.pdf> by Passman. I think he published this years later in Communications in Algebra.
| 7 | https://mathoverflow.net/users/15934 | 247335 | 112,551 |
https://mathoverflow.net/questions/247284 | 3 | Clairaut's relation for a great circle parametrized by $t$ is $r(t)\cos\gamma(t)=\text{Const}$ where $r$ is the distance to the $z$-axis and $\gamma$ is the angle with the latitude. The implicit equation of great circle in spherical coordinates $(\theta,\phi)$ is $\cot \phi= a\cos(\theta-\theta\_0)$ where $\phi$ is the... | https://mathoverflow.net/users/28128 | Clairaut's relation and the equation of great circle in spherical coordinates | A great circle is the intersection of the sphere with a plane through the origin. Let a unit normal to that plane be ${\bf u} = [-\sin(\gamma), 0,\cos(\gamma)]$, where for convenience we choose our $x$ and $y$ axes so that $u\_2 = 0$. Then in spherical coordinates, the equation ${\bf u} \cdot [\sin(\phi) \cos\theta), \... | 7 | https://mathoverflow.net/users/13650 | 247336 | 112,552 |
https://mathoverflow.net/questions/247338 | 1 | Let $p \ge 11$ be a prime number, $k,n$ be positive integers such that $n|gcd(p-1,k-1)$ and $p > k > n \ge 5$. Let $s \in \mathbb Z\_p$ such that $ord\_p(s) = n$. Is it possible that the sets $A = \{1,2,3,\dots,k-1\}$ and $B = \{k,k+1,k+2,\dots,p-1\}$ of classes modulo $p$ satisfy $sA = A$ and $sB = B$, simultaneously?... | https://mathoverflow.net/users/89000 | Invariant complementary sets modulo $p$ | No.
I'm going to use $\bar{s} \in \mathbb{Z}\_p, s \in \mathbb{Z}$ its minimal positive representative.
We can assume WLOG that the initial set we're looking at has size less than $\frac{p - 1}{2}$ as $\{k, k + 1, ..., p - 1\}$ is $s$-invariant if and only if $\{1, 2, ..., p - k\}$ is. Then $s \leq k - 1 \leq \fr... | 1 | https://mathoverflow.net/users/44191 | 247339 | 112,553 |
https://mathoverflow.net/questions/247317 | 4 | Given a category, one is often interested in the category of (abelian) group and (commutative) ring objects in it. I would like to know what exactness properties such categories and their simplicial analogues have, e.g what can be said about the simplicial abelian groups and commutative ring objects in some category $\... | https://mathoverflow.net/users/69037 | Properties of category of (simplicial) algebraic objects | If $C$ is a regular or [(Barr-)exact category](https://ncatlab.org/nlab/show/exact+category) and $T$ is a [Lawvere algebraic theory](https://ncatlab.org/nlab/show/Lawvere+theory), then the category $Alg\_T$ of $T$-algebras is also regular or exact, respectively. Reference: Exact Categories by Michael Barr, theorem 5.11... | 3 | https://mathoverflow.net/users/2926 | 247342 | 112,555 |
https://mathoverflow.net/questions/247341 | 2 | For a closed manifold $X$, suppose $b\_2(X)=0$.
**Question:** For any complex line bundle $L\to X$, can we always find a flat connection, i.e. does there exist a connection $\nabla$, such that $\nabla^2=0$?
| https://mathoverflow.net/users/95296 | Cohomology Torsion and Flat connection | The structural group of the bundle is $C-\{0\}$. Since $S^1$ is the maximal compact subgroup of $C-\{0\}$, you have a $S^1$-reduction of the bundlle. Consider a trivialization $(U\_i)\_{i\in I}$ of the bundle. The connection is defined by $d+\omega\_i$ where $\omega\_i$ is a $1$-form which takes its values in the Lie a... | 7 | https://mathoverflow.net/users/80891 | 247343 | 112,556 |
https://mathoverflow.net/questions/247290 | 0 | In a set of notes, I came across the following few lines involving the covariant derivative, and just wanted to make sure I understood the notation correctly:
Let $\lbrace F\_{1},F\_{2},F\_{3},F\_{4}\rbrace$ be a null tetrad. Then from Newman Penrose formalism we have an equation
$\nabla\_{F\_{4}}F\_{3}=(\alpha-\ov... | https://mathoverflow.net/users/51137 | Clarification on tetrad indices | $$F\_4 = F\_4^l \frac{\partial}{\partial x^l}$$
$$\nabla\_{F\_4}F\_3 = \nabla\_{\left(F\_4^l \frac{\partial}{\partial x^l}\right)}F\_3 = F\_4^l \nabla\_{\frac{\partial}{\partial x^l}}F\_3 = F\_4^l \nabla\_lF\_3 = F\_4^l \Big(\nabla\_lF\_3\Big)^k\frac{\partial}{\partial x^k}$$
$$ F\_4^l \Big(\nabla\_lF\_3\Big)^k\frac{\p... | 4 | https://mathoverflow.net/users/75853 | 247346 | 112,557 |
https://mathoverflow.net/questions/245764 | 13 | Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes\_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping algebra, so that $k$-central $(A,A)$-bimodules are the same as left $A^e$-modules.
Recall that a finite-dimensional $k$-a... | https://mathoverflow.net/users/778 | Is a "smooth" finite-dimensional algebra separable modulo its radical? | Let $K$ be an algebraic closure of $k$.
The following lemma must surely be well-known, but I haven't found an explicit reference, so I'll include a proof at the end of this post.
>
> **Lemma.** If $S$ is a finite dimensional semisimple $k$-algebra, then $S\_K=S\otimes\_kK$ is a finite product of matrix rings over... | 11 | https://mathoverflow.net/users/22989 | 247352 | 112,559 |
https://mathoverflow.net/questions/247357 | 14 | In Chapter One of his notes (March 2002) Thurston says:
>
> If $K$ is the trivial knot the cyclic branched covers are $S^3$. It seems intuitively obvious (but it is not known) that this is the only way $S^3$ can be obtained as a cyclic branched covering of itself over a knot.
>
>
>
This sentence sounds a litt... | https://mathoverflow.net/users/58934 | $S^3$ as cyclic branched cover of itself | The statement that for arbitrary K in $S^3$, if for some $n \ge 2$, the n-fold cyclic branched cover is $S^3$ (or in some versions, a homotopy 3-sphere) then K is the unknot, was known as the [Smith conjecture](https://en.wikipedia.org/wiki/Smith_conjecture). It was proved around 1979, combining work of many authors: T... | 22 | https://mathoverflow.net/users/3460 | 247363 | 112,562 |
https://mathoverflow.net/questions/247358 | 4 | Say $E$ is a $(\kappa,\lambda)$-extender on some model $\mathcal M$ of set theory, and consider the ultrapower embedding $i:\mathcal M\to\text{Ult}(\mathcal M,E)$. Now recall that a **generator** of $E$ is an ordinal $\xi\in[\kappa,\lambda)$ such that $[a,f]\_E\neq\xi$ for every $a\in[\xi]^{<\omega}$. That is, it can't... | https://mathoverflow.net/users/38602 | Are generators of extenders cardinals? | The answer is not necessarily.
Here is one easy way to make a counterexample. Assume the GCH and
let $j:V\to M$ be a $\kappa^+$-supercompactness embedding with
critical point $\kappa$. So $M^{\kappa^+}\subset M$ and $\kappa^{++}<j(\kappa)$.
Consider the collection of functions $f:\kappa^{<\omega}\to\kappa$ in $V$; ... | 6 | https://mathoverflow.net/users/1946 | 247367 | 112,564 |
https://mathoverflow.net/questions/247347 | 4 | Let $G$ be a semi-simple algebraic group over $\mathbb C$, $B$ be a Borel and P is be parabolic containing $B$ and let $\mathcal L$ be an ample line bundle on $G/P$. Is it true that $H^0(G/B, L^{\otimes d})=H^0(G/P, L^{\otimes d})$ for all d > 0?
From the projection map from $G/B \rightarrow G/P$ we get an injection ... | https://mathoverflow.net/users/97163 | Global sections of G/B and G/P | Q. Why is it a surjection?
A. It's a nonzero $G$-equivariant map to (by Borel-Weil) a $G$-irrep.
(Of course this has punted the difficulty, solved geometrically by the commenters, into the Borel-Weil theorem.)
| 1 | https://mathoverflow.net/users/391 | 247368 | 112,565 |
https://mathoverflow.net/questions/247348 | 2 | After reading:
[Detection of Redundant Constraints](https://mathoverflow.net/questions/69662/detection-of-redundant-constraints)
It appears that linear-programming is the most commonly known way to remove ALL redundant constraints from a system of inequalities of the form
$$ Ax \le b$$
Short of some heuristic ... | https://mathoverflow.net/users/46536 | Better tactics for removing redundant constraints than Linear Programming? | I don't know why you find this surprising. Saying that a particular linear inequality $a x \le b$ is non-redundant is exactly the statement that the linear programming problem, maximize $a x$ subject to the other constraints, does not have an optimal solution with objective value $\le b$. Linear programming is the natu... | 5 | https://mathoverflow.net/users/13650 | 247378 | 112,569 |
https://mathoverflow.net/questions/247382 | 1 | *Definition* A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T|y<x\}$ is well-ordered.
**Question** Assume CH fails and let $\aleph\_1\le\kappa< 2^{\aleph\_0}$. Is it possible that a tree of countable height and countable levels has **exactly** $\kappa$ many maxim... | https://mathoverflow.net/users/13694 | Tree with countable height and exactly $\aleph_1$-many maximal branches | No, and this is because every uncountable closed set of reals contains a perfect set. This is how Cantor proved that the continuum hypothesis holds for closed sets.
The easiest way to prove it is by using the [Cantor-Bendixson analysis](https://en.wikipedia.org/wiki/Derived_set_(mathematics)#Cantor.E2.80.93Bendixson... | 7 | https://mathoverflow.net/users/1946 | 247384 | 112,572 |
https://mathoverflow.net/questions/247334 | 3 | I know Dold-Kan actually holds for any abelian category, i.e gives an equivalence between simplicial objects in a fixed abelian category and the connective chain complexes over it.
I've never seen an actual proof of this fact though, so I was wondering whether in the case of a general symmetric monoidal abelian categ... | https://mathoverflow.net/users/69037 | Are there Alexander-Whitney and shuffle maps for Dold-Kan for abelian categories? | §1.2.3 in Lurie's Higher Algebra establishes the Dold—Kan correspondence for idempotent complete additive categories
and constructs the Alexander—Whitney maps in this generality.
| 4 | https://mathoverflow.net/users/402 | 247386 | 112,573 |
https://mathoverflow.net/questions/247376 | 1 | Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be an isofibration of small categories (fibration of categories in the canonical sense). Under what conditions is the nerve $N(F):N(\mathcal{C})\rightarrow N(\mathcal{D})$ a Kan fibration of simplicial sets?
| https://mathoverflow.net/users/84563 | When is an isofibration a Kan Fibration? | This follows from HTT, Prop 2.1.1.3. A map of categories is a Grothendieck fibration that is (co)fibered in groupoids if and only if the map on nerves is a left (right) Kan fibration, so by loc. cit. an isofibration induces a Kan fibration after taking nerves if and only if it's fibered and cofibered in groupoids, as D... | 3 | https://mathoverflow.net/users/nan | 247390 | 112,574 |
https://mathoverflow.net/questions/247385 | 10 | This question is related to [my previous question](https://mathoverflow.net/q/247301/4149), to which I got a partial answer.
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta\_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta\_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta\_8]$ denote the ring... | https://mathoverflow.net/users/4149 | Does every Shimura variety contain a generic point defined over a number field? | The Answer to Question 1 is YES. It follows from Serre's variant of Hilbert's irreducibility theorem (for infinite Galois extensions) combined with the Tate conjecture on homomorphisms of abelian varieties in char 0 (proven by Faltings); see Prop. 1.3 and Cor. 1.5 of [1995 Compositio paper by Rutger Noot (vol 97, Oort ... | 10 | https://mathoverflow.net/users/9658 | 247391 | 112,575 |
https://mathoverflow.net/questions/247301 | 11 | Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta\_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta\_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta\_8]$ denote the ring of integers of $L$.
>
> **Question 1.** Does there exist an abelian variety $A$ over ${{\mathbb{C}}}$ of dimension **two**... | https://mathoverflow.net/users/4149 | Abelian variety with prescribed endomorphism ring | The answer to your Question 3 is YES with the ground field $\mathbb{Q}$.
Here is a sketch of the proof. For each positive integer $q$ and a "parameter" $t$ (in char 0) consider the smooth projective model $C\_{q,t}$ of an affine curve $y^q=x^3-x-t$. Let $P\_{8,t}$ be the Prym variety of the double cover
$$C\_{8,t}\to... | 10 | https://mathoverflow.net/users/9658 | 247393 | 112,576 |
https://mathoverflow.net/questions/247402 | 25 | Any finite group $G$ can be embedded into $A\_{|G|+2}$ via Cayley's theorem ($G\hookrightarrow S\_{|G|}\hookrightarrow A\_{|G|+2}$). If $G$ is not assumed to be finite, is it still always possible to embedd it into a simple group?
| https://mathoverflow.net/users/83073 | Can any group be embedded in a simple group? | Yes. Assume $G$ is infinite. Cayley still embeds $G$ into the group $S\_G$
of permutations of $G$. This group is no longer simple: there is a normal
subgroup, call it $N\_G$, consisting of all permutations that fix
the complement of a subset of $G$ of cardinality smaller than that of $G$.
But by a theorem of Baer, Sch... | 45 | https://mathoverflow.net/users/14830 | 247403 | 112,580 |
https://mathoverflow.net/questions/247407 | 11 | Let $E\_d$ be the group of rational points on the elliptic curve $$y^2=(x-d)x(x+d)$$
It is well-known – and easy to verify by elementary algebra – that $(x,y)\in E\_d$ belongs to $2E\_d$ precisely when all three of $x-d$, $x$, $x+d$ are squares.
This seems too neat to be a mere coincidence.
Is there a natural gen... | https://mathoverflow.net/users/8217 | Why does doubling move from product-is-square to each-is-square? | It looks as if you're working over $\mathbb Q$. And you are using an elliptic curve in which all of the 2-torsion is rational, so $E[2]$ is isomorphic to (say) $\boldsymbol\mu\_2^2$ as a Galois module, where $\boldsymbol\mu\_2=\{\pm1\}$ is the group of square roots of 1. Okay, now consider the injection (this comes fro... | 13 | https://mathoverflow.net/users/11926 | 247409 | 112,581 |
https://mathoverflow.net/questions/247415 | 3 | Suppose $\alpha >1$ such that the distance from $\alpha^n$ to the integers tends to zero as $n$ tends to infinity.
Question 1: Can $\alpha$ be a rational number?
Question 2: The property is satisfied if $\alpha$ is a root of a monic polynomial with integer coefficient that has all the other roots with module less t... | https://mathoverflow.net/users/97209 | On the number $\alpha>1$ such that $lim_{n \to \infty}{d(\alpha^n, \mathbb{Z})}$ = 0 | 1) No, it can not unless it is integer.
2) This is an open problem known as Pisot-Vijayaraghavan problem. For algebraic numbers the answer is positive, that implies the answer to 1).
See
<https://en.m.wikipedia.org/wiki/Pisot-Vijayaraghavan_number>
Short self-contained answer to 1): if $\alpha^n=A\_n+\delta\_n$ for... | 11 | https://mathoverflow.net/users/4312 | 247418 | 112,583 |
https://mathoverflow.net/questions/247419 | 1 | I'm looking for a source of properties for semisimple Lie algebra elements, specifically finite dimensional classical Lie algebras.
I start with the assumption that I have a complexified Lie algebra $\mathfrak{g}^\mathbb{C}$, with a semisimple element $\Lambda \in \mathfrak{g}^\mathbb{C}$, and that it's conjugate $\b... | https://mathoverflow.net/users/97218 | Properties of semisimple Lie algebra elements | First the counterexample: Let $\mathfrak g=\mathfrak{sl}(3,\mathbb C)$ and
$$
\Lambda:=\left[ \begin {array}{ccc} 1&-3\,i&0\\3\,i&1&0
\\0&0&-2\end {array} \right]
$$
Then $\Lambda$ and $\overline\Lambda$ commute and are semisimple. But only $\Lambda$ commutes with
$$
\left[ \begin {array}{ccc} 0&0&i\\0&0&1
\\0&0&0\end ... | 4 | https://mathoverflow.net/users/89948 | 247427 | 112,584 |
https://mathoverflow.net/questions/246961 | 7 | The grassmannian space $G(n,m)$ may be identified with the quotient space $O(n)/(O(m)\times O(n-m)$. As such, it is endowed with a natural invariant probability measure which I call "Haar measure on $G(n,m)$". Let $\gamma\_{n,m}$ be this measure. Let us endow $G(n,m)$ with the usual metric $d(V,W)=\|P\_V-P\_W\|$, where... | https://mathoverflow.net/users/46931 | Haar measure on the Grassmannian space | Yes, this is true. The answer by Ben Webster gives the idea, but for completion let me give you a list of easy facts that combined together answer the question.
1. The inequalities you seek are correct for any Riemannian manifold with bounds on the curvature (upper/lower bounds for corresponding sides of the inequali... | 2 | https://mathoverflow.net/users/89334 | 247429 | 112,585 |
https://mathoverflow.net/questions/247437 | -1 | Let $1<p<\infty$. Let $(f\_{n})\_{n}$ be a sequence in the unit sphere $S\_{L\_{p}(\mu)}$ of $L\_{p}(\mu)$ such that $(f\_{n})\_{n}$ weakly converges to $f\in S\_{L\_{p}(\mu)}$. Does $(f\_{n})\_{n}$ necessarily converge to $f$ in norm? It seems that it is true. But I am not sure. Thank you!
| https://mathoverflow.net/users/41619 | Weak convergent sequences and norm convergent sequences in $L_{p}(\mu)$ | Consider $g\_n=(f+f\_n)/2$ and let $f^\*$ be the supporting functional of $f$. Then $f^\*(g\_n)\to 1$, hence $||g\_n||\to 1$. By the uniform convexity of $L\_p$ for $1<p<\infty$ this implies $||f-f\_n||\to 0$.
| 1 | https://mathoverflow.net/users/85406 | 247450 | 112,594 |
https://mathoverflow.net/questions/247243 | 0 |
>
> Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka
> fibration. Consider the following singular hermitian metric
> $$h(\sigma,\sigma)=\left(\int\_{X\_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}$$
> where $y\in Y$ and $\sigma$ is a section of $$\frac{1}{m!}f\_\*\mathcal
> O\_X(m!K\_{X/Y})|\_{X\_y}$$ ... | https://mathoverflow.net/users/nan | Lelong number of curvature of Kawamata's hermitian metric | By theorem 2 and Poroposition 6, of the paper, Georg Schumacher and Hajime Tsuji, Quasi-projectivity of moduli spaces of polarized varieties, Annals of Mathematics,159(2004), 597–639
the Weil-Petersson metric has zero Lelong number.
<http://annals.math.princeton.edu/wp-content/uploads/annals-v159-n2-p03.pdf>
So... | 1 | https://mathoverflow.net/users/nan | 247459 | 112,600 |
https://mathoverflow.net/questions/247465 | 1 | From the article [Weyl's law](http://www-personal.umich.edu/~stevmatt/weyl_law.pdf) of Matt Stevenson page $3$, there's a lemma where I'd like some help to understand the proof.
>
> **Lemma.** Let $\mu\_n$ be the Dirichlet eigenvalues on $\Omega$ and $\nu\_n$ be the Neumann eigenvalues on $\Omega$, then $\nu\_n \l... | https://mathoverflow.net/users/96956 | Why the Minimax Principle says that $\nu_n$ is given by minimizing the same expression as for $\mu_n$ over a larger space | To save my fingers some typing, denote by $\mathcal{R}$ the Rayleigh quotient, $\mathcal{R}(u) = \|\nabla u\|^2/\|u\|^2$. Let
$$R(H) = \{\sup\_{u\in X^n}\mathcal{R}(u)\ |\ X\in\Phi(H)\}$$
where $H\in \{H^1\_0,H^1\}$. Notice that $R$ is the image of a function $\Phi(H)\to\mathbb{R}$ taking $X\mapsto\sup\_{u\in X}\mathc... | 0 | https://mathoverflow.net/users/20796 | 247467 | 112,604 |
https://mathoverflow.net/questions/247477 | 0 | For every $k\in\Bbb N$ is it possible to find a graph $G$ on large enough $n\in\Bbb N$ vertices such that $\big|Aut(G)\big|=(n-k)!$? Is it possible to construct it quickly?
| https://mathoverflow.net/users/10035 | Graphs with prescribed Automorphism groups | For $k=0$, take a complete graph and for $k=1$, a star. For $k\geq 2$, take a path with $k$ vertices, and a complete graph with $n-k$ vertices, then join one end of the path with all the vertices in the complete graph.
It is not hard to see that the resulting graph has the required property. For large enough $n$, the... | 4 | https://mathoverflow.net/users/22377 | 247481 | 112,608 |
https://mathoverflow.net/questions/238313 | 6 | Endow the real projective plane with the distance defined by $d(L,L')$ := "the angle between the lines $L$ and $L'$ ".
It is the case that every isometry from $RP^2$ onto $RP^2$ is induced by an element of the orthogonal group $O(3)$.
I want to use this fact in a research paper. But it turns out that all my conside... | https://mathoverflow.net/users/56097 | Elementary reference for the isometry group of $\mathbb{RP}^2$ | Robert Bryant's post provides a neat proof: it is a good reference for the result I am interested in. I found another (more conventional: published book vs MO answer) elementary reference for it: 9.7.1 and 19.1.2.2 in Berger's *Geometry*. This proof is less trickily efficient than that of Robert Bryant, but it is also ... | 2 | https://mathoverflow.net/users/56097 | 247487 | 112,610 |
https://mathoverflow.net/questions/247475 | 2 | I'm a novice to homotopical algebra, but I've found myself confronted with it by necessity and have some basic questions...
I'm going to consider chain complexes over a field $F := \mathbb{F}\_2$. Given a chain complex $C$, I'm interested in two operations:
* the ``homotopy Sym'', where I form $(C \otimes C \otim... | https://mathoverflow.net/users/84144 | homotopy Sym and quotients | If $C$ has an action of $\mathbb Z$, then $C\otimes C$ has an action of the wreath product ${\mathbb Z}\wr {\mathbb Z}/2$. This is the split group extension $1\to {\mathbb Z}\times{\mathbb Z}\to {\mathbb Z}\wr {\mathbb Z}/2\to {\mathbb Z}/2\to 1$ associated with the natural action of ${\mathbb Z}/2$ on ${\mathbb Z}\tim... | 5 | https://mathoverflow.net/users/6668 | 247490 | 112,611 |
https://mathoverflow.net/questions/247488 | 1 | Let $T=\{\alpha\_1,\dots,\alpha\_t\}$ be a set of distinct integers.
If $A\in T^{n\times m}$ is an $n\times m$ matrix with entries from $T$ then
>
> does $rank\_{\Bbb R\_+}A=rank\_{\Bbb Q\_+}A$ hold true?
>
>
>
| https://mathoverflow.net/users/10035 | On ranks of finite element matrices | Yes it's true. The inequality $\le$ is trivial. For the nontrivial inequality, we have some family of nonnegative rational vectors $v\_1,\dots,v\_n$ (by nonnegative vectors I mean with nonnegative coefficients, I apology to use this poor and absurd English terminology), we assume that there exist nonnegative real vecto... | 2 | https://mathoverflow.net/users/14094 | 247493 | 112,613 |
https://mathoverflow.net/questions/247484 | 2 | Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = aZ + \epsilon$. It can be show that
\begin{align}
\mathbb{E} [ W^2 (Z^2 - 1)] = 2 a^2.
\end{align}
Now suppose I truncate random variable $W$.
I was wondering what truncation level $R$ shou... | https://mathoverflow.net/users/81633 | Expectation of Truncated Bivariate Gaussian Random Variables | There is a simple but often efficient trick to facilitate such computations. Let $H\_k$ be the normalized Hermite polynomials and let $\Phi,\Psi$ be any two functions square integrable with respect to the Gaussian measure $\gamma$. Write $\Phi=\sum\_k\varphi\_kH\_k$, $\Psi=\sum\_k\psi\_kH\_k$. Then, if $Y,Z$ are jointl... | 4 | https://mathoverflow.net/users/1131 | 247501 | 112,614 |
https://mathoverflow.net/questions/247500 | 3 | **The short version of my question:** Suppose $T\in\mathcal{L}(X,Y)$ is strictly cosingular. Must $T^\*$ be strictly singular?
**The long version.**
Let $X$ and $Y$ be Banach spaces, and denote by $\mathcal{SS}(X,Y)$ the space of strictly singular operators in $\mathcal{L}(X,Y)$. (An operator is *strictly singular*... | https://mathoverflow.net/users/73784 | Example of a strictly cosingular operator whose dual is not strictly singular? | The inclusion operator $\iota \colon c\_0\to \ell\_\infty$ is strictly cosingular but $\iota^\*$ is not strictly singular. Indeed, $\iota$ is not weakly compact, so neither is $\iota^\*$ (Gantmacher's theorem). As $c\_0^\*=\ell\_1$, by Pełczyński's theorem, $\iota^\*$ fixes a copy of $\ell\_1$.
| 3 | https://mathoverflow.net/users/15129 | 247503 | 112,615 |
https://mathoverflow.net/questions/247513 | 1 | I have been stuck in this question for a while, and I would appreciate any new ideas. I have been considering the inequality
$$
\|f\|^2\_{H^{-1}(\mathbb{T})}\lesssim \int\_\mathbb{T} |\sin(x)f(x)|^2 \; dx,
$$
assuming that $\int\_\mathbb{T} f(x)=0$.
The inequality has the Fourier series analogue
$$
\sum\_{j\in\mathbb... | https://mathoverflow.net/users/39062 | $\|f\|^2_{H^{-1}(\mathbb{T})}\lesssim \int_\mathbb{T} |\sin(x)f(x)|^2 \; dx$? | The inequality does not hold. This becomes clear if we rewrite the discrete version as
$$
\sum |f\_n|^2/n^2 \lesssim \|Lf\|^2 ,
$$
with $(L f)\_n=f\_{n+1}-f\_{n-1}$. Notice that $iL$ is self-adjoint on $\ell^2$ and has spectrum $[-2,2]$ (because it becomes multiplication by $2\sin x$ after taking Fourier transforms, an... | 2 | https://mathoverflow.net/users/48839 | 247519 | 112,620 |
https://mathoverflow.net/questions/247515 | 0 | I'm trying to show that a connected graph which has order >=3, and having the following inequality is Hamiltonian:
```
degG(x) + degG(y) >= n x and y are two non-adjacent vertices
```
I know that Dirac's theorem will be helpful here, but I really don't know how to apply it. (Dirac's theorem: Let G be a graph of... | https://mathoverflow.net/users/97268 | degG(x) + degG(y) >= n, show that the graph is hamiltonian | Your result is true. It is a generalization of Dirac's Theorem known as [Ore's Theorem](https://en.wikipedia.org/wiki/Ore%27s_theorem).
| 3 | https://mathoverflow.net/users/51668 | 247520 | 112,621 |
https://mathoverflow.net/questions/247495 | 2 | Let $C$ be a site. Localizing either the projective or injective model structure on simplicial presheaves at the Cech nerves of covers in $C$ gives a new model structure on simplicial presheaves which presents the $(\infty,1)$-category of simplicial sheaves on $C$. Passing to pointed simplicial presheaves, one can mimi... | https://mathoverflow.net/users/43687 | Symmetric spectra for simplicial sheaves | Under some mild conditions, symmetric spectra in a monoidal model category
admit a monoidal model structure with many nice properties
(in particular, algebras over operads in symmetric spectra
have a model structure of their own),
see <http://arxiv.org/abs/1410.5699>.
In particular, as explained there, one can use th... | 2 | https://mathoverflow.net/users/402 | 247522 | 112,622 |
https://mathoverflow.net/questions/247351 | 4 | There are a lot of articles which explain what filtered colimits are (e.g. <https://ncatlab.org/nlab/show/filtered+limit>), but I couldn't find why they are named "filtered colimits".
It doesn't look like they have something to do with filters (in order theory), so I'm wondering.
Any ideas?
| https://mathoverflow.net/users/89916 | Where does the name "filtered colimit" come from? | Filtered categories are meant to generalize (upward) directed posets, i.e. nonempty posets $P$ such that for all $p,q \in P$, there is $r \in P$ with $p \leq r \geq q$. In particular, a poset $P$ is directed if and only if it is filtered when regarded as a category. The main point of directed posets (and filtered categ... | 5 | https://mathoverflow.net/users/2362 | 247524 | 112,624 |
https://mathoverflow.net/questions/247529 | 1 | Let $x\_n$ be the continuous function on $[0, 1]$ that is $1$ at $2^{-n}$ and $0$ outside $[2^{-n-1}, 2^{-n+1}]$. For an arbitrarily given $\epsilon >0$, is there a finite dimensional subspace $V$ of $C([0, 1])$ such that the distance from $x\_n$ to $V$ is $<\epsilon$ for all $n$?
| https://mathoverflow.net/users/97269 | Uniform finite dimensional approximation of continuous function sequence | No, there isn't.
Note first that $\|x\_n\|=1$ and $\|x\_n - x\_m\| = 1$ for all $n \ne m$.
Suppose there were, so that for each $n$ there is $y\_n \in V$ with $\|x\_n - y\_n\| < \epsilon$. Let's say $\epsilon < 1/3$. Then $\|y\_n\| < 4/3$ for all $n$, and $$\|y\_n - y\_m\| \ge \|x\_n - x\_m\| - \|y\_n - x\_n\| - \|... | 0 | https://mathoverflow.net/users/4832 | 247530 | 112,625 |
https://mathoverflow.net/questions/239008 | 10 | Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x\_1,\ldots, x\_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\mathcal{M}=\{p\, q: p,q\in\mathcal{H}\}$ is dense in $C(K)$ under supremum norm, where $K$ is a compact set in $\mathbb{R}... | https://mathoverflow.net/users/85652 | Density of the linear span of products of harmonic polymomials | Induction on $n$. Base $n=2$ is clear, as you said. Let $\nu$ be a non-trivial finite compactly supported (complex) measure in $\mathbb R^n$ orthogonal to any product $uv$. Then we can smear it a bit and get a non-trivial continuous compactly supported function $f$ orthogonal to each product. Now choose any $n-1$ dimen... | 3 | https://mathoverflow.net/users/1131 | 247534 | 112,626 |
https://mathoverflow.net/questions/247521 | 6 | Where can I find a proof of the following fact:
If $X\_1$ and $X\_2$ are subvarieties of $\mathbb{G}(k,n)$ of codimension $c\_1$ and $c\_2$ satisfying $c\_1+c\_2<n+1-2k$, then the intersection $X\_1\cap X\_2$ is nonempty.
(This is Lemma 3.13 in the paper *Hypersurfaces of Low Degree*, and it was said that it is a s... | https://mathoverflow.net/users/16356 | Nonempty intersection in Grassmannian | By recursion on $k$.
If $k=1$, this is Bezout's theorem in projective space. Now let $k>1$. We have the Pl\"ucker embedding $\mathbb{G}(k,n) \subset \mathbb{P}(\bigwedge^k \mathbb{C}^n)$. Let $p$ be a general point in $\mathbb{C}^n$. Let $Y\_i = \mathbb{P}(\bigwedge^{k-1} \mathbb{C}^n/\langle p\rangle) \cap X\_i$.
... | 6 | https://mathoverflow.net/users/37214 | 247535 | 112,627 |
https://mathoverflow.net/questions/247528 | 0 | Could anybody point to me a good reference for defining the $\Gamma$-category associated to a permutative category (better also with some illustrative examples)?
Dan Freed in his [notes](https://www.ma.utexas.edu/users/dafr/M392C-2012/Notes/lecture19.pdf) does provide a definition but many details are left out.
| https://mathoverflow.net/users/82645 | The $\Gamma$-category associated to a permutative category | There are several different, provably equivalent, definitions. Construction 10 in <http://www.math.uchicago.edu/~may/PAPERS/23.pdf> is one example. It is used to prove the uniqueness of a machine taking permutative categories to spectra. Therefore, any two reasonable constructions of $\Gamma$-categories from permutativ... | 3 | https://mathoverflow.net/users/14447 | 247537 | 112,628 |
https://mathoverflow.net/questions/247479 | 4 | I want to prove or find a counterexample to the following statement.
Let $G$ be a finite group. If for every minimal subgroup $L$ (= cyclic subgroup of prime order), there exists a minimal subgroup $H$ such that $\langle L,H\rangle=G$. Then one of the following cases occurs:
a) $G$ is a Frobenius group whose kernel... | https://mathoverflow.net/users/97247 | Characterization of finite groups generated by one additional element of prime order for every element of prime order | $\mathrm{P\Sigma L}(2,8)$ is a counter-example.
---
A corrected version of the statement is that, under the hypothesis, either
a) $G\cong Z\_p$ for some prime $p$,
b) $G\cong Z\_p\times Z\_q$ for some primes $p$ and $q$,
c) $G$ is a Frobenius group with kernel an elementary abelian $p$-group and the compl... | 5 | https://mathoverflow.net/users/22377 | 247543 | 112,630 |
https://mathoverflow.net/questions/247206 | 12 | Let $G$ be a connected, finite graph. (For me a graph is undirected, and it possibly has multiple edges, although the latter is not really crucial for this question). The *complexity* $c(G)$ (also known as the tree-number of $G$) is defined to be the number of spanning trees of $G$.
There is another number $k(G)$ tha... | https://mathoverflow.net/users/23434 | A different avatar of the complexity of a graph | You're looking at the "integral break divisors" of the graph $G$. These were introduced by Mikhalkin and Zharkov. For a very clear presentation check out the paper "Canonical representatives for divisor classes on tropical curves and the Matrix-Tree Theorem" by An, Baker, Kuperberg, and Shokrieh, available online at <h... | 10 | https://mathoverflow.net/users/25028 | 247554 | 112,633 |
https://mathoverflow.net/questions/246855 | 0 | Suppose $N \ge 3$ and let $\Phi(x):= C\_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$.
Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \partial \Omega$. Let $ \delta(x):=dist(x,\partial \Omega)$.
Question I am interested in. I am interested in obtaining boun... | https://mathoverflow.net/users/66623 | Gradient bounds on Newtonian potentials | Take a look at
[1] J. I. Díaz and J.-M. Rakotoson, “On the differentiability of very weak solutions with right hand side data integrable with respect to the distance to the boundary,” J. Funct. Anal., vol. 257, no. 3, pp. 807–831, 2009.
| 1 | https://mathoverflow.net/users/10834 | 247568 | 112,639 |
https://mathoverflow.net/questions/247578 | 15 | Let $D$ be a divisor on a compact Riemann surface of genus $g$. The inequality
$$
l(D)\geq {\textrm {deg}}(D)-g+1
$$
is called Riemann's inequality.
$ \phantom{aaaaaaaa}$**In which of Riemann's papers did this inequality first appeared?**
| https://mathoverflow.net/users/12204 | Original reference for Riemann's inequality | B. Riemann, [Theorie der Abel'schen Functionen](http://www.maths.tcd.ie/pub/HistMath/People/Riemann/AbelFn/), Journal für die reine und angewandte Mathematik **54**, 101–155 (1857).
Here is a description of this contribution, by [Jeremy Gray](https://eudml.org/doc/223239):
>
> In this 1857 paper Riemann establish... | 14 | https://mathoverflow.net/users/11260 | 247583 | 112,644 |
https://mathoverflow.net/questions/247592 | 2 | The SDEs \begin{equation}
dZ\_t = \mu(t,Z\_t)dt + \sigma(t,Z\_t)dW\_t
\end{equation} with prescribed initial conditions are well studied. My question came up in my research and I have not found much on the subject. I was wondering under what conditions does there exist a solution to the above SDE on $[t\_1,t\_2]$ with ... | https://mathoverflow.net/users/36886 | Existence of solution to SDE with perscribed initial and terminal conditions | The keywords are **diffusion bridges** or **conditioned diffusions**. I like to think of the distribution of these bridges as the stationary distribution of an SPDE on the path space of the diffusion. For an intro to this viewpoint and practical methods for sampling from this distribution, check out the following refer... | 4 | https://mathoverflow.net/users/64449 | 247597 | 112,648 |
https://mathoverflow.net/questions/247600 | 4 | I am looking for a reader-friendly proof of the following theorem:
let $A$ be a special $\Gamma$-space then $\pi\_0(A(S^0))$ is a commutative monoid (I have proved up to this), if further it is an abelian group then the adjoint map $t: A(S^0) \to \Omega{BA(S^0)}$ for the structure map $s: \Sigma(A(S^0)) \to BA(S^0)$ ... | https://mathoverflow.net/users/82645 | Adjoint map of $\Gamma$-space prespectrum | It is proposition 1.4 in Segal's [*Categories and cohomology theories*](http://www.sciencedirect.com/science/article/pii/0040938374900226) (a paper I love and I strongly encourage everyone interested in homotopy theory to read).
| 6 | https://mathoverflow.net/users/43054 | 247601 | 112,651 |
https://mathoverflow.net/questions/247517 | 8 | Let $\Pi$ be a Poisson structure on a manifold $M$. Then we can define a differential $d$ on the complex $\Lambda^{\bullet}M$
$$
C^{\infty}M \to TM \to...\Lambda^kTM \to...
$$
in the following way:
$$
d = [\Pi,\cdot],
$$
where $[\cdot,\cdot]$ is the Schouten-Nijenhuis bracket. The cohomology of this complex is called P... | https://mathoverflow.net/users/88385 | Poisson cohomology | First of all, one can think of Poisson cohomology (or, rather, Poisson cochains) as functions on the $QP$-manifold $T^\*[1]M$ with the differential as you describe. In particular, it has a natural odd Poisson ($\mathbb{P}\_2$) structure given by the Schouten bracket.
1. You can split $T^\*[1](\mathbb{R}^{2d+s}) = T^\... | 9 | https://mathoverflow.net/users/18512 | 247605 | 112,653 |
https://mathoverflow.net/questions/247614 | 6 | Let $G$ be a semi-simple algebraic group over $\mathbb{C}$ with Borel subgroup $B$ and consider the flag variety $G/B$. If $G\_0 \subset G$ is some real form, then $G\_0$ acts on $G/B$ and decomposes the space into a disjoint union of orbits. In the case of $SL\_2$ one gets 2 open orbits while in $SL\_3$ my calculation... | https://mathoverflow.net/users/97316 | Real orbits of Complex Flag Varieties | [Aomoto (1966)](http://www.ams.org/mathscinet-getitem?mr=191994) gives a formula for the number of open orbits (page 15, between (46) and (47)), and specializes it to various special cases (just search the paper for "open").
| 7 | https://mathoverflow.net/users/19276 | 247615 | 112,655 |
https://mathoverflow.net/questions/247193 | 6 | This question is connected with my previous question: [Union of Hamming balls](https://mathoverflow.net/questions/247159/union-of-hamming-balls)
Let $V \subseteq \{0,1\}^n$, $\log|V| = k < 0.9n$.
Harper's theorem states that the set $V\_r:= \bigcup\_{x \in V} V\_r(x)$ has rather large cardinality ($V\_r(x)$ is a H... | https://mathoverflow.net/users/31356 | Union of almost Hamming balls | Unfortunately the life is not that good. Take the Hamming ball $B\_R$ of radius $R=\beta n$ (where $\beta$ is small but positive) centered at $(0,0,\dots,0)$ and take the union of the Hamming balls of radius $r\ll n$ centered at the points from that ball. Not surprisingly you'll get the Hamming ball of radius $R+r$ and... | 4 | https://mathoverflow.net/users/1131 | 247619 | 112,658 |
https://mathoverflow.net/questions/247553 | 26 | Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex function on $[0,+\infty)$?
Updates:
1) It was pointed out by @user44191 that, observing $\binom{x}{i}=\binom{x-1}{i}+... | https://mathoverflow.net/users/97277 | $\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$ is a convex function on $[0,+\infty)$? | *This is not an answer to your question, is only an equivalent reformulation that seems promising. I write it as an answer only because of space constraints.*
For any nonnegative integer $n$ and any $\newcommand{\bR}{\mathbb{R}}$ $x\in\bR$ we define
$$ a\_n(x)=\sum\_{k=0}^n \binom{x}{2k}, \;\;b\_n(x)=\sum\_{k=0}^n ... | 10 | https://mathoverflow.net/users/20302 | 247634 | 112,661 |
https://mathoverflow.net/questions/247640 | 0 | I would like to compose/decompose an $n$-dimensional orthogonal rotation matrix (restricting to simple planar rotations, which rotates in the specified plane of rotation, and fixes in the plane orthogonal to the plane of rotation) from/to the two basis vectors of the plane of rotation, and the angle of rotation.
The ... | https://mathoverflow.net/users/97344 | Compose/decompose rotation matrix from/to plane of rotation and angle | I may be misunderstanding the question, but to construct the matrix of a rotation in a given plane, complete the two vectors to a basis using Gram-Schmidt, then conjugate the obvious matrix consisting of a two-by-two block in your question (and identity outside the $xy$ plane). To find the rotation, diagonalize over $\... | 2 | https://mathoverflow.net/users/11142 | 247653 | 112,665 |
https://mathoverflow.net/questions/247626 | 18 | I asked this at [mathstackexchange](https://math.stackexchange.com/questions/1886719/in-which-sense-the-gns-construction-is-a-functor) a week ago, without success.
I think the [Gelfand–Naimark–Segal construction](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction) must be a functor in s... | https://mathoverflow.net/users/18943 | In which sense the GNS-construction is a functor? | If you want $\tilde{\varphi}$ to be normal then this is false. But first let me point out that there is a sense in which the GNS construction is a functor. Note that $\varphi$ induces an isometric embedding of $H\_{f\circ\varphi}$ into $H\_f$ (the respective GNS Hilbert spaces). So define a morphism between two represe... | 8 | https://mathoverflow.net/users/23141 | 247657 | 112,668 |
https://mathoverflow.net/questions/247652 | 2 | Let $\vec{x} \in \mathbb{R}^n$ be a fixed vector and suppose that we are given an isotropic random vector $\vec{a} = (a\_1, \dots, a\_n)^T$ in $\mathbb{R}^n$ (i.e., the covariance matrix of $\vec{a}$ is the identity matrix).
Are there some general conditions on $\vec x$ and the distribution of $\vec a$ such that the ... | https://mathoverflow.net/users/97349 | Independent Decomposition of a Random Vector | This will not work in general. Let's take $a$ as a vector with iid $\pm 1$ entries, with probability $1/2$ for each of $\pm 1$, and $x=(1,1,\ldots ,1)$ (normalize this if you prefer). Then if $\langle a, x\rangle = n$, we know for sure that $a=(1,1,\ldots , 1)$, so $P(a)$ would have to be constant almost surely if we w... | 1 | https://mathoverflow.net/users/48839 | 247661 | 112,669 |
https://mathoverflow.net/questions/247639 | 1 | I try to find some relations for the irreducible character degrees of the extensions of the groups. For example:
Let $ G $ be a finite group of order $1800$ such that $ G $ has a normal subgroup of order $30$ and $ G/N $ is isomorphic to the alternating group $ A\_5$. How we can prove that there is no irreducible cha... | https://mathoverflow.net/users/31045 | About the character degree of an extension | The argument below is mainly group theoretic. I think it should be possible to give a more direct character-theoretic proof.
Let $H = O^{2}(N)$ which is cyclic of order $15$, and let $X = O^{2}(G).$ Then it is enough to prove that $X$ has no irreducible character of degree divisible by $15$, since $X \lhd G$ with $[G... | 2 | https://mathoverflow.net/users/14450 | 247665 | 112,672 |
https://mathoverflow.net/questions/247659 | 9 | Following [this](https://math.stackexchange.com/questions/1860288/completion-and-algebraic-closure) question:
Given a valued field $K$, denote with $\bar{K}$ its algebraic closure and with $\hat{K}$ the completion. Then both $\hat{\bar{K}}$ and $\hat{\bar{\hat{K}}}$ are complete and algebraically closed. If $K=\mathb... | https://mathoverflow.net/users/50081 | Completion and algebraic closure | First you have to observe that since all extensions of the valuation to $\bar{K}$ are conjugate, $\hat{\bar{K}}$ is well-defined up to (non-unique) isomorphism.
Now, since $\hat{\bar{K}}$ is complete and $K$ is dense in $\hat{K}$, the inclusion $K\subset \hat{\bar{K}}$ extends continuously to $K\subset \hat{K}\subse... | 17 | https://mathoverflow.net/users/7666 | 247666 | 112,673 |
https://mathoverflow.net/questions/247130 | 4 | Assume $S$ is a scheme over $\mathbb{C}$ (as nice as you want), $\mathcal{E}$ is a locally free $\mathcal{O}\_S$-module and $\mathcal{A}$ is a coherent $\mathcal{O}\_S$-algebra, locally free of finite rank, not neccesarily commutative. We assume that $\mathcal{E}$ has the structure of an $\mathcal{A}$-module.
Accord... | https://mathoverflow.net/users/70593 | Relative tangent bundle of a twisted Grassmann variety | I think the answer is yes. I will work out the proof when $S = \textrm{Spec}(\mathbb{C})$, but it certainly also works for $S$ being any scheme (stack?).
So let $\mathcal{A}$ be a finitely generated algebra over $\mathbb{C}$ and let $\mathcal{E}$ be a finite dimensional vector space with a structure of (left) $\mathc... | 1 | https://mathoverflow.net/users/37214 | 247668 | 112,674 |
https://mathoverflow.net/questions/247656 | 6 | I have following questions about the complexified octonion algebra $\mathbb C \otimes\_{\mathbb R} \mathbb O$. Zero divisors are of shape $p+i\otimes q$ (shortly $p+iq$) where $p$, $q$ are perpendicular octonions of the same length. We denote octonion conjugation by an overbar, so $\overline{z \otimes x} = z \otimes \b... | https://mathoverflow.net/users/nan | Properties of complexified octonions | Here are some answers, but let me introduce some notation: What you are calling the *norm* of $x\in\mathbb{C}{\otimes}\mathbb{O}$, I will denote by $N(x) = x\bar x = \bar x x\in \mathbb{C}$. The complexified octonions satisfy $N(xy) = N(x)N(y)$, and the norm is a nondegenerate quadratic form on $\mathbb{C}{\otimes}\mat... | 5 | https://mathoverflow.net/users/13972 | 247670 | 112,675 |
https://mathoverflow.net/questions/247633 | 7 | Kelly describes a constructive procedure for building the algebraically free monad on a pointed endofunctor. Garner gives a concise summary, which I partially review here for convenience.
---
Let $V$ be a cocomplete category (feel free to make other strong assumptions about $V$ if I have omitted them) and $(S,\et... | https://mathoverflow.net/users/3075 | Free monad sequence versus colimit over injections of ordered sets | Let's talk about free monoid constructions in a monoidal category $(\mathcal{V}, \otimes, I)$. Free monad constructions are a special case, when we take $\mathcal{V}$ to be a monoidal category of endofunctors. There are at least two flavors: the free monoid on an object $X$, and the free monoid on a pointed object $I \... | 4 | https://mathoverflow.net/users/2362 | 247692 | 112,681 |
https://mathoverflow.net/questions/247685 | 5 | I am currently trying to understand the first chapter of the HoTT book and for 1.2 Functions Types of the book,
>
> Since it is by definition ``the function that applies $f$ to its argument'' we consider it to be definitional equal to $f$:
> $$
> f\equiv (\lambda x.f(x))
> $$
> This equality is the **uniqueness p... | https://mathoverflow.net/users/95722 | Uniqueness Principle for function types | It is not true "by construction". Remember that at this point in the theory a "function" is an abstract undefined thing, not something "defined by its action on inputs" as it is in set theory. The only thing we can do with a "function" is apply it to an argument; the only way we have to make "functions" is to $\lambda$... | 11 | https://mathoverflow.net/users/49 | 247693 | 112,682 |
https://mathoverflow.net/questions/247542 | 5 | If $C\subset\mathbb{P}^2$ is an integral curve of degree $d$, do its singularities deform independently as we vary $C$ over degree $d$ curves? If not, what about in the case $C$ is a nodal curve or $C$ is a general nodal curve?
Edit: In other words, is the composition $H^0(C,N\_{C/\mathbb{P}^2})={\rm Hom}(N\_{C/\math... | https://mathoverflow.net/users/16356 | Do singularities of plane curves deform independently? | A good survey on this problem is the paper by Greuel, Lossen and Shustin
*Equisingular families of projective curves*, see <http://arxiv.org/pdf/math/0612310.pdf>.
In particular, at page 5 one can find the reference [Sev68] to the work of Severi showing that singular points of a nodal curve,
irreducible or not, can b... | 5 | https://mathoverflow.net/users/7460 | 247702 | 112,684 |
https://mathoverflow.net/questions/247695 | 5 | Let $A$ be a finite alphabet, let $S = A^{\mathbb{Z}}$ be the set of bi-infinite sequences of characters from $A$, where $A$ is given the discrete topology and $S$ is given the corresponding product topology, and let $\sigma$ denote the right shift operator.
In order to solve some other problem, I have recently shown... | https://mathoverflow.net/users/39374 | Reference request: shift invariant measures are (locally exactly) approximable by periodic ones | I don't know wheter this counts as standard, but...
There is a paper by Krystyna Ziemian:
*Rotation sets for subshifts of finite type*. Fund. Math. **146** (1995), no. 2, pp. 189--201
containing some general results which after some work yield the conclusion. The reduction of your question to Ziemian's framewor... | 3 | https://mathoverflow.net/users/24676 | 247703 | 112,685 |
https://mathoverflow.net/questions/247566 | 2 | We can define set truncation as a higher inductive type with the following constructors:
* $|-|\_0 : A \to ||A||\_0$
* trunc : $(a\ a' : ||A||\_0)\ (p\ p' : a = a') \to p = p'$
If we replace the type of trunc with $(a\ a' : A)\ (p\ p' : a = a') \to \text{pmap}\ (|-|\_0)\ p = \text{pmap}\ (|-|\_0)\ p'$, then this de... | https://mathoverflow.net/users/62782 | Set truncations in homotopy type theory | In order for this to be a higher inductive type, we also need an induction principle. By analogy with the induction principle for the usual set truncation in section 6.9 of [the HoTT book](https://hott.github.io/book/nightly/hott-online-1040-gd35275f.pdf), we might use the following:
Suppose we are given $B : \| A \|... | 2 | https://mathoverflow.net/users/30790 | 247729 | 112,692 |
https://mathoverflow.net/questions/247743 | 20 | I have seen this enough times that I thought it was common practice, but I recently saw someone on a website about writing tips (can't find the link now) say that this was not good. I am hoping to get opinions from people who review or make publication decisions about whether this is viewed negatively.
Examples:
In... | https://mathoverflow.net/users/19088 | Is it acceptable to use the citation references like [1] or [Joh] as nouns in mathematical writing? | First, I do it all the time and don't really see the objections. A phrase like "In [S] it was shown..." is a good alternative to "Siegel showed, [S], that ...".
Out of curiosity I did some cursory research and looked up the citation habits in Annals of Mathematics 1958. There one author (R.D. Anderson) does use "In [... | 23 | https://mathoverflow.net/users/89948 | 247746 | 112,697 |
https://mathoverflow.net/questions/247745 | 0 | Are there any homological checks I can use to check if a projective morphism is flat? For example, I would expect the following projective morphism to be flat
$$
\textbf{Proj}\left( \frac{\mathbb{C}[s][x,y,z]}{x^3 + y^3 - xyz + sz^3} \right) \to \textbf{Spec}(\mathbb{C}[s])
$$
while
$$
\textbf{Proj}\left( \frac{\mathbb... | https://mathoverflow.net/users/78824 | How can I show flatness for projective morphisms? | For a scheme to be flat over $A^1\_s = Spec(C[s])$ is equivalent to not having irreducible (or embedded) components supported over (closed) points of $A^1\_s$. Since your schemes are hypersurfaces in $A^1\_s \times A^3\_{x,y,z}$, having such a component is equivalent to containing $\{a\} \times A^3\_{x,y,z}$ as a subva... | 5 | https://mathoverflow.net/users/4428 | 247747 | 112,698 |
https://mathoverflow.net/questions/247755 | 33 | We say two topologies $\tau$ and $\rho$ on $X$ are similar if the set of continuous functions $f:(X,\tau) \rightarrow (X,\tau)$ is the same as the set of continuous functions $f:(X,\rho)\rightarrow (X,\rho)$.
Does there exist a topology $\tau$ that is similar to the euclidean topology on $\mathbb R$?
This was asked... | https://mathoverflow.net/users/24478 | can another topology be given to $\mathbb R$ so it has the same continuous maps $\mathbb R\rightarrow \mathbb R$? | The only topology similar to the Euclidean topology on $\mathbb{R}$ is the Euclidean topology.
---
Suppose there is such a topology $\tau$. I'll use "open," "continuous," etc. to mean with respect to the Euclidean topology and "$\tau$-open" etc. for $\tau$.
Since $\tau$ is a refinement of the Euclidean topolog... | 33 | https://mathoverflow.net/users/3075 | 247760 | 112,701 |
https://mathoverflow.net/questions/247762 | 2 | Let $L=\mathbb{Q}(\zeta\_r , a^{1/s})$ where $s|r$. Note that it is a splitting field of $f(x)= x^r-a^{r/s}$ over $\mathbb{Q}$ and thus a Galois extension of $\mathbb{Q}$.
I want to estimate : $$\pi\_L(x)=\#\{p \le x : \mbox{ p splits completely in }L \}$$
Chebotarev's density theorem implies that since $L$ is a Ga... | https://mathoverflow.net/users/70612 | Counting number of primes that split completely in a number field | The unconditional counterpart to that estimate is
$$\pi\_L(x)=\frac{\mathrm{Li}(x)}{[L:\mathbb{Q}]}+\frac{\mathrm{Li}(x^\beta)}{[L:\mathbb{Q}]}+c\_1|\tilde{C}|x\exp(-c\_2 n\_L^{-1/2} \log^{1/2}x)$$
for all $x\geq 2$ such that $\log x \geq c\_3 n\_L\log^2 d\_L$, and where $\beta$ is the possible exceptional zero and... | 2 | https://mathoverflow.net/users/43108 | 247768 | 112,703 |
https://mathoverflow.net/questions/247763 | 8 | At 31:37 in his lecture titled *[What is a manifold?](https://youtu.be/u5DLpAqX4YA?t=31m37s)* posted on Youtube, Mikhail Gromov states that if we do not allow generic functions to exist then the continuum hypothesis is "obviously" true, and that if we do allow generic functions to exist, then the continuum hypothesis i... | https://mathoverflow.net/users/93694 | Does anyone understand this comment about the continuum hypothesis? | I remember reading in Proof from the Book (starting p. 119) that Erdős proved the equivalence between the continuum hypothesis and the following problem :
Let $(f\_\alpha)$ be a pairwise distinct family of analytic functions over $\mathbb{C}$, sur that for any $z\in \mathbb{C}, (f\_\alpha(z))$ is countable. Is the fami... | 1 | https://mathoverflow.net/users/97424 | 247776 | 112,708 |
https://mathoverflow.net/questions/247782 | 6 | Let $X$ be a complex manifold and let $$f\_X^n : \bigoplus\_{p + q = n} H^{p, q}(X) \to H^n\_{\mathrm{DR}}(X, \, \mathbb{C})$$ be the natural map from the Dolbeault cohomology to the de Rham cohomology with complex coefficients.
>
> **Question**. Is there an example of a compact complex manifold $X$ which is not K... | https://mathoverflow.net/users/nan | Examples of non-Kähler compact complex manifolds which satisfy the Dolbeault isomorphism | A compact complex manifold $X$ satisfies the Hodge decomposition $$H^k\_{\mathrm{DR}}(X, \, \mathbb C) = \bigoplus\_{p+q=k}H^{p, q}(X)$$ (possibly without Hodge symmetry) if its Frölicher spectral sequence
$$E\_1^{p,q} = H^{p,q}(X) \Rightarrow H^{p+q}(X)$$
degenerates at $E\_1$. This happens for instance if $X$ satisf... | 12 | https://mathoverflow.net/users/7460 | 247783 | 112,711 |
https://mathoverflow.net/questions/247784 | 2 | Let $X$ be a nonsingular complex toric variety with moment map $\mu : X \to P$ over a convex polytope $P$. Given a facet $F$ of $P$, its preimage $\mu^{-1}(F)$ is a complex codimension 1 subvariety of $X$, so by Poincare duality, induces a cohomological class $\mathfrak{z}\_F \in H^2(X; \mathbb{C})$.
Is $\mathfrak{z}... | https://mathoverflow.net/users/nan | In a nonsingular complex toric variety, is an algebraic cycle over a facet of the quotient polytope integral? | Yes, and in fact the classes $\mathfrak{z}\_F$ generate the integral cohomology ring. See, for instance, Section 5.7 of [these notes](http://pages.uoregon.edu/njp/sg.pdf) by Nick Proudfoot. He proves a stronger result: these classes generate the $T$-equivariant integral cohomology ring.
| 4 | https://mathoverflow.net/users/12840 | 247797 | 112,716 |
https://mathoverflow.net/questions/247794 | 0 | <https://youtu.be/co78AEqsv3s?t=1901>
Within this video at 31:41, Professor Ronald Brown handles knots algebraically, I am confused to how he deals with the crossing indices. He substitutes and simplifies a way of interlacing a separate rope in the trefoil and then joining the ends. Algebraically it equals "1", which... | https://mathoverflow.net/users/97431 | What is this type of knot algebra called within knot theory? | He is computing the [fundamental group](https://en.wikipedia.org/wiki/Fundamental_group) of the complement of the knot. This is also known as the [knot group](https://en.wikipedia.org/wiki/Knot_group) and is a key invariant in knot theory.
More specifically, he is computing the [Wirtinger presentation](https://en.wik... | 2 | https://mathoverflow.net/users/3121 | 247801 | 112,718 |
https://mathoverflow.net/questions/247799 | 2 | In the [following link](http://user.math.uzh.ch/ayoub/PDF-Files/periods-GKZ.pdf)$^{[1]}$, page $2$, we find the following question :
>
> Let $X$ be a smooth $ \mathbb{Q} $ - variety and let $\mathcal{P} \mathrm{er} (X)$ be the subfield of $\mathbb{C}$ generated by the image of the
> pairing $(3)$. What is the tran... | https://mathoverflow.net/users/89900 | What is explicitly, $ \mathcal{P} \mathrm{er} ( X ) / \mathbb{Q} $? | In your definition of $\mathcal P er(X)/\mathbb Q$, replace $\mathbb Q[t\_1,\dots,t\_m]$ with $\mathbb Q(t\_1,\dots,t\_m)$. Since periods are often transcendental, you need denominators to make it a field. It is a finitely generated extension of $\mathbb Q$ because $m$ is finite. This brings us to your second question.... | 4 | https://mathoverflow.net/users/30554 | 247815 | 112,723 |
https://mathoverflow.net/questions/119493 | 3 | Is there a standard test for the recognition of toroidal graphs? I have been using the Boyer-Myrvold planarity algorithm, which has a MATLAB and [C++](http://www.boost.org/doc/libs/1_36_0/libs/graph/doc/boyer_myrvold.html) implementation, and I would like to know if there is something similar for toroidal graphs.
| https://mathoverflow.net/users/22051 | Toroidality testing | Currently there is no *standard* algorithm for testing if a graph can be embedded in a torus. It looks like this is because (1) there isn't too widespread of a need for one yet, and (2) among the available algorithms, there is a huge trade-off between algorithmic complexity and ease-of-implementation so there is no obv... | 4 | https://mathoverflow.net/users/64073 | 247820 | 112,724 |
https://mathoverflow.net/questions/247777 | 2 | Let $X$ be a complete and barrelled (DF)-space. It is known that the bidual $X''$ is a complete (DF)-space. Does it follow that it is also barrelled? More generally, is the bidual of a barrelled space again barrelled?
| https://mathoverflow.net/users/75127 | Biduals of barrelled spaces | According to Grothendieck, a Frechet space $E$ is distinguished if its dual (endowed with the strong topology $\beta(E',E)$ of uniform convergence on all bounded subsets of $E$) is barrelled (or, by a theorem of him equivalently, bornological or ultrabornological). $E'$ is thus a barrelled (DF)-space. [Here](http://lin... | 3 | https://mathoverflow.net/users/21051 | 247831 | 112,727 |
https://mathoverflow.net/questions/247837 | 5 | The semi-classical Einstein equations (without a cosmological constant) are $G^{\mu \nu} = 8\pi \langle T^{\mu \nu} \rangle$.
I am told that there are serious objections as to why these equations cannot be a completely correct description of nature. (If they are correct, then there is no need for quantum gravity.) Fro... | https://mathoverflow.net/users/3709 | Singularity theorems for semiclassical gravity | In terms of singularity theorems:
The Hawking-Penrose singularity theorems require certain *energy conditions* be satisfied; the theorems are in particular **not-sensitive** to the underlying fields being classical or quantum. So the trivial answer to your question is:
>
> **As stated, the singularity theorems ho... | 8 | https://mathoverflow.net/users/3948 | 247853 | 112,733 |
https://mathoverflow.net/questions/247830 | 7 | Is there a Tychonoff space $X$ without isolated points with the following property:
For any $a\in X$ and any function $f : X\longrightarrow \mathbb{R}$, if $f$ is continuous on $X\backslash \{a\}$ then we can redefine $f$ at $x=a$ such that (the new) $f$ is continuous on $X$.
| https://mathoverflow.net/users/72273 | Topological spaces with too many open sets | Consistently, $\omega^\*$ has this property. In the paper
>
>
> >
> > E. van Douwen, K. Kunen, and J. van Mill, "There can be $C^\*$-embedded dense proper subspaces of $\beta\omega - \omega$," *Proc. Amer. Math. Soc.* **105** (1989), pp. 462-470, available [here](https://staff.fnwi.uva.nl/j.vanmill/papers/papers... | 4 | https://mathoverflow.net/users/70618 | 247854 | 112,734 |
https://mathoverflow.net/questions/247866 | 4 | It is known (as a slogan) that the "existential fragment of second-order logic (ESO) is compact".
My first question is:
(1) Is ESO compact for:
(a) uncountable languages
(b) languages with constants or just for relational vocabularies?
The possible answers are obviously:
1. ESO is compact only for countable... | https://mathoverflow.net/users/97463 | Compactness of existential second order logic and definability of certain quantifiers | Existential second-order logic is indeed compact for arbitrary languages. The proof I know is via ultraproducts, which I'll sketch here. (I once saw a Henkinization-style argument, but it was quite messy.)
As in the usual case, suppose $\Gamma$ is a set of ESO sentences which is finitely satisfiable. Let $F$ be the s... | 5 | https://mathoverflow.net/users/8133 | 247867 | 112,740 |
https://mathoverflow.net/questions/247868 | 3 | Let $M$ be a manifold and let $p \mapsto F\_p$ be a foliation, i.e. each $p$ is a point of $M$ and each $F\_p \subset T\_p M$ is a linear subspace of $T\_p M$.
Fix a Riemannian metric on $M$. Is the family of perpendicular subspaces $p \mapsto F\_p^\perp$ a foliation of $M$ as well?
| https://mathoverflow.net/users/97469 | Perpendicular foliation | Take any non-integrable distribution $Q\subseteq TM$ of codimension $1$, for example the kernel of a [contact form](https://en.wikipedia.org/wiki/Contact_geometry) on a $3$-manifold. The distribution $F:=Q^{\perp}$ has dimension $1$. Now in every $1$-dimensional distribution the local sections are closed under Lie brac... | 5 | https://mathoverflow.net/users/4721 | 247872 | 112,743 |
https://mathoverflow.net/questions/247765 | 1 | Let $\sigma \in C(\mathbb R)$, and $X$ be a solution of
\begin{equation}\label{eq:1}
X\_{t} = x + t + \int\_{0}^{t} \sigma(X\_{s}) dB\_{s}
\end{equation}
where $B$ is 1-d Brownian motion under filtered probability space $(\Omega, \mathcal F, \mathbb P, \{\mathcal F\_{t}\}\_{t\ge 0})$.
We denote by $\mathbb P^{x}$ the... | https://mathoverflow.net/users/5656 | Sufficient condition of continuity of the expected stopping time | Referring to $X(t)$ and $\tau(x) = \inf \{ t>0 : X(t)<0 \mid X(0) = x \}$ stated above, a Feynman-Kac formula implies that the function $u(x) = \mathbb{E}^x \exp(-\tau)$ satisfies a second-order, linear differential equation:
$$
\begin{cases}
\frac{1}{2} \sigma(x)^2 u''(x) + u'(x) - u(x) = 0 \\
u(0)=1\;, \quad u(\inft... | 3 | https://mathoverflow.net/users/64449 | 247878 | 112,746 |
https://mathoverflow.net/questions/247838 | 1 | I'm fairly familiar with the literature dealing with convergence of SDEs in $\mathbb{R}^d$ but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However I am quite unfamiliar with the subject.
My question what is a good *reference* treating the following question:
If ... | https://mathoverflow.net/users/36886 | Reference for convergence of Hilbert-space valued SDEs | Check out Chapter 5 for projection-based methods for SDEs on Hilbert spaces:
**Second Order PDE's in Finite and Infinite Dimension: A Probabilistic Approach**
Springer-Verlag
Authors: Cerrai, Sandra
ISBN: 9783540421368
Check out Chapter 10 of the following textbook for a basic intro to convergence of discretization... | 1 | https://mathoverflow.net/users/64449 | 247879 | 112,747 |
https://mathoverflow.net/questions/247631 | 3 | I think [this MSE thread](https://math.stackexchange.com/questions/1887055/measure-theory-and-probability-theory-on-generalizations-of-topological-spaces) is more suitable for the MO community, so I copy it here.
Given a set $X$ and a topology $\tau$ on $X$ the definition of the Borel $\sigma$-algebra $B(X)$ makes us... | https://mathoverflow.net/users/58682 | Measure theory and Probability Theory on generalizations of topological spaces | For question 1: There is another way to generalize the notion of a topology, different from (i), (i') and (ii): Extract an abstract notion of a compact-like class of sets.
The measures that are approximated from within by such compact-like classes have been studied, with interesting non-trivial results. A good entry... | 5 | https://mathoverflow.net/users/95282 | 247881 | 112,748 |
https://mathoverflow.net/questions/247844 | 2 | Let $f:M\rightarrow\mathbb{R}$ be a smooth function, where $M$ is a closed
manifold. I have the feeling that the [Lusternik-Schnirelmann category](https://en.wikipedia.org/wiki/Lusternik%E2%80%93Schnirelmann_category) of $M$
is an estimate from below of the number of *components* of the critical
points set of $f$, rath... | https://mathoverflow.net/users/94090 | Is L-S category estimating the number of components of the critical set? | Doesn't the height function on a torus laying down on its side give a counter-exsmple? There are two circles of critical points, but $\mathrm{cat}(T^2) =3$.
*Added:* A constant function on a connected non-contractible manifold gives an even easier example.
| 3 | https://mathoverflow.net/users/8103 | 247888 | 112,749 |
https://mathoverflow.net/questions/247898 | 8 | In a recent research work, I have come across the following nice identity, where the entries $a,b,x$ belong to an arbitrary commutative unital ring:
$$\begin{vmatrix}
2 & a & b & ab-x \\
a & 2 & x & b \\
b & x & 2 & a \\
ab-x & b & a & 2
\end{vmatrix}=(x^2-abx+a^2+b^2-4)^2.$$
Note that if the ring has characteristi... | https://mathoverflow.net/users/34951 | Determinant of a specific $4 \times 4$ symmetric matrix | Here's a method for calculating the determinant, explaining at least why it ends up as a product. I don't know if there's any significance to your determinant being a square.
Define
$$H=
\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 & 0 & 0 \\
1 & -1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & -1 \\
\end{pmatrix}.
$$
(The ten... | 12 | https://mathoverflow.net/users/35453 | 247900 | 112,752 |
https://mathoverflow.net/questions/247899 | 4 | When constructing a classifying space $BPL$ for piecewise linear microbundles,
one would like it to be a polyhedron, i.e. a locally finite simplicial complex.
Milnor solved this by showing that the isomorphism classes of pl microbundles $mb\_{PL}(S^n)$ over each sphere are countable. Then some special version of Browns... | https://mathoverflow.net/users/91925 | A CW is of countable type, iff all its homotopy groups are countable? (References?) | If $X$ is simply-connected, then the homotopy groups will be countable iff the homology groups are countable; and then one can build $X$ by a homology resolution using Moore spaces for countable groups, resulting in countably many cells.
| 3 | https://mathoverflow.net/users/3634 | 247907 | 112,755 |
https://mathoverflow.net/questions/243809 | 4 | While studying a portfolio optimization problem, I came across the process
$$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1\_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$
which has a discountinuous drift: once the process crosses a pre-defined and fixed threshold $C>0$, it obtains an additional negativ drift of $-\al... | https://mathoverflow.net/users/66399 | Stochastic process with discontinuous drift | You are absolutely on the right track, and in fact, your change of measure is well-defined. In order to fully leverage this change of measure, one must *eliminate* the time integrals that appear in it. This is nicely done in **Brownian Motion and Stochastic Calculus** by Ioannis Karatzas and Steven E. Shreve (1991); se... | 2 | https://mathoverflow.net/users/64449 | 247928 | 112,764 |
https://mathoverflow.net/questions/247884 | 3 | Consider the longest runs $\ell\_\sigma(x)$ of the pattern $\sigma$ for $\sigma\in \{0, 1, 01, 10, 001,\dots\}$ etc. in a binary sequence $x=x\_1\dots x\_n$.
For example, $\ell\_{001}(0001110010011001)=2$ since 001001 is a contiguous subsequence of that sequence but 001001001 is not.
And $\ell\_0$ is approximately ... | https://mathoverflow.net/users/4600 | Longest runs and concentration of measure | You can certainly prove the following: for each $L>0$, for all $\epsilon>0$, there exists an $n\_0$ such that if $n>n\_0$, then with probability at least $1-\epsilon$,
$$
\Big|\frac{\ell\_\sigma(x\_1\ldots x\_n)}{\log\_2 n/|\sigma|}-1\Big|<\epsilon
\text{ for each $\sigma$ such that $|\sigma|\le L$.}
$$
I wouldn't say... | 3 | https://mathoverflow.net/users/11054 | 247929 | 112,765 |
https://mathoverflow.net/questions/247935 | 2 | "Is there a topological groupoid structure on the pair $(Gl(n,\mathbb{R}), O(n))$, with their standard topologies?"
This is already asked here but this linked question is a very general question, so we consider its special case about general linear group, as an independent question.
[A topological groupoid structur... | https://mathoverflow.net/users/36688 | The pair $(Gl(n,\mathbb{R}), O(n) )$ as a groupoid | There is an easy differentiable groupoid structure: Use the Iwasawa decomposition
$G=KAN= K\times A\times N$. In the case of the question this is the Gram-Schmidt orthogonalization proceedure for the rows and remembering the coefficients as an upper triangular matrix with positive entries on the main diagonal: $GL(n)\n... | 6 | https://mathoverflow.net/users/26935 | 247939 | 112,767 |
https://mathoverflow.net/questions/247938 | 1 | Let $f:X \to Y$ be a birational morphism between projective, irreducible surfaces. Assume $X$ is non-singular and $Y$ is a hypersurface in $\mathbb{P}^3$ (not necessarily smooth). Is the arithmetic genus of $X$ equal to that of $Y$?
More generally, given a smooth projective irreducible surface $X$, is it possible to ... | https://mathoverflow.net/users/43198 | Birational morphism and invariance of arithmetic genus | These are two different questions.
1) No. The arithmetic genus of a degree $d$ surface $Y\subset\mathbb{P}^3$ is $\chi (\mathcal{O}\_Y)-1=\binom{d-1}{3}$, regardless of the singularities of $Y$. If $Y$ has a $(d-1)$-uple point, it is rational, hence any desingularization $X$ of $Y$ has arithmetic genus $0$.
2) No. ... | 6 | https://mathoverflow.net/users/40297 | 247940 | 112,768 |
https://mathoverflow.net/questions/247930 | 1 | Inspire from [Kantorovich Inequality](http://mathworld.wolfram.com/KantorovichInequality.html) and [my previous question](https://mathoverflow.net/questions/245092/a-conjecture-generalization-of-karamata-inequality). I am looking for a proof of the nice inequality as following:
Let $f(x)$ is a real continuous functio... | https://mathoverflow.net/users/76698 | Inspire from Kantorovich Inequality | This is true. Choose a linear function $\ell(x)$ such that $\ell(m)=f(m),\ell(M)=f(M)$. Denoting $g=f-\ell$ we get $g(m)=g(M)=0$ and have to prove $$ng\left(\frac{x\_1+\cdots+x\_n}{n}\right)-2ng\left(\frac{M+m}{2}\right) \geqslant g(x\_1)+\cdots+g(x\_n).$$
But $g$ is convex, thus $g$ is non-positive on $[m,M$]. So, RHS... | 2 | https://mathoverflow.net/users/4312 | 247941 | 112,769 |
https://mathoverflow.net/questions/247673 | 6 | Let $A$ be a real $n \times n$ matrix. Denote by $\operatorname{cof} A$ The [cofactor matrix](https://en.wikipedia.org/wiki/Adjugate_matrix) of $A$. By definition, $A (\operatorname{cof} A)^T=\det A \cdot I$.
Thus, it is immediate that $A \in \operatorname{SO}\_n$ if and only if $$ (\*\*) \operatorname{cof} A =A,\det... | https://mathoverflow.net/users/46290 | What is special in dimension $2$ (When characterizing isometries using the cofactor matrix)? | This is somewhat informal. My argument is in the spirit of dimensional analysis in physics.
The essential point is that the cofactor matrix encapsulates $n-1$ dimensional volume, while the dimension of the matrix $A$ in this sense is $1$. (Think the of operator norm, for instance. The "size" of the elements of $A$ ro... | 1 | https://mathoverflow.net/users/46290 | 247945 | 112,771 |
https://mathoverflow.net/questions/243956 | 3 | Consider a diffusion given by,
$d X\_t = \mu(X\_t) dt + \sigma(X\_t) dB\_t$
$X\_0 = x$.
Suppose the functions $\mu$ and $\sigma$ are as follows -
$f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \text{ if } x \ge 0 \\
1 & \text{ if } x < 0 \end{cases}
$
The purpose of $f(x)$ will be clear in a moment.
By Na... | https://mathoverflow.net/users/78761 | Differentiability of a simple value function driven by a diffusion | Here is a rough probabilistic argument for differentiability of $\nu(x)$. At least formally, we have:
$$
\nu^{\prime}(x) = \mathbb{E} \int\_0^{\infty} e^{-t} \delta(X\_t(x)) X\_t^{\prime}(x) dt \tag{$\star$}
$$ since the derivative of a step function, in the distributional sense, is a Dirac delta function $\delta(\cd... | 2 | https://mathoverflow.net/users/64449 | 247948 | 112,772 |
https://mathoverflow.net/questions/247944 | 3 | Let E be a (Mitchell-Steel) extender over some M. Recall that the natural length of E, $\nu\_E$, is the strict sup of the generators of E and $\kappa^{+M}$. It is claimed in both "Fine structure and iteration trees" and Steel's handbook article that if the natural length is a limit ordinal then it's also a cardinal in ... | https://mathoverflow.net/users/38602 | Natural length is a cardinal if it's a limit ordinal | it should follow from the fact that generators are critical points. if $ \xi<\nu\_E$ is a generator of E then $\xi$ is a critical point of the canonical factor map
$ k: Ult(M, E|\xi)\rightarrow Ult(M, E)$
Let $\alpha$ be the index of $E$, we claim that if $\nu\_E$ is a limit ordinal then $J\_\alpha^M\models ``\nu$ ... | 4 | https://mathoverflow.net/users/20584 | 247952 | 112,773 |
https://mathoverflow.net/questions/247954 | 1 | In the book of Olivier Debarre (Tores et variétés abéliennes complexes), at the moment where he defines divisors (p. 40), he says that one could define a divisor of a meromorphic function as a formal sum of codimension 1 sets (which we can define easily with the associated Hausdorff measure on the complex manifold) at ... | https://mathoverflow.net/users/56191 | Definition of divisors on complex manifolds | I don't see the role of Hausdorff measure here. You can define codimension 1 sets by their being the closed sets which are locally identified in charts with codimension 1 sets, i.e. with zero loci of analytic functions. This is standard, but be careful of the distinction between Weil divisors and Cartier divisors on si... | 3 | https://mathoverflow.net/users/13268 | 247956 | 112,774 |
https://mathoverflow.net/questions/247733 | 3 | What are explicit expressions for the operators $I,J$ and $K$ on the hyperkähler manifold $TS^2$ (or $TRP^2$), say, in the coordinates $(x\_1,x\_2, \alpha\_1, \alpha\_2)$ where a line $m\in TS^2$ is given parametrically by $(x\_1+\alpha\_1t,x\_2+\alpha\_2t,t)$? Are they known? I did not find them in
the book "The geom... | https://mathoverflow.net/users/97381 | Hyperkähler structure on $TS^2$ | I will explain how you can get the formulae for the closed $2$-forms that define the hyperKähler structure without introducing coordinates. Once you know the $2$-forms, you can find the $I$, $J$, and $K$ by well-known formulae.
Start with the fact that we can think of $TS^2$ as the space of pairs $(a,b)$ where $a$ an... | 5 | https://mathoverflow.net/users/13972 | 247963 | 112,777 |
https://mathoverflow.net/questions/247968 | 0 | I'm reading a few books on category theory, and they talk about a functor from Set to List, with a object s in Set mapped to a list of elements of s. However, there are many lists possible from s, whereas I thought a functor should take one object to one object.
Am I missing something? I'm a programmer, and this func... | https://mathoverflow.net/users/97540 | Cannot understand the functor from Set to List | As a functor, applyall should be a functor $L$ from Set to Set (not to some other category List --- I'm not even sure what category is meant by "List"), and it should take any object $s$ of Set (i.e., a set $s$) to the set $L(s)$ of **all** lists of elements of $s$ (not to any particular list). Its action on functions ... | 5 | https://mathoverflow.net/users/6794 | 247974 | 112,784 |
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