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https://mathoverflow.net/questions/295034 | 3 | Let $n \in \mathbb{N}$. Let $A,B,C$ real matrices of size $n \times n$. Let $\alpha,\beta,\gamma,\delta \in \mathbb{R}^{4}$ such that $\alpha,\beta$ are non-zero.
I an looking for two matrices $T\_1$ and $T\_2$ belonging to $\mathbb{R}^{n \times n}$ such that:
\begin{eqnarray}
\alpha A T\_1 + T\_1 B &=& \gamma C T\_... | https://mathoverflow.net/users/101003 | Coupled Sylvester equations | There is a simpler criterion that does not involve constructing a $2n^2 \times 2n^2$ matrix and checking its singularity (which is a costly operation).
Your equation is equivalent to the standard Sylvester equation (with rectangular unknown)
$$
M \begin{bmatrix}T\_1\\T\_2\end{bmatrix} + \begin{bmatrix}T\_1\\T\_2\end{... | 4 | https://mathoverflow.net/users/1898 | 295102 | 129,744 |
https://mathoverflow.net/questions/295080 | 1 | Let $f$ be a rational map of degree $d\geq 2$, and $B$ is a simply connected immediate basin of an supper-attracting fixed point of $f$. I want to know whether there exists a fixed point of $f$ contained in $\partial B$. Any hint will be welcome! Or some similar result can be given!
| https://mathoverflow.net/users/22498 | Does the boundary of immediate basin contain a fixed point? | Yes. This was proved by Fatou,
Sur les equations fonctionnelles, Bull SMF, 48 (1920) on p. 81.
See also:
MR1295160
F. Przytycki, A. Zdunik,
Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique,
Fund. Math. 145 (1994), no. 1, 65–77.... | 2 | https://mathoverflow.net/users/25510 | 295110 | 129,749 |
https://mathoverflow.net/questions/295089 | 8 | EFA can prove the exponential function to be total, but it cannot prove the superexponential function to be total. Is there an analog of Tennenbaum's theorem (which states the PA has no recursive non-standard models) for EFA which states that EFA has no "sub-superexponential" non-standard models? (Here "sub-superexpone... | https://mathoverflow.net/users/20781 | Analog of Tennenbaum's theorem for EFA | Tennenbaum’s theorem applies unrestricted to EFA, and even to the much weaker theory $IE\_1$ (Robinson’s arithmetic + induction for bounded existential formulas):
* $IE\_1$ has no recursive nonstandard models (Wilmers [1]).
On the other hand, $\mathit{IOpen}$ (induction for quantifier-free formulas) does have recur... | 12 | https://mathoverflow.net/users/12705 | 295115 | 129,750 |
https://mathoverflow.net/questions/295114 | 3 | I'd like to prove the following:
>
> $$ a + b + 4 \sqrt{1 + a^{2} + b^{2}} \leq 4 \sqrt{a^{2} + b^{2}} +
> \sqrt{1+b^{2}} + \sqrt{1+a^{2}} + 2 $$
>
>
>
for all $a, b \in \mathbb{R}\_{>0}$.
**Question**: is there a way to prove this inequality?
**Motivation**: In a [paper](https://pure.uvt.nl/ws/files/6... | https://mathoverflow.net/users/93724 | How to prove that $a + b + 4 \sqrt{1 + a^{2} + b^{2}} \leq 4 \sqrt{a^{2} + b^{2}} + \sqrt{1+b^{2}} + \sqrt{1+a^{2}} + 2 $ for all $a, b > 0$? | This rewrites as $4f(a^2 +b^2)\le f(a^2) +f(b^2 )+2f(0) $ for a decreasing function $f(x)=\sqrt{x+1}-\sqrt{x}=1/(\sqrt{x+1}+\sqrt{x})$.
| 25 | https://mathoverflow.net/users/4312 | 295116 | 129,751 |
https://mathoverflow.net/questions/281632 | 20 | Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 19... | https://mathoverflow.net/users/66889 | Open subsets of Euclidean space in dimension 5 and higher admitting exotic smooth structures | I finally talked to Rob and did some literature search. Here are some examples of open subsets of Euclidean spaces which are homeomorphic but not diffeomorphic.
Let $\Sigma$ be an exotic $(n-1)$-dimensional sphere which can be realized as a Brieskorn variety (see [here](http://www.map.mpim-bonn.mpg.de/Exotic_spheres... | 7 | https://mathoverflow.net/users/21684 | 295119 | 129,753 |
https://mathoverflow.net/questions/295141 | 2 | Let $(X^{n,m}, d'^{n,m}, d''^{n,m})$ be a double complex. In Kashiwara, Schapira *Categories and Sheaves* as a corollary of
*Theorem.* Let $f: X \rightarrow Y$ be a morphism of bounded double complexes such that $f$ induces an isomorphism $f: H\_{II}(H\_{I}(X)) \rightarrow H\_{II}(H\_{I}(Y))$, then $f$ induces an is... | https://mathoverflow.net/users/91572 | Double complexes with exact rows | After taking the cohomology of the rows, only one row survives. Then taking cohomology of the columns has no effect. Thus, in this case
$H\_{II}(H\_I(X))$ is isomorphic to the cohomology of this row.
But the fact that all the rows except the $p$-th one are exact is true also for the stupid truncation, because we only... | 2 | https://mathoverflow.net/users/115052 | 295143 | 129,762 |
https://mathoverflow.net/questions/294932 | 5 | *(Background, may be skipped by the knowledgeable reader: A Galois-Tukey connection between two partial orders $(P,\le)$ and $(Q,\le)$ is a pair of maps $\varphi^+:P\to Q$ and $\varphi^-:Q\to P$ satisfying
$$ \forall p\in P \ \forall q\in Q: \left[ \varphi^-(q)\le p \Rightarrow q\le \varphi^+(p)\right]$$
for all $p\in ... | https://mathoverflow.net/users/14915 | A "strong" Galois-Tukey connection between orders with suborders | Martin, first definition is just J.W.Tukey cited in my <http://www.ksi.mff.cuni.cz/~vojtas/MathPub/1993_GeneralizedGaloisTukey.pdf> . We have to be careful the implication should not be true in a void sense (false implies \* is true) - so some restrictions on $P\_0, \phi^- , ...$ are necessary. In my Galois-Tukey relat... | 3 | https://mathoverflow.net/users/121845 | 295145 | 129,764 |
https://mathoverflow.net/questions/206448 | 2 | When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere.
Then when the list "**Questions that may already have your answer**" appeared, the first on the list was [Constructing Markov traces simply](https://mathoverflow.net/q/9... | https://mathoverflow.net/users/41291 | Are Markov traces matrix traces? | You can construct Markov traces explictly in tensor product representations of braid groups. If $V$ is a finite dimensional vector space and $R\in{\rm End}(V\otimes V)$ an invertible solution on the Yang-Baxter equation, represent the generators of the $n$ strand braid group $B\_n$ on $V^{\otimes n}$ by $R\_k:={\rm id}... | 1 | https://mathoverflow.net/users/23753 | 295160 | 129,769 |
https://mathoverflow.net/questions/295135 | 2 | I am looking at the Hilbert Projection Theorem, which states that every non-empty closed convex set in a Hilbert space admits a unique element that has the minimum norm in the set.
The proof involves taking the infimum of the norm on the set and constructing a seqeuence in the set with decreasing norm. However, this ... | https://mathoverflow.net/users/94546 | Constructive version of Hilbert Projection Theorem | A constructive equivalent might be: Any *located* closed convex set has a unique element of minimal norm. This is roughly the version in chapter 7, problem 11 of Bishop and Bridges’s 1985 *Constructive Analysis.*
Here, locatedness is a standard constructive assumption, meaning that the distance to the set exists, or ... | 2 | https://mathoverflow.net/users/nan | 295164 | 129,771 |
https://mathoverflow.net/questions/247552 | 21 | On chapter III.4 ("Metrized $\mathcal{o}$-modules") of this book on algebraic number theory, Neukirch credits his treatment of the theory of finitely generated $\mathcal{o}$-modules to the course "Arakelov theory and Grothendieck-Riemann-Roch" taught by Günter Tamme. He continues:
>
> There, however, proofs were no... | https://mathoverflow.net/users/43108 | Günter Tamme's course "Arakelov theory and Grothendieck-Riemann-Roch" | Here is [a link](https://drive.google.com/file/d/1GLqlREiVA1eNbsC3axBY9IhJXiSbsQB-/view?usp=sharing) of Gunter Tamme's course. I take a brief look and it seems centering around proving Gronthendieck-Riemann-Roch using K-theory machinery. I did not see Arakelov theory anywhere. The course notes at here are incomplete (t... | 3 | https://mathoverflow.net/users/18850 | 295170 | 129,773 |
https://mathoverflow.net/questions/295181 | 1 | Define the Cartan permanent of a finite dimensional algebra as the permanent of the Cartan matrix.
Is the Cartan permanent of a finite dimensional algebra with finite global dimension always an odd integer?
This is true for acyclic quiver algebras and experiments suggest that it is also true for all Nakayama algebras.... | https://mathoverflow.net/users/61949 | Is the Cartan permanent odd for finite global dimension? | A theorem of Eilenberg asserts that the Cartan determinant of a finite-dimensional algebra of finite global dimension is equal to $\pm 1$. Hence the Cartan determinant and the Cartan permanent are both congruent to $1$ modulo $2$ in this situation.
The result is Proposition 21 in the following reference.
* Samuel ... | 6 | https://mathoverflow.net/users/296 | 295195 | 129,782 |
https://mathoverflow.net/questions/295185 | 28 | Is there (if not, why?) a software where I can input a sequence of integers, like into the [OEIS](https://oeis.org/), and then it makes some simple transformations on it to check whether the sequence can be obtained from some other sequence(s)?
For example, if I enter 2, 4, 6, 8, 10, then currently OEIS returns A0058... | https://mathoverflow.net/users/955 | Advanced software for OEIS? | There has been some previous discussion on the [OEIS mailing list](http://list.seqfan.eu/pipermail/seqfan/) about similar topics. For instance, about [the sum of two sequences](http://list.seqfan.eu/pipermail/seqfan/2015-July/015023.html). (If I recall correctly, there was a university project that performed a more in ... | 15 | https://mathoverflow.net/users/51771 | 295196 | 129,783 |
https://mathoverflow.net/questions/295191 | 4 | Let $X$,$Y$ be compact, connected, smooth manifolds of the same dimension. Suppose you have a surjective smooth map $f : X \rightarrow Y$, such that $|f^{-1}(p) | \leq k$ for all $p \in Y$.
Let $U \subset X$ be an open dense subset.
**Question**: Is it possible to find $p \in Y$ such that $f^{-1}(p) \subset U$?
| https://mathoverflow.net/users/99732 | Map between manifolds and open dense subsets | Yes. I think you do not even have to assume that the preimages are finite. $f$ can be any $C^1$ mapping from $X$ onto $Y$. Let $y\in Y$ be a regular value of the mapping $f$. Then for some $\epsilon>0$
$$
f^{-1}(B(y,\epsilon))=\bigcup\_{i=1}^m U\_i,
$$
and $f|\_{U\_i}$ is a diffeomorphism of $U\_i$ onto $B(y,\epsilon)$... | 7 | https://mathoverflow.net/users/121665 | 295202 | 129,788 |
https://mathoverflow.net/questions/295154 | 19 | How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of showing $100$% of elliptic curves over $\mathbf{Q}$ satisfy the BSD conjecture? (which, of course, would still not prove th... | https://mathoverflow.net/users/nan | Bhargava's work on the BSD conjecture | This was meant to be a comment, but it won't fit, so here we go.
Just a few naive observations.
For completeness:
>
> **Weak BSD** For any global field $K$ and any abelian variety $A$ defined over $K$, we have:
> $$\text{ord}\_{s=1}L(A/K,s) = \text{rk}\_{\mathbf{Z}}A(K).$$
>
>
> **BSD** For any global field ... | 8 | https://mathoverflow.net/users/nan | 295204 | 129,789 |
https://mathoverflow.net/questions/295061 | 5 | Definition of $C\_L$ for people who like number theory:
Let $m$ be a number with prime factorisation $m=p\_1^{n\_1} ... p\_r^{n\_r}$ with $n\_i>0$. Define $I\_m$ to be the incidence algebra of the divisor lattice of $m$. Up to isomorphism this just depence on the $n\_i$ and not on the primes so lets define with $L=[n... | https://mathoverflow.net/users/61949 | Periods in the trivial extension algebra of the incidence algebra of the divisor lattice | Calculating periods of simple modules for the trivial extension algebra $TA$ can be reduced to a calculation with $A$-modules (at least if $A$ has finite global dimension), which is much easier.
The starting point is a result of Happel, that for a finite-dimensional algebra $A$ with finite global dimension, the bound... | 5 | https://mathoverflow.net/users/22989 | 295211 | 129,792 |
https://mathoverflow.net/questions/295210 | 8 | Classical formulation
=====================
Consider the (untwisted) Dijkgraaf-Witten invariant, defined for an oriented, connected, closed manifold $M$ and a finite group $G$:
$$DW\_G(M) := \lvert \operatorname{Hom}(\pi\_1(M), G )\rvert $$
We're counting group homomorphisms from the fundamental group of the mani... | https://mathoverflow.net/users/13767 | Is Yetter's invariant multiplicative under connected sum? | As Kevin Walker pointed out in a comment, Dijkgraaf-Witten invariants are weighted by $1/\mathrm{Stab}(\rho)$. In
the same way, the Yetter invariant for $\mathcal T$ and $M$ is generally defined such that it's weighted using the
2-groupoid cardinality of $\pi\_{\le 2}\mathrm{Map}(M, \mathcal T)$, so that the invariant ... | 3 | https://mathoverflow.net/users/97265 | 295219 | 129,794 |
https://mathoverflow.net/questions/238113 | 8 | For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.
Specifically I want find the Toeplitz matrix such that the $L^2$ norm between the eigenvectors of the Toeplitz matrix and eigenvectors of the matrix $A$ is m... | https://mathoverflow.net/users/71105 | Finding Toeplitz matrix nearest to a given matrix | The set of $n \times n$ symmetric Toeplitz matrices is
$$\left\{ x\_1 \mathrm M\_1 + x\_2 \mathrm M\_2 + \cdots + x\_n \mathrm M\_n \mid x\_1, x\_2, \dots, x\_n \in \mathbb R \right\}$$
where $\mathrm M\_1, \mathrm M\_2, \dots, \mathrm M\_n$ are $n \times n$ symmetric Toeplitz **basis** matrices. Let $\mathrm M\_1 ... | 7 | https://mathoverflow.net/users/91764 | 295232 | 129,800 |
https://mathoverflow.net/questions/295229 | 11 | Let $V$ be an algebraic variety. If there is a finite ascending chain of Zariski closed sets $\emptyset=V\_0\subset V\_1\subset \cdots \subset V\_n=V$ such that $V\_i-V\_{i-1}$ is a fintie disjoint union of copies of affine space $\mathbb{A}^i$ we say $V$ is *affine paved* (so $V$ is "algebraically cellular").
Note:... | https://mathoverflow.net/users/12218 | Affine Paved Affine Varieties; not Affine Space | One example is $uw = v(v+1)$. The equations $u=v=0$ cut out an $\mathbb{A}^1$, and the complement is isomorphic to $\mathbb{A}^2$ by the map $(x,y) \mapsto (y, xy-1, x(xy-1))$. Note that the hypersurface $u=v=0$ is not principal; by dhy's comment, it can't be.
More generally, any of the Danielewski surfaces $u w^k = ... | 14 | https://mathoverflow.net/users/297 | 295236 | 129,801 |
https://mathoverflow.net/questions/295230 | 8 | $\newcommand{\nat}{\mathbb{N}}$
$\newcommand{\then}{\ \Longrightarrow\ }$
A partial function $f : \mathbb{N} \to \mathbb{N}$ is said to be $\lambda$-*definable* if there is a term $F \in \Lambda$ such that
$$ \forall n \in \nat: \quad f(m)\!\downarrow = n \then
F c\_n =\_\beta c\_m $$
where $=\_\beta$ is beta conversio... | https://mathoverflow.net/users/39329 | Is every total computable function definable by a normalizing lambda term? | Yes, you can actually encode all computable functions by normalizable terms since it is always possible to turn a term into a normal form which computes the same function on Church's numerals. We just need to put $n (\lambda x. x)$ in front of every lambda in your term, where $n$ is the argument of the function. Formal... | 7 | https://mathoverflow.net/users/62782 | 295238 | 129,802 |
https://mathoverflow.net/questions/295233 | 2 | If we work in [General Extensional Atomic Mereology](https://plato.stanford.edu/entries/mereology/) [without bottom], so the primitives of the language are $P$ standing for "is a part of", and equality, now we add to it membership $\in$ relation symbol.
I observed that all rules of $\text{ZF}$ can be derived from the... | https://mathoverflow.net/users/95347 | Can rules of set theory be founded by paralleling parts of atomic Mereology? | Many people find it natural to consider the set-theoretic analogue of mereology to be the inclusion relation $\subseteq$ rather than $\in$, since after all, $\subseteq$ is reflexive and transitive and admits relative complements, and these are all axioms listed on the mereology page to which you link, but $\in$ does no... | 10 | https://mathoverflow.net/users/1946 | 295246 | 129,804 |
https://mathoverflow.net/questions/295244 | 5 | Let $S$ be an inverse semigroup. Define a category $C(S)$ as follows:
* the objects of $C(S)$ are the elements of $S$
* for any $a,b,e\in S$ let $e\colon a\to b$ be a morphism of $C(S)$ iff $aa^{-1}eb^{-1}b=e$
* if $e\colon a\to b$ and $f\colon b\to c$ are morphisms, let $f\circ e\equiv eb^{-1}f\colon a\to c$ (this i... | https://mathoverflow.net/users/7227 | a category associated with an inverse semigroup | I think this is isomorphic to what Alfredo and I call the Schutzenberger category of a semigroup in [this paper (arXiv link)](https://arxiv.org/abs/1408.1615) .
We define it for semigroups in general but it should boil down to what you wrote for inverse semigroups. We however would say your arrow e goes from b to a.... | 5 | https://mathoverflow.net/users/15934 | 295252 | 129,805 |
https://mathoverflow.net/questions/295251 | 2 | Let $S$ be a smooth connected noetherian scheme (not necessarily over a field) with residue fields that are all of finite type over their prime field.
Let $f: \mathcal{X}\to S$ be a smooth projective morphism.
Let $s'\to s$ be a specialization of points in $S$, ie. $s\in\overline{\{s'\}}$.
>
> (1) If the Lefsch... | https://mathoverflow.net/users/nan | Lefschetz standard conjecture under specialization/generization | If your Weil cohomology theory is étale cohomology, then (2) is easy. Indeed, we have specialisation isomorphisms (for $\ell \neq \operatorname{char} \kappa(s)$) [SGA4$\_3$, Exp. XVI, Cor. 2.2]
$$H^i\_{\text{ét}}(\mathcal X\_{s'},\mathbb Q\_\ell) \stackrel\sim\to H^i\_{\text{ét}}(\mathcal X\_s, \mathbb Q\_\ell)$$
as we... | 5 | https://mathoverflow.net/users/82179 | 295259 | 129,808 |
https://mathoverflow.net/questions/294910 | 13 | A while ago I asked [a question](https://mathematica.stackexchange.com/q/143766/35000) at Mathematica.SE about how to factorize a polynomial into terms with as few monomials as possible each. I now realized that I actually do not know what is rigorous mathematics behind this.
In fact that one was about univariate pol... | https://mathoverflow.net/users/41291 | Factorization of polynomials into "shortest possible" factors | This is essentially a question about algebraic circuit complexity. Namely, we can consider formulas which are products of sums of powers of variables (which, using standard notation from circuit complexity, would be called "$\Pi\Sigma\wedge$ formulas"), and you are asking for the smallest expression which evaluates to ... | 7 | https://mathoverflow.net/users/38434 | 295264 | 129,810 |
https://mathoverflow.net/questions/295225 | 3 | Let $\Sigma$ be a genus $g$ closed orientable surface and let $\alpha = \{ \alpha\_1,...,\alpha\_g\}$ and $\beta = \{ \beta\_1 ,..., \beta\_g \}$ be two cut systems (i.e. nonintersecting homologically linearly independent sets of $g$ curves) on $\Sigma$. Is there an algorithm to tell if $\alpha$ is handleslide equivale... | https://mathoverflow.net/users/99414 | An algorithm to tell if two cut systems are handle slide equivalent? | Yes, here is a sketch of one such proof. Let $V = V\_\alpha$ be the genus $g$ handlebody determined by $\alpha$. The curves $\beta$ now give conjugacy classes in $\pi\_1(V)$. Any one of the $\beta\_i$ bounds a disk if and only if its class is trivial. (Strictly speaking, this requires the disk theorem, but there is a e... | 3 | https://mathoverflow.net/users/1650 | 295275 | 129,815 |
https://mathoverflow.net/questions/291422 | 8 | Working on some problems in the $C\_p$-theory I discovered the following simple but amazing
**Fact.** *For any subset $A\subset \mathbb R^n$, non-zero vector $a\in \bar A\subset\mathbb R^n$ and $\varepsilon>0$ there are points $a\_1,\dots,a\_n\in A$ and real numbers $t\_1,\dots,t\_n$ such that $a=\sum\_{i=1}^nt\_ia\... | https://mathoverflow.net/users/61536 | Almost convex combinations in $\mathbb R^n$ | I do not know the reference, but it looks that even more may be achieved: $t\_1,\dots,t\_{n-1}$ may be chosen close to 0 and $t\_n$ close to 1. Indeed, if the span of $a\cup A$ is spanned by linearly independent vectors $a, a\_1,\dots,a\_k$, $k\leqslant n-1$, then choose $a\_{n} $ close to $a$, we get $a\_{n}=c a+c\_1a... | 3 | https://mathoverflow.net/users/4312 | 295286 | 129,820 |
https://mathoverflow.net/questions/288358 | 6 | Let us consider polynomials as functions on $[0,1]$, and so define
\begin{align\*}
\|f\|\_2 &= \sqrt{\int\_0^1f(x)^2\,dx} \\
\|f\|\_\infty &= \max\{|f(x)|: 0 \leq x\leq 1\}.
\end{align\*}
I am interested in the ratio of these norms. It is easy to see that $\|f\|\_2\leq\|f\|\_\infty$, with equality only for constant ... | https://mathoverflow.net/users/10366 | Reference request: maximal ratio of different norms of polynomials | So, I write this as an answer rather than a comment to close the question.
This is problem VI.103 in volume 2 of Polya and Szego. For the interval $[-1,1]$, the extremal polynomial is
$$\frac{P\_{n}(x)-P\_{n+1}(x)}{1-x},$$
where $P\_{n}$ is the Legendre polynomials of degree $n$ (with normalization $\int P\_{n}^{2}(... | 1 | https://mathoverflow.net/users/89429 | 295291 | 129,823 |
https://mathoverflow.net/questions/295262 | 7 | If a positive Radon measure $\mu$ and the Lebesgue measure $\lambda$ are singular, can we show that the derivative of $\mu$ with respect to $\lambda$ is $\infty$, $\mu$-a.e.? Namely, can one show that
$$\frac{\mu(B(x,r))}{\lambda(B(x,r))}\to\infty \quad \mbox{ as } \ r\to0, \quad \mu\mbox{-a.e.?}$$
Does this result a... | https://mathoverflow.net/users/121733 | If a measure $\mu$ and Lebesgue measure $\lambda$ are singular, is the derivative of $\mu$ with respect to $\lambda$ $\infty$, $\mu$-a.e.? | Let A be such that $\mu(A) = 1, \lambda(A) = 0$. There is an open set $O$ with $ A \subset O, \lambda(A) < \epsilon$. Suppose for convenience that $$\liminf\frac{\mu(B(x,r))}{\lambda(B(x,r))}< a $$ on $A$, otherwise replace $A$ with the set on which it is true. For each x in A pick $r\_x$ so that $$B(x, r\_x) \subset O... | 6 | https://mathoverflow.net/users/nan | 295297 | 129,826 |
https://mathoverflow.net/questions/295292 | 4 | I am currently working on some problems related to Grassmann manifolds and eventually come to the following question.
>
> Let $S$ be a subset of $G\_1(\mathbb{R}^n)$ such that any element of $G\_{n-1}:=G\_{n-1}(\mathbb{R}^n)$ contains a line in $S.$ Is it true that there exists a hyperplane in $G\_{n-1}$ that is co... | https://mathoverflow.net/users/62818 | Subset of $G_1(\mathbb{R}^n)$ having a line in common with every hyperplane of $G_{n-1}(\mathbb{R}^n)$ | **The answer is in the negative** and a construction of a counterexample is provided here. I wrote this argument quickly, but I hope it is correct. Let me know if it is not.
We will construct a family of lines passing through the origin in $\mathbb{R}^3$ so that:
**1. Every plane passing through the origin contains a... | 1 | https://mathoverflow.net/users/121665 | 295304 | 129,830 |
https://mathoverflow.net/questions/295248 | 6 | Let $\mathcal{C}$ and $\mathcal{D}$ be symmetric monoidal categories and assume that the symmetric monoidal product $\otimes\_{\mathcal{D}}$ on $\mathcal{D}$ preserves colimits in both variables.
Then the Day convolution $\*$ defines a symmetric monoidal structure on the category of functors $Fun(\mathcal{C},\mathcal{D... | https://mathoverflow.net/users/91925 | Does the Day convolution induce the structure of a bimonoidal category on $Fun(C,D)$? | I don't know in general whether $\otimes\_D$ distributes over $\ast$. However, I think it is unlikely, because there is in general a different compatibility between them: they are a [duoidal category](https://ncatlab.org/nlab/show/duoidal+category#examples_2). This is stated without proof on the nLab page, probably as ... | 4 | https://mathoverflow.net/users/49 | 295308 | 129,833 |
https://mathoverflow.net/questions/295310 | 2 | Suppose $M$ is a smooth manifold and $P$ is a principal bundle on $M$. Let $U^\prime\Subset U\Subset M$ be strictly contained precompact open subsets. Let $g\in C^\infty(U, \hbox{Ad}P|\_U)$ be a gauge transformation of $P$ on $U$, is it always possible to extend $g|\_{U^\prime}$ to be a global gauge transformation, wha... | https://mathoverflow.net/users/98536 | Extend a gauge transformation | This is not always possible, even if the structure group is compact.
Let $M = \mathbb{R}^4, U' = \{x\mid \|x\|\in (2,3)\}, U = \{x\mid \|x\|\in (1,4)\}$, let $P = M\times SU(2)$ be the trivial $SU(2)$-bundle, and let $g: x\mapsto \frac{x}{\|x\|}$, using the diffeomorphism $S^3\cong SU(2)$. This fulfills all of your a... | 2 | https://mathoverflow.net/users/35687 | 295312 | 129,834 |
https://mathoverflow.net/questions/295214 | 4 | It is well known that a vector space over an infinite field [cannot be a finite union of its proper subspaces](https://mathoverflow.net/questions/26/can-a-vector-space-over-an-infinite-field-be-a-finite-union-of-proper-subspaces).
Does this fact have an immediate and obvious generalization to modules over infinite di... | https://mathoverflow.net/users/36688 | Modules over infinite rings which can not be a finite union of their proper submodules | The two questions can be answered in the positive.
>
>
> >
> > **Claim 1.** Let $R$ be an infinite division ring. Then no left $R$-module can be the union of finitely many of its proper left $R$-submodules.
> >
> >
> >
>
>
>
The proof is a straightforward generalization of the classical pigeonhole argumen... | 6 | https://mathoverflow.net/users/84349 | 295330 | 129,841 |
https://mathoverflow.net/questions/295337 | 3 | Is it true that $$\int\_{\mathbb{R}^N}(-\Delta)^su(x) dx = 0,$$
where $(-\Delta)^s$ is the [fractional Laplacian](https://www.ma.utexas.edu/mediawiki/index.php/Fractional_Laplacian)?
| https://mathoverflow.net/users/nan | Integral of fractional Laplacian is zero | Yes, under certain assumptions.
For $0<\alpha<2$ and $\varphi\in\mathscr{S}\_n$ we have
\begin{eqnarray\*}
(-\Delta)^{\alpha/2}\varphi
& = &
\frac{1}{2\gamma(-\alpha)}
\int\_{\mathbb R^n}
\frac{\varphi(x+y)+\varphi(x-y)-2\varphi(x)}{|y|^{n+\alpha}}\, dy,
\end{eqnarray\*}
where
$$
\gamma(-\alpha)=
\frac{\pi^{\frac{n}{... | 2 | https://mathoverflow.net/users/121665 | 295339 | 129,846 |
https://mathoverflow.net/questions/295289 | 7 | In Rudyak's 'On Thom Spectra, Orientability, and Cobordism', the following fact is used:
>
> Let $\mathbb{Z}\_{(p)}$ be $\mathbb{Z}$ localized at the ideal $(p)$. Let $\pi,\tau$ be two cyclic $\mathbb{Z}\_{(p)}$-modules, that is, $\pi,\tau=\mathbb{Z}/p^m$ or $\mathbb{Z}\_{(p)}$. Then the stable cohomology groups o... | https://mathoverflow.net/users/119553 | Reference request: cohomology of Eilenberg Maclane spaces with $p$-local groups | Here is a sketch proof.
**Step 1:** For sensible spaces or spectra (connected, finite type) $X$, $H^\*(X;\tau) $ will have exponent $p$ for all the coefficient groups $\tau$ you list exactly when the image of the Bockstein $\beta: H^\*(X;\mathbb Z/p) \rightarrow H^{\*+1}(X;\mathbb Z/p)$ equals the kernel of $\beta$.... | 9 | https://mathoverflow.net/users/102519 | 295347 | 129,849 |
https://mathoverflow.net/questions/295349 | 11 | Let $n\in\mathbb{N}$ be a positive integer. We say that an $n\times n$-matrix $A$ with all entries in $\{0,1\}$ is *$k$-regular* for some $k\in \{0,\ldots,n\}$ if the sum of every row and the sum of every column of $A$ equals $k$. Let $M(n, k)$ be the number of $k$-regular $n\times n$-matrices with all entries in $\{0,... | https://mathoverflow.net/users/8628 | Sum-regular $\{0,1\}$-matrices | I'm pretty sure this is unknown, though it would be great if I'm wrong.
Asymptotically there is a function $A(n)$ such that $M(n,n/2+t)\sim e^{-2t^2}A(n)$ for $t=o(n^{1/2})$, which follows from [this paper](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v12i1r29) of Canfield and McKay.
It is also know... | 14 | https://mathoverflow.net/users/9025 | 295354 | 129,852 |
https://mathoverflow.net/questions/295043 | 5 | I'm aware that there are a lot of counterexamples to show that distributional solutions for hyperbolic (scalar) conservation laws are not unique.
However, I'd like to ask:
>
> 1. Conceptually, at which point of a proof of uniqueness is the definition of distributional solution not enough to go on?
> 2. Why is the... | https://mathoverflow.net/users/nan | Why is the definition of entropy solution necessary to prove uniqueness for hyperbolic conservation laws? | 1. Unless an evolution PDEs be linear, a uniqueness proof is always nonlinear in essence: you prove that some distance $d(u(t),v(t))$ between two solutions is bounded in terms of $d(u(0),v(0))$. To carry out the proof, you need to be able to compute the time derivative of $d(u,v)$, and for this you need the chain rule.... | 3 | https://mathoverflow.net/users/8799 | 295358 | 129,853 |
https://mathoverflow.net/questions/215795 | 6 | As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u\_t+f(u)\_x=0$. But if I understood correctly there are some works which use it for $N\times N$ systems; for instance
1- Hyperbolic to Parabolic Relaxation Theory fo... | https://mathoverflow.net/users/75491 | Compensated compactness for system of conservation laws? | So far, only the div-curl Lemma could be used when applying compensated compactness to systems of conservation laws. But it has two flaws.
1. Because we only have divergences but no curls in general, we cannot handle multi-dimensional systems. Only one-dimensional systems can be considered by this method.
2. The idea... | 4 | https://mathoverflow.net/users/8799 | 295363 | 129,855 |
https://mathoverflow.net/questions/295319 | 13 | Suppose $(A,\mathfrak m)$ is a Neotherian local $k$-algebra with residue field $k$. Then, we define (the coordinate ring of) its algebraic tangent cone to be the $k$-algebra $A\_c = \sum\_{i\ge 0} \mathfrak m^i/\mathfrak m^{i+1}$.
On the other hand, we also have the Zariski tangent space $(\mathfrak m/\mathfrak m^2)^... | https://mathoverflow.net/users/110236 | Geometric interpretation of algebraic tangent cone | You may wish to read the discussion on pages 106-108 of Eisenbud/Harris "The Geometry of Schemes", where they explicitly compute the tangent cone of $(y^2-x^3)$ as a degeneration of the tangent cones in the family $(y^2 - tx^2 - x^3)\_{t \in \mathbb{A}^1}$.
In a direction close to what @JasonStarr said, though one r... | 5 | https://mathoverflow.net/users/39777 | 295375 | 129,858 |
https://mathoverflow.net/questions/295387 | 2 | $\text{Ordinal Replacement:}$ if $\phi(x,y)$ is a formula in two free variables $x,y$, then:
$\forall x \ [ordinal(x) \to \exists! y \ (ordinal (y) \wedge \phi(x,y)) ]
\to
\forall A \ (\forall x \in A (ordinal(x)) \to \exists B \ \forall y \ (y \in B \leftrightarrow \exists x \in A \ \phi(x,y))) $
is an axiom.
... | https://mathoverflow.net/users/95347 | Is full Replacement provable in Z + Ordinal Replacement? | No. Consider the model $\langle V\_{\omega\_1},\in\rangle$. This is a model of Zermelo's theory, but all ordinals in it are countable. Thus, it satisfies the ordinal-replacement axiom, since the image of a countable ordinal under any function will be countable, and $V\_{\omega\_1}$ contains all its countable subsets, s... | 11 | https://mathoverflow.net/users/1946 | 295388 | 129,861 |
https://mathoverflow.net/questions/295405 | 8 | The conjecture says that for any a, b belong to the the set of non-negative integers ($a$ and $b$ are not necessarily distinct), taking any natural value of $c$; we have always that $$(10^c-1) \cdot \frac{10^a+10^{2a}+1}{3} \cdot \frac{10^b+10^{2b}+1}{3}$$ is palindromic in base $10$. This conjecture was experimented w... | https://mathoverflow.net/users/113991 | simple conjecture on palindromes in base 10 | Using
$$\frac{10^c-1}{9}=\sum\_{m=0}^{c-1} 10^m,$$
the product in question equals
$$\sum\_{n=0}^{2a+2b+c-1}r(n)10^n,$$
where $r(n)$ is the number of times $n$ occurs among the numbers (counted with multiplicity)
$$\begin{matrix}
m+2a+2b,&m+2a+b,&m+2a,\\
m+a+2b,&m+a+b,&m+a,\\
m+2b,&m+b,&m,\\
\end{matrix}$$
for some $m\i... | 16 | https://mathoverflow.net/users/11919 | 295407 | 129,871 |
https://mathoverflow.net/questions/295383 | 6 | The notion of [Eulerian lattice](https://en.wikipedia.org/wiki/Eulerian_poset) generalizes the notion of [face lattice](https://en.wikipedia.org/wiki/Convex_polytope#The_face_lattice) of a [convex polytope](https://en.wikipedia.org/wiki/Convex_polytope).
(Bruggesse-Mani): The boundary complex of a convex polytope is... | https://mathoverflow.net/users/34538 | Is an Eulerian lattice shellable? | The simplest example of a nonshellable (even non-Cohen Macaulay) Eulerian lattice is the disjoint union of two boolean algebras of rank three with their bottom elements identified and their top elements identified. The total number of elements is 14.
| 9 | https://mathoverflow.net/users/2807 | 295410 | 129,873 |
https://mathoverflow.net/questions/295088 | 1 | Let $S$ be a irreducible scheme over a field $k$ (for example a smooth projective curve over algebraically closed field). Denote by $k(S)$ its field of fractions. Let $K$ be a(n algebraically closed) field. Given an embedding $k(S) \hookrightarrow K$ we can construct a $K$-point of $S\otimes K$ in the following way: ch... | https://mathoverflow.net/users/88385 | Embeddings of fields and rational points | So I am at the airport with some time to kill, so I will answer this question when $S$ is the spectrum of $k[t]$, i.e., is the affine line. The base change is the spectrum of $K[t]$. A $K$-point of this is given by an element of $K$, namely the image of $t$ under the corresponding $K$-algebra map. Now I leave it up to ... | 1 | https://mathoverflow.net/users/118654 | 295413 | 129,875 |
https://mathoverflow.net/questions/295411 | 7 | The Hirzebruch surfaces are the $\mathbb{P}^1$ bundles $\mathbb{F\_n}$ ($n\geqslant 0$) which can be obtained projectivizing the rank $2$ vector bundles $\mathcal{O}\_{\mathbb{P}^1}\oplus \mathcal{O}\_{\mathbb{P}^1}(n)$. Those surfaces are famous because they exhaust ($n\neq 1$) the minimal rational (smooth, projective... | https://mathoverflow.net/users/50468 | Automorphism groups of Hirzebruch surfaces | For $n \ge 2$ the surface $\mathbb{F}\_n$ is the blowup of the weighted projective plane $\mathbb{P}(1,1,n)$ at its singular point. Because of that
$$
Aut(\mathbb{F}\_n) \cong Aut(\mathbb{P}(1,1,n)).
$$
The automorphism group of $\mathbb{P}(w\_1,w\_2,\dots,w\_m)$ is the group of non-degenerate $m$-by-$m$ matrices with ... | 12 | https://mathoverflow.net/users/4428 | 295418 | 129,876 |
https://mathoverflow.net/questions/295404 | 3 | Let $A$ be a finite set in $\mathbb{R}^2$ of $k^2$ elements and consider a set $B=\{x\_1,x\_2,x\_3,x\_4\}$ such that the points in $B$ are in general position (no three points on a line).
Question 1: Is it true that $|A+B|\geq (k+1)^2$?
Question 2: Is it true that $|\cap\_{i}(A+x\_i)|\leq (k-1)^2$?
Question 3: I... | https://mathoverflow.net/users/24494 | Bounding the size of certain sumsets in the plane | As shown by Gardner and Gronchi ("[A Brunn-Minkowski Inequality for the Integer Lattice](http://www.ams.org/journals/tran/2001-353-10/S0002-9947-01-02763-5/S0002-9947-01-02763-5.pdf)", equality (8) / Theorem 6.6), if $A,B\subset\mathbb R^n$ are finite sets such that $B$ has full dimension, then
$$ |A+B|^{1/n} ≥ |A|^{1... | 4 | https://mathoverflow.net/users/9924 | 295422 | 129,877 |
https://mathoverflow.net/questions/295420 | 0 | I have a very simple linear first order ODE.
$$v(x) = c x + A - B x(1-x) v'(x)$$
$c, A ,B \in(0,1)$. The domain is $(\underline{x}, 1)$. where $\underline x > 0$.
I am guessing that for any initial condition $v(\underline x)$ there will be a unique solution. In particular, suppose I give an initial condition $... | https://mathoverflow.net/users/78761 | Convexity of a solution of a first order linear ODE | If I did not make any mistake, $v(x)$ need not be convex.
---
We find that
$$B v'(x) = \frac{c x + A - v(x)}{x (1 - x)} ,$$
and therefore
$$B v''(x) = \frac{(c - v'(x)) x (1 - x) - (c x + A - v(x)) (1 - 2 x)}{x^2 (1 - x)^2} .$$
Plugging in the expression for $v(x)$, we get
$$B v''(x) = \frac{(c - v'(x)) x (1 - x)... | 2 | https://mathoverflow.net/users/108637 | 295426 | 129,878 |
https://mathoverflow.net/questions/295429 | 0 | Given $J\_\alpha$ denotes the $1+\alpha$th term in the Jensen Hierarchy and $J=\bigcup\_{\alpha\in On}J\_\alpha$, are there any known large cardinals $\mathfrak{K}$ such that $\text{ZFC}+\mathfrak{K}$ is consistent and
$\text{ZFC}+\mathfrak{K}\vdash J\neq L$? (assuming ZFC is consistent)
| https://mathoverflow.net/users/120848 | Large cardinals disproving J=L | Unfortunately, your question seems to be based on a misunderstanding of the $J$ hierarchy. There is no such large cardinal hypothesis.
The reason is that already ZFC and indeed much weaker theories such as KP prove $J=L$ outright. The point of the $J$ hierarchy was not to construct a different universe, but rather to... | 8 | https://mathoverflow.net/users/1946 | 295438 | 129,881 |
https://mathoverflow.net/questions/295441 | 0 | Is there algorithm for 3SUM which have complexity O(n) or O($n^{3/2}$) for randomly chosen input with bit length of maximum number approximately equal to count of input numbers?
| https://mathoverflow.net/users/50088 | Is there fast algorithm for 3SUM? | Yes, here it is (code in Python):
```
from collections import Counter
from math import ceil, log
from random import randint
def rnd(n):
return randint(0, n - 1)
def get_bit_len(n):
if n == 0:
res = 1
else:
res = int(ceil(log(n + 1) / log(2)))
return res
def rev_bits_to_int(bits):
... | 0 | https://mathoverflow.net/users/50088 | 295442 | 129,883 |
https://mathoverflow.net/questions/295446 | 0 | Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with no-zero finite length $L$. Then, does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and satisfies $lth\_t(g) = tL ~\forall ~t \in [0,1]$, where $lth\_t(g)$ represent the length o... | https://mathoverflow.net/users/42134 | does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and ..? | This is a standard result that every rectifiable curve in a metric space admits an arc length parametrization. The proof can be found in many sources. For examples Theorem 3.2 in
[these](http://www.pitt.edu/~hajlasz/OriginalPublications/Hajlasz-SobolevSpaces-ContempMath-338-2003-173-218.pdf) notes. Then the arc-length ... | 2 | https://mathoverflow.net/users/121665 | 295448 | 129,885 |
https://mathoverflow.net/questions/295001 | -1 | Informally the axiom *schema* of accessibility states that for each unary function $F$ that is definable over the whole universe of discourse "in the language of set theory", like the powerset function $P$, or the singleton function $\iota$, or the set Union function $\bigcup$, etc.. for each such a function $F$ we can... | https://mathoverflow.net/users/95347 | What is the consistency strength of Z+ Accessibility? | A limited form of Replacement is provable in this theory! That of Replacing elements of a set of ordinals by ordinals, where an ordinal is defined in the usual manner after von Neumann.
Lemma: $\forall ordinals \ \alpha, \beta \ (\alpha \ ..\leq \ \beta \lor \beta \ ..< \ \alpha)$
The proof of this lemma is obvious... | 0 | https://mathoverflow.net/users/95347 | 295449 | 129,886 |
https://mathoverflow.net/questions/295450 | 4 | The ideal of the $10$-dimensional Spinor variety $S^{10}\subset\mathbb{P}^{15}$ is generated by $10$ quadrics.
Does anyone know a reference where these 10 quadratic equations are written down explicitly in the homogeneous coordinates $[x\_0:\dots:x\_{15}]$ of $\mathbb{P}^{15}$ ?
| https://mathoverflow.net/users/nan | Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$ | Mukai, [CURVES AND SYMMETRIC SPACES, I](https://www.jstor.org/stable/2375032), (0.1)
| 7 | https://mathoverflow.net/users/4428 | 295452 | 129,887 |
https://mathoverflow.net/questions/128581 | 6 | I already asked this on [M.SE](https://math.stackexchange.com/questions/365692/a-finely-open-set-not-open-up-to-polar-set), but get no answers.
Is there a (simple) example of a finely open set (i.e. w.r.t. the [fine topology in potential theory](http://en.wikipedia.org/wiki/Fine_topology)) $O$ in $\mathbb R^n$, $n \g... | https://mathoverflow.net/users/32507 | A finely open set, not open up to polar set? | I like the following abstract construction, which relies on a connection between quasi-continuity and the fine topology and also works in the setting of $p\ne 2$. It is debatable whether the resulting example is “simple”. The standard reference for the required techniques is Adams, Hedberg: Function spaces and potentia... | 3 | https://mathoverflow.net/users/14849 | 295459 | 129,891 |
https://mathoverflow.net/questions/89362 | 6 | I've been trying without success to find the paper
Dickman, Karl, "On the frequency of numbers containing prime factors of a certain relative magnitude." Ark Mal., Astronomi och Physik, 22A (10), 1930. I checked at Rice U. and it was not available.
Actually,I would be satisfied with an exposition of the information... | https://mathoverflow.net/users/8955 | Reference request: Dickman, On the frequency of numbers containing prime factors | Answering from the far future! I present some literature on this topic:
1) K. Dickman in his original paper of 1930 gave an heuristic argument that can be found in pages 382-383 of *The art of computer programming, volume 2* (third edition) by Knuth.
2) V. Ramaswami made the argument rigorous in his 1949 paper *[On... | 4 | https://mathoverflow.net/users/1234 | 295468 | 129,894 |
https://mathoverflow.net/questions/295469 | 2 | 1. In 'Computational Experience with Lenstra's Algorithm' by L Gao, Y Zhang it is claimed that they have an implementation of Lenstra's fixed dimension integer programming algorithm. Is this available online anywhere?
2. Is Barvinok's algorithm available online?
| https://mathoverflow.net/users/10035 | On software for ILP | This may be of some help:
>
> De Loera, Jesús A., David Haws, Raymond Hemmecke, Peter Huggins, and Ruriko Yoshida. "A computational study of integer programming algorithms based on Barvinok's rational functions." *Discrete Optimization* 2, no. 2 (2005): 135-144.
> ([Elsevier link](https://www.sciencedirect.com/sci... | 1 | https://mathoverflow.net/users/6094 | 295470 | 129,895 |
https://mathoverflow.net/questions/295433 | 3 | The problem is:
For any closed 2-form in the Minkowski space $\mathbb{R}^{3,1}$ satisfiying $dF=0$ and $\delta F \ne 0$ (with $\delta$ denoting the codifferential), does there exist a Lorentz manifold $(M,g)$ and a corrsponding diffeomorpism $f:M\to\mathbb{R}^{3,1}$ such that the pullback of $F$ on $M$, $f^{\*}F$, sa... | https://mathoverflow.net/users/56185 | A problem about closed 2-forms on Minkowski space | Your question can be equivalently phrased as:
>
> Given a closed two-form $F$ on $\mathbb{R}^4$, is it always possible to find a Lorentzian metric on $\mathbb{R}^4$ such that $\delta F = 0$.
>
>
>
Then the answer is, in general, no.
First, consider the case if you require $(\mathbb{R}^4,g)$ to be globally h... | 5 | https://mathoverflow.net/users/3948 | 295472 | 129,896 |
https://mathoverflow.net/questions/295481 | 11 | Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL\_n(\mathbb{Q}\_p)$ and $\pi: G \to GL\_n(\mathbb{F}\_p)$. My question is:
>
> When does there exist a representation $\rho: G \to GL\_n(\mathbb{Z}\_p)$ such that $\sigma$ is isomorphic to the base change of $\rho$ to $... | https://mathoverflow.net/users/121 | Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters? | The condition on Brauer characters is not sufficient.
Let $G$ be a $p$-group, $\pi$ any nontrivial representation over $\mathbb{F}\_p$, and $\sigma$ the trivial representation over $\mathbb{Q}\_p$ of the same degree as $\pi$. Then a representation over $\mathbb{Z}\_p$ whose base change to $\mathbb{Q}\_p$ is isomorphi... | 13 | https://mathoverflow.net/users/22989 | 295485 | 129,901 |
https://mathoverflow.net/questions/181863 | 10 | CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering the other direction,
Quesion: Given a CAT(0) group G , will one always be able to find a CAT(0) cubical complex such ... | https://mathoverflow.net/users/4760 | CAT(0) groups that does not act on CAT(0) cubical complex | Many CAT(0) groups cannot act geometrically on CAT(0) cube complexes. For instance:
* CAT(0) groups satisfying Kazhdan's property (T), eg. uniform lattices in simple Lie groups of higher rank or in quaternionic hyperbolic spaces (as mentioned by Jean Raimbault in the comments).
* Some Kähler groups (as mentioned by M... | 10 | https://mathoverflow.net/users/122026 | 295487 | 129,902 |
https://mathoverflow.net/questions/295222 | 3 | Let $A\subset \mathbb{R^2}$ be a finite set such that $|A|=k^2$. Let $x\_i\in \mathbb{R^2}$, $i=1,2,3,4$, be four points in the plane in general position (no three lie on any line).
Let us form the multiset of cardinality $4k^2$ out of the four translates $A+x\_i$ and call it $M$.
Question: Can we partition $M$ in... | https://mathoverflow.net/users/24494 | Separating points of shifts of a finite set in the plane | Writing $B:=\{x\_1,x\_2,x\_3,x\_4\}$ and denoting by $S\_i(A,B)$ the set of all those $c\in\mathbb R^2$ with at least $i$ representations as $c=a+b$ with $a\in A$ and $b\in B$, your question can be equivalently restated as follows: Is it true that
\begin{align\*}
|S\_1(A,B)| &\ge (k+1)^2, \\
|S\_1(A,B)|+|S\_2(A,B)| &... | 4 | https://mathoverflow.net/users/9924 | 295499 | 129,907 |
https://mathoverflow.net/questions/295503 | 4 | For affine schemes, there are contravariant maps V(-) and I(-) that create a bijection between irreducible closed subsets of the affine scheme and prime ideals of the ring that is the ring of global sections of that scheme (i.e. points on the affine scheme).
What happens to V(-) and I(-) when we take an arbitrary sch... | https://mathoverflow.net/users/93576 | Generalization of V(-) and I(-) to arbitrary schemes | Given any scheme $X$, there is a bijective correspondence between quasi-coherent ideal sheaves on $X$ and closed subschemes of $X$. (See Hartshorne's *Algebraic Geometry*, Proposition II.5.9.) When we restrict to affine schemes, this correspondence is given by the usual $V(-)$ and $I(-)$ once we identify quasi-coherent... | 10 | https://mathoverflow.net/users/31308 | 295504 | 129,909 |
https://mathoverflow.net/questions/295516 | 6 | Do $2^{n-1}\equiv1\pmod n$ and $(n-1)/2$ prime imply $n$ prime?
Equivalently: Does $n$ being a Fermat pseudoprimes to base 2 ([OEIS A001567](https://oeis.org/A001567)) imply that $(n-1)/2$ is composite? That holds for all $n<2^{64}$, based on [Jan Feitsma's table](http://www.cecm.sfu.ca/Pseudoprimes/).
Motivation i... | https://mathoverflow.net/users/122065 | Do $2^{n-1}\equiv1\pmod n$ and $(n-1)/2$ prime imply $n$ prime? | We have $2^{\varphi(n)}\equiv 1 \pmod n$, thus $2^k\equiv 1$, where $k=\text{gcd}(\varphi(n),n-1)$. Note that $k$ is even, since both $n-1$ and $\varphi(n)$ are even. If $n=2p+1$ for prime $p=(n-1)/2$, then even divisor of $n-1$ is either 2 or $2p$. If $k=2$, we get $n|2^2-1=3$; if $k=2p=n-1$, we get $\varphi(n)\geqsla... | 20 | https://mathoverflow.net/users/4312 | 295517 | 129,916 |
https://mathoverflow.net/questions/295421 | 0 | Transferred from MSE where it now received a complete answer.
Maybe the following is easy, but I am not an expert in finite-dimensional Lie algebras and was stuck on the following problem.
>
> Can you prove or disprove the following statement ?
>
>
> **S)** *Let $V$ be an indecomposable (finite-dimensional) mod... | https://mathoverflow.net/users/25256 | About indecomposability and nilpotence | The answer is positive. Details and link to Bourbaki, Lie groups and Lie algebras, Chap VII, are given in [the answer here](https://math.stackexchange.com/a/2694748/35400).
| 2 | https://mathoverflow.net/users/14094 | 295521 | 129,917 |
https://mathoverflow.net/questions/295522 | 11 | I'm wondering if there is existing terminology to describe fields $F$ with the properties below. I don't have a completely precise description of the concept I have in mind, but hopefully this will be enough.
1. Every element of $F$ requires only a finite amount of information to describe (so that it can be stored in... | https://mathoverflow.net/users/46987 | Representing field elements in a computer | You are looking at a computable field (if your focus is on the field), or a computable presentation of a field (if your focus is on the details of how elements and operations are coded). These objects are studied in computable structure theory.
| 16 | https://mathoverflow.net/users/15002 | 295524 | 129,918 |
https://mathoverflow.net/questions/295523 | 2 | Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$?
Clearly the analogous question for a triangle in $\Bbb CP^1 \cong S^2$ has affirmative answer: it is a maximal equilateral triangle.
More gen... | https://mathoverflow.net/users/23193 | Very symmetric quadrangle in $\Bbb CP^2$ | That is true, since this is so for $\mathbb RP^n$ - take $n+2$ vertices of a "regular simplex" in it -- i.e. take the regular simplex in $S^n$ and project its vertices to $\mathbb RP^n$.
In coordinates, take points $(1,0,\ldots,0)$, ... $(0,\ldots,0,1)$ in $\mathbb R^{n+2}$, take the hyperplane $\sum\_i x\_i=1$ and ... | 8 | https://mathoverflow.net/users/943 | 295525 | 129,919 |
https://mathoverflow.net/questions/295526 | 5 | Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf\_0(Z):\dots:pf\_n(Z)]$ where the $pf\_i(Z)$ are the principal Pfaffians of order $n$ of $Z$.
Is the closure of the image in $\mathbb{P}^n$ of this rational map a known variety?
| https://mathoverflow.net/users/nan | Rational map given by pfaffians | At least in characteristic zero, the image of this map is all of $\mathbb P^n$. In finite characteristic, I am not sufficiently familiar with Pfaffians to say whether the argument carries through.
Indeed, start by noting that the map is only well-defined for $Z$ of rank $n$, since otherwise all principal Pfaffians of... | 4 | https://mathoverflow.net/users/300 | 295528 | 129,920 |
https://mathoverflow.net/questions/295424 | 3 | Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, let $\textbf{w}$ and $\textbf{x}$ be vectors in $\mathbb{R}^d$, and let $\epsilon \in (0,1)$. In [Shi et al. 2012](https://icml.cc/2012/papers/327.pdf "Shi et al. (2012)"), it is claimed in Theor... | https://mathoverflow.net/users/122021 | Help with understanding a proof on angle preservation | *Disclaimer.* I am unable to follow the proof of Theorem 5 in [Shi et al. 2012](https://icml.cc/2012/papers/327.pdf).
But it is not difficult to show the following *angle preservation* statement based on the **Tail Bound Lemma**, that is, Lemma 10 of the article.
>
>
> >
> > **Claim.** Let $\mathbf{x}, \mathbf... | 2 | https://mathoverflow.net/users/84349 | 295532 | 129,923 |
https://mathoverflow.net/questions/295531 | 0 | Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p\_1,\ldots, p\_n\}) \ \mbox{with at worst a simple pole at}\ p\_i \} $$ where $\mathbb M(U)$ means the set of meromorphic functions on $U$ and $\mathbb O(U)$ means the set of holomorphic functions on $U$. What is the cohomology... | https://mathoverflow.net/users/122085 | The cohomology of meromorphic functions | The cohomology of these sheaves are always 0. This follows for example from the case $\{p\_i\} = \emptyset$ by induction and using the sequences
$A\_{p\_1,..,p\_{n-1}} \to A\_{p\_1,...,p\_n} \to \mathbb{C}\_{p\_n}$.
In general, the isomorphism class of this sheaf depends only on the number of points, and a representa... | 3 | https://mathoverflow.net/users/115052 | 295533 | 129,924 |
https://mathoverflow.net/questions/295535 | 0 | 7 is a Heegner number. Therefore the integer ring $O\_K$ corresponding to $K=\mathbb{Q}[\sqrt{-7}]$ is a unique factorization domain. Now, it is easy to show that $\mathbb{Z}[\sqrt{-7}]\subset O\_K$, i.e. $O\_K \neq \mathbb{Z}$, because $x=\sqrt{-7}$ is a solution to $x^2+7=0$. On the other hand, $8=2^3=(1+\sqrt{-7})(1... | https://mathoverflow.net/users/122087 | Why is this not a non-unique factorization in the integer ring for $\mathbb{Q}[\sqrt{-7}]$ when 7 is a Heegner? | The ring of integers of $\mathbb{Q}(\sqrt{-7})$ is not $\mathbb{Z}[\sqrt{-7}]$ but $\mathbb{Z}\left[\frac{1+\sqrt{-7}}{2}\right]$. In this ring, the ideal $(2)$ factors into prime ideals as follows:
$$ (2)=\left(\frac{1+\sqrt{-7}}{2}\right)\left(\frac{1-\sqrt{-7}}{2}\right). $$
For further information, see [here](https... | 11 | https://mathoverflow.net/users/11919 | 295536 | 129,925 |
https://mathoverflow.net/questions/294519 | 6 | **Question**: Let $X$ be a continuum and $p \in X$. Under what conditions does there exist a compactification $\gamma (X-p)$ with $\gamma (X-p) - (X-p)$ connected and nondegenerate?
Throughout, $X$ is a continuum. That is to say, a compact connected Hausdorff space. (I'd be interested on any info for non-connected sp... | https://mathoverflow.net/users/58082 | Is there a compactification with nontrivial connected remainder? |
>
> Given a compact Hausdorff space $X$ (not necessarily connected) and a point
> $p \in X$, the following are equivalent:
>
>
> 1. $X - p$ has a compactification with non-trivial connected remainder.
> 2. There is a continuous surjection $q: (X - p)^\* \to I$, where $(X-p)^\*$
> denotes the Stone-Čech remainder ... | 2 | https://mathoverflow.net/users/10075 | 295537 | 129,926 |
https://mathoverflow.net/questions/295549 | 2 | The sequence $(y\_n)$ satisfies the relationship ${y\_{n-1}}{y\_{n+1}} + y\_n = 1$ for all $n \ge 2$. If $y\_1= 1$ and $y\_2= 2$, What can you say about the sequence? What happens for other starting values?
| https://mathoverflow.net/users/122068 | Sequence problem find its character | For arbitrary Initial values $y\_1=a,y\_2=b$ (with $ab$ not zero) you get $a,b, \frac{1-b}{a},\frac{a+b-1}{ab},\frac{1-a}{b}$ . Then the sequence is periodic with period 5.
Thus for $y\_1=1,y\_2=2$ the sequence should be $1,2,-1,1,0,1,2,-1,1,0,\dots.$
| 3 | https://mathoverflow.net/users/5585 | 295552 | 129,930 |
https://mathoverflow.net/questions/295553 | 4 | Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a *global complete intersection* inside a smooth projective toric variety?
| https://mathoverflow.net/users/nan | Complete intersections in toric varieties | Any smooth projective toric variety is rational, in particular simply connected.
Then, by the [Lefschetz hyperplane theorem](https://www.encyclopediaofmath.org/index.php/Lefschetz_theorem) for global complete intersections, if $\dim X \geq 3$ is a smooth complete intersection into a smooth toric variety then $\pi\_1... | 12 | https://mathoverflow.net/users/7460 | 295557 | 129,932 |
https://mathoverflow.net/questions/295561 | 0 | Let $n>1$ be an integer and $[n] = \{1,\ldots,n\}$. What is an example of non-isomorphic simple, undirected graphs $G\_i = ([n], E\_i)$ for $i=1,2$ with the following properties?
1. both $G\_1$ and $G\_2$ have [diameter](http://mathworld.wolfram.com/GraphDiameter.html) two,
2. $\chi(G\_1)=\chi(G\_2)$.
| https://mathoverflow.net/users/8628 | Non-isomorphic graphs with diameter two | $K\_n$ without edge and $K\_n$ without two edges which share a common vertex (well, if $n>3$). In both cases $\chi=n-1$.
| 4 | https://mathoverflow.net/users/4312 | 295562 | 129,935 |
https://mathoverflow.net/questions/295519 | 1 | Let $R$ be a commutative ring with unity. Let $\alpha$ be an infinite cardinal . Let $M$ be an $R$-module such that $\mu(M)< \alpha$ . Let $N$ be a submodule of $M$ and $m\in M$ and $r\in R$ be such that $rm \in N$, $\mu (N+rM) < \alpha$ and $\mu (N+Rm) < \alpha$ . Then is it true that $\mu (N) < \alpha$ ?
If this is... | https://mathoverflow.net/users/nan | On cardinality of generating subsets of some submodules | This is not true for any cardinal.
For any cardinal $\alpha$, there is a commutative ring $R$ with elements $x,y\in R$ such that $xy=0$ and $\mu(Rx\cap Ry)=\alpha$. So you can take $M=R$, $m=x$, $r=y$ and $N=Rx\cap Ry$.
To construct such a ring, it's sufficient to find a commutative ring $S$ and an $S$-module $A$ w... | 5 | https://mathoverflow.net/users/22989 | 295563 | 129,936 |
https://mathoverflow.net/questions/295571 | 5 | This question concerns two definitions of the reflection principle. One of them known to be a consequence of the other one. I would like to understand if the reverse is true.
Let us state the first definition of reflection.
***Definition(Reflection):*** A stationary set $S$ in $[H\_\lambda]^\omega$ reflects at $X\... | https://mathoverflow.net/users/38866 | Variants of reflection principle | The answer is "yes". The principle you call $\text{RP}^\*$ is called "reflection to internally club sets" in the literature; as far as I know this terminology first appeared in Foreman-Todorcevic's ``A new Lowenheim-Skolem Theorem". See their paper for the definition of internally stationary, internallly club, and inte... | 8 | https://mathoverflow.net/users/26319 | 295573 | 129,938 |
https://mathoverflow.net/questions/295158 | 4 | [Abhyankar-Moh theorem](https://eudml.org/doc/151610) says that if $L$ is a complex line in the complex affine plane $\mathbb{C}^2$, then every embedding of $L$ into $\mathbb{C}^2$ extends to an automorphism of the plane.
It seems that one can replace $\mathbb{C}$ by any algebraically closed field $k$ of characteristic... | https://mathoverflow.net/users/72288 | Generalizations of Abhyankar-Moh theorem (embeddings of the line in the plane) | The general question that one can ask is if embeddings $\iota:\mathbb{A}^k\hookrightarrow\mathbb{A}^n$ can be rectified, i.e., if there is an automorphism of $\mathbb{A}^n$ which takes $\iota$ to the standard embedding of a coordinate $k$-plane. The Abhyankar-Moh(-Suzuki) theorem is the case where $k=1$ and $n=2$. The ... | 6 | https://mathoverflow.net/users/50846 | 295575 | 129,939 |
https://mathoverflow.net/questions/295551 | 6 | This is a cross-post from [math.stackexchange.com](https://math.stackexchange.com/questions/2696441/symmetric-c-property)
What can be said about elements $a$ of a C\*-algebra which fulfil the 'symmetric C\* property'
$$\| a^\ast a+ aa^\ast\|=2\| a\|^2$$
I'd guess that this is not a general property, but I don't hav... | https://mathoverflow.net/users/89934 | Symmetric C* property | This is hardly a complete answer.
As you say, all normal operators $a \in B(\mathcal H)$ satisfy this equality. Moreover, so does any element $c = a\oplus b \in B(\mathcal H\oplus \mathcal H)$ where $a$ is normal and $\|b\| \leq \|a\|$. This is because
$$ \|c^\*c + cc^\*\| = \max\{\|a^\*a + aa^\*\|, \|b^\*b + bb^\*\|... | 7 | https://mathoverflow.net/users/76593 | 295578 | 129,941 |
https://mathoverflow.net/questions/295465 | 1 | I am reading Shurygin's survey "Smooth Manifolds over Local Algebras and Weil Bundles" (Journal of Math. Sciences, Vol. 108, No. 2, 2002) and it mentions the following basic fact which I don't quite understand:
>
> Let $A$ be an $n$-dimensional commutative associative unital local $\mathbb{R}$-algebra with max. ide... | https://mathoverflow.net/users/1849 | Jordan-Hölder series of $k$-subalgebras? | Since $A$ is an $n$-dimensional commutative associative unital $\Bbb R$-albebra, it has finite length as a module over itself.
>
> **Claim:** The length $\ell\_A(A)$ as a module over itself equals $n$.
>
>
> *$\because$)* It's immediate that $\ell\_A(A)\leq n$. To see that $\ell\_A(A)=n$, let $F$ be an $A$-module... | 2 | https://mathoverflow.net/users/36720 | 295589 | 129,944 |
https://mathoverflow.net/questions/295592 | 1 | Let $D(\epsilon,C)$ be the collection of all random variables $X$ on $\mathbb{R}$ such that $E[X]=0$, $E[X^2]=1$, and $E[|X|^{2+\epsilon}]\leq C$. Define a function $L\_{\epsilon,C}(n)$ by
$$L\_{\epsilon,C}(n) = \sup\_{X\in D(\epsilon,C)} \sup\_{x\in\mathbb{R}} \left\vert P\left(\frac{X\_1+X\_2+\ldots+X\_n}{\sqrt{n}}<x... | https://mathoverflow.net/users/108574 | Berry-Esseen type bounds with lower moment assumptions | $\newcommand{\ep}{\epsilon}$
To answer your question, one can use e.g. a result by [Bikelis](https://books.google.com/books?id=zSDqCAAAQBAJ&q=bikelis#v=snippet&q=bikelis&f=false), which states the following: Let $X\_1,\dots,X\_n$ be independent zero-mean random variables such that $E|X\_i|^{2+\ep}<\infty$ for some $\... | 2 | https://mathoverflow.net/users/36721 | 295595 | 129,945 |
https://mathoverflow.net/questions/295599 | 0 | In this [page](https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action) there are calculations of variation of Einstein-Hilbert action.
I see variations of terms like this:
>
> $\delta {R^{\rho }}\_{{\sigma \mu \nu }}$
>
>
>
where the term is not a functional, and
>
> $\frac {\delta {\mathcal {L}... | https://mathoverflow.net/users/103977 | Variation in Einstein-Hilbert action | It is a notational short hand. (See, e.g. Appendix E in Wald's *General Relativity*).
Given a function $\psi$ and a one-parameter family of functions $\psi\_{\lambda}$ with $\psi\_0 = \psi$, the notation $\delta \psi$ refers to the short hand $\delta\psi := \frac{d}{d\lambda} \psi\_\lambda \Big|\_{\lambda = 0}$.
| 3 | https://mathoverflow.net/users/3948 | 295600 | 129,947 |
https://mathoverflow.net/questions/295564 | 9 | Is the hyperbolic or spherical analogy of the following Euclidean fact, true?
>
> Two triangles with equal corresponding medians are congruent.
>
>
>
More precisely: Assume that $\Delta ABC$ and $ \Delta A'B'C'$ are two triangles in the hyperbolic space $\mathbb{H}^2$ or elliptic space $\mathbb{S}^2$ such that... | https://mathoverflow.net/users/36688 | Are two triangles with equal corresponding medians, congruent? | **This is not true for $S^2$.** Namely, one can construct two convex non-isometric spherical triangles such that all medians have length $\frac{3\pi}{4}-\varepsilon$.
1) The first triangle is the standard equilateral triangle. Clearly medians of such triangles all have the same length and their length vary from $0$ t... | 7 | https://mathoverflow.net/users/943 | 295602 | 129,948 |
https://mathoverflow.net/questions/295606 | 4 | Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $S$ and $T$ be two disjoint infinite subsets of $X(k)$. Does there exist a morphism $f:X \to \mathbb{P}^1$ and two infinite subsets $A$ and $B$ of $\mathbb{P}^1(k)$ such that the preimages of $A$ and $B$ by $f$ lie inside $S$ and $T$ respe... | https://mathoverflow.net/users/4069 | Partitions of algebraic curves and maps to $\mathbb{P}^1$ | No in case $k$ is countable of characteristic zero by a simple diagonal argument. Assume $X$ does not have genus $0$ for simplicity. Number the maps $f\_1, f\_2, \ldots$. For each $n$ by induction choose $S\_n$ and $T\_n$ finite disjoint subsets of $X(k)$ such that $f\_i$, $i \leq n$ maps an element of $S\_n$ and $T\_n... | 4 | https://mathoverflow.net/users/122145 | 295609 | 129,950 |
https://mathoverflow.net/questions/295622 | 1 | Lebesgue theorem says that a bounded function $f$ is Riemann Integerable if and only if $f$ continuous almost everywhere.
Unfortunately, we know a function has antiderivative has no relation to Riemann Integerable.
>
> **Q.** Is there some nice equivalent condition to the fact that a real function has antiderivat... | https://mathoverflow.net/users/115376 | Equivalent conditions for a real function to have antiderivates | There are various characterizations of real functions which have antiderivatives. See, for example, Theorem 4 in M. W. Botsko: Exactly which bounded Darboux functions are derivatives?, Amer. Math. Monthly 114 (2007), 242-246.
| 1 | https://mathoverflow.net/users/11919 | 295623 | 129,956 |
https://mathoverflow.net/questions/295624 | 11 | By Brill-Noether theory, the generic genus $6$ curve is birational to a sextic plane curve in $\mathbb{P}^2$. I was wondering if there is a direct/natural construction of this birational map. In other words,
>
>
> >
> > do we know a divisor $D$ on the curve inducing this map? Is it given by a complete system, or... | https://mathoverflow.net/users/118889 | Natural model for genus $6$ curves | On the canonical model $C\_{10} \subset \mathbb{P}^5$ of a smooth curve of genus six there exist five special $g\_4^1$, obtained as follows: we take an arbitrary point on the curve, and the remaining three are determined so that the four points lie on a plane. We can also see these five $g^1\_4$ as cut on the $C\_{10}$... | 12 | https://mathoverflow.net/users/7460 | 295626 | 129,957 |
https://mathoverflow.net/questions/295625 | 3 | Given any relation $R$, that is, any set of ordered pairs, we can associate a unique digraph $D$ to our relation $R$ by setting $D=(\text{fld}(R),R)$ where $\text{fld}(R)=\text{dom}(R)\cup\text{rng}(R)=\bigcup\_{(x,y)\in R}\{x,y\}$. Therefore, in this way the study of binary relations can be put on an equal footing wit... | https://mathoverflow.net/users/38626 | In the context of directed graphs is it standard notation to allow an element of an independent vertex set to be contained in a loop? | Well, I personally don't think there is much to say here; regarding "are there any authors who define independent vertex-sets in directed graphs where vertices contained in loops are permitted": I know one. E.g.,
>
> **[BJG2009]** Jørgen Bang-Jensen, Gregory Z. Gutin, *Digraphs: Theory, Algorithms and Applications... | 3 | https://mathoverflow.net/users/108556 | 295628 | 129,958 |
https://mathoverflow.net/questions/295577 | 2 | Let $R$ be an integral domain. Let $\alpha$ be an infinite cardinal . Let $M$ be a faithful $R$-module such that $\mu(M)< \alpha$ . Let $N$ be a submodule of $M$ and $m\in M$ and $r\in R$ be such that $rm \in N$, $\mu (N+rM) < \alpha$ and $\mu (N+Rm) < \alpha$ . Then is it true that $\mu (N) < \alpha$ ?
If this is no... | https://mathoverflow.net/users/nan | Cardinality of generating sets of faithful modules over integral domain | Let $W$ be a well-ordered set with cofinality at least $\alpha$ (for example, $W$ could be the cardinal $2^\alpha$).
Let $\mathbb{Z}^W$ be the group of functions $W\to\mathbb{Z}$, ordered lexicographically using the well-order on $W$ (i.e., $f<g$ if and only if $f(w)<g(w)$ for the least $w\in W$ with $f(w)\neq g(w)$)... | 2 | https://mathoverflow.net/users/22989 | 295631 | 129,959 |
https://mathoverflow.net/questions/295621 | 8 | Is there an example of a presentable, stable, $k$-linear $\infty$-category which is dualizable but not compactly generated, where $k$ has characteristic zero, and which is $\text{QCoh}(X)$ (by which I mean the derived dg category of quasicoherent sheaves on $X$) for some prestack $X$?
Or, perhaps by removing some of ... | https://mathoverflow.net/users/6059 | Example of a (presentable $k$-linear $\infty$-)category which is dualizable but not compactly generated? | If $X$ is a locally compact topological space, then
$\mathrm{Shv}(X, \mathrm{Mod}\_{k} )$ is a presentable $k$-linear stable $\infty$-category which is dualizable (in fact, self-dual), but is rarely compactly generated (for example, this fails for $X = \mathbf{R}$).
| 11 | https://mathoverflow.net/users/7721 | 295635 | 129,960 |
https://mathoverflow.net/questions/295643 | 1 | Suppose $(X, \Sigma\_X)$ and $(Y, \Sigma\_Y)$ are measurable spaces such that $\Sigma\_Y$ is generated by a set $B$.
Suppose $k : X \times \Sigma\_Y \to [0, 1]$ has the property that $k(x, -)$ is a (sub-)probability measure for each $x \in X$, and $k(-, E)$ is measurable for every $E \in B$, wrt the Borel $\sigma$-a... | https://mathoverflow.net/users/114537 | Measurability of kernel on generating set | $\newcommand{\N}{\mathbb N}
\newcommand{\R}{\mathbb R}
\newcommand{\B}{\mathcal B}
\newcommand{\F}{\mathcal F}
\newcommand{\X}{\mathcal X}
\newcommand{\ep}{\epsilon}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\renewcommand{\c}{\circ}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\E}{\... | 1 | https://mathoverflow.net/users/36721 | 295650 | 129,966 |
https://mathoverflow.net/questions/295619 | 4 | Let $(X,\mu)$ be a standard probability space and let $T:X \to X$ be a measure-class preserving transformation such that there is no $T$-invariant measure absolutely continuous with respect to $\mu$. (Such transformations are called type $\mathrm{III}$.) For $1 \leq p < \infty$ let $U\_{T,p}$ be the isometry of $L^p(X,... | https://mathoverflow.net/users/30721 | Invariant function for Koopman operator of measure-class preserving tranformation | No, because if $f$ is invariant under $U\_{T,2}$ then $|f|^2$ is invariant under $U\_{T,1}$.
More generally the Mazur map $f \in L^p(X,\mu) \mapsto sgn(f) |f|^{p/q} \in L^q(X,\mu)$ is a homeomorphism intertwining the Koopman operators $U\_{T,p}$ and $U\_{T,q}$, so most properties of the Koopman operators do not depen... | 2 | https://mathoverflow.net/users/10265 | 295652 | 129,967 |
https://mathoverflow.net/questions/295634 | 1 | Suppose there is a group of $n$ people giving gifts to one another. Everybody brings a gift but we want the gifts to be "well-distributed" in the group. By this I mean the following:
In how many ways can the gifts be given such that
- nobody receives their own gift, and in addition
- there is no $k$-element subset ... | https://mathoverflow.net/users/12824 | Sub-circle-free Christmas-gift-giving | Everybody sits down. The host initiates the exchange of gifts. He stands up and chooses one of the $(n-1)$ others to give his gift to. The person that received the gift stands up and gives his gift to someone who is currently sitting, then that person gets up, etc., until the last one stands up and gives his gift to th... | 4 | https://mathoverflow.net/users/37432 | 295653 | 129,968 |
https://mathoverflow.net/questions/295659 | 1 | Let $(B, \Sigma\_B)$ and $(C, \Sigma\_C)$ be standard Borel spaces and let $\mu$ be a sub-probability measure on $C$.
Given $Y\in \Sigma\_{B\times C}$, I would like to use the following function:
$$
f: B \to [0, 1] \\
b \mapsto \mu(\{c \in C \mid (b, c) \in Y\})
$$
But I am stuck trying to prove that it is well-... | https://mathoverflow.net/users/114537 | Is this function measurable? | Yes, it is a part of Fubini theorem for the characteristic function of $Y$.
| 2 | https://mathoverflow.net/users/4312 | 295660 | 129,969 |
https://mathoverflow.net/questions/295640 | 3 | For concreteness let's assume that $u\in W^{1,2}(\Bbb R^2).$ It is well known that
$$
\|u\|\_4\le C \|u\|\_2^{\frac 12} \|\nabla u\|\_2^{\frac 12}.
$$
This is also true if $u\in W^{1,2}\_0(\Omega)$ for a bounded domain $\Omega$ in $\Bbb R^2$.
>
> Is it still true if we modify this in the same way as Poincare ineqau... | https://mathoverflow.net/users/80191 | Gagliardo-Nirenberg inequality for bounded domain | If you assume that $\Omega$ is a bounded uniform extension domain, then your desired inequality holds true. By uniform extension domain, I mean that there exists a linear extension operator $E$ which maps $L^p(\Omega)$ to $L^p(\mathbb{R}^n)$, where $1<p<\infty$, and $W^{1,2}(\Omega)$ to $W^{1,2}(\mathbb{R}^n)$ at the s... | 4 | https://mathoverflow.net/users/85906 | 295668 | 129,971 |
https://mathoverflow.net/questions/295676 | 4 | I'm familiar with [Newman's](https://projecteuclid.org/euclid.mmj/1028999029) rational approximation of the absolute value function via rational functions. Are there other explicit functions that approximate $|x|$ with exponential error? I was under the impression that Chebhyshev polynomials also can give such good app... | https://mathoverflow.net/users/119834 | approximating the $|x|$ function | Denote the minimal approximation error to the function $f(x)=|x|$ in the uniform norm on $[-1,1]$ by
$$
E\_{mn}(f,[-1,1])=\inf\_{r\in\mathcal{R}\_{mn}}\|f-r\|\_{\infty,[-1,1]},
$$
where $\mathcal{R}\_{mn}$ denotes the set of rational functions with numerators of degree at most $m$ and denominators of degree at most $n... | 11 | https://mathoverflow.net/users/89429 | 295684 | 129,978 |
https://mathoverflow.net/questions/295687 | 5 | [All references are wrt to Lurie's "Higher Topos Theory" in its latest online available version (March 10, 2012)]
**Definition 7.2.1.8**:
An ∞-topos $X$ is locally of homotopy dimension $\leq n$ if there exists a collection $\{U\_\alpha\}$ of objects of $X$ which generate $X$ under colimits, such that each $X\_{/U\_\... | https://mathoverflow.net/users/26470 | Is an ∞-topos of local homotopy dimension $\leq n$ of homotopy dimension $\leq n$? | Let $\mathcal{X}$ denote the $\infty$-topos $\mathcal{S}\_{/S^1}$, whose objects are spaces $X$ with a map $X \rightarrow S^1$. Then $\mathcal{X}$ is generated under colimits by the object given by the base point inclusion $\ast \rightarrow S^1$, and is therefore locally of homotopy dimension $0$. However, it is not of... | 15 | https://mathoverflow.net/users/7721 | 295692 | 129,980 |
https://mathoverflow.net/questions/295706 | 15 | Let $\beta \mathbb{N}$ denote the Stone-Cech compatification of the natural numbers and $\beta \mathbb{N} \setminus\mathbb{N}$
denote the reminder of this compactification. I wonder if there is a characterization in ZFC of the continuous images of $\beta \mathbb{N} \setminus\mathbb{N}$. I mean: Which compact Hausdorff ... | https://mathoverflow.net/users/122189 | Continuous images of $\beta \mathbb{N} \setminus\mathbb{N}$ | This is a great question. There has been quite a bit of work done to figure out what the continuous images of $\beta \mathbb N \setminus \mathbb N$ are. I'll do my best to summarize some of that work here.
In some sense the answer to your question is yes: a space $X$ is a continuous image of $\beta \mathbb N \setminu... | 23 | https://mathoverflow.net/users/70618 | 295710 | 129,989 |
https://mathoverflow.net/questions/295670 | 2 | Let $X$ be a geodesic metric space. Are there known local obstructions to the existence of a (bi-Lipschitz) homeomorphism between $X$ and $X\times \mathbb{R}$?
| https://mathoverflow.net/users/nan | Obstructions to being a trivial line bundle over itself | A homological obstruction that I mentioned in the comments can be extracted from [On homotopical and homological $Z\_n$-sets](https://arxiv.org/abs/1002.2802) by Taras Banakh, Robert Cauty, and Alex Karassev.
This is a long paper but we won't need to know much.
Given an abelian group $G$, a point $x\in X$ is a *$... | 2 | https://mathoverflow.net/users/1573 | 295716 | 129,993 |
https://mathoverflow.net/questions/295718 | 0 | Expanding on a previous [post](https://mathoverflow.net/questions/294972/polynomials-that-preserve-nonnegativity) I made recently, let
$$
\mathscr{P}:= \{ p(x) \in \mathbb{R}[x] \mid p(x) \ge 0,~\forall x\ge 0\}.
$$
The Pòlya-Szegö theorem (see Theorem 3.21 [here](https://link.springer.com/chapter/10.1007/978-0-387-096... | https://mathoverflow.net/users/104633 | Polynomials $p$ such that $p$ and $p'$ preserve nonnegative numbers | The equivalent condition is $p(0)\geqslant 0$, $p'(x) \geqslant 0$ for $x\geqslant 0$.Thus this class reduces to $\mathscr {P} $ by taking antiderivative.
The inequality $|p(z)|\leqslant p(|z|)$ fails for $p(z) =z^2 - z^3 +z^4$ and $z=-1$. Also $p'''\notin \mathscr{P}$.
| 6 | https://mathoverflow.net/users/4312 | 295719 | 129,994 |
https://mathoverflow.net/questions/295738 | 2 | Motivation:
1. I want to compute $$E[g(X)] := \int\_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{\*}$$ without needing change of variable formula.
2. I want to prove the change of variable formula (you know the one I don't want to use for #1) without 'standard machine' (indicator, simple, nonnegative, integrable).
... | https://mathoverflow.net/users/69696 | Explicitly representing a random variable in terms of indicator functions | $\newcommand{\ep}{\epsilon}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}}$
For any random variable (r.v.) $X$, you can write $X=X\_+-X\_-$, where $X\_+:=\max(0,X)$ and $X\_-:=\max(0,-X)$. Then you can write
$$X\_+=\int\_0^\infty I\{X>x\}dx,\quad X\... | 3 | https://mathoverflow.net/users/36721 | 295742 | 130,006 |
https://mathoverflow.net/questions/295735 | 2 | Given two nilpotent matrix B1 and B2 over complex numbers which commute i.e. [B1,B2]=0, we know that they can be conjugated to upper-triangular ones (even strictly-triangular since they're nilpotent).
But, can we conjugate them to upper-triangular ones so that one of them e.g. B1 gets into its Jordan normal form?
T... | https://mathoverflow.net/users/114985 | Simultaneous upper-triangularization for two nilpotent commuting matrix | Yes, we can conjugate $B\_1$ and $B\_2$ simultaneously by a matrix $S$, whose columns are a basis of generalized eigenvectors of $B\_1$, such that $S^{-1}B\_1S$ is in Jordan normal form, and both $S^{-1}B\_1S$, $S^{-1}B\_2S$ are upper-triangular. This is possible, because commuting operators leave invariant generalized... | 5 | https://mathoverflow.net/users/32332 | 295748 | 130,009 |
https://mathoverflow.net/questions/295772 | 4 | Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is
**Given an element as a product of Dehn twist, is it possible to write down the corresponding Nielsen-Thurston decomposition?**
Explicitly, suppose $\phi\in Mod(S)$ such that $\p... | https://mathoverflow.net/users/9485 | Nielsen-Thurston decomposition from the product of Dehn twists | There isn’t a simple recipe. Bell and Webb recently gave [a polynomial-time algorithm](http://arxiv.org/abs/1609.09392v2) to compute the Nielsen—Thurston type of a mapping class. Their paper also contains a summary of previously known algorithms.
| 2 | https://mathoverflow.net/users/1463 | 295783 | 130,022 |
https://mathoverflow.net/questions/295775 | 4 | Let $R\_1, R\_2, \cdots$ be i.i.d. Rademacher random variables (taking values $-1,+1$ w.p. $0.5$). At time $k$, their average is $\frac{1}{k}\sum\_{i=1}^k R\_i$. One can imagine after $k\geq n$ for some $n$, this average becomes quite close to zero. It is interesting to characterize the maximum deviation of the average... | https://mathoverflow.net/users/91110 | Concentration inequalities on the supremum of average after time $n$ | $\newcommand{\ep}{\epsilon}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}}
\newcommand{\PP}{\operatorname{\mathsf P}}$
Let $S\_k:=\sum\_{i=1}^k R\_i$ and $K\_j:=\{n\_j,\dots,n\_{j+1}\}$, where $n\_j:=n2^{j-1}$. For $x>0$,
\begin{align\*}
\PP(Y\_n> x)=\PP(\sup\_{k\ge n}\frac{S\_k}k> x)
&=\P... | 4 | https://mathoverflow.net/users/36721 | 295784 | 130,023 |
https://mathoverflow.net/questions/295773 | 6 | If $\Gamma$ is a finite group with action on two vector spaces $\mathbb R^n$ and $\mathbb R^m$ denoted by $\gamma\_n$ and $\gamma\_m$ respectively, the fundamental equivariants are the polynomials $f: \mathbb R^n \rightarrow \mathbb R^m$ that commute with $\Gamma$, i.e. $$\gamma\_m \cdot f(x) = f(\gamma\_n^{-1} \cdot x... | https://mathoverflow.net/users/46236 | Software for computing equivariants | The canonical reference appears to be Karin Gatermann's book:
*Gatermann, Karin*, [**Computer algebra methods for equivariant dynamical systems**](http://dx.doi.org/10.1007/BFb0104059), Lecture Notes in Mathematics. 1728. Berlin: Springer. xv, 153 p. (2000). [ZBL0944.65131](https://zbmath.org/?q=an:0944.65131).
In ... | 2 | https://mathoverflow.net/users/11142 | 295789 | 130,025 |
https://mathoverflow.net/questions/295642 | 2 | I have a question about Feller property of reflecting Brownian motions.
Let $D \subset \mathbb{R}^2$ be a domain. Assume $D$ is represented as
\begin{equation\*}
D=\{(x,y) \in \mathbb{R} \times \mathbb{R} \mid |y|<H(|x|)\}.
\end{equation\*}
Here, $H$ denotes a smooth function on $[0,\infty)$.
Let $X\_t$ be the ref... | https://mathoverflow.net/users/68463 | Reflecting Brownian motions on horns are not Feller? | You are right: $X\_t$ is not a Feller process. The following argument is somewhat sketchy, but it should not be too difficult to make it complete.
---
Let us consider the function $$u(x,y) = \tfrac{1}{4} (1 - x^{-2} - 4 y^2) $$ Observe that:
* the normal derivative of $u$ on the boundary of $D = \{(x, y) : |y| ... | 4 | https://mathoverflow.net/users/108637 | 295794 | 130,027 |
https://mathoverflow.net/questions/215401 | 4 | What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible?
Recall, $M$ is said to be *topologically/smoothly irreducible* if it is not homeomorphic/diffeomorphic to a product of two topological/smooth manifolds.
This question is a parti... | https://mathoverflow.net/users/36688 | No irreducible parallelizable manifold of a given dimension | I don't know the answer for general dimensions, but here is an argument that shows that the four-dimensional manifolds suggested by valeri fit your criteria (provided $n > 0$).
The Dold-Whitney Theorem states that if $E\_1$ and $E\_2$ are orientable rank four vector bundles over an oriented four-dimensional manifold ... | 2 | https://mathoverflow.net/users/21564 | 295799 | 130,029 |
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