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https://mathoverflow.net/questions/281069 | 2 | We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds.
We have $\mathcal L\_\Bbb Z\subseteq \mathcal L\_\Bbb Q\subseteq\mathcal L\_\Bbb R$ where
$$\mathcal L\_\Bbb Z=\{uB\in\Bbb Z^n:u\in\Bbb Z^k\},\quad\mathcal L\_\Bbb Q=\{uB\in\Bbb Z^n:u\in\Bbb Q^k\},\quad\mathcal L\_\Bbb R=\{uB\... | https://mathoverflow.net/users/10035 | Integer points spanned by real, rational and integer combination of integer vectors | We always have $\mathcal L\_\Bbb Q=\mathcal L\_\Bbb R:$ For any particular $v \in\Bbb Z^n$ consider how we determine $\{u \in \Bbb R^n \mid uB=v\}.$ The method, if the set is nonempty, will yield one or more solutions with $u \in \Bbb Q^n.$
The Smith Normal Form of $B$ is a certain integer diagonal matrix. I'd expect... | 2 | https://mathoverflow.net/users/8008 | 281094 | 124,440 |
https://mathoverflow.net/questions/281088 | 11 | I wonder if there are natural examples of sets whose cardinality is $\frak p$
and $\frak t$?
Examples that could provide intuition for non- set-theory experts?
I find the definitions in the Malliaris & Shelah paper that established
$\frak p=t$ difficult to grok, so to speak.
I am not asking to understand the proof, jus... | https://mathoverflow.net/users/6094 | Intuition for the infinite cardinals p and t (now that p = t)? | Each element of **p** and each element of **t** is a countable family of infinite sets of natural numbers.
Let's start by giving an example of such family.
A simple example of a countable family of infinite sets of natural numbers would be $$E = \bigotimes\_{j\in \Bbb N} \{k \in \Bbb N : \exists m | k = jm \}$$ so th... | 4 | https://mathoverflow.net/users/82067 | 281098 | 124,442 |
https://mathoverflow.net/questions/280875 | 4 | Let $Q(x,y,z)$ be a geometrically irreducible quadratic form in $x,y,z$ with real coefficients, such that $z^2$ appears with non-zero coefficient. Define the region $\mathcal{R}(X)$ by
$$\displaystyle \mathcal{R}(X) = \{(x,y,z) \in \mathbb{R}^3 : |x|, |y|, |Q(x,y,z)| \geq 1, 1 \leq |x||y|Q(x,y,z)^2 \leq X\}.$$
This... | https://mathoverflow.net/users/10898 | Estimating the volume of a region bounded by polynomial inequalities | Since we are restricted now to $Q(x,y,z)=z^2-4xy$, it makes sense to use the symmetry and consider $x,y\ge 1$ once with the same $Q$ and once with $Q(x,y,z)=z^2+4xy$. Let now $w=xy$. Then, if we switch to the new variable $w$, we shall get the regions $1\le w\le X, 1\le |z^2\pm 4w|\le \sqrt{X/w}$ with the measure that,... | 3 | https://mathoverflow.net/users/1131 | 281106 | 124,444 |
https://mathoverflow.net/questions/281105 | 3 | Let $K$ be a knot, $G=\pi\_1(S^3 - K)$ and $G^{(n)}$ is the $n$-th term
of the derived series. Then $G/G^{(n)}$ is poly-torsion-free abelian since it is known that each $G^{(i)}/G^{(i+1)}$ is torsion-free. Suppose each $G^{(i)}/G^{(i+1)}$ is infinite for $i\geq 0$. Consider a tower of iterated universal abelian covers ... | https://mathoverflow.net/users/114032 | Does there exist a limit of (infinite) iterated universal abelian covers of a knot group? | If you take a non-trivial cable of a knot with trivial Alexander polynomial, then I think $G^{(i)}/G^{(i+1)}$ will be infinite for $i\geq 0$, yet one will never have $G^{(\alpha)}=1$ for any ordinal $\alpha$.
A knot with Alexander polynomial $=1$ (such as a Whitehead double) has infinite cyclic cover acyclic, and he... | 6 | https://mathoverflow.net/users/1345 | 281109 | 124,446 |
https://mathoverflow.net/questions/281111 | 12 | Let $n\geq 1$ be an integer and suppose $S\subseteq {\mathbb R}^n$ is countable and dense. Do we have $S \cong {\mathbb Q}^n$ where both sets carry the topology inherited from the Euclidean topology on ${\mathbb R}^n$?
| https://mathoverflow.net/users/8628 | Are countable dense subspaces of $\mathbb{R}^n$ homeomorphic to ${\mathbb Q}^n$? | According to <https://arxiv.org/abs/1210.1008>
Example 2(c)... yes, they are all homeomorphic to $\mathbb Q$!
| 20 | https://mathoverflow.net/users/4600 | 281112 | 124,447 |
https://mathoverflow.net/questions/281113 | 8 | What is the standard reference for the provability of the cut-elimination theorem in PRA?
Update: Rasmus Blanck has offered a reference for a system other than Gentzen's $\mathfrak L \mathfrak K$. The bounty is for a reference for the provability of the cut-elimination theorem for the sequent calculus $\mathfrak L \m... | https://mathoverflow.net/users/66833 | cut-elimination provable in PRA | My suggestion is Theorem 5.17 in section V.5.(d) of Hájek and Pudlák's *Metamathematics of First-Order Arithmetic*, Springer-Verlag, 1993. There, cut-elimination is shown to be provable in the even weaker theory $I\Delta\_0 + \text{superexp}$.
| 11 | https://mathoverflow.net/users/103227 | 281116 | 124,449 |
https://mathoverflow.net/questions/281075 | 18 | In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons functional of the hyperbolic connection (see section 5.3.4).
Let me state this more precisely.
Let $M$ be a three dimension... | https://mathoverflow.net/users/114489 | Hyperbolic Volume and Chern-Simons | This is a Theorem of Yoshida, the reference is
* Yoshida, Tomoyoshi: ''The η-invariant of hyperbolic 3-manifolds.'' Invent. Math. 81, 473-514 (1985). <http://mathlab.snu.ac.kr/~top/articles/Yoshida.pdf>
The proof is by explicit computation and comparison of the Chern-Simons form and the volume form.
There is an... | 9 | https://mathoverflow.net/users/39082 | 281126 | 124,456 |
https://mathoverflow.net/questions/281124 | 41 | Consider a set of fractions $\left\{1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{n}\right\}$. How many subsets of this set have sum at most 1? I'm interested in the asymptotics of this number.
Clearly, any subset of $\left\{\frac{1}{\lceil n/2 \rceil}, \ldots, \frac{1}{n}\right\}$ works, hence the answer is $\Omega(... | https://mathoverflow.net/users/106512 | Sets of unit fractions with sum $\leq 1$ | Let $n\_0$ be the smallest number such that the sum of the reciprocals of the integers from $n\_0+1$ to $n$ is $<2$. It is easy to see that $n\_0 \approx n/e^2$, since $\sum\_{j>n/e^2}^{n} 1/j \approx \log n - \log (n/e^2)
=2$. Now for any subset $A$ of $\{n\_0 +1, \ldots, n\}$ either the sum of the reciprocals of ele... | 36 | https://mathoverflow.net/users/38624 | 281131 | 124,460 |
https://mathoverflow.net/questions/281139 | 3 | I am not sure if this is the right place to ask but since I don't get many answers on this kind of questions on SE I might as well try it here. I also posted this question on SE.
I have the following problem and would be grateful if somebody could point me to a reference or give a brief explanation:
>
> I have a ... | https://mathoverflow.net/users/89264 | Automorphism of semisimple algebraic group of order 2 | It is well-known that the fixed point group $H=G^\sigma$ for an involution is reductive and that it is of finite index in its normalizer. Search for "symmetric varieties". Therefore $K$ has a reductive subgroup of finite index and is therefore reductive itself.
The reductivity is best seen by observing that
$\mathfra... | 4 | https://mathoverflow.net/users/89948 | 281169 | 124,472 |
https://mathoverflow.net/questions/281167 | 0 | Let $(X, \leq)$ be a partial order and $\Sigma\_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra?
| https://mathoverflow.net/users/114537 | Ordered measurable spaces | This can fail, even if you assume that the lower cones of the order are measurable with respect to the algebra $\Sigma$. For example, consider the real numbers $X=\mathbb{R}$ and let $\Sigma$ be the usual Borel algebra. Let's assume the continuum hypothesis, and let $\leq$ be a well-order of the reals in order type $\o... | 4 | https://mathoverflow.net/users/1946 | 281171 | 124,474 |
https://mathoverflow.net/questions/281157 | 3 | My question is not typical for MathOverflow, and arises in my teaching rather than research, but I think there will be readers who can give interesting answers.
Identify $\{\mathrm{A}, \ldots, \mathrm{Z}\}$ with $\mathbb{Z}/26\mathbb{Z}$. Define a bijection $R$ on the set of words over $\{\mathrm{A}, \ldots, \mathrm... | https://mathoverflow.net/users/7709 | Breaking the rotate-then-substitute alphabetic cipher | As you point out, frequency analysis will yield (for cipher texts long enough, say longer than Shannon's *unicity distance*) a non-English text which has been transformed from English via the position dependent transformation $R$.
But I think a differential analysis can break this with not too much extra effort. If y... | 5 | https://mathoverflow.net/users/17773 | 281176 | 124,478 |
https://mathoverflow.net/questions/281128 | 8 | Let $f\colon X\to \mathbb{A}^1$ be a smooth projective morphism of complex algebraic manifolds, where the target $\mathbb{A}^1$ is the affine line. **Are there any restrictions on the Hodge structures on the cohomology groups of fibers of $f$ over different complex points of $\mathbb{A}^1$? (Say are there examples wher... | https://mathoverflow.net/users/16183 | Variations of Hodge structures over the line | See Theorem 11 on page 191 of [these notes](https://www-fourier.ujf-grenoble.fr/~peters/Articles/PubSteen2.pdf). A special case is as follows.
**Theorem of the fixed part.** Let $S$ be a smooth quasiprojective variety, and $V$ a variation of $\mathbb{Q}$-Hodge structures on $S$ (for example, $R^i\pi\_\*\underline{\ma... | 9 | https://mathoverflow.net/users/6950 | 281177 | 124,479 |
https://mathoverflow.net/questions/281182 | 5 | For what kind of topological vector spaces (separable maybe?) are the closed convex subsets *countable* intersections of halfspaces.
I've seen somewhere that it's true for separable Hilbert spaces, but without proof or reference. Is there a reference on this fact (and related questions)?
| https://mathoverflow.net/users/103312 | When are the closed convex subsets countable intersections of halfspaces | This question is addressed in:
*Borwein, Jonathan M.; Vanderwerff, Jon D.*, [**Constructible convex sets**](http://dx.doi.org/10.1023/B:SVAN.0000023393.75251.05), Set-Valued Anal. 12, No. 1-2, 61-77 (2004). [ZBL1059.46056](https://zbmath.org/?q=an:1059.46056).
| 6 | https://mathoverflow.net/users/11142 | 281184 | 124,480 |
https://mathoverflow.net/questions/281180 | 2 | Suppose $\circ:G \times X \to X$ with $X$ countable. There is a measure preserving dynamical system $([0,1]^X, \lambda^X, G)$ where $\lambda$ is the uniform measure on $[0,1]$ and for $(a\_x)\_{x \in X} \in [0,1]^X$, $g$ takes $(a\_x)\_{x \in X}$ to $(a\_{\circ(g,x)})\_{x \in X}$.
If $X$ has a finite orbit $Y$ then ... | https://mathoverflow.net/users/8106 | Is there always an G-invariant uniform measure preserving map for countable G? | I presume that by an "invariant" map you mean one which is constant on the orbits of the action of $G$ on the configuration space. Therefore, it should factorize through the space of ergodic components of your action, and your question is equivalent to asking whether the space of ergodic components is purely non-atomic... | 3 | https://mathoverflow.net/users/8588 | 281185 | 124,481 |
https://mathoverflow.net/questions/281194 | 6 | Let $H$ be a hereditary algebra of Dynkin type. There is a cluster category $\mathcal{C}\_H$ defined by Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov in [Tilting theory and cluster combinatorics](https://arxiv.org/abs/math/0402054).
In [the file](http://www2.math.uni-paderborn.de/fi... | https://mathoverflow.net/users/11877 | What is the status of a problem about cluster categories? | For $H$ the path algebra of a 'star-shaped' quiver having three legs with lengths $r$, $s$, $t$, an answer seems to be implicit in Lamberti's combinatorial model for the cluster category:
<https://arxiv.org/abs/1403.0549>
but she doesn't make it explicit. This includes all Dynkin types, where $(r,s,t)\in\{(1,p,q),(... | 6 | https://mathoverflow.net/users/21483 | 281202 | 124,487 |
https://mathoverflow.net/questions/281205 | 3 | Let $U \subset A \subset X$ be spaces (in the sense of homotopy theory).
For every pointed space $Y$ restriction maps induce the following canonical map between mapping spaces:
$$fiber(Map(X,Y) \to Map(A, Y)) \to fiber(Map(X-U,Y) \to Map(A-U,Y))$$
**Is this map a homotopy equivalence?** If not, can we give necessa... | https://mathoverflow.net/users/22810 | A "non-abelian excision" statement for mapping out of a space | It depends on what exactly you mean by "subspace" and "fiber". Let me put some theorems down for you:
**Theorem**: Let $U,V\subseteq X$ open subspaces. Then the following is a homotopy pushout square:
$$\require{AMScd}
\begin{CD}
U\cap V @>>> U\\
@VVV @VVV \\
V @>>> U\cup V
\end{CD}\,.$$
**Proof:** This is a partic... | 8 | https://mathoverflow.net/users/43054 | 281206 | 124,489 |
https://mathoverflow.net/questions/251605 | 6 | In [KwoM] it is proved that the the core model K exists under the assumption that there is no inner model having a Woodin cardinal and satisfying ZFC. Furthermore, they also mention that the result is the "best possible", in the sense that we cannot weaken this anti-large-cardinal hypothesis without strengthening the r... | https://mathoverflow.net/users/38602 | Core model with $\omega$ Woodin cardinals | Dan, the answer is "no." You might be in a situation where Kc is a mouse (i.e., its ctble. substructures are ω1+1 iterable), but Kc is not fully iterable. E.g. it could be that Kc has M2F,#, where F is the operator simultaneously closing under all Mn#, n<ω, but for some θ, V = LF(Vθ) (same F, V could even be an extende... | 3 | https://mathoverflow.net/users/114509 | 281207 | 124,490 |
https://mathoverflow.net/questions/281200 | 1 | Can we introduce independent coordinates on a sphere such that **any** great circle could be represented as a linear equation (like line on the plane)? If yes, what is a generalization for higher dimensions?
Thank you in advance.
| https://mathoverflow.net/users/112492 | Linear equation for a great circle on a (multidimensional) sphere | This can be done locally by projection from the center of the sphere. Great circles go to lines, so if you pull back the cartesian coordinates from the plane to a hemisphere, every great half-circle is described by a linear equation.
You cannot extend this beyond the hemisphere, because two great circles meet in two ... | 2 | https://mathoverflow.net/users/98590 | 281208 | 124,491 |
https://mathoverflow.net/questions/281199 | 1 | Let $Z\_n$ denote the population of the $n$-th generation of a Galton Watson Process, with
* $Z\_0=N>1$
* $p\_0 \in (0,1)$
* supercritical, i.e. the mean of descendeants is above $1$
* $(Z\_n$) is conditioned on survival
Now let $K\in \mathbb{N}$. Uniformly choose an individual of the $K$-th generation, kill it and... | https://mathoverflow.net/users/60768 | Extinguish a blood line | If I understand correctly, your problem is equivalent to the following one. Take a random number $M$ (in the original problem $M$ is the size of the $K$-th generation), consider $M$ independent G-W processes $Z\_n^{(j)}$, $j = 1, 2, \ldots, M$, and condition on event that at least one of them never dies out. What is th... | 3 | https://mathoverflow.net/users/108637 | 281213 | 124,493 |
https://mathoverflow.net/questions/281203 | 5 | Let $R$ be a commutative ring and let $\mathfrak{a}\subseteq R$ be an ideal. The ideal $\mathfrak{a}$ is called *T-nilpotent* if for every sequence $(r\_i)\_{i\in\mathbb{N}}$ in $\mathfrak{a}$ there exists $n\in\mathbb{N}$ such that $\prod\_{i=0}^nr\_i=0$, and *quasinilpotent* if there exists $n\in\mathbb{N}$ such that... | https://mathoverflow.net/users/11025 | T-nilpotency and quasinilpotency of ideals | No need to specify the ambient ring since every commutative associative ring $I$ (possibly without unit) is an ideal in a ring (namely $I\oplus\mathbf{Z}$ with multiplicative law $(a\_1,n\_1)(a\_2,n\_2)=(a\_1a\_2+n\_1a\_2+n\_2a\_1,n\_1n\_2)$).
Now call a X-ring a ring (possibly without unit) satisfying the axioms: a... | 4 | https://mathoverflow.net/users/14094 | 281215 | 124,494 |
https://mathoverflow.net/questions/281183 | 2 | Let $X$ be a smooth weak Fano but not Fano $3$-fold ($-K\_X$ is nef and big but not ample). Then the anti-canonical morphism $\phi:X\rightarrow W$ (the morphsim induced by the linear system $|-mK\_X|$ some $m\gg 0$) induces is a birational contraction.
Assume that $\phi$ contracts a divisor. Could such divisor be co... | https://mathoverflow.net/users/nan | Anti-canonical divisorial contractions of weak Fano $3$-folds | What about taking the following example: consider $\mathbb P^1$ bundle over $\mathbb P^2$ given by the projectivisation of $\mathbb P(O\oplus O(-3))$ over $\mathbb P^2$. The linear system $-mK\_X$ contracts $\mathbb P^2$ to a point.
| 2 | https://mathoverflow.net/users/13441 | 281228 | 124,500 |
https://mathoverflow.net/questions/280626 | 11 | **Background.** As we know from Fred Cohen's [Thesis](https://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf), taking homology of the little 2-discs operad $\mathcal{D}\_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\mathcal{Gerst}$. Algebras over that operad have a commutativ... | https://mathoverflow.net/users/105615 | What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically? | Well, for your question 1 you presumably may ask yourself first about a relationship between (shifted) Lie algebras and Lie 2-algebras. Lie 2-algebras of Hanlon and Wachs can be viewed as $L\_\infty$-algebras where all operations except for the ternary one vanish, and this is the only relationship to Lie algebras that ... | 4 | https://mathoverflow.net/users/1306 | 281230 | 124,502 |
https://mathoverflow.net/questions/281115 | 2 | Consider the multivariate case for the question "Approximation theory reference for a bounded polynomial having bounded coefficients" ([Approximation theory reference for a bounded polynomial having bounded coefficients](https://mathoverflow.net/questions/97769/approximation-theory-reference-for-a-bounded-polynomial-ha... | https://mathoverflow.net/users/113059 | A bounded polynomial having bounded coefficients: several variables | Bounds in the univariate case, see e.g. [here](https://mathoverflow.net/questions/257909/find-the-maximum-of-a-p-if-a-0a-1x-dotsa-nxn-1-1-mapsto-1-1/257918#257918), were established by V.A. Markov in 1892.
S.N. Bernstein has given an extension of the result to the multivariate case in
>
> S.N. Bernstein, On cer... | 4 | https://mathoverflow.net/users/89429 | 281245 | 124,508 |
https://mathoverflow.net/questions/280664 | 10 | Let $A \cong L^\infty[0,1]$ be a non-atomic maximal abelian \*-subalgebra
in $M \cong B(L^2[0,1])$ (or any von Neumann algebra $M$).
Is the following true?
For every $T \in M$ and $\epsilon>0$, there are non-zero
projections $p,q \in A$ such that $\| qTp \| \le \epsilon \| T \|$.
| https://mathoverflow.net/users/7591 | A small corner w.r.t. a masa in a von Neumann algebra | This is not true. Here's an example for which it fails:
Let $\{ r\_n \}\_{n \in \mathbb N}$ be an enumeration of the rationals. Construct an orthogonal set $\left\{ f\_{j, k}^{l, m} \right\}\_{j, k, l, m \in \mathbb N} \subset L^2 (\mathbb R)$ such that each $f\_{j, k}^{l, m}$ is valued in $\{ -1, 0, 1 \}$, has only ... | 10 | https://mathoverflow.net/users/6460 | 281248 | 124,509 |
https://mathoverflow.net/questions/281246 | 22 | The largest prime in the order of the Monster group is $71$. This number $71$ shows up at various places:
* The minimal faithful representation has dimension $196883 = 47.59.71$
* The Monster group can be [realised](http://ac.els-cdn.com/002186938490228X/1-s2.0-002186938490228X-main.pdf?_tid=716c0f7a-9a42-11e7-bbbd-0... | https://mathoverflow.net/users/111498 | 71, the Monster, and c = 24 CFTs | Schellekens' enumeration is exhaustive in the following sense: the degree 1 subspace of the meromorphic CFT/vertex algebra is naturally a Lie algebra, and it is known that this Lie algebra must be one of the 71 that Schellekens wrote down.
Each of these 71 Lie algebras is realised as the weight 1 piece of some holomo... | 16 | https://mathoverflow.net/users/114577 | 281255 | 124,513 |
https://mathoverflow.net/questions/281260 | 2 | Given an injective linear map $T$ between Banach spaces $X$ and $Y$, let
\begin{equation} d(T) = \sup \left \{ \frac{||x||\_X}{||Tx||\_Y}: x \in X \mbox{ is nonzero } \right\} \cdot ||T||\_{\mathrm{op}} \end{equation}
Let \begin{equation} c(X,Y) = \inf \Bigl\{ d(T): T \mbox{ is an injective linear map from } X \mb... | https://mathoverflow.net/users/30721 | Distortion of embedding in Hilbert space | You are asking what is the Banach-Mazur distance between $\ell\_p^n$ and the Hilbert space of dimension $n$. The answer is $n^{|1/p-1/2|}$. You can find this in many books; in particular, the book of Tomczak-Jaegermann. It does not change anything if you use the Lipschitz analogue of the Banach-Mazur distance.
It is... | 2 | https://mathoverflow.net/users/2554 | 281264 | 124,518 |
https://mathoverflow.net/questions/281242 | 8 | If $X$ and $Y$ are two sets of $n$ independent, uniformly sampled points in the unit square, then standard methods can show that the expected minimum distance between points in $X$ and $Y$ is proportional to $1/n$, that is, $$E(\min\_{i,j}\|x\_i - y\_j \|)\sim 1/n$$as $n\to\infty$. Is there anything similar that can be... | https://mathoverflow.net/users/70190 | The minimum-perimeter triangle of three sets of points | Partition the unit square into small squares of area roughly $a$. Your question is equivalent to asking for which $a$ do we typically see about 1 small square with points from each of $X$,$Y$ and $Z$? this probability is roughly $(an)^3$ since the probability of seeing a point from each set is about $an$ and they're in... | 9 | https://mathoverflow.net/users/1061 | 281272 | 124,521 |
https://mathoverflow.net/questions/281291 | 2 | The following are subgroup embedding properties introduced by Bah and Borevich.
**Definition 1**: A subgroup $H$ of $G$ is said to be paranormal if for each $g \in G$, we have that $H^{\langle H, H^g \rangle}= \langle H, H^g \rangle$.
**Proposition 1**: A subgroup $H$ of $G$ is paranormal if and only if, for any $K... | https://mathoverflow.net/users/92488 | Subgroup embedding properties paranormality and polynormality | One way to ensure that a subgroup $H$ of $G$ is polynormal is to find an example in which there do not exist unequal $K$ and $L$ with $H \le K \lhd L \le G$. We could could look for an example of a simple group $G$ with subgroups $H <K < G$, where $H$ is maximal in $K$ and $K$ is maximal in $G$, so there are unlikely t... | 2 | https://mathoverflow.net/users/35840 | 281296 | 124,532 |
https://mathoverflow.net/questions/281295 | 8 | Regular graphs are the graphs in which the degree of each vertex is the same. The [Weisfeiler-Lehman algorithm](https://link.springer.com/chapter/10.1007/978-3-319-57586-5_22) fails to distinguish between the given two non-isomorphic regular graphs.
Is there a fastest known algorithm for regular graph isomorphism? A... | https://mathoverflow.net/users/111831 | Are regular graphs the hardest instance for graph isomorphism? | The first place to look at is [E.M. Luks algorithm for polynomial GI for graphs of bounded degree](http://ieeexplore.ieee.org/abstract/document/4567803/?reload=true). The algorithm is group-theoretic, and utilizes the fact that subgroups of products of $S\_k$ for bounded $k$ always have subgroups of bounded index, whic... | 7 | https://mathoverflow.net/users/106512 | 281300 | 124,533 |
https://mathoverflow.net/questions/281299 | 2 |
>
> Do we know, or are there any References that list down complete oriented and unoriented Bordism Group $Ω\_{n,O}(pt)$ and $Ω\_{n,SO}(pt)$ of points $pt$ for dimensions $n=1,2,...,10$?
>
>
>
Here are some informations I know. And a useful Ref is [here](http://www.maths.ed.ac.uk/~aar/papers/cobord.pdf).
$$Ω\_... | https://mathoverflow.net/users/27004 | Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$ | Unoriented cobordism: can be read off from the structure of the [unoriented cobordism ring](http://www.map.mpim-bonn.mpg.de/Unoriented_bordism#Construction_and_examples) (calculated in Thom's thesis): $\Omega\_6^O = (\mathbb Z/2)^3$, $\Omega\_7^O = \mathbb Z/2$, $\Omega\_8^O = (\mathbb Z/2)^5$, $\Omega\_9^O = (\mathbb ... | 12 | https://mathoverflow.net/users/97265 | 281302 | 124,534 |
https://mathoverflow.net/questions/281273 | 4 | Let $a\in\mathbb N^\mathbb Z$ be a bounded sequence of positive integers. Define another bounded sequence of positive integers $b=b(a)\in\mathbb N^\mathbb Z$ by
$$
b\_n=\sum\_{l=0}^\infty \max\{0,a\_{n+l}-l\}.
$$
For a constant sequence $a\equiv k\in\mathbb N$, then $b(a)\equiv \frac {k(k+1)}{2}$. Is the converse al... | https://mathoverflow.net/users/85550 | A certain operator on integer sequences | Yes, this is true if specifically $b\_n=k(k+1)/2$; however, in general, the sequences $a\_n$ with $b\_n=c$ are exactly the ones that take only two consecutive values $M,M-1$, repeated in an $M$ periodic pattern.
When we sum the $a\_{n+j}-j$ over $j\ge 0$ to compute $b\_n$, we can impose the restriction $a\_{n+j}-j\ge... | 5 | https://mathoverflow.net/users/48839 | 281305 | 124,536 |
https://mathoverflow.net/questions/281257 | 9 | The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk.
What is the analogue of this, over positive characteristic? Specifically, what replaces the Maurer-Cartan equation $... | https://mathoverflow.net/users/56594 | deformation theory in positive characteristic | Since the philosophy that deformation theory is controlled by DGLAs long predates the abstract characterisation of formal moduli problems, I'll break the answer in two. My answer will also hold in mixed characteristic, where things are complicated by the absence of a basepoints for the deformation functor (objects over... | 8 | https://mathoverflow.net/users/103678 | 281307 | 124,538 |
https://mathoverflow.net/questions/281301 | 4 | Epsilon Calculus is a formalism developed by Hilbert adding his $\epsilon$ operator to predicate logic. $\epsilon x. A(x)$ is a term such that $\exists x.A(x) \implies A(\epsilon x.A(x))$. In can actually replace the quantifiers, since $\exists x. A(x) \iff A(\epsilon x. A(x))$ and $\forall x. A(x) \iff A(\epsilon x.\l... | https://mathoverflow.net/users/65915 | Can adding Hilbert's epsilon to a theory, and then expanding that theories axiom schema to include the new language, cause it to become inconsistent? | Even if you're looking at extensions of ZFC, the answer is still yes. Consider the theory of ZFC plus the scheme asserting that there is no definable global choice function. That is, the formulas "the class defined by $\varphi(x,y) $ is not a global choice function."
This theory is equiconsistent with ZFC, since if w... | 8 | https://mathoverflow.net/users/1946 | 281319 | 124,540 |
https://mathoverflow.net/questions/281289 | 4 | Let $\phi:{\mathbb{N}}\rightarrow {\mathbb{N}}$ be a bijection. Can we extend the notion of parity (of a finite permutation) to $\phi$ ?
In other words, Can we define a group homomorphism $\Lambda $, between the group of all bijections of $\mathbb{N}$ to the group {1,-1,$\cdot$} such that $\Lambda(\psi)=-1$ if $\psi... | https://mathoverflow.net/users/114589 | Defining parity on the set of all bijections of ${\mathbb{N}}$ | It would be easier, and equivalent, to consider $\mathbb{Z}.$ But I'll stick to $\mathbb{N}.$
Consider $\sigma=(2\ 3)(4\ 5)(6\ 7)\cdots$ (so $0$ and $1$ are fixed points) and $(0\ 1)\sigma.$ They have the same cycle type but should not have the same parity. Note that fixed points and the particular ordering of $\mat... | 6 | https://mathoverflow.net/users/8008 | 281329 | 124,546 |
https://mathoverflow.net/questions/281324 | 6 | Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg\_LX = c\_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $V$, considered as a prime cycle of dimension $\dim{X}$.
**Question.** *For subvarieties $X$ and $Y$ of $V$ with $\de... | https://mathoverflow.net/users/26522 | The kernel of a nef line bundle | Consider $V=\mathbb{P}^1\times \mathbb{P}^2$ with projections $p\_1\colon V \rightarrow \mathbb{P}^1 \text{ and } p\_2\colon V \rightarrow\mathbb{P}^2.$ Let $L = p\_1^\*(\mathcal{O}\_{\mathbb{P}^1}(1))$, let $X = p\_2^{-1}(\ell\_1)$ and $Y=p\_2^{-1}(\ell\_2)$ for two distinct lines $\ell\_1$ and $\ell\_2\subset \mathbb... | 7 | https://mathoverflow.net/users/114605 | 281331 | 124,547 |
https://mathoverflow.net/questions/281341 | 4 | I am limitedly aware of some works done in connecting semigroups and computer sciences. And, indeed, there are many works in this area.
>
> May I ask you what would the ingeniously work be with this goal?
>
>
>
I am asking here because the giants may want to pay an attention to my little request. :-)
| https://mathoverflow.net/users/13898 | Applications of semigroups | Here is a question of mine along similar lines:
[Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?](https://mathoverflow.net/questions/243022/have-semigroups-with-actions-on-themselves-that-have-a-dual-to-the-compatibility/243106#243106)
The answer by Joseph V... | 4 | https://mathoverflow.net/users/9909 | 281352 | 124,554 |
https://mathoverflow.net/questions/281156 | 3 | Let $(M,g)$ be a Riemannian manifold of dimension at least $4$.
We consider the differential operator $$D:\Gamma(TM)\to \Gamma (TM)$$
with $$D(X)=\nabla \circ Div(X)$$.
The principal symbol $\delta (D): \pi^\* TM \to \pi^\* TM$, as a bundle morphism, has constant fiber wise rank $1$ where $\pi$ is the natural pr... | https://mathoverflow.net/users/36688 | A second cohomology class associated to a Riemannian manifold | Up to sign, the principal symbol of the operator in question is the projection onto the tautological line bundle in $\pi^\*TM$. To see this, identify $TM$ with $T^\*M$ using the Riemannian metric, and compute the symbol at $w\in S(TM)$. The symbol of the divergence at $w$ is $\langle-,w\rangle$. The symbol of the gradi... | 2 | https://mathoverflow.net/users/70808 | 281353 | 124,555 |
https://mathoverflow.net/questions/281356 | 16 | Let $G$ be a connected Fréchet-Lie group and let $\mathfrak g$ be its Lie algebra. Does the image $\exp(\mathfrak g) \subset G$ of the exponential map generate $G$?
| https://mathoverflow.net/users/23500 | Does the image of the exponential map generate the group? | I believe that this is open. In fact, even for the very special case where $G = \text{Diff}\_0(M)$ for a smooth manifold $M$, the only proof I know that $G$ is generated by the image of the exponential map uses a very deep theorem of Thurston that says that in this case $G$ is a simple group (this implies that $G$ is g... | 19 | https://mathoverflow.net/users/317 | 281362 | 124,557 |
https://mathoverflow.net/questions/281363 | 2 | I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we assume that $V$ is well-behaved (a bounded smooth function for instance).
The question I have is whether it follows t... | https://mathoverflow.net/users/114633 | Meromorphic extension of solutions to ODEs | As Mateusz remarked, $V$ must be assumed to have a meromorphic extension to $\mathbb C$, and $V = \lambda + \psi''/\psi$. Now if $\psi$ has a pole or zero at $z=p$, so that $\psi(z) \sim c (z-p)^d$ as $z \to p$ (with $d \ne 0$), then
$V(z) \sim d (d-1) (z-p)^{-2}$. Thus the only poles that $V$ may have are
poles of or... | 5 | https://mathoverflow.net/users/13650 | 281365 | 124,558 |
https://mathoverflow.net/questions/281359 | 9 | $\newcommand\Z{\mathbb{Z}}$
Let $K$ be an algebraically closed field of characteristic $\ne 2$. We have the unit sphere $S^n:~x\_0^2 + \ldots + x\_n^2 = 1$.
**What are the $\Z/2\Z$ cohomology groups of the sphere - $H^i\_{et}(S^n,\Z/2\Z)$?**
If $K$ is of characteristic 0, then by comparison with the complex number... | https://mathoverflow.net/users/26923 | Computing the etale cohomology of spheres | This is true over any algebraically closed field $k$ of characteristic different from $2$. More generally, if $\ell$ is invertible in $k$, then
$$H^i(X,\mathbb Z\_\ell) = \left\{\begin{array}{ll}\mathbb Z\_\ell & i = 0, n, \\ 0 & i \neq 0, n. \end{array}\right.$$
See for example [SGA 7$\_\text{II}$, Exp. XII, Table 3.7... | 13 | https://mathoverflow.net/users/82179 | 281366 | 124,559 |
https://mathoverflow.net/questions/281315 | 4 | Let
$$\begin{eqnarray}\nonumber
f(y, t) &=& \frac{C}{\sigma ^2 t} \left[\frac{(1-\alpha) (b-y)}{\alpha t^{\alpha}} \, \_1F\_1\left[\frac{\alpha+1}{2 \alpha};\frac{3}{2};-\frac{ (b-y)^2}{2 \sigma^2 t^{2 \alpha}}\right]- \sqrt{2} \sigma \frac{\Gamma \left(\frac{3}{2}-\frac{1}{2 \alpha}\right)}{\Gamma \left(1-\frac{1}{2 ... | https://mathoverflow.net/users/18929 | How to show that this limit converges in the distributional sense to a dirac delta function | Here is an attempt at a solution.
Continuing with your idea: Substitute $\xi:=\frac{b-y}{\sqrt{2}\sigma t^\alpha}$ to make everything nicer:
$$\begin{align}
f\_{t}(\xi) &= \frac{\sqrt{2}C}{\sigma t} \left[\frac{(1-\alpha)}{\alpha} \xi\, \_1F\_1\left(\frac{\alpha+1}{2 \alpha};\frac{3}{2};-\xi^2\right)-\frac{\Gamma \le... | 3 | https://mathoverflow.net/users/3041 | 281367 | 124,560 |
https://mathoverflow.net/questions/281101 | 2 | I am reading a paper by Adams and Frazier (namely *Adams, Frazier, Composition operators on potential spaces. Proc. Amer. Math. Soc. 114 (1992), no. 1, 155–165*, available [here](http://www.ams.org/journals/proc/1992-114-01/S0002-9939-1992-1076570-5/home.html)), whose main purpose is to show that, whenever $H\in C^\inf... | https://mathoverflow.net/users/36952 | On a paper by Adams and Frazier | I'm not quite confident about what I'll write, but I think that there is an issue here. I haven't read the paper, but I'd suggest the following amendment:
By Hölder, with exponents $q$ and $q'$, to be chosen later to make everything works fine, I'd write, just after "proof of Lemma 2.4."
$
S\_{\theta}(f\_1f\_2)(x)\... | 1 | https://mathoverflow.net/users/90189 | 281368 | 124,561 |
https://mathoverflow.net/questions/281290 | 2 | This post is improved from [Determine binary function $f(x)$ by partial observation of $x$](https://mathoverflow.net/questions/280737/). Since the form of the problem is changed in a great extent. I would like to create a new post rather than edit the old one. If it is inappropriate, feel free to edit these two.
--... | https://mathoverflow.net/users/22954 | Derive a theoretical bound about coding with a partial eavesdropper | (comment) If all subsets of size k leak $\frac{\epsilon}{n}$ bits, then you are looking at threshold (secret-sharing) schemes. In that specific case, it is tight: the bound is the *threshold gap* and the construction is a ramp scheme. Your leak condition being quite loose, other structures usually require a different b... | 1 | https://mathoverflow.net/users/70713 | 281388 | 124,563 |
https://mathoverflow.net/questions/281342 | 6 | (Cross-posted to <https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion>.)
It seems strange but, even after consulting several books, and hours spent on google, nothing came out about a law of iterated logarithm for the fractional Brownian motion.
I just need a p... | https://mathoverflow.net/users/70148 | Reference for LIL for fractional Brownian motion | This has been proved by S. Orey in this article:
Steven Orey. Growth rate of certain Gaussian processes. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II : Probability theory. Berkeley, Calif. : Univ. California Press, 443-451, 1972.
Note that the growth rate is equiv... | 4 | https://mathoverflow.net/users/69642 | 281395 | 124,567 |
https://mathoverflow.net/questions/281387 | 7 | I've been reading Girard et al's 'Proofs and Types', which in Chapter 6 presents a proof of strong normalisation for the simply typed lambda calculus with products and base types. The proof is based on Tait's method of defining a set of 'reducible terms for type $T$' by induction on the type $T$. For example a term $t$... | https://mathoverflow.net/users/114652 | How to handle sums in Tait's reducibility proof of strong normalisation? | The typical strategy when you're building a model to perform normalisation by evaluation (i.e. the computational part of Tait's method, see [1]) with sum types is to have as a semantics for `A + B`:
* either the semantics of `A`
* or the semantics of `B`
* or a neutral term of type `A + B` (i.e. a head variable with ... | 7 | https://mathoverflow.net/users/14972 | 281399 | 124,569 |
https://mathoverflow.net/questions/281389 | 1 | Let $X$ be an infinite set, and let $E \subseteq {\cal P}(X)$ be a collection of subsets of $X$. We say that $E$ is $T\_1$ (with respect to $X$) if for all $x\neq y\in X$ there is $e\in E$ with $x\in e$ and $ y\notin e$.
The set $E := \{X\setminus \{y\}: y\in X\}$ is an easy example for a $T\_1$-set $E$ with $|E| = |... | https://mathoverflow.net/users/8628 | Infinite "$T_1$"- hypergraphs | It depends on the cardinality of $X$:
>
> The answer to your question is no if and only if $|X|$ is a strong limit cardinal.
>
>
>
Recall that $|X|$ is a [strong limit cardinal](https://en.wikipedia.org/wiki/Limit_cardinal) iff $2^\lambda < |X|$ for every $\lambda < |X|$.
If $E \subseteq \mathcal P(X)$ is "$... | 3 | https://mathoverflow.net/users/70618 | 281404 | 124,571 |
https://mathoverflow.net/questions/281398 | 0 | I just saw the following question that was asked yesterday on math overflow [on meromorphic solutions to ODEs](https://mathoverflow.net/questions/281363/meromorphic-extension-of-solutions-to-odes)
Although, I understand the answers and comments to the questions, I did not understand how this applies to some of the stan... | https://mathoverflow.net/users/112877 | Meromorphic solutions to Legendre's equation | Solutions of the equation you wrote are not polynomials. Legendre polynomials
are solutions of the equation
$$(1-x^2)y''-2xy'+l(l+1)y=0.$$
This equation is written in the same article you refer to. It has two regular singularities: at $1,-1$, so this is a relatively simple case. The general theory
only says that at eac... | 2 | https://mathoverflow.net/users/25510 | 281410 | 124,575 |
https://mathoverflow.net/questions/281377 | 6 | Given an ordinal $\alpha$, I define $F\_{n}(\alpha)$ as follows:
* $F\_0(\alpha)=\alpha$
* $F\_{n+1}(\alpha)$ is the smallest $\beta$ such that no first-order $\phi$ in the
language of $\{\in\}$ has $(\mathrm{V}\models\phi(S,F\_0(\alpha),F\_1(\alpha)...F\_{n}(\alpha)))\Leftrightarrow S=\beta$
* $F\_\omega(\alpha)=\m... | https://mathoverflow.net/users/111429 | Do these ordinals exist? | For the definition of $F\_{n+1}$ to make sense, we need, in addition to the usual axiomatic apparatus of ZFC, a notion of "satisfaction of formulas in $V$." If we have this additional notion and if we allow it to occur in replacement axioms, then we can prove that $F\_n(\alpha)$ exists and is countable for every $n$ an... | 11 | https://mathoverflow.net/users/6794 | 281421 | 124,579 |
https://mathoverflow.net/questions/281414 | 12 | Given (finite, simple) graphs $G$, $H$ and $K$ and a [homomorphism](https://en.wikipedia.org/wiki/Graph_homomorphism)
$$
G+K\to H+K
$$
where $+$ denotes the [join](http://mathworld.wolfram.com/GraphJoin.html), does it follow that there also exists a graph homomorphism $G\to H$?
If this is known, I'd also appreciate a... | https://mathoverflow.net/users/27013 | Cancelling a graph join from a graph homomorphism | If $|K|=\infty$, then this is false, as a counterexample $G=K\_2$, $H=K\_1$, $K=K\_\infty$ shows.
Let us prove that the claim is true if $K$ is finite (with no such assumption for $G$ and $H$). Induction on $|K|$; if $|K|=0$, the claim is trivial.
For the inductive step, consider a homomorphism $\psi\colon G+K\to H... | 13 | https://mathoverflow.net/users/17581 | 281425 | 124,580 |
https://mathoverflow.net/questions/281426 | 10 | I'm trying to find generating manifolds for the cobordism group $\mathit{MO}\_5(K(\mathbb Z/2, 2))\cong (\mathbb Z/2)^4$, which can be represented as the cobordism group of closed 5-manifolds $M$ together with a class $B\in H^2(M;\mathbb Z/2)$. I've found three of the four generators; the fourth should be a 5-manifold ... | https://mathoverflow.net/users/97265 | Is there a closed 5-manifold $M$ with $w_1(M)w_2(M)\ne 0$? | Note that the third Wu class is $\nu\_3 = w\_1w\_2$, so on a closed connected smooth $n$-manifold $M$, $\operatorname{Sq}^3 : H^{n-3}(M; \mathbb{Z}\_2) \to H^n(M; \mathbb{Z}\_2)$ is given by $\operatorname{Sq}^3(x) = w\_1(M)w\_2(M)x$.
As $\operatorname{Sq}^i(x) = 0$ if $i > \deg x$, when $n = 5$ we see that $\operat... | 13 | https://mathoverflow.net/users/21564 | 281440 | 124,586 |
https://mathoverflow.net/questions/281434 | 6 | After reading many textbooks I still can't get the jargon correct. Given a spherical harmonic $u \in L^2(S^n)$ one could construct a theta function:
$$ \theta (z;u) = \sum\_{ m \in \mathbb{Z}^3} u (m) e^{2\pi i \, |m|^2\,z}$$
with $z\in \mathfrak H$. The situation of interest is that $u$ is not constant. This thin... | https://mathoverflow.net/users/1358 | Are theta functions cuspidal representations? | "Cusp form" means "cuspidal automorphic form" by definition. So yes, $\theta(z;u)$ is an automorphic form. But it is not a form lying in a cuspidal automorphic representation, because it is not a Hecke eigenform. (Actually, this also depends on the context, see the Added section for clarification.)
Cuspidal automorph... | 9 | https://mathoverflow.net/users/11919 | 281443 | 124,587 |
https://mathoverflow.net/questions/281439 | 0 | Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write
$$v(x) = \sum\_{k=0}^\infty a\_k(r) \psi\_k(\theta)$$
where $ r=|x|$ and $ \theta = \frac{x}{|x|}$ and $ \psi\_k$ are the $L^2(S^{N-1})$ normalized eigenfunctions.
We now suppose $u(x)$ satisfies
$$ ... | https://mathoverflow.net/users/66623 | Improved maximum principle estimates (deleting first mode) | Yes, one can get a constant better than $1/(2N)$. Simply take a look at the proof of the original bound $\|u\|\_\infty \le \|f\|\_\infty / (2N)$: one writes
$$ u(x) = \int\_B G\_B(x, y) f(y) dy , $$
estimates $|f(y)|$ by $\|f\|\_\infty$ under the integral, and uses $$\int\_B G\_B(x, y) dy = (1 - |x|^2) / (2 N) \le \fra... | 3 | https://mathoverflow.net/users/108637 | 281446 | 124,588 |
https://mathoverflow.net/questions/281432 | 4 | Let $Z$ be a subvariety of an irreducible projective variety $X$, and let $i:Z\rightarrow X$ be the inclusion.
Let $N\_1(X),N\_1(Z)$ be the $\mathbb{Q}$-vector spaces of curves in $X$ and $Z$ respectively, and let $i\_{\*}:N\_1(Z)\rightarrow N\_1(X)$ be the pushforward.
Assume that for a general curve $C\subset Z... | https://mathoverflow.net/users/nan | Pushforward of curves | You can prove this using the "Moving Lemma." Let $k$ be a field. Let $Z$ be a **projective** $k$-scheme that is integral of dimension $d$. Let $D\subset Z$ be a proper closed subset of $Z$. Let $C\subseteq Z$ be a closed subscheme that is integral of dimension $d-e$. Let $W\_1,\dots,W\_r\subseteq Z$ be integral closed ... | 4 | https://mathoverflow.net/users/13265 | 281456 | 124,592 |
https://mathoverflow.net/questions/281444 | 5 | Let a group $G$ act on a (not necessarily irreducible) algebraic variety over ${\bf C}$.
It seems to be well-known that the quotient in the sense of geometric invariant theory (i.e., the categorical quotient in the category of algebraic varieties) agrees with the largest Hausdorff quotient (i.e., the categorical quo... | https://mathoverflow.net/users/39082 | GIT quotient vs. largest Hausdorff quotient | I recommend reading the very nice exposition here (which is where I learned this fact):
Schwarz, Gerald W. *The topology of algebraic quotients.* Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), 135–151, Progr. Math., 80, Birkhäuser Boston, Boston, MA, 1989.
In particular, see the ... | 6 | https://mathoverflow.net/users/12218 | 281457 | 124,593 |
https://mathoverflow.net/questions/281386 | 2 | I have asked this question on stackexchange and have not received any answers or comments after 2 days of it being there.
I read somewhere that the following statement is correct. A proof or any hint as to how to prove it would be helpful.
Let $G$ be a connected reductive group defined over $k$(maynot be of charact... | https://mathoverflow.net/users/58056 | Field of definition of a normal subgroup of Reductive group | First, some easy reduction steps to pass to the case where $G$ is semisimple. Let $T$ be a maximal $k$-torus in $G$, and pass to a finite separable extension of $k$ so that $T$ is split. We know that $H$ is the almost-direct product $\mathscr{D}(H) \cdot Z$ for a subtorus $Z \subset T\_{\overline{k}}$ that is the maxim... | 5 | https://mathoverflow.net/users/81332 | 281461 | 124,596 |
https://mathoverflow.net/questions/72098 | 6 | Let $A$ be an associative algebra over a field. Then $A$ can be regarded as a Lie algebra via the Lie bracket defined by $[a,b]=ab-ba$ for every $a,b\in A$.
The algebra $A$ is called Lie locally nilpotent if it is locally nilpotent as a Lie algebra. Also, $A$ is said to be locally Lie nilpotent if every finitely genera... | https://mathoverflow.net/users/14653 | Lie locally nilpotent associative algebras | I have to mention that for associative algebras over fields of characteristic not 3 the question has now a positive answer. This is a consequence of a very recent paper by Dias and Krasilnikov: <https://arxiv.org/pdf/1709.05728.pdf>
| 3 | https://mathoverflow.net/users/14653 | 281477 | 124,602 |
https://mathoverflow.net/questions/281478 | 4 | I have the following situation: a smooth compact manifold $M$, without boundary and furnished with a smooth volume $\mu $.
I also have an operator defined on the space of smooth functions with $0$ average w.r.t $\mu$, let's call the space $C^{\infty}\_{\mu}(M)$.
This operator is an isomorphism and extends to an iso... | https://mathoverflow.net/users/69474 | when is an operator on smooth functions realized by pull-back | If (and only if) $T$ is also an algebra homomorphism, then it is a pullback by a diffeomorphism. See the introduction to chapter VIII of [here](http://www.mat.univie.ac.at/~michor/kmsbookh.pdf).
One might be able to replace algebra homomorphism by order respecting properties of $T$.
| 4 | https://mathoverflow.net/users/26935 | 281492 | 124,607 |
https://mathoverflow.net/questions/281438 | 6 | Let $c: S^1 \to M$ be a smooth contractible loop (not necesarily an embedding, or even an immersion) on the connected, compact symplectic manifold $(M,\omega)$ (if this helps somehow, $c$ is a $1$-periodic orbit of a $1$-periodic time-dependent Hamiltonian $H\_t$). The [book by Audin, Damian](https://rads.stackoverflow... | https://mathoverflow.net/users/48745 | Every contractible smooth loop has a neighbourhood with $H^2=0$ | Inductively isotope the skeleta of a triangulation of $M$ so that the $k$-skeleton is transverse to $c$. (We may use an isotopy to achieve transversality because the simplices are already embedded, and we don't need to homotope the boundary at all; a small homotopy of only the interior of the simplex doesn't change the... | 4 | https://mathoverflow.net/users/40804 | 281495 | 124,609 |
https://mathoverflow.net/questions/281494 | 3 | This is kind of a continuation of a recent [(closed) question](https://mathoverflow.net/questions/280964/is-there-a-continuous-surjection-omega-omega-to-mathbbr).
Is there an order-preserving surjective function $f:{\mathbb N}^{\mathbb N}\to [0,\infty)$ (where for $a,b\in {\mathbb N}^{\mathbb N}$ we have $a\leq b$ if... | https://mathoverflow.net/users/8628 | Order-preserving surjection ${\mathbb N}^{\mathbb N}\to [0,\infty)$ | Yes. Let us isomorphically identify the poset of functions $\omega \to \omega$ (under the pointwise order) with the set of functions $\omega \to \mathbb{N}\_2 = \{n \in \mathbb{N}: n \geq 2\}$, again ordered pointwise.
Now in fact there is an *isomorphism* of posets $\mathbb{N}\_2^\omega \to [1, \infty)$ given by co... | 10 | https://mathoverflow.net/users/2926 | 281498 | 124,610 |
https://mathoverflow.net/questions/281497 | 5 | Suppose that $X$ is a finite 2-dimensional CW-complex with free fundamental group and $\phi :K \longrightarrow X$ is a map which induces an isomorphism of fundamental groups, where $K$ is a finite bouquet of circles with the wedge point $a$. Consider the mapping cylinder $M=X\bigcup\_{\phi} (K ... | https://mathoverflow.net/users/114476 | A question about a statement in the paper of C.T.C. Wall | The attaching map for a 2-dimensional cell is a map $f: S^1 \to K$, and it determines an element $[f]$ of $\pi\_1(K)$ (up to conjugacy). What Wall means by trivial attaching maps is that these elements of $\pi\_1(K)$ must be trivial. If they weren't, then the map $\pi\_1(K) \to \pi\_1(K \cup\_f D^2) \to \pi\_1(X)$ that... | 4 | https://mathoverflow.net/users/360 | 281503 | 124,615 |
https://mathoverflow.net/questions/281507 | 4 | Suppose we are given two integer numbers $p$ and $q$ such that $p+q\equiv 0 \pmod{12}$. There is a result saying that for every such pair there exists a non necessarily connected almost complex manifold of dimension $4$ such that its Chern numbers are equal to $p$ and $q$. Could you give me a reference for the proof of... | https://mathoverflow.net/users/88385 | Chern numbers of almost complex manifolds | A. Van de Ven: [**On the Chern numbers of certain complex and almost complex manifolds**](http://dx.doi.org/10.1073/pnas.55.6.1624), *Proc. Natl. Acad. Sci. USA* **55**, 1624-1627 (1966). [ZBL0144.21003](https://zbmath.org/?q=an:0144.21003).
| 4 | https://mathoverflow.net/users/7460 | 281509 | 124,618 |
https://mathoverflow.net/questions/281508 | 3 | [This paper](https://hal.inria.fr/file/index/docid/241199/filename/ajp-jphystap_1907_6_202_0.pdf) by Hadamard is often cited as being the source of the definition of well-posed and ill-posed problems.
However, it is in French so I cannot verify that claim.
Is there an English translation?
| https://mathoverflow.net/users/6618 | Is there an English translation of Hadamard's classic French paper on well-posed problems? | No, there is no "official" English translation, however, [Google translate](https://translate.google.com/?hl=en&tab=TT) should work just fine, here is the translation of the first paragraph, *without any corrections from my side*:
>
> The general question that we propose to study is the determination of
> the solu... | 5 | https://mathoverflow.net/users/11260 | 281513 | 124,619 |
https://mathoverflow.net/questions/281504 | 1 | Let $\mathcal{M}(S)$ be a space of finite signed measures on a metric space $S$ ($=\mathbb{R}^2$ in my case) equipped with the total variation norm. Let
$\mathcal{M}\_1(S)=\{\mu \in \mathcal{M}(S):\int\_S d(e,x) \mu(dx)<\infty \}$ where $d$ is a distance. It implicitly follows from [this question](https://mathoverflow.... | https://mathoverflow.net/users/114695 | Proof that the subspace of signed measures integrating d(x,e) is closed | Let's try this. I think this is false, since the total variation has no connection with the distance $d$. (That other question does not seem to mention the total variation norm at all.)
**counterexample**
Our metric space $S$ is $\mathbb R$ with the usual distance. For $a \in S$, let $\epsilon\_a$ be the unit poin... | 2 | https://mathoverflow.net/users/454 | 281514 | 124,620 |
https://mathoverflow.net/questions/281385 | 1 | Originally I meant to ask this question [here](https://mathoverflow.net/questions/280562/criterion-of-reflexivity), but got confused and ended up asking another question, which had some mathematical meaning, but was not what I vaguely had in mind.
Let me restate the motivation.
Let $E$ be a Banach space.
It is kn... | https://mathoverflow.net/users/53155 | Criterion of reflexivity 2 | Your strong version has a strongly negative answer. Let $F$ be a non reflexive Banach space, let $E=F^{\*}$, and regard $F\subset E^\* = F^{\*\*}$. On $F$, the weak and weak$\*$ topologies are the same, so every norm closed convex subset of $F$ is relatively weak$^\*$ closed in $F$.
Your weak version has a strongly ... | 1 | https://mathoverflow.net/users/2554 | 281525 | 124,629 |
https://mathoverflow.net/questions/281517 | 1 | Is there any singular compactification of $PGL(2,\mathbb{C})$? Only compactification is known to me is $\mathbb{P}^3$. Does anyone know any other compactification?
| https://mathoverflow.net/users/nan | Compactification of $PGL(2,\mathbb{C})$ | The answer depends on how much symmetry of $PGL(2)$ you want to retain.
If the compactification is to be equivariant for left and right multiplication then, indeed, $\mathbb P^3$ is the only normal compactification. This follows easily from the embedding theory of spherical varieties.
If no symmetry is to be preser... | 8 | https://mathoverflow.net/users/89948 | 281538 | 124,632 |
https://mathoverflow.net/questions/278492 | 2 | Let $A$ be a non-negative (all entries $\geq 0$) square matrix. Is it always true that
$$
(a\_{11}+a\_{12}+a\_{21}+a\_{22})^2\geq 4a\_0a\_2
$$
where
$a\_{ij}$ is the permanent of a matrix obtained by deleting $i$-th row and $j$-th column from $A$.
$a\_0=\rm{perm}(A)$
$a\_2$ is the permanent of $A$ without th... | https://mathoverflow.net/users/16678 | An inequality between permanents of non-negative matrices | Notice that
$$
(a\_{11}+a\_{22}+a\_{12}+a\_{21})^2\geq (a\_{11}+a\_{22})^2+(a\_{12}+a\_{21})^2\geq 4a\_{11}a\_{22}+4a\_{12}a\_{21}.
$$
Thus, it suffices to show
$$
a\_{11}a\_{22}+a\_{12}a\_{21}\geq a\_0a\_2.
$$
Expand each permanent and perform multiplication to obtain the sums of products of $2n-2$ entries on both s... | 4 | https://mathoverflow.net/users/17581 | 281541 | 124,634 |
https://mathoverflow.net/questions/281140 | 2 | Edit: After helpful comments, I now know that I am concerned with flat fiber bundles **up to fiber preserving homotopy.**
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are nice spaces (say smooth manifolds). Then $E$ has the form of a twisted product
(i) $E \cong \widetilde{B} \... | https://mathoverflow.net/users/114528 | simplicial structure on a flat fiber bundle | Up to homotopy, everything works. For simplicity, assume that $F$ is connected. Let $G=\pi\_1$ with discrete topology, $EG$ the classifying space for $G$; then first replace $F$ with $EF= F\times EG$, the action $G\times F\to F$ with $G\times EF\to EF$. The latter is now properly discontinuous, $EF/G$ has homotopy type... | 1 | https://mathoverflow.net/users/21684 | 281552 | 124,641 |
https://mathoverflow.net/questions/281560 | 3 | Let $F$ be a non-Archimedean local field of characteristic $0$ and $K/F$ be a finite extension. Let $D\_F$ be the central division algebra of dimension $n^2$ over $F.$ Write $D\_K=D\_F\otimes\_FK$, which is again a central division algebra over $K$ of dimension $n^2$. Does there exist an idea of base change for divisio... | https://mathoverflow.net/users/56638 | Idea of base change for Division algebras over local field | An obvious idea to get a commutative square diagram, where $GL(n,F)$ and $GL(n,K)$ are respectively replaced by $D\_F^\times$ and $D\_K^\times$, is to use the Jacquet-Langlands transfer between representations of $D\_F^\times$ (resp. $D\_K^\times$) and square integrable representations of $GL(n,F)$ (resp. $GL(n,K)$) (c... | 3 | https://mathoverflow.net/users/4767 | 281564 | 124,644 |
https://mathoverflow.net/questions/281536 | 6 | This is an open question: given a sequence of $n$ real numbers $x\_1<x\_2<\dots<x\_n$, does there always exist a probability distribution, such that $\{x\_i\}$ happens to be the $n$ *expected order statistics* of this distribution?
In other words, can we always "reverse engineer" the distribution from its expected o... | https://mathoverflow.net/users/110654 | Existence of distribution for certain order statistics | The answer is **no** when $n = 4$.
Let $X\_1, X\_2, X\_3, X\_4$ be a sample from a distribution with CDF $F$. Denote by $Y\_1, Y\_2, Y\_3, Y\_4$ the order statistics (that is, the non-decreasing rearrangement) of $X\_1, X\_2, X\_3, X\_4$, and let $x\_j = \mathbb{E} X\_j$. We will show that $3(x\_4-x\_1) \leqslant 7(x... | 4 | https://mathoverflow.net/users/108637 | 281567 | 124,645 |
https://mathoverflow.net/questions/281556 | 1 | Given data points $(x\_i,y\_i)\in \mathbb{R}^m\times \mathbb{R}^n$ with $n>m$ satisfying $y\_i=f (x\_i)$ with a sufficiently smooth injective unknown function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ and a value $y \in Im (f)$, I want to find an estimate for $x\in \mathbb{R}^m$ s.t. $f (x)=y$. Does a problem of this so... | https://mathoverflow.net/users/35593 | inverse interpolation | After having calculated an "explicit" interpolating function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$, satisfying $y\_i=f (x\_i)$, you can calculate the local inverse via evaluation of the [implicit taylor series](http://pdfs.semanticscholar.org/7362/e8c3ce5fc3c10403c5e93aa29e4a79333b0a.pdf) or some other series expans... | 2 | https://mathoverflow.net/users/31310 | 281570 | 124,648 |
https://mathoverflow.net/questions/281471 | 3 | Suppose $P=L\ltimes U$ is an $F$-stable parabolic subgroup of a finite group of Lie type $G$, with $F$-stable Levi complement $L$. Here $F$ is a Frobenius endomorphism, and $G^F$ is the subgroup of fixed points under $F$, etc. Let $\Lambda$ be a commutative ring of some appropriate characteristic. The parabolic inducti... | https://mathoverflow.net/users/81993 | Translating parabolic induction as $\Lambda G^F/U^F\otimes_{\Lambda L^F}-$ to $\hom_{\Lambda L^F}(\Lambda U^F/G^F,-)$? | I think it's easiest to break this up into the composition of three simpler isomorphisms.
First, for any ring $A$, if $M\_A$ is a right $A$-module and $\_AX$ is a left $A$-module, there is a natural map
$$M\otimes\_AX\to\text{Hom}\_A\left(\text{Hom}\_A(M,A),X\right)$$
given by
$$m\otimes x\mapsto[\varphi\mapsto \varp... | 3 | https://mathoverflow.net/users/22989 | 281578 | 124,650 |
https://mathoverflow.net/questions/281566 | 3 | Consider the following theorem by Aupetit.
>
> Let $A$ and $B$ be two von-Neumann algebras and let $\phi$ be a spectrum-preserving linear mapping from $A$ onto $B$. Then $\phi$ is a Jordan isomorphism.
>
>
>
The statement can be relaxed further by allowing $B$ be to be any semi-simple Banach algebra without al... | https://mathoverflow.net/users/60913 | $C^*$-algebra which is not von Neumann, but satisfies the property that ever self-adjoint element is | Isn't $c\_0$ already a counterexample?
| 3 | https://mathoverflow.net/users/23141 | 281587 | 124,655 |
https://mathoverflow.net/questions/281543 | 5 | In Woodin's book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal" Remark 2.55 (5), it states SRP by Todorcevic (defined below) is consistent with the existence of a Suslin tree (hence it does not imply MM). Is there any reference about this? My guess is we could probably get a restricted version ... | https://mathoverflow.net/users/23835 | Consistency of Strong reflection principle with the existence of a Suslin tree | This appears to be proven in the following paper of Miyamoto:
*Miyamoto, Tadatoshi*, [**On iterating semiproper preorders**](http://dx.doi.org/10.2178/jsl/1190150293), J. Symb. Log. 67, No. 4, 1431-1468 (2002). [ZBL1050.03034](https://zbmath.org/?q=an:1050.03034).
In section 5 of the paper, he introduces a forcing ... | 3 | https://mathoverflow.net/users/26002 | 281589 | 124,657 |
https://mathoverflow.net/questions/281590 | 7 | What is the maximum number of concurrent bitangents, i.e. all intersecting at the same point, of a smooth complex projective quartic curve? Can the number of concurrent bitangents be six?
| https://mathoverflow.net/users/88058 | Concurrent bitangents of a quartic curve | I think the maximum number of concurrent bitangents is at most 4. Consider the double cover $S\rightarrow \mathbb{P}^2$ branched along your quartic curve. Fix one point $q$ of $S$ above the point of intersection; each bitangent lift to a "line" in $S$ passing through $q$. The description of these lines is well-known. W... | 9 | https://mathoverflow.net/users/40297 | 281602 | 124,659 |
https://mathoverflow.net/questions/281609 | 8 | A split monomorphism is a morphism $m \colon A \to B$ for which there exists a morphism $e \colon B \to A$ such that $e \circ m = \mathrm{id}\_A \colon A \to A$. Is there an elegant description of a universal construction that splits monomorphisms in a given category, i.e. that adds a new morphism $e$ for every monomor... | https://mathoverflow.net/users/10368 | How to formally split monomorphisms nicely? | Here are three possibilities:
**First:** you have the obvious 'universal solution':
You start from $C$ any category and $J$ a set of maps, you can consider the category $C'$ freely generated from $C$ by adding a retract $r\_i$ to each arrow $i \in J$.
$C'$ has the same objects as $C$ and rather complicated arrows... | 8 | https://mathoverflow.net/users/22131 | 281614 | 124,664 |
https://mathoverflow.net/questions/281611 | 3 | (Note: I asked [this question on math stackexchange](https://math.stackexchange.com/questions/2282504/volume-ratio-of-ell-1-balls-and-ell-1-surfaces) but did not get an answer. So I decide to ask this question here and hopefully somebody would know the answer/how to approach).
Consider the $d$-dimensional $\ell\_1$-b... | https://mathoverflow.net/users/84299 | Volume ratio of $\ell_1$ balls and $\ell_1$ surfaces | The $d$-dimensional $\ell\_1$-ball is a [cross polytope](https://en.wikipedia.org/wiki/Cross-polytope) which has $2^d$ facets that in this case are all regular $(d-1)$-simplices with side length $\sqrt{2}$. Since [the volume of a regular $(d-1)$-simplex with unit side lengths](https://en.wikipedia.org/wiki/Simplex#Volu... | 4 | https://mathoverflow.net/users/353 | 281616 | 124,665 |
https://mathoverflow.net/questions/281547 | 1 | Assume an objective function $f = g (X\_{t\_1},Y\_{t\_1}) + g (X\_{t\_2},Y\_{t\_2}) + g (X\_{t\_3},Y\_{t\_3}) + \dots+g (X\_{t\_n},Y\_{t\_n})$ where $g$ is a convex function. $X\_{t\_i}$ and $Y\_{t\_i}$ are sets of real numbers where $1 \le i \le n$ and the data points in $X\_{t\_i}$ and $Y\_{t\_i}$ arrives sequentiall... | https://mathoverflow.net/users/92342 | Optimization over sliding windows | It's not clear to me from the problem description what's being optimized, so I'm going to try to rephrase the problem, then answer the rephrased version. Also, the presence of two streams ($X$ and $Y$) seems superfluous to the underlying question, so I'll just pose the question in the context of a single stream.
Let ... | 2 | https://mathoverflow.net/users/8938 | 281626 | 124,670 |
https://mathoverflow.net/questions/281608 | 2 | Denote by $B(\mathbb{R})$ the set of all functions on $\mathbb{R}$ which are representable in the form $f(x)=\int\_{\mathbb{R}}e^{itx}d\mu(t)$, where $\mu$ is a finite complex-valued Borel measure.
Question: Is there a description of all functions $\alpha:\mathbb{R}\to\mathbb{R}$ such that $f\circ\alpha\in B(\mathbb{... | https://mathoverflow.net/users/114756 | Function in $B(\mathbb{R})$ | You may be aware of this already, but a similar (if not the same) question is addressed in Rudin's *Fourier Analysis on Groups*. Material on the same subject can be found in Katznelson's *Harmonic analysis*. I think pdfs of both of these books are available free online.
The relevant theorem is (I think) that of Beurl... | 1 | https://mathoverflow.net/users/61771 | 281633 | 124,673 |
https://mathoverflow.net/questions/281600 | 8 | I can see three ways to "do formal category theory":
1. Yoneda structures à la Street-Walters.
2. cosmoi à la Street.
3. proarrow equipments.
It seems to me that 2. is older than 1., and a particular case thereof (you impose more requests on $y\_A : A \to PA$ asking that there is an adjoint to the pseudofunctor $P ... | https://mathoverflow.net/users/7952 | How do the existing ways to do formal CT relate to each other? | Street's "fibrational cosmoi" are indeed a special case of, or rather a particular way to construct, a Yoneda structure. They are substantially less general, since Yoneda structures and equipments include the case of enriched categories, but fibrational cosmoi do not. (Street also used the word "cosmos" later for a bic... | 5 | https://mathoverflow.net/users/49 | 281640 | 124,675 |
https://mathoverflow.net/questions/281601 | 2 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We call it *Hausdorff* if for all $x\neq y \in V$ there are $e\_1,e\_2\in E$ with $e\_1\cap e\_2 = \emptyset$ such that $x\in e\_1$ and $y\in e\_2$. We say that $D\subseteq V$ is *dense* if $D\cap e \neq \emptyset$ for all $e\in E\setminus\{\emp... | https://mathoverflow.net/users/8628 | Small dense subsets in "Hausdorff" hypergraphs | Take $V$ to be the set of ultrafilters on $\omega$ and let $E \subset P(V)$ be the smallest subset closed under arbitrary unions containing all sets of the form $[X]= \{ p \in V: X \in p \}\subset V$ where $X \subset \omega$ is infinite and co-infinite.
**Fact:** The hyper-graph $(V,E)$ is Hausdorff.
*Proof:* Let ... | 1 | https://mathoverflow.net/users/8843 | 281642 | 124,676 |
https://mathoverflow.net/questions/281442 | 1 | Assume $x$ is a variable belongs to $\mathbb R \setminus \{ 0,-1,+1 \}$ and consider for all $i, j \in \mathbb N$,
$$a(i,j) = \frac{(x^{i+1} + 1)^{j-1} + (x-1)}{x}$$
then for all $n \in \mathbb N$ the solution set of the matrix system $[a(i,j) \mid 1 \leq i \leq n, 1 \leq j \leq (1+n)]$ exists and is unique with re... | https://mathoverflow.net/users/113991 | Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets | Let me denote $u\_i = x^{i+1} + 1$. As I learned from the [previous question](https://mathoverflow.net/q/280058), under the "matrix system" OP understands the following matrix equation (up to a factor $\frac{1}{x}$):
$$My = b,$$
where
$$M:=\begin{bmatrix}
u\_1^0 + x - 1 & u\_1^1 + x - 1 & \cdots & u\_1^{n-1} + x - 1 \... | 4 | https://mathoverflow.net/users/7076 | 281647 | 124,680 |
https://mathoverflow.net/questions/281653 | 4 | Is there an infinite, connected space $(X,\tau)$ such that the only Hausdorff topology $\tau\_2$ with the property that $\tau\subseteq \tau\_2$ is the discrete topology?
| https://mathoverflow.net/users/8628 | Refining topologies to make them Hausdorff | **Remark.** Every non-discrete $\text T\_1$ topology has a non-discrete Hausdorff refinement.
**Proof.** Let $(X,\tau)$ be a non-discrete $\text T\_1$ space. Choose $p\in X$ such that $\{p\}\notin\tau,$ and let $\tau\_2=\{A\cup B:A\in\tau,\ B\subseteq X\setminus\{p\}\}.$ Then $\tau\_2$ is a Hausdorff topology on $X$ ... | 12 | https://mathoverflow.net/users/43266 | 281654 | 124,682 |
https://mathoverflow.net/questions/281655 | 12 | I am a PhD student in Physics. This problem is motivated by representing a spin-$j$ state by Majorana's stellar representation. There are $2j$ points (Majorana stars) on the two-dimensional sphere (Bloch sphere) where the wave function vanishes. Hence, a spin state corresponds to a configuration of points on the sphere... | https://mathoverflow.net/users/114787 | How to classify continuous maps from 2-spheres to 2-spheres with n fixed points? | Let $x\_1, \ldots, x\_n$ be your points on the sphere. Choose an $n+1$-th point $z$ on the target sphere, and paths from $z$ to each one of the $x\_i$s. By moving along these paths you can construct a homotopy from the space of maps that fix the $x\_i$s to the space of maps that send all the points $x\_i$ to $z$.
Th... | 18 | https://mathoverflow.net/users/6668 | 281657 | 124,684 |
https://mathoverflow.net/questions/281610 | 4 | I've come to the conclusion that what is universal, in the statistics of high Reynolds number turbulence of viscous incompressible fluids, could be modelled exactly only with Alfred Renyi's concept of a "conditional probability field" defined from an **unbounded** measure (on a set of velocity or vorticity fields on $\... | https://mathoverflow.net/users/75422 | Renyi's conditional probability fields and turbulence | An early contribution along these lines is by Mandelbrot, [Sporadic random functions and conditional spectral analysis: self-similar examples and limits](https://projecteuclid.org/euclid.bsmsp/1200513626) (1967):
>
> There is, however, a simple way of generalizing the concept of a
> stationary random function, so ... | 4 | https://mathoverflow.net/users/11260 | 281659 | 124,685 |
https://mathoverflow.net/questions/281668 | 2 | Let us consider the equation:
\begin{align\*}
(\partial\_t - \Delta - b(t,x) \partial\_x) u(t,x)& = f(t,x) \\
u(0,x) & = u\_0
\end{align\*}
defined on the whole real line (so in one dimension - but this is only for simplicity) and with a possibly finite time horizon, so say on $[0,T]\times \mathbb{R}.$ Suppose that $b$... | https://mathoverflow.net/users/33717 | Growth at infinity of a solution to a parabolic PDE | What about $u(t,x) = x e^{t x^2}$, which is a solution of $$\partial\_t u = \Delta u + b(t,x) \partial\_x u$$ with $$b(t,x) = \frac{x^3 - 6 t x - 4 t^2 x^3}{1 + 2 t x^2}$$ with initial value $u\_0(x) = x$?
| 5 | https://mathoverflow.net/users/108637 | 281670 | 124,690 |
https://mathoverflow.net/questions/93224 | 11 | Is there an algorithmic (or other) way to prove that a (projective)
variety is not isomorphic to a toric variety?
I'd be happy with an algebraic answer (for affine or projective varieties),
using the fact that toric ideals are binomial prime ideals. There ne could
use that the coordinate rings are characterized as th... | https://mathoverflow.net/users/5495 | Proving that a variety is not (isomorphic to) a toric variety | The question of algorithmically deciding if an ideal is binomial after a (suitable, e.g. linear) automorphism of affine space is decidable and various algorithms are discussed in ["When is a polynomial ideal binomial after an ambient automorphism?"](https://arxiv.org/abs/1706.03629) by Lukas Katthän, Mateusz Michałek, ... | 6 | https://mathoverflow.net/users/5495 | 281679 | 124,694 |
https://mathoverflow.net/questions/281658 | 0 | Suppose $L$ is a complete lattice, $P$ is a poset, and $f: L \to P$ is a surjective order-preserving map. If ${\bf DM}(P)$ is the [Dedekind MacNeille completion](https://en.wikipedia.org/wiki/Dedekind%E2%80%93MacNeille_completion) of $P$, is there necessarily a surjective order-preserving map ${\bar f}: L\to {\bf DM}(P... | https://mathoverflow.net/users/8628 | Order-preserving surjections on the Dedekind MacNeille completion | No.
Take a poset $P$ on the ground set $S=\{0,1,a,b,c,d\}$ determied by $a>c$, $a>d$, $b>c$, and $b>d$ (with $0$ the minimal and $1$ the maximal element). Let $L$ be a complete lattice on $S$ with $\{a,b,c,d\}$ being an antichain (and the same minimum and maximum). Then the identical map on $S$ is an order preserving... | 2 | https://mathoverflow.net/users/17581 | 281681 | 124,696 |
https://mathoverflow.net/questions/281637 | 10 | I recall having read, some time ago, a beautiful and poetic opening of an article (or was it a book?). From memory, it was by Shigefumi Mori, and talked about the (mathematical) universe consisting of two hemispheres, a real and a p-adic; somehow meeting at the equator. We mere humans should strive to contemplate both ... | https://mathoverflow.net/users/10481 | Mori: p-adic and real hemispheres of the mathematical universe? |
>
> As the night sky, mathematics has two hemispheres; the archimedean hemisphere and the non-archimedean hemisphere. For some reasons, the latter hemisphere is usually under the horizon of our world, and the study of it is historically behind the study of the former. [...] The aim of this paper is [...] to discuss t... | 12 | https://mathoverflow.net/users/26522 | 281683 | 124,697 |
https://mathoverflow.net/questions/280627 | 10 | One way to construct an $E\_\infty$-algebra is to consider the cochain complex $C^\*(X;M)$ for $X$ a topological space and $M$ a module over some ring $\Lambda$. From what I can recall, the $E\_\infty$-algebra structure should contain the homotopy type of $X$. If $X$ is a $K(G,n)$, can the $E\_\infty$-structure on
$$
C... | https://mathoverflow.net/users/78824 | Is the $E_\infty$-structure on the cochain complex of a $K(G,n)$ readily understandable? | There is a very general framework that gives the $E\_\infty$ structure on cochains in McClure and Smith's "[Multivariable cochains and little $n$-cubes](http://www.ams.org/journals/jams/2003-16-03/S0894-0347-03-00419-3/)", with formulas comparable to the Alexander-Whitney diagonal. Roughly, the $E\_\infty$ structure is... | 9 | https://mathoverflow.net/users/360 | 281697 | 124,703 |
https://mathoverflow.net/questions/281375 | 4 | $\newcommand{\R}{\mathrm{R}} \newcommand{\N}{\mathrm{N}}\newcommand{\DD}{\mathrm{D}}\newcommand{\dd}{\mathrm{d}}$
**Prerequisites**: Let $\mathrm{T}: C^\infty(\Omega) \rightarrow
C^\infty(\Omega), u(\cdot)\mapsto F(\cdot, \{\partial^\alpha
u(\cdot)\}\_{\vert{\alpha}\vert\leq k})$ be a non-linear differential
operator... | https://mathoverflow.net/users/109191 | Principal symbol for non-linear differential operators | I've seen only the first. It is indeed used mostly for identifying whether a nonlinear PDE is elliptic, hyperbolic, or parabolic. If so, one can use the respective linear theory, along with the appropriate implicit function theorem to prove existence theorems. Look up fully nonlinear elliptic PDEs for one well studied ... | 4 | https://mathoverflow.net/users/613 | 281700 | 124,705 |
https://mathoverflow.net/questions/281451 | 3 | Let $C$ be a curve in a smooth $3$-fold $X$ with an ordinary node $p\in X$. Blow-up $p$ let $E$ be the exceptional divisor, and $\widetilde{C}$ the strict transform of $C$. Furthermore let $L$ be the line in $E$ through the two points $E\cap \widetilde{C}$.
Now, let us blow-up $\widetilde{C}$ (with exceptional divis... | https://mathoverflow.net/users/nan | Is this divisorial contraction a blow-up? | The variety $Y$ you constructed is terminal. It is singular at the point where $\mathbb{P}^1\times\mathbb{P}^1$ is contracted. However this singularity is normal.
Now, you can write $C$ as $\{y^2-x^3-x^2=z=0\}\subset\mathbb{A}^3$. The blow-up $Z$ of $\mathbb{A}^3$ along $Z$ is defined by $\{s(y^2-x^3-x^2)-tz=0\}\subs... | 1 | https://mathoverflow.net/users/14514 | 281702 | 124,707 |
https://mathoverflow.net/questions/281715 | 1 | I am a PhD student in Physics. Let us consider a vector in an infinite dimensional Hilbert space as
\begin{equation}
|f\rangle\equiv
\begin{bmatrix}
1 \\
z \\
z^2 \\
\vdots
\end{bmatrix},
\end{equation}
where $z$ is a complex number. The norm of the vector $|f\rangle$ is
\begin{equation}
\left\lVert\langle f|f\... | https://mathoverflow.net/users/114787 | How to prove or disprove a type of states form an overcomplete basis in the Hilbert space? | As suggested by Yemon Choi, the mapping $\hat{\psi}(z) = \langle \psi | f \rangle$ is an analytic function on the unit disc, a member of the $H^2$ complex Hardy space. Even though $f$ is not a member of the $\ell^2$ Hilbert space when $|z| = 1$, the mapping $\hat{\psi}$ extends to the boundary of the unit disk (in the ... | 1 | https://mathoverflow.net/users/108637 | 281717 | 124,713 |
https://mathoverflow.net/questions/281727 | 3 | Given a poset $(P,\leq)$ the *interval topology* $\tau\_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
If we look at infinite lattices, the interval topology need not be Hau... | https://mathoverflow.net/users/8628 | Hausdorff interval topology on distributive lattices | The countable atomless Boolean algebra is a counterexample. See
E.S. Northam, [The interval topology of a lattice](http://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0057534-2/S0002-9939-1953-0057534-2.pdf), 1953 (Propositions 2 and 3).
| 4 | https://mathoverflow.net/users/4600 | 281734 | 124,718 |
https://mathoverflow.net/questions/281730 | 23 | I asked my advisor the question in the title. He told me it was a stupid question and that I should focus on my research. Thus we're asking here.
The statement of Fontaine-Winterberger, per their groundbreaking 1979 paper **Extensions algébrique et corps des normes des extensions APF des corps locaux**, is as follows... | https://mathoverflow.net/users/114832 | Geometric intuition for Fontaine-Wintenberger? | First, the Fontaine-Winterberger isomorphism can also be recovered from a theorem of Deligne, namely Thm 2.8 [here](http://alpha.math.uga.edu/~lorenz/Deligne.pdf). Deligne showed that if two local fields $K\_1$ and $K\_2$ (possibly of different characteristic) were such that $O\_{K\_1}/\mathfrak{m}\_{K\_1}^N \simeq O\_... | 13 | https://mathoverflow.net/users/21724 | 281736 | 124,719 |
https://mathoverflow.net/questions/281735 | 0 | I am looking for proofs of or counterexamples to the following assumptions:
if the corners of a triangle are chosen from a compact subset $\mathcal{S}$ of the Euclidean plane then all three corners of the following triangles are elements of the convex hull $CH(\mathcal{S})$ of $\mathcal{S}$
* the triangles with c... | https://mathoverflow.net/users/31310 | Extremality of Triangles with Corners from Planar Convex Hulls | 1) No. Consider the square $ABCD$. Then the triangle $ABC$ does not have the largest incircle. Indeed, it has smaller incircle than $ABM$, where $M$ is the midpoint of $CD$. This follows from the formula $r(\Delta)=S(\Delta)/p(\Delta)$, where $r,S,p$ denote the inradius, area and semi-perimeter of a triangle $\Delta$. ... | 2 | https://mathoverflow.net/users/4312 | 281737 | 124,720 |
https://mathoverflow.net/questions/273812 | 8 | Let $k$ be a local field of residue characteristic $p$, and let D be a central
division algebra over $k$ of index $n>2$. How to determine the irreducible complex representations of the group $SL\_1(D)$?
Suggest some reference regarding this.
| https://mathoverflow.net/users/56638 | Representations of $SL_1(D),$ where $D$ a division algebra over a local field | An article by Shai Shechter recently appeared on Math ArXiv:
"Characters of the Norm-One Units of Local Division Algebras of Prime Degree"
<https://arxiv.org/abs/1512.02448>
| 3 | https://mathoverflow.net/users/4767 | 281741 | 124,722 |
https://mathoverflow.net/questions/281749 | 12 | Let $\xi = \pi \colon E \to B$ a topological fiber bundle with connected base $B$, $E\_x = \pi^{-1}(x)$ the fiber at $x \in B$, $j \colon E\_x \hookrightarrow E$ the canonical injection, and let suppose that there exists a retraction $r \colon E \to E\_x$, i.e. $r◦j=Id\_{E\_x}$. Can we conclude that $\xi$ is trivial ?
... | https://mathoverflow.net/users/74372 | Is a topological fiber-bundle, whose total space admits a retraction onto a fiber, trivial? | The statement is not true. Let $\pi:V\to M$ be a vector bundle over a manifold which is non-trivial as a fiber bundle. Let $U$ be an open neighborhood of $M$ over which $V$ is trivial, fix $x\in U$, and pick a local trivialization
$$
T:\pi^{-1}(U)\xrightarrow{\sim}U\times V\_x.
$$
Let $f:M\to\mathbb{R}$ be a continuous... | 16 | https://mathoverflow.net/users/5263 | 281751 | 124,724 |
https://mathoverflow.net/questions/279381 | 0 | Let $V$ be a set and let $V^V$ denote the set of all functions $f:V\to V$. Suppose that $F\subseteq V^V$. Let $[V]^2 = \big\{\{x,y\}: x, y\in V \land x\neq y\big\}$. We say $E\subseteq [V]^2$ is *$F$-compatible* if all members of $F$ are [graph homomorphisms](https://en.wikipedia.org/wiki/Graph_homomorphism) from $(V,E... | https://mathoverflow.net/users/8628 | Largest edge set compatible with graph endomorphisms | No. In [this paper](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1446788708000815) the define the *hull* of a graph $G$. The hull of $G$ has the same vertices of $G$ and has an edge between any pair of vertices that cannot be identified by any endomorphism of $G$. In other words, two vertice... | 1 | https://mathoverflow.net/users/18606 | 281756 | 124,727 |
https://mathoverflow.net/questions/281772 | 8 | The branching laws for the restricted representation of $SO(n)$ with respect to the subgroup $SO(n-1)$ are discussed in this Wikipedia [article](https://en.wikipedia.org/wiki/Restricted_representation). Am I correct in reading from this that any given representation of $SO(n-1)$ can appear in a representation of $SO(n)... | https://mathoverflow.net/users/89074 | The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$ | Yes, the branching is multiplicity free. See e.g. Theorem 8.1.3 and Theorem 8.1.4 in *Symmetry, Representations, and Invariants* by Nolan Wallach and Roe Goodman. The quotient space is a sphere $S^{n-1}$ which you can see for example by calculating the stabilizer of $e\_n.$
For orthogonal groups you basically have ex... | 7 | https://mathoverflow.net/users/6818 | 281778 | 124,730 |
https://mathoverflow.net/questions/281669 | 2 | Consider $G^{\rtimes k} := ((G \rtimes \dots ) \rtimes G)\rtimes G$ with diagonal action by inner automorphisms, $G^{\rtimes 1} = G$. Let $\mathcal P$ be a collection of groups. Is it true that
* $G$ residually $\mathcal P$ $\Rightarrow $ $G^{\rtimes \geq 2}$ residually $\mathcal P$? (very unlikely, but I can't find ... | https://mathoverflow.net/users/81055 | Residual properties of iterated semidirect products | The group law of $G^{\rtimes 2}$ is given by
$$(g, h)(g',h') = (ghg'h^{-1}, hh')$$ and more generally for $G^{\rtimes k}$, we have
$(g\_1, \dots, g\_{k -1}, g\_k)^{(1, \dots,1, g)} = (g\_1^g, \dots,g\_{k - 1}^g, g\_k^{g})$ where $x^y \Doteq y^{-1}xy$.
**Therefore $G^{\rtimes k}$ is just $G^k$ in disguise**.
Indeed... | 3 | https://mathoverflow.net/users/84349 | 281787 | 124,734 |
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