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https://mathoverflow.net/questions/266900 | 5 | Let $\mathbb{G}$ be a connected reductive group over $\mathbb{F}\_q$, and let $G$ be the base change to an algebraic closure of the base field. Denote by $F$ the associated geometric Frobenius.
Let $T$ be an $F$-rational (i.e. $FT=T$) maximal torus of $G$. As $T$ is $F$-rational, the Frobenius $F$ acts on the finite... | https://mathoverflow.net/users/56217 | When the longest element of Weyl group is rational? | Let $B$ be a Borel subgroup containing $T$. As $F(B)$ and $B$ are both Borel subgroups containing $T$ there exists an element $n \in N\_G(T)$ such that ${}^nF(B) = B$. Thus the Frobenius endomorphism $F' : G \to G$ defined by $F'(g) = nF(g)n^{-1}$ induces an automorphism $F' : W \to W$, where $W = N\_G(T)/T$, and this ... | 8 | https://mathoverflow.net/users/22846 | 266911 | 119,802 |
https://mathoverflow.net/questions/266903 | 8 | Let $(M,g)$ be a time-oriented smooth Lorentzian manifold, with Lorentzian metric $g$. In the following thread:
<https://physics.stackexchange.com/questions/228669/why-pseudo-riemannian-metric-cannot-define-a-topology/228676?noredirect=1#comment730891_228676>
physicist @ValterMoretti makes the following claim:
"A... | https://mathoverflow.net/users/66688 | On the topology induced by a Lorentzian metric | In general, this topology is coarser than the original topology of the manifold, and, without further assumptions, strictly coarser. It coincides with the original one iff the Lorentz manifold is strongly causal, see e.g. Prop 3.11 in Beem-Ehrlich-Easley. To get counter-example consider the cylinder $\mathbb{S}^1 \time... | 9 | https://mathoverflow.net/users/12482 | 266912 | 119,803 |
https://mathoverflow.net/questions/266919 | 1 | For a positive constant $C$:
\begin{align}
y(x)+C\ln y(x)=f(x).
\end{align}
At least from specific $f(x)$, such as piece-wise linear function, is there an explicit solution for $y(x)$?
| https://mathoverflow.net/users/62302 | Does this equation have an explicit solution? | The solution can be expressed in terms of the Lambert W-function: according to Maple, $y(x) = e^{-W(e^{f(x)})+f(x)}$.
| 3 | https://mathoverflow.net/users/10744 | 266922 | 119,804 |
https://mathoverflow.net/questions/266925 | 7 | Consider the Banach algebra $\ell\_1(\mathbb{N}\_0)$ (with convolution / Cauchy product of series). I am looking for an elementary proof of the fact that the group of invertible elements in this algebra is not dense.
If one wishes to use some hammers, here is a way to do so. Call this algebra $A$; it is not too hard ... | https://mathoverflow.net/users/15129 | Non-density of invertible elements in $\ell_1(\mathbb{N}_0)$ | Q1 can be answered with the following result.
>
> An element in the boundary of the invertible group of a unital Banach algebra is a topological divisor of zero.
>
>
>
I must confess I remembered the statement but not the proof, but it can be shown using elementary arguments related to the fact that the invert... | 6 | https://mathoverflow.net/users/763 | 266930 | 119,808 |
https://mathoverflow.net/questions/266913 | 15 | Apologise in advance if this problem isn't research-level (I'm quite certain it isn't). It's just I found it quite intriguing because it turned out to be much more subtle than it appeared at my first glance. Also, this problem didn't get an answer so far (nor much attention) from MSE, so I decided to post it here in ho... | https://mathoverflow.net/users/64010 | Is the following series consisting of equally distributed $\pm 1$ bounded? | The sequence $\sum a\_n$ is unbounded.
This is a consequence of a general result from Kesten,
[On a conjecture of Erdös and Szüsz related to uniform distribution mod 1](https://eudml.org/doc/204796), Acta Arithmetica (1966). The proof is not very long but quite computational, using properties of continued fraction... | 17 | https://mathoverflow.net/users/6129 | 266940 | 119,814 |
https://mathoverflow.net/questions/266934 | 4 | Take some 4 consecutive primes $p\_n,p\_{n+1},p\_{n+2},p\_{n+3}$ where $p\_n \geq 5$.
Now form two products: $p\_n \cdot p\_{n+3}$ and $p\_{n+1} \cdot p\_{n+2}$.
Is there always at least one prime in the interval $[p\_n \cdot p\_{n+3},p\_{n+1} \cdot p\_{n+2}]$ (it does not matter if we have that $p\_n \cdot p\_{n+3... | https://mathoverflow.net/users/108263 | Is there always at least one prime in intervals of this form? | It is very likely that there are infinitely many counterexamples. More precisely, by a special case of the prime tuple conjecture (due to Dickson and Hardy-Littlewood), there are infinitely many consecutive primes of the form $p<p+2<p+6<p+8$. The corresponding difference then equals
$$ (p+2)(p+6)-p(p+8)=12, $$
and the ... | 12 | https://mathoverflow.net/users/11919 | 266942 | 119,816 |
https://mathoverflow.net/questions/266805 | 2 | The following counts Cohen-Macaulay modules in a certain Gorenstein algebra. I search for a closed formula, see also [Elementary interpretation of a homological result](https://mathoverflow.net/questions/266800/elementary-interpretation-of-a-homological-result) .
Let $n \geq 4$ and $w >3$ and let $w$ be an unit in $\ma... | https://mathoverflow.net/users/61949 | Counting certain tuples | First, notice that $s\equiv (-w)^{-1}\pmod{n}$ and thus all elements in the set
$$M = \{ (-wk)\bmod n\mid 1\leq k\leq s \}$$
are distinct (i.e., $|M|=s$).
Second, the restrictions on $a,b$ can be restated as
\begin{split}
a &\not\equiv -wk\pmod{n},\\
a+b &\not\equiv -wk\pmod{n}
\end{split}
for all $k\in\{1,2,\dots,s\... | 2 | https://mathoverflow.net/users/7076 | 266960 | 119,820 |
https://mathoverflow.net/questions/266897 | 0 | In following [question](https://mathoverflow.net/questions/264832/24-vectors-in-leech-lattice-having-scalar-product-frac14-pairwise) on MathOverflow I received construction of new Leech lattice provided by Noam Elkies. Let's call it $(E)$. This Leech lattice has nice feature that there is easy to see $24$ vectors havin... | https://mathoverflow.net/users/nan | Map homemade Leech lattice to classic one | I applied following method to map Elkies Leech lattice to the classic one. I describe first general algorithm and next concrete realization and GAP code.
**General algorithm**
Let $p$ be order $23$ automorphism of Leech lattice $L$. There is only one vector fixed, let's call it $u\_0$. There is also one vector "typ... | 0 | https://mathoverflow.net/users/nan | 266969 | 119,822 |
https://mathoverflow.net/questions/266965 | 3 | Let $R$ be a commutative ring with $1$ such that every ideal containing $J(R)$, the intersection of all maximal ideals, is an intersection of maximal ideals. Is there any characterization for such a ring or is there any geometric interpretation for this property?
| https://mathoverflow.net/users/108286 | When every ideal containing $J(R)$ is an intersection of maximal ideals | The condition is equivalent to $R/J$ being von Neumann regular.
Set $S = R/J$. Then the condition requires that no factor of $S$ have any nilpotent elements, and thus the square of any ideal in $S$ is the ideal itself. In particular, $rS = (rS)^2 = r^2 S$ for all $r \in S$, and thus $r = r^2 x$ for some $x \in S$, p... | 7 | https://mathoverflow.net/users/42278 | 266975 | 119,826 |
https://mathoverflow.net/questions/266954 | 2 | Let $(u,v)$ be a pair of non-zero integers. We say that $(u,v)$ is a pair of *simultaneous squares* if for all primes $p$ dividing $u$, we have $\left(\frac{v}{p}\right) = 1$ and for all primes $q$ dividing $v$, we have $\left(\frac{u}{q} \right) = 1$. Here $\left(\frac{\cdot}{m}\right)$ denotes the Legendre symbol.
... | https://mathoverflow.net/users/10898 | Density of "simultaneous squares" | This should be asymptotic to an expression of the form
$$\frac{cX^2}{\log X}$$
as $X \to \infty$, for some $c > 0$ (this should be interpreted as $(X/\sqrt{\log X}\,)^2$). Proving an upper bound of the correct order of magnitude in this case should not be too difficult using the large sieve (I can provide more details ... | 5 | https://mathoverflow.net/users/5101 | 266984 | 119,828 |
https://mathoverflow.net/questions/266948 | 1 | This is a problem occurs in my research. For any algebraically closed field $k$ of characteristic $p$. I want to show that $\sum\_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}$ is a separable polynomial over this field. I am trying to prove that $p$ does not divide the discriminant of the pol... | https://mathoverflow.net/users/108278 | Show that $\sum_{i=0}^{\frac{p-1}{2}} {{\frac{p-1}{2}}\choose {i}}^2 x^{\frac{p-1}{2}-i}$ is separable | Let me elaborate on Noam D. Elkies' comment. If we denote $n=(p-1)/2$, the discriminant of this polynomial $g(x)$ is non-zero modulo $p$ if and only if the discriminant of Legendre's polynomial $f(x)=2^{-n}\sum\_{k=0}^n \binom{n}{k}^2(x-1)^{n-k}(x+1)^k=2^{-n}(x-1)^ng((x+1)/(x-1))$ is non-zero modulo $p$ (the roots of $... | 4 | https://mathoverflow.net/users/4312 | 266985 | 119,829 |
https://mathoverflow.net/questions/267005 | 3 | Let $\kappa\geq \aleph\_0$ be a cardinal. Is there a simple undirected graph $G=(V,E)$ such that every simple undirected graph on $\kappa$ vertices is isomorphic to a minor of $G$?
| https://mathoverflow.net/users/8628 | "Universal" infinite graph (with respect to minors) | Yes. In fact, a stronger statement is true: for every cardinal $\kappa$, there is a simple undirected graph $G = (V,E)$ such that every simple undirected graph on $\kappa$ vertices is isomorphic to an induced subgraph of $G$.
There are several ways to see this. My favorite is as follows. First, assume that $\kappa$ i... | 11 | https://mathoverflow.net/users/70618 | 267009 | 119,837 |
https://mathoverflow.net/questions/266842 | 2 | In Feferman's paper ([located here](http://www.ams.org/journals/tran/1962-104-01/S0002-9947-1962-0142453-3/S0002-9947-1962-0142453-3.pdf) (pg. 2 of the PDF)), he begins to outline various methods used for attacking the problem of classifying the computable functions by means of hierarchies. In particular, he discusses ... | https://mathoverflow.net/users/64167 | A "Folkloric" Result for Classes of Computable Functions | Let $E=\{g\_0,g\_1\ldots\}$ be uniformly computable. Then the function $f$ defined by letting $f(n)=\textrm{max}\{g\_i(n):i<n\}+1$ is computable and majorizes every member of $E$. As to continuing constructions of this kind through computable ordinals, you might want to look at fast-growing hierarchies (see e.g. <https... | 1 | https://mathoverflow.net/users/47312 | 267011 | 119,838 |
https://mathoverflow.net/questions/267002 | 21 | Let $G(V,E)$ be a graph. I am searching for graphs with only **disjoint** [perfect matchings](https://en.wikipedia.org/wiki/Matching_(graph_theory)) (i.e. every edge only appears in at most one of the perfect matchings).
Examples:
* [Cyclic graph](https://en.wikipedia.org/wiki/Cycle_graph) $C\_n$ with even $n$, wit... | https://mathoverflow.net/users/63938 | Graphs with only disjoint perfect matchings | $m=3$ is indeed the maximum, and $K\_4$ is the only example for this value of $m$.
Two perfect matchings form a disjoint union of cycles. If there is more than one cycle, then you may swap one of them, obtaining a third matching on the same edges. So any two of the $m$ matchings form a Hamiltonian cycle.
Assume tha... | 22 | https://mathoverflow.net/users/17581 | 267013 | 119,840 |
https://mathoverflow.net/questions/267018 | 8 | Does there exist asymptotic formula for ways to write n as sum of four squares? Or can this be proved impossible? I can only find reference for sums of five squares.
| https://mathoverflow.net/users/108308 | Asymptotic formula for sums of four squares? | The function you are asking for is $r\_4(n)$, the number of ways to write $n$ as a sum of four squares. The exact formula was discovered by Jacobi, and is given as
$$\displaystyle r\_4(n) = 8\sum\_{\substack{d | n \\ 4 \nmid d}} d.$$
| 11 | https://mathoverflow.net/users/10898 | 267020 | 119,842 |
https://mathoverflow.net/questions/266872 | 3 | This problem arises when minimizing the operator equation $X P X^\* + X Q + R$ over positive $X$ with respect to the positive cone on a Hilbert space $\mathcal{H}$.
**The (reduced) task**:
Given $P$ and $Q$ are positive (bounded) linear operators on $\mathcal{H}$, respectively, find a positive $X\in\mathcal{L}\_+(\... | https://mathoverflow.net/users/108236 | Solving Matrix/Operator Equation $H P X + X P H + HQ = 0$ | The following was resolved through personal communication, but I thought I'd post it here in case anyone has a similar problem in the future.
---
I'm going to assume $P$ is nonsingular (strictly positive) so that $P^{-1}$ exists since $P$ is bounded.
As seen above $Q$ must commute with every *positive* operator... | 0 | https://mathoverflow.net/users/108236 | 267026 | 119,846 |
https://mathoverflow.net/questions/267010 | 2 | I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in $n$-dimensional space with n even, I am thinking in the phase space of a system with $f$-degrees of freedom that have dimensionality $2f,$ the aim is picking phase-space points that have the same to... | https://mathoverflow.net/users/108306 | Sampling point from the surface of an n-dimensional ellipsoid with uniform distribution | If your ellipsoid is not too squashed, the method described in the first answer to [this MSE question](https://math.stackexchange.com/questions/973101/how-to-generate-points-uniformly-distributed-on-the-surface-of-an-ellipsoid) should work decently. If it *IS* very squashed, it won't, but an ellispoid with $a \gg b > c... | 0 | https://mathoverflow.net/users/11142 | 267027 | 119,847 |
https://mathoverflow.net/questions/125596 | 32 | At the end of his excellent article, "The Emergence of Descriptive Set Theory" (<http://math.bu.edu/people/aki/2.pdf>), Kanamori writes:
>
> "Another mathematical eternal return: Toward the end of his life, Godel regarded the question of whether there is a linear hierarchy for the recursive sets as one of the big ... | https://mathoverflow.net/users/8133 | Godel on recursion-theoretic hierarchies | I don't mean to give this as a complete answer, but regarding question 2, there have been a great deal of negative results surrounding this problem.
In fact, most of them take the form of showing that any ''reasonably constructive'' (read ''effectively generated'') subrecursive hierarchy of computable functions index... | 10 | https://mathoverflow.net/users/64167 | 267028 | 119,848 |
https://mathoverflow.net/questions/267019 | 4 | For a Banach algebra $A$ the bidual $A^{\*\*}$ may be given two natural products called the Arens products. By local reflexivity, there is an ultrafilter $U$ so that $A^{\*\*}$ embeds into the ultrapower $A^U$ isometrically via some map $h$. This map has a one-sided inverse $\sigma\colon A^U\to A^{\*\*}$ given by $\sig... | https://mathoverflow.net/users/106520 | Biduals of Banach algebras | In general, no: for certain $A$ one can get non-zero $c\in A$ and sequences $(a\_n)$ and $(b\_n)$ in $A$ that converge weakly to zero, such that $a\_n b\_n=c$ for all $n$; these will show that $\ker\sigma$ is not even closed under multiplication, let alone an ideal.
We can arrange for $A$ to be Arens regular, so that... | 6 | https://mathoverflow.net/users/763 | 267029 | 119,849 |
https://mathoverflow.net/questions/267031 | 3 | I read the following paragraph from Serre's book (Topics in Galois Theory).
>
> Although the proof of the classification theorem has been announced, described, and advertised since 1980, it is not yet clear whether it is complete or not: the part on "quasi-thin" groups has never been published
>
>
>
I am conf... | https://mathoverflow.net/users/92070 | Quasi-thin groups and classification theorem | I think this question is answered by the [Wikipedia page](https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups) quoted in the question:
>
> Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of ... | 8 | https://mathoverflow.net/users/11919 | 267039 | 119,851 |
https://mathoverflow.net/questions/266999 | 6 | This is a problem that I encountered in my research and have no clues to fully
resolve it. Basically, I need large (or moderate) deviation bounds on the
difference between an order statistic of independent and identically
distributed (i.i.d.) random variables on the compact interval $\left[
0,1\right] $ and the expecta... | https://mathoverflow.net/users/14390 | Rate of convergence of uniform order statistics to their expectations | For your first question, Rigollet has a series of notes(with minor typos) that dicusses basics of this kind of tail bounds. The result you mentioned in the "introduction" is actually the classic Hoeffding bound that mentioned by Rigollet in this set of notes.
>
> High Dimensional Statistics, Philippe Rigollet (2015... | 3 | https://mathoverflow.net/users/25437 | 267046 | 119,855 |
https://mathoverflow.net/questions/267048 | -4 | Suppose $k$ and $n$ are natural numbers such that $2^{2^k} \lt n \lt 2^{2^{k+1}}$. I am curious how many integers are there in the interval $\left[2^{2^k}, 2^{2^{k+1}}\right]$ in terms of $n$.
I need to know this because some student claims doing a binary search for number $n$ inside the interval above takes $O(log n... | https://mathoverflow.net/users/108321 | How many integers between $\left[2^{2^k}, 2^{2^{k+1}}\right]$? | In the worst case, $n=2^{2^k}$, which would mean that there are $n^2-n$ integers to check, as $2^{2^{k+1}}=2^{2^k+2^k}=(2^{2^k})^2$.
So the student is validated.
| 0 | https://mathoverflow.net/users/36155 | 267050 | 119,857 |
https://mathoverflow.net/questions/267045 | 43 | The Fibonacci recurrence $F\_n=F\_{n-1}+F\_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question might even be known. Yet, if true, I like to ask for alternative proofs.
>
> **Question.** Does the following iden... | https://mathoverflow.net/users/66131 | Fibonacci series captures Euler $e=2.718\dots$ | It follows from the identity
$$F\_{n+k} = \sum\_{j=0}^k {k \choose j} F\_{n-j}$$
which is obtained by applying the standard recurrence $k$ times to the left side, each time splitting up each term into two terms.
Indeed, this gives
$$\sum\_{k=0}^{\infty} \frac{F\_{n+k}}{k!} = \sum\_{k=0}^{\infty} \frac{ \sum\_{... | 78 | https://mathoverflow.net/users/18060 | 267053 | 119,859 |
https://mathoverflow.net/questions/267064 | 1 | I just wanted to ask wether this problem has already been proved or not.
I know that there are 2 other posts that deal with exactly the same question, but I decided to ask it again, since they are too old (more than 5 years) and a new paper was published in 2015 on arxiv:
<https://arxiv.org/abs/1601.02890>
In sp... | https://mathoverflow.net/users/100873 | Proof claimed of Gauss' Circle Problem | The Gauss circle problem is open (as of now).
| 7 | https://mathoverflow.net/users/11919 | 267065 | 119,863 |
https://mathoverflow.net/questions/266970 | 8 | Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic group over $k$.
>
> Is the following statement true: $H\_{et}^1(X,F) = H^1(\pi\_1^{et}(X), F(\bar{k}))$?
>
>
>
| https://mathoverflow.net/users/108289 | Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true? | This is true, yes. More generally, if $X$ is a scheme and $F$ is a locally constant étale sheaf of finite abelian groups on $X$, then
$$
H^1\_{et}(X,F) = H^1(\Pi\_1^{et}(X), \tilde F),
$$
where $\Pi\_1^{et}(X)$ is the étale fundamental pro-groupoid of $X$ and $\tilde F$ a certain local system on $\Pi\_1^{et}(X)$ corres... | 10 | https://mathoverflow.net/users/20233 | 267069 | 119,864 |
https://mathoverflow.net/questions/266813 | 5 | The classification of finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero and whose group of group-like elements is abelian is very much completed. However, from a more categorical point of view it would make sense to classify such Hopf algebras up to gauge equivalence,i.e... | https://mathoverflow.net/users/103448 | Classification of pointed Hopf algebras up to gauge equivalence | There's been quite a lot of interest in classifying fusion categories of (low) dimensions, and by [reconstruction](https://ncatlab.org/nlab/show/reconstruction+theorem) this effectively classifies the represented objects up to gauge equivalence. A few Arxiv preprints on this:
* [Classification of fusion categories of... | 2 | https://mathoverflow.net/users/41862 | 267070 | 119,865 |
https://mathoverflow.net/questions/267055 | 23 | Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds.
Then, if the answer is *No*, one can remove various conditions on the dimension,
and allow non-compact examples and disconnected examples, to realize a (perhaps) wider range of rationals.
This came up after a class I'm teach... | https://mathoverflow.net/users/6094 | Is every rational realized as the Euler characteristic of some manifold or orbifold? | Products of 2-orbifolds with manifolds will do the trick. There are 2-orbifolds of Euler characteristic $1/n$ (take a quotient of $S^2$ by a rotation of order $2n$). Then take a product with a manifold of Euler characteristic $m \in \mathbb{Z}$ to get all rationals.
| 34 | https://mathoverflow.net/users/1345 | 267075 | 119,867 |
https://mathoverflow.net/questions/266921 | 13 | **The question is clarified by Prof.V.Vovk. See his answer below for discussion.**
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a promising candidate towards distribution-free inference coming along with confidence level guarantee.
Given the curre... | https://mathoverflow.net/users/25437 | How is the "conformal prediction" conformal? | Thanks for your interest. The term “conformal prediction” was suggested by Glenn Shafer, and at first I did not like it exactly for the reason that you mention: it has nothing (or very little) to do with conformal mappings in complex analysis. But then I discovered other meanings, even in maths; e.g., Wikipedia has fiv... | 15 | https://mathoverflow.net/users/108333 | 267081 | 119,869 |
https://mathoverflow.net/questions/267074 | 1 | [Ian Morris](https://mathoverflow.net/questions/242295/continuous-upper-envelope-of-upper-semicontinuous-function) quoted the following:
>
> For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\subset C(X,\math... | https://mathoverflow.net/users/42411 | Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable? | Based on the comment by Nik Weaver, the answer to my question is negative, that is, there exists a function $f:X \rightarrow \mathbb{R}$ such that it is not an infimum of any upper-semicontinuous functions:
Define $f:X \rightarrow \mathbb{R}$ such that $f(x) = 0$ if $x \leq 0$ and $f(x) =1$ if $x>0.$ Clearly $f$ is l... | 0 | https://mathoverflow.net/users/42411 | 267103 | 119,874 |
https://mathoverflow.net/questions/266876 | 2 | Take a linear ordinary differential equation of the form :
$$
\sum\_{k=0}^n p\_{n-k}(z)(z (z-1))^k \partial\_k f = 0
$$
Where $p\_i$ is a polynomial fraction of degree $i$, without zeros at $0$ or $1$, $p\_{0} = 1$
Using the Frobenius method, we can find a basis of solution around $0$ (a regular singularity) of the ... | https://mathoverflow.net/users/104920 | Relation between different basis of solutions of an ODE | I don't know of any general method to do it *exactly*. More specifically, I don't believe it is know whether testing if such a connection constant is zero is decidable.
However, it is possible to find *rigorous numerical enclosures* of the connection constants. In fact, I am developing [code](http://marc.mezzarobba.n... | 1 | https://mathoverflow.net/users/108360 | 267109 | 119,876 |
https://mathoverflow.net/questions/267094 | 6 | I am trying to see if, for the complete graph $K\_{2n}$, there exists a labelling of the vertices with two labels $a$ and $b$ (each used exactly $n$ times), such that we can decompose the graph into $n$ hamiltonian paths that have the same labelling.
For example, if I take n=3, I numerate my vertices from $1$ to $6$,... | https://mathoverflow.net/users/105215 | Existence of a 2-labelled Hamiltonian Path decomposition of $K_{2n}$ | I think I have a proof that such a labelling cannot exist if $n$ is even.
Suppose we have a labelling $\ell : V(K\_m) \to \{ a, b \}$ and a decomposition of $K\_{m}$ into a family $\mathcal{P}$ of Hamiltonian paths. It is clear by counting edges that $p := |\mathcal{P}| = m/2$, so in particular $m$ must be even. Ever... | 4 | https://mathoverflow.net/users/37432 | 267110 | 119,877 |
https://mathoverflow.net/questions/267091 | 7 | In the paper *On the Castelnuovo-Mumford regularity of the cohomology ring of a group*, Symonds describes the following space.
Let $G = (\mathbb{Z}/2\mathbb{Z})^2 = \{1,a,b,ab\}$ be an elementary abelian $2$-group, and let $Z = \{ z\_1, \dots, z\_6\}$ be a discrete space with six elements. Let $G$ act on $Z$ such tha... | https://mathoverflow.net/users/36146 | Computing the equivariant cohomology of a specific $(\mathbb{Z}/2\mathbb{Z})^2$-space | The standard way to compute equivariant cohomology of a $G$-space $X$ is to use the spectral sequence of the fibration
$$X\to EG\times\_G X\to BG,$$
where the projection is induced by $X\to \ast$. With $\mathbb{F}\_2$ coefficients this takes the form
$$
E\_2^{p,q} = H^p(BG; H^q(X;\mathbb{F}\_2))\Rightarrow H^\*\_G(X;\m... | 6 | https://mathoverflow.net/users/8103 | 267111 | 119,878 |
https://mathoverflow.net/questions/267138 | 5 | Is there exist a similar conjecture to the famous [Jacobian Conjecture](https://en.wikipedia.org/wiki/Jacobian_conjecture) with $\mathbb{C}[x\_1,\ldots,x\_n,x\_1^{-1},\ldots,x\_n^{-1}]$ instead of
$\mathbb{C}[x\_1,\ldots,x\_n]$?
Namely, let $f$ be $\mathbb{C}$-algebra endomorphism of $\mathbb{C}[x\_1,\ldots,x\_n,x\_... | https://mathoverflow.net/users/72288 | "Jacobian Conjecture" for $k[x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}]$? | Counterexample: the endomorphism of the product of two punctured lines (complement of the curve of equation $xy=0$ in the plane) given by $$(x,y)\mapsto f(x,y)=\left(\frac{x}{y},y^2\right)$$
We have
$$\begin{pmatrix}\partial\_1f\_1 & \partial\_2f\_1\\ \partial\_1f\_2 & \partial\_2f\_2\end{pmatrix}=\begin{pmatrix}\fra... | 15 | https://mathoverflow.net/users/14094 | 267142 | 119,887 |
https://mathoverflow.net/questions/262967 | 7 | I'm concerned with the following
Proposition: If a compact manifold $M$ satisfies $$Rc + \textstyle\frac{1}{2}\mathcal{L}\_Xg = \lambda g $$
where $\lambda$ is a constant (i.e. $M$ is a compact Ricci soliton), then in fact $$Rc + \nabla^2f = \lambda g $$
so $X = \nabla f + K$ where $K$ is a Killing vector.
(Qui... | https://mathoverflow.net/users/47391 | "Elliptic" proof that Compact Ricci Solitons are Gradient Ricci Solitons | There is an "elliptic" proof in [this paper by Eminenti, La Nave, Mantegazza](http://dx.doi.org/10.1007/s00229-008-0210-y), see Theorem 3.1. It does still use Perelman's $\mathcal{W}$ entropy (but it does not use the Ricci flow, it just uses a minimization argument).
The authors there ask in Problem 3.2 whether ther... | 3 | https://mathoverflow.net/users/13168 | 267146 | 119,888 |
https://mathoverflow.net/questions/267147 | 2 | Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=h\circ g$ where $g:Y\rightarrow X$, $h:X\rightarrow Z$ are two generically finite morphism.
Can we say something abou... | https://mathoverflow.net/users/nan | Singularities of $3$-folds | It can have bad singularities and dimension doesn't matter. Take an arbitrarily singular $X$ of dimension $d$. There is always a finite generic projection $X \to Z = {\mathbb{P}}^d$ (and $Z$ is smooth and Fano). On the other hand let $Y \to X$ be a resolution of singularities.
Then $Y \to X \to Z$ is generically fin... | 5 | https://mathoverflow.net/users/3521 | 267150 | 119,891 |
https://mathoverflow.net/questions/266846 | 5 | It is well known that the set of all polygons with consecutive side lengths $l\_1, \dots, l\_n$ in $\mathbb{R}^3$, considered up to rigid motions, is a compact complex manifold. Of course, I am assuming, for the sake of simplicity that there are no "straight line" polygons i.e., the length vector $L:= (l\_1,\dots, l\_n... | https://mathoverflow.net/users/7494 | Is there a relationship between the moduli space of spatial polygons and the moduli space of labeled points? | 1. Yes.
2. I assume that your $M\_n$ is what is more usually denoted $\overline{M\_{0,n}}$. Then the answer is yes, there is a natural map $\overline{M\_{0,n}} \twoheadrightarrow M\_L$, for each $L$. Specifically, let $\gamma$ be a tree of $\mathbb P^1$s glued along nodes and with $n$ labeled points (not at the nodes).... | 5 | https://mathoverflow.net/users/391 | 267159 | 119,893 |
https://mathoverflow.net/questions/267134 | 8 | This question arises from a comment by user nfdc23 on an unrelated recent MO question [*here*](https://mathoverflow.net/questions/266956/the-image-of-a-base-of-absolute-roots-is-a-base-of-relative-roots). It concerns textbook treatments of what has been called the "Theorem of Kostant-Rosenlicht", stated as Theorem 2 in... | https://mathoverflow.net/users/4231 | Unipotent algebraic group action on quasi-affine (vs. affine) variety? | Since the quasi-affine case is so easily reduced to the affine case, one doesn't really get much extra mileage out of it.
After checking my papers I am pretty sure that I never used the quasi-affine case seriously. To the contrary, in some cases I had to reduce to the affine case anyway because one is actually *losin... | 6 | https://mathoverflow.net/users/89948 | 267168 | 119,898 |
https://mathoverflow.net/questions/262879 | 8 | I'm looking for a proof to the following statement:
>
> Let G be a simple connected graph
>
>
> If $\chi''(G)=\chi'(G)+\chi(G)$ holds then the graph should be bipartite,
>
>
>
where $\chi''(G)$ is the total chromatic number $\chi'(G)$ the chromatic index and $\chi(G)$ the chromatic number of a graph.
I was... | https://mathoverflow.net/users/14726 | Total chromatic number and bipartite graphs | Let $a=\chi(G)\geq 3$ and $b=\chi'(G)$. Paint the vertices in $a$ colors and edges in $b$ colors properly. Now choose one color class of edges. Repaint each of them into one of the first $a$ colors, distinct from the two colors of the edge's endpoints. Since the repainted edges were pairwise non-adjacent,we get a prope... | 3 | https://mathoverflow.net/users/17581 | 267173 | 119,901 |
https://mathoverflow.net/questions/267079 | 19 | Let $C$ be a category with binary products. The product functor $\times : C^2 \to C$ is *right* adjoint to the diagonal $\Delta: C \to C^2$. If $C$ has biproducts, then $\times$ is also *left* adjoint to $\Delta$. But from the fact that $\times$ is left adjoint to some functor $R$, can we conclude that $R = \Delta$?
... | https://mathoverflow.net/users/2362 | If a right adjoint to the product functor exists, must it be the diagonal? | For the first question: Yes.
Let $C$ be a category with finite products and let $\Delta=(\Delta\_1,\Delta\_2):C\to C\times C$ s.t. $$\times\dashv \Delta.$$
More specifically we find $$[X\times Y, Z]\simeq[(X,Y),\Delta Z]=[X,\Delta\_1 Z]\times[Y,\Delta\_2 Z].$$ Note that this isomorphism is functorial in all arguments... | 10 | https://mathoverflow.net/users/1261 | 267176 | 119,903 |
https://mathoverflow.net/questions/267175 | 4 | This is motivated by [this](https://math.stackexchange.com/questions/1923064/generalizing-an-inequality-involving-a-convex-function) question on math.SE.
One way to think about Jensen's inequality is that it says that if we have some probability distribution $\mu$ (continuous or discrete) over a space $X$ and $f:X\t... | https://mathoverflow.net/users/85349 | Mass-redistribution generalization of Jensen's inequality | There is a conditional version of the Jensen inequality that may be what you are looking for.
$$
f(E(X\mid {\cal F})) \leq E(f(X) \mid {\cal F})
$$
Taking the expectation, this gives
$$
E(f(E(X\mid {\cal F}))) \leq E(f(X)).
$$
So you can replace the expectation $E(X) = \int X \, d\mu$ by any averages $E(X\mid {\cal F}... | 5 | https://mathoverflow.net/users/6129 | 267179 | 119,905 |
https://mathoverflow.net/questions/266841 | 8 | Let $G$ be a split reductive algebraic group (over a local field if you like), $B$ be a fixed Borel subgroup, and $P$ be a fixed standard parabolic subgroup. Let $W$ be the Weyl group of $G$. For $w\in W$, denote $C(w)=BwB$.
Given Weyl elements $w>w'>w\_1$ (Bruhat order), if $C(w)\subset Pw\_1P$, do we know that $C(... | https://mathoverflow.net/users/13466 | There are no "holes" in the Bruhat decomposition of parabolic cell $Pw_1P$ | Your condition $C(w)\subseteq Pw\_1P$ implies $PwP=Pw\_1P\Rightarrow\overline{PwP}=\overline{Pw\_1P}$.
Moreover $w\ge w'\Rightarrow\overline{BwB}\supseteq\overline{Bw'B}\Rightarrow\overline{PwP}\supseteq \overline{Pw'P}$.
Similarly $w'\ge w\_1\Rightarrow\overline{Bw'B}\supseteq\overline{Bw\_1B}\Rightarrow\overline{... | 4 | https://mathoverflow.net/users/89948 | 267185 | 119,907 |
https://mathoverflow.net/questions/267198 | 4 | Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$.
Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ respectively. There
exists a natural homomorphism of $k$-algebras:
$$
\theta \colon \mathcal{O}(X) \otimes\_k \mathcal{O}(Y) \to \... | https://mathoverflow.net/users/108431 | Regular functions on a product of varieties | This is true for $X$ and $Y$ any quasi-compact quasi-separated schemes over any field $k$. Consider a finite open cover $\{Y\_i\}$ of $Y$ with quasi-compact $Y\_i$, so the overlaps $Y\_{ij} = Y\_i \cap Y\_j$ are quasi-compact since $Y$ is quasi-separated. Then we have an evident left-exact sequence of $k$-vector spaces... | 9 | https://mathoverflow.net/users/81332 | 267202 | 119,912 |
https://mathoverflow.net/questions/267219 | 1 | I'm interesting in finding analytical solutions for the equation
$$\alpha K x^\alpha + x -N = 0,$$
where $\alpha$ is a positive integer and both $K$ and $N$ are positive real constants.
Based on the meaning of the equation (derived from a problem in chemistry), there must be one (and only one) real solution between... | https://mathoverflow.net/users/108404 | Are there analytical solutions for the polynomial $\alpha K x^\alpha + x -N = 0$? | Dividing by $\alpha K$ we obtain a polynomial equation of the form $$x^{\alpha} + d\_1x + d\_0=0.$$
Already for $\alpha=5$ it is known that such an equation (called in *Bring-Jerrard normal form*) is *not* solvable by radicals for general $d\_0$, $d\_1$. A solution of the Bring-Jerrard quintic in terms of hypergeomet... | 1 | https://mathoverflow.net/users/7460 | 267221 | 119,919 |
https://mathoverflow.net/questions/48909 | 13 | **Noether-Deuring theorem** (not in the strongest form, but in the one I usually need)**:**
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over $K$. Let $U$ and $V$ be two left $A$-modules which are finite-dimensional as vector spaces over $K$. If $U\otim... | https://mathoverflow.net/users/2530 | Noether-Deuring for injections and surjections? | Hendrik W. Lenstra has just informed me that the answer to my question is "no", both in the case of $r = \dim U$ (so we are looking at injective $A$-linear maps) and in the case of $r = \dim V$ (so we are looking at surjective $A$-linear maps). Counterexamples can be found in Exercise 21 (c) and Exercise 22 of his note... | 6 | https://mathoverflow.net/users/2530 | 267227 | 119,923 |
https://mathoverflow.net/questions/267238 | 2 | Does anyone on here know of a reference that explicitly computes a conversion formula between the drift terms in multidimensional Ito and Stratonovich SDEs?
In particular, given a solution $(X\_t)$ of an N-dimensional Stratonovich SDE
$$
dX\_t=b(X\_t)dt+\sigma(X\_t)\circ dB\_t
$$
what is the drift term $\tilde{b}(X\_... | https://mathoverflow.net/users/106811 | Conversion formula between multidimensional Ito and Stratonovich SDEs | Here is a reference for the multidimensional Itô-Stratonovich conversion: [pages 137 and 138](https://books.google.nl/books?id=_WfwCAAAQBAJ&pg=PA137&lpg=PA137) of *Theory and Numerics of Differential Equations*, by James Blowey, John P. Coleman, Alan W. Craig (Springer, 2013).
| 2 | https://mathoverflow.net/users/11260 | 267239 | 119,925 |
https://mathoverflow.net/questions/267235 | 6 | This question is probably obvious to experts but I couldn't find the answer in the literature...
**Background:** Consider the mapping class group $Mod\_g$ of the closed genus $g$ surface. There are many nice sets of generators (i.e. Humpreys famous $2g+1$ Dehn twists or [Wajnryb's 2-element generating set](http://www... | https://mathoverflow.net/users/105615 | Explicit description (=pictures!) of elements in $Mod_g[k]$? | Generators for the higher terms in the Johnson filtration are not known.
Probably the best way to find explicit elements is to use the fact that the Johnson filtration forms a central filtration, and thus the kth term of the lower central series of the Torelli group lies in the kth term of the Johnson filtration. You... | 6 | https://mathoverflow.net/users/317 | 267243 | 119,928 |
https://mathoverflow.net/questions/266597 | 8 | Let $\mathcal{K}$ be a category. I denote by $\mathcal{D}\mathcal{K}$ the category of all small diagrams over $\mathcal{K}$: an object is a functor $F:I\to \mathcal{K}$ from a small category $I$ to $\mathcal{K}$ and a morphism from $F:I\to \mathcal{K}$ to $G:J\to \mathcal{K}$ is a functor $\phi:I\to J$ together with a ... | https://mathoverflow.net/users/24563 | About the category of all small diagrams | If $\mathcal{K}$ is assumed both complete and cartesian closed, then $\mathcal{DK}$ is also complete and cartesian closed. Since completeness is not in question for the OP, I'll skip over that part and focus on cartesian closure, although we will need to recall the structure of finite cartesian products.
So: if $g: ... | 5 | https://mathoverflow.net/users/2926 | 267247 | 119,929 |
https://mathoverflow.net/questions/267236 | 4 | Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three.
Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}^2\setminus C$ the fiber $f^{-1}(p)$ is a smooth curve and when $p\in C$ then $f^{-1}(p) = A\_p\cup B\_p$ where $A\_p,... | https://mathoverflow.net/users/14514 | Singularities of fibrations | This is true. Actually, even better, these singularities will be terminal and Gorenstein, so as mild as it can get. Well, at least if we assume that you are working over an algebraically closed field, but otherwise you would have to be more careful about what exactly do you mean by these assumptions and questions, so I... | 3 | https://mathoverflow.net/users/10076 | 267255 | 119,930 |
https://mathoverflow.net/questions/267252 | 22 | Alternate formulation of the question (I think): What's a precise version of the statement: "In a stable $\infty$-category, finite limits and finite colimits coincide"?
Recall that a stable $\infty$-category is a type of finitely complete and cocomplete $\infty$-category characterized by certain exactness conditions.... | https://mathoverflow.net/users/2362 | What are _all_ of the exactness properties enjoyed by stable $\infty$-categories? | I don't think this "finite limits and finite colimits coincide" business can be taken very far. If you take any small category $S\_0$ you can add an initial and a terminal object to form $S = \mathrm{pt} \ast S\_0 \ast \mathrm{pt}$. A diagram of shape $S$ could potentially be both a colimiting cocone and a limiting con... | 23 | https://mathoverflow.net/users/644 | 267265 | 119,932 |
https://mathoverflow.net/questions/267248 | 4 | Let $x, y \in \mathbb{R}^{n}$ be two fixed unit vectors with angle $\alpha \in (\frac{\pi}{2}, \frac{3\pi}{4})$. Define the positive half space associated with a vector $z$ to be $\mathcal{H}(z) = \{h : z^\top h \geq 0\}$.
Choose $m$ unit vectors $\{a\_i\}\_{i=1}^{m}$ uniform over the set $\mathcal{H}(x) \cap \{h : ... | https://mathoverflow.net/users/108479 | Union of random half spaces cover a ray | You should look instead at the probability $p\_m$ that
$$
y\notin \bigcup\_{i=1}^m \mathcal H(a\_i) \;,
$$
or, in other words, that $\langle y, a\_i \rangle < 0$ for all $a\_i$. Since $a\_i$ are independent,
$$
p\_m = \left( \mathbf P \{ \langle y, a \rangle < 0 \} \right)^m \;,
$$
where $a$ is uniformly distributed ... | 2 | https://mathoverflow.net/users/8588 | 267281 | 119,936 |
https://mathoverflow.net/questions/267068 | 2 | Is every [semi-stratifiable](https://topospaces.subwiki.org/wiki/Semi-stratifiable_space) space $\omega$-monolithic?
---
Definitions
-----------
A topological space $(X,\tau)$ is called **semi-stratifiable** if there exists a function $g:\omega\times X\to\tau$ such that:
1. for any point $x$ of $X$ holds $\{x... | https://mathoverflow.net/users/39873 | Is every semi-stratifiable space $\omega$-monolithic? | As a counterexample to this question we can consider the Katetov extension $\kappa\omega$ of the discrete space of all finite ordinals $\omega$.
By definition, $\kappa\omega$ is the space of all ultrafilters on $\omega$ with the topology in which a neighborhood base of an ultrafilter $\mathcal U$ consists of the set... | 3 | https://mathoverflow.net/users/61536 | 267287 | 119,939 |
https://mathoverflow.net/questions/267288 | 1 | Let $X$ be a metric space.
In Borel hierarchy, $\Sigma\_{1}^0$ is the set of all open sets in $X$ while $\Pi\_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma\_{2}^0 = \{ \cup\_{n \in \mathbb{N}} A\_n : A\_n \in \Pi\_1^0 \}$, that is, elements of $\Sigma\_2^0$ are $F\_{\sigma}$ sets. ... | https://mathoverflow.net/users/42411 | Definition of $F_{\sigma}$ sets in terms of $\varepsilon$? | Suppose that $(X,d)$ is a complete metric space. Then a subset $G\subseteq X$ is a $G\_{\delta}$-set precisely when $G$ can be given a complete metric which induces the subspace topology on $G$. The notions of completeness and compatibility can easily be written in terms of $\epsilon,\delta$ and a new metric. I therefo... | 3 | https://mathoverflow.net/users/22277 | 267297 | 119,943 |
https://mathoverflow.net/questions/266964 | 3 | For $\mathsf{V}$ a closed monoidal category, it is canonically [powered](https://ncatlab.org/nlab/show/power) (or cotensored) and [copowered](https://ncatlab.org/nlab/show/copower) (or tensored) over itself with respect to the internal hom and tensor product.
Likewise, any (co)complete category is canonically (co)po... | https://mathoverflow.net/users/56938 | Examples of enriched categories which are (co)powered or (co)tensored | To answer the question raised in the most recent edit: abelian sheaves are tensored and powered over abelian groups.
First of all, abelian presheaves have tensors and powers that are computed *pointwise*. By "abelian presheaves" I mean the $\textbf{Ab}$-category of functors $F: C \to \textbf{Ab}$ where $C$ is a smal... | 3 | https://mathoverflow.net/users/2926 | 267303 | 119,946 |
https://mathoverflow.net/questions/267295 | 5 | Consider the following function defined for complex numbers $z\in\mathbb{C}$ with $\Re(z)\geq \frac{1}{2}$:
$$F(z)=\frac{1}{5^{\Re(z)}}\int\_0^\infty \left| \frac{\Gamma(z+ix)\Gamma(z-ix)}{\Gamma(z)^2} \cdot \exp\left( \frac{\pi}{2} x\right)\right| dx.$$
I am wondering about the behavior of $F(z)$ for $\vert z \ver... | https://mathoverflow.net/users/106571 | Asymptotic behavior of integral with gamma functions | If I just insert the large-$z$ asymptotics of $\Gamma(z)\rightarrow \sqrt{2 \pi } e^{-z} z^{z-\frac{1}{2}}$, and take $z>1/2$ real for simplicity, I find
$$5^z\,F(z)\rightarrow \int\_0^\infty \left(1+x^2/z^2\right)^{z-\frac{1}{2}} e^{\pi x/2-2 x \arctan \left(x/z\right)}\,dx$$
$$\qquad = z\int\_0^\infty(1+x^2)^{-1/2}\e... | 7 | https://mathoverflow.net/users/11260 | 267304 | 119,947 |
https://mathoverflow.net/questions/267319 | 4 | Let $I$ be the category with objects points of $[0,1]$ with unique morphism for every pair of objects.
Let $C$ be a complete category, suppose that there is an isomorphism $m: a \to b$, $m \in C$ when does there exist a functor $F: I \to C$, with $I(0)=a$, $I(1)=b$, and such that the following holds. Let $D(t)$ denote ... | https://mathoverflow.net/users/16877 | Existence of "Continuous paths" in categories as directed systems | This is a bit of a boring answer, but such a functor always exists. We can use the functor that sends $0$ to $a$, every thing else in $[0,1]$ to $b$ and then sends the morphisms to $\mathrm{id}\_a$, $\mathrm{id}\_b$ or $m$ as appropriate.
| 5 | https://mathoverflow.net/users/4613 | 267322 | 119,953 |
https://mathoverflow.net/questions/267316 | 17 | It is a well-known fact that in ZF, the axiom of choice is equivalent to the statement that every commutative ring has a maximal ideal. On the other hand, for Noetherian rings, this is not necessary (either in the sense that it only requires dependent choice or that it requires no choice at all, depending on your defin... | https://mathoverflow.net/users/108284 | Artin Rings, Noetherian Rings, and the Axiom of Choice | Suppose $A$ is a nonzero artinian ring. Then the collection of all nonzero (possibly improper) ideals of $A$ has a minimal element, say $I\subseteq A$. Then $I$ has no nonzero proper submodules, and so is a simple $A$-module. In particular, it must be cyclic (generated by any nonzero element) and the kernel of a surjec... | 16 | https://mathoverflow.net/users/75 | 267325 | 119,956 |
https://mathoverflow.net/questions/267306 | 6 | Consider the quotient space of $\mathbf{CP}^2$ obtained by collapsing a line (a $\mathbf{CP}^1$) to a point. Is this a complex analytic space (in a natural way)?
| https://mathoverflow.net/users/101909 | Is $\mathbb{CP}^2$ with a line collapsed a complex analytic space? | The answer is *no*, because of the following general result.
>
> **Theorem (Grauert's contractibility criterion).** Let $X$ be a smooth complex surface and let $E \subset X$ be a connected curve in $X$, with irreducible components $E\_i$. Then there exists an analytic contraction $$\pi \colon X \to Y$$
> of $E$ to... | 10 | https://mathoverflow.net/users/7460 | 267326 | 119,957 |
https://mathoverflow.net/questions/267300 | 9 | Define a group to be 2-locally finite if, for any two elements, the subgroup generated by them is finite.
Define a group to be locally finite if the subgroup generated by any finite subset is finite.
I want an example of a 2-locally finite group that is not locally finite.
This paper <https://arxiv.org/pdf/1403.0... | https://mathoverflow.net/users/3040 | Example of 2-locally finite group that is not locally finite | Golod ([MR link](http://www.ams.org/mathscinet-getitem?mr=238880); *Some problems of Burnside type. (Russian) 1968 Proc. Internat. Congr. Math. (Moscow, 1966) pp. 284–289 Izdat. "Mir'', Moscow; English translation Amer. Math. Soc. Transl. (2) 70 (1968), 49*) produced infinite $n$-generator groups in which all $(n-1)$-g... | 10 | https://mathoverflow.net/users/14094 | 267328 | 119,958 |
https://mathoverflow.net/questions/267331 | 2 | Some propositions in math can be modeled as a physical system. Has anyone done this for RH?
| https://mathoverflow.net/users/108527 | What is the physical interpretation of the Riemann Hypothesis? | You can try '[Inxeplicable Secrets of Creation](http://empslocal.ex.ac.uk/people/staff/mrwatkin/isoc/)' by Matthew Watkins.
| -2 | https://mathoverflow.net/users/70355 | 267332 | 119,959 |
https://mathoverflow.net/questions/267329 | 9 |
>
> ***Q***. Do there exist De Bruijn tori in dimension $d > 2$?
>
>
>
A [De Bruijn torus](https://en.wikipedia.org/wiki/De_Bruijn_torus)
is a two-dimensional generalization of a
[De Bruijn sequence](https://en.wikipedia.org/wiki/De_Bruijn_sequence).
A De Bruijn sequence is, for two symbols,
a cyclical bit-stri... | https://mathoverflow.net/users/6094 | De Bruijn tori in higher dimensions? | Yes. See ["New constructions for de Bruijn tori" by Hurlbert and Isaak](https://dx.doi.org/10.1007/BF01390770)
| 9 | https://mathoverflow.net/users/1847 | 267337 | 119,963 |
https://mathoverflow.net/questions/267307 | 2 | I want a comprehension principle to capture $\Pi^1\_1$-sets from a domain as well as sets that are relative complements of or finite unions of sets already defined by comprehension. I want to use as little of the Analytical Hierarchy as possible, so is there a natural way to restrict comprehension to obtain such a set-... | https://mathoverflow.net/users/37385 | Can a Boolean Set Algebra be Restricted in the Analytical hierarchy? | Remember that $RCA\_0$ already proves that the class of sets is closed under Boolean combinations:
* Given a set $A$, its complement is computable relative to $A$.
* Given sets $A$ and $B$, their union is computable relative to $A\oplus B$ (and $RCA\_0$ stipulates closure under joins).
So if $M$ is any model of $R... | 3 | https://mathoverflow.net/users/8133 | 267343 | 119,966 |
https://mathoverflow.net/questions/267354 | 1 | Two polynomials $f,g$ are isomorphic iff $f(x\_1,\ldots x\_n)=g(\pi(x\_1, \ldots x\_n))$ for a permutation $\pi$.
$f,g$ are equivalent if there exists invertible linear transormation
$A$ such that $f(X)=g(A\cdot X)$.
Assume $f,g$ are quadratic.
[Paper](https://www.microsoft.com/en-us/research/wp-content/uploads/2... | https://mathoverflow.net/users/12481 | Complexity of quadratic polynomials isomorphism | It is equivalent to (coloured) graph isomorphism problem.
To see this,
one writes $f(X)=X^\top A\_f X$ for $A\_f$ a symmetric $n\times n$ matrix. Then $f$ and $g$ are isomorphic if $\pi A\_f \pi^\top=A\_g$ for a permutation $\pi$. To see it is also only if, suppose $f(X)=g(\pi(X))$, but $A':=\pi A\_f \pi^\top\not=A\... | 1 | https://mathoverflow.net/users/11100 | 267357 | 119,971 |
https://mathoverflow.net/questions/267363 | 2 | Consider the following ring: $\mathbb{Z}\_p[[X,Y]]\otimes\_{\mathbb{Z}\_p}\mathbb{Q}\_p$
(this is the power series ring over $\mathbb{Z}\_p$ of two indeterminates tensored with $\mathbb{Q}\_p$). Is it the case that all finitely generated projective modules over it are free?
| https://mathoverflow.net/users/108548 | Are all finitely generated projective modules over the following ring free? | Quillen's question is whether every finite projective module over $R\_f$ is free, when $R$ is a regular local ring and $f \in \mathfrak m \setminus \mathfrak m^2$. There are many results of this nature in the literature. In particular, this is proved by Gabber for all three dimensional regular local rings in "Some theo... | 6 | https://mathoverflow.net/users/108245 | 267368 | 119,973 |
https://mathoverflow.net/questions/267355 | 2 | Let $H\_i = (V\_i, E\_i)$ be [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) for $i=1,2$. Then we say that $H\_1\cong H\_2$ if there is a bijection $\varphi:V\_1\to V\_2$ such that $A\in E\_1$ implies $\varphi(A) \in E\_2$ and $B\in E\_2$ implies $\varphi^{-1}(B)\in E\_1$.
Is there a collection $\cal C$of pai... | https://mathoverflow.net/users/8628 | Non-isomorphic hypergraphs on $\omega$ | The answer is yes.
Consider the collection $\mathcal C$ of hypergraphs of the following form. They have underlying set $\omega$ as the vertices, the natural numbers. The finite edges in the hypergraph are all and only the sets of the form $\{0,1,\ldots,n\}$. And then the hypergraph can have any desired collection of... | 4 | https://mathoverflow.net/users/1946 | 267369 | 119,974 |
https://mathoverflow.net/questions/267364 | 3 | I know, that field ${\mathbb{Q}\_p}$ (field of p-adic numbers) has the same cardinality as $\mathbb{C}$. Taking algebraic closure doesn't change the cardinality of infinite field, so cardinality $\overline{{\mathbb{Q}}\_p}$ is also equal to ${|\mathbb{C}|}$.
Why taking completion (passing to $\mathbb{C}\_p : = \widehat... | https://mathoverflow.net/users/108551 | Cardinality of ${\mathbb{C}_p}$ | Not only does $\mathbb C\_p$ have the same cardinality as $\mathbb C$, but the larger field $\Omega\_p$, the spherical completion of $\overline{\mathbb Q}\_p$, also has this cardinality. Further, one can explicitly describe $\Omega\_p$ as the set of series $$\sum\_{r\in\mathbb Q} c\_rp^r$$
with coefficients given by Te... | 13 | https://mathoverflow.net/users/11926 | 267370 | 119,975 |
https://mathoverflow.net/questions/267366 | 4 | Setup & question
----------------
Let $C \hookrightarrow \mathbb{P}^{g-1}$ be a general canonical curve of genus $g \ge 4$ and let $Y\_1,Y\_2 \subset \mathbb{P}^{g-1}$ be codimension 2 linear subspaces such that $Y\_i \cap C = \emptyset$ for $i=1,2$. Given any pair of points $(p,q) \in C^2$ we will denote by $L\_{p,q... | https://mathoverflow.net/users/45609 | Do only finitely many bisecants of a canonical curve intersect two distinct codimension 2 spaces simultaneously? | For every curve $C$ of genus $g\geq 5$ ~~that is neither hyperlliptic nor trigonal and that admits no morphism of degree greater than 1 to a curve of positive genus~~ that has general moduli, there exists no such pair $(Y\_1,Y\_2)$ of distinct linear spaces.
For every point $p$, for every codimension $2$ linear space... | 3 | https://mathoverflow.net/users/13265 | 267372 | 119,977 |
https://mathoverflow.net/questions/267378 | 9 | I was going through these notes <https://www.dpmms.cam.ac.uk/~ty245/2008_AGR_Fall/2008_agr_week1.pdf> . There, Theorem 9.2 states that: If $\pi ^{\infty}$ is a cuspidal automorphic representation of $\text{GL}\_2(\mathbb A^{\infty})$ (on $V$), then there exists $N \in \mathbb Z \_{>0}$ with $V^{U\_1(N)} \ne 0$ and for ... | https://mathoverflow.net/users/23927 | Newform of a cuspidal Automorphic Representation | Yes, the classical version of this adelic newform is the newform in the sense of Atkin-Lehner, and vice versa. See Casselman: On some results of Atkin and Lehner (Math. Ann. 201 (1973), 301-314), especially Theorem 4 there. Another important reference is Miyake: On automorphic forms on GL\_2, and Hecke operators (Ann. ... | 6 | https://mathoverflow.net/users/11919 | 267388 | 119,981 |
https://mathoverflow.net/questions/267411 | 0 | Let $\omega\_n=e^{\frac{\pi i}{2n+1}}$. I've an experimental encounter with certain relation involving roots of unity.
>
> **Question.** Is this true? If yes, any proof? For $p\geq0$ an integer, we have the identity
> $$\sum\_{j=1}^n\left\vert\frac{1-\omega\_n^{2j}}{1+\omega\_n^{2j}}\right\vert^{2p}=
> \sum\_{j=1}... | https://mathoverflow.net/users/66131 | modulus identity with roots of unity | Notice that
$$\frac{1-\omega\_n^{2j}}{1+\omega\_n^{2j}}\cdot\frac{1-\omega\_n^{2n-2j+1}}{1+\omega\_n^{2n-2j+1}}=\frac{(1+\omega\_n^{2n+1})-(\omega\_n^{2j}+\omega\_n^{2n-2j+1})}{(1+\omega\_n^{2n+1})+(\omega\_n^{2j}+\omega\_n^{2n-2j+1})} = -\frac{\omega\_n^{2j}+\omega\_n^{2n-2j+1}}{\omega\_n^{2j}+\omega\_n^{2n-2j+1}}=-1... | 4 | https://mathoverflow.net/users/2384 | 267426 | 119,992 |
https://mathoverflow.net/questions/267400 | 7 | First of all, I am so sorry if this question is not appropriate to be here. I tried to ask something similar on Math Stack Exchange but it didn't have much attention. Any comment and I delete the question.
I was reading the classical paper from Milnor entitled Curvature of Left Invariant Metrics on Lie Groups. The ap... | https://mathoverflow.net/users/94097 | Geodesics equation on Lie groups with left invariant metrics | Let $\nabla$ be the connection induced by the Levi-Civita connection. $\nabla$ is left invariant. It thus defines a bilinear product $b$ on ${\cal G}$ the Lie algebra of $G$. Let $c(t)$ be a geodesic. We can write $\dot c(t)=dL\_{c(t)}(x(t))$ where $x(t)\in {\cal G}$. It you write the equation $\nabla\_{\dot c(t)}\dot ... | 7 | https://mathoverflow.net/users/80891 | 267427 | 119,993 |
https://mathoverflow.net/questions/267429 | 6 | Let $\mathfrak{S}\_n$ denote the permutation group, and $I\_0(n)=\sum\_{j\geq0}\binom{n}{2j}\frac{(2j)!}{2^jj!}$ stand for involutions [see A000085 for more interpretations](https://oeis.org/A000085). There is also these numbers $I\_1(n)=\sum\_{j\geq0}\binom{n}jI\_0(j)I\_0(n-j)$ described [in A000898 by several means](... | https://mathoverflow.net/users/66131 | Provoking involutions further | The generating function for involutions with respect to the number of fixed points is given by an evaluation of [Hermite polynomials](https://en.wikipedia.org/wiki/Hermite_polynomials) . The bilinear generating function of Hermite polynomials is given by ["Mehler's formula"](https://en.wikipedia.org/wiki/Mehler_kernel... | 9 | https://mathoverflow.net/users/2384 | 267431 | 119,994 |
https://mathoverflow.net/questions/267408 | 5 | My question concerns the Mumford-Tate conjecture for abelian varieties over number fields.
Most proven cases (that I am familiar with) show that the l-adic monodromy group is as large as it can possibly get because of the conditions imposed and has the same rank as the Mumford-Tate group. A notable exception is the p... | https://mathoverflow.net/users/99726 | Mumford-Tate conjecture cases with small $l$-adic monodromy groups | This is not a clear answer, but let me attempt to clarify the question a little, and also explain why the problem, properly interpreted, is so difficult:
Deligne's theorem that Hodge classes are absolute Hodge shows that the identity component of the $\ell$-adic monodromy group is a subgroup of the Mumford-Tate group... | 3 | https://mathoverflow.net/users/18060 | 267435 | 119,996 |
https://mathoverflow.net/questions/267420 | 3 | Let $X$ be a smooth, projective curve (over $\mathbb{C}$) of genus at least $2$ and $E$ be a globally generated sheaf on $X$. I am looking for conditions/examples such that there exists a closed point $x \in X$ for which the natural morphism from $H^0(\mathcal{E}nd(E))$ to $\mathcal{E}nd(E)\_x$ is surjective, where $\m... | https://mathoverflow.net/users/43198 | Endomorphism of globally generated sheaves on curves |
>
>
> >
> > **Theorem.** Let $\mathcal E$ be a vector bundle on a smooth projective curve $C$ over an algebraically closed field $k$, and let $x \in C$. If $H^0(\mathcal End(\mathcal E))$ surjects onto $\mathcal End(\mathcal E)\_x$, then $\mathcal E \cong \mathcal L^n$ for some line bundle $\mathcal L$ and some nat... | 5 | https://mathoverflow.net/users/82179 | 267437 | 119,998 |
https://mathoverflow.net/questions/267439 | 7 | Consider an entire function $\ f:\mathbb C\rightarrow\mathbb C.\ $ Let $\ (a\_n\in\mathbb C:n=0\ 1\ \ldots)\ $ be an infinite sequence, where $\ a\_k\ne a\_n\ $ whenever $\ k\ne n.\ $ Let $\ L\_n\ $ be the the degree $\le n$ polynomial $\ L\_n\ $ such that $\ L\_n(a\_k) = f(a\_k)\ $ for every $\ k=0\ \ldots\ n.\ $ What... | https://mathoverflow.net/users/8385 | Convergence of Lagrange interpolation polynomials to entire functions | There are too many results to survey them here. The principal books addressing this question are:
B. Levin, Distribution of zeros of of entire functions,
A. Gelfond, Calculus of finite differences,
J. M. Whittaker, Interpolatory function theory.
All these books exist in multiple editions, and can be found on In... | 10 | https://mathoverflow.net/users/25510 | 267443 | 119,999 |
https://mathoverflow.net/questions/267442 | 11 | Suppose $a,b\in\Bbb N$ are odd coprime with $a,b>1$ then is it true that if all four of $$x\_1a+x\_2b,\mbox{ }x\_2a-x\_1b,\mbox{ }x\_1\frac{(a+b)}2+x\_2\frac{(a-b)}2,\mbox{ }x\_2\frac{(a+b)}2-x\_1\frac{(a-b)}2$$ are in $\Bbb Z$ for some $x\_1,x\_2\in\Bbb R$ then $x\_1,x\_2\in\Bbb Z$ should hold?
| https://mathoverflow.net/users/10035 | A simple number theory confirmation | Yes. If we set $\alpha = (a+b)/2$ and $\beta=(a-b)/2$, then the lattice generated by the four vectors
$$
\binom{\alpha}{\beta},\binom{-\beta}{\alpha},\binom{\alpha+\beta}{\alpha -\beta},\binom{\beta -\alpha}{\alpha +\beta}
$$
is contained in the set of $(y\_1,y\_2) \in \mathbb{Z}^2$ such that $x\_1 y\_1 + x\_2 y\_2 \... | 14 | https://mathoverflow.net/users/21724 | 267445 | 120,000 |
https://mathoverflow.net/questions/267392 | 0 | Several [sources](https://www.google.it/search?client=opera&q="left+t-structure"&sourceid=opera&ie=UTF-8&oe=UTF-8) I see speak of a "left t-structure", but lack a precise definition. Where can I find a reference for this?
| https://mathoverflow.net/users/7952 | Left and right $t$-structures | When you try to put a t-structure, say, on the homotopy category of complexes $H(\mathcal{A})$, where $\mathcal{A}$ is "almost an abelian category" (more than additive, but the coimage must not be isomorphic to the image; I do not remember all the assumptions, but they are easy to find), there are two ways to do that. ... | 4 | https://mathoverflow.net/users/10941 | 267447 | 120,001 |
https://mathoverflow.net/questions/267428 | 2 | I am looking for ways to solve the following system of boundary value implicit ODEs over the interval $t \in [0, 1]$:
\begin{equation}
\lambda (Fg - Gf)^3 + 4 FGfg(g-f) = 0 \\
fg(Fg-Gf) + 2FG(gf' - fg') = 0,
\end{equation}
where $f = F'$ and $g=G'$ and $f, -g \geq 0$. The boundary values are $F(0) = G(1) = 0$ and $F(1)... | https://mathoverflow.net/users/4923 | Numerical or exact solution for a system of differential algebraic equations | If you assume that $f>0$ and $g < 0$ on $[0,1]$, then one can integrate the equations explicitly.
Assume $0<t<1$, so that $F$ and $G$ are positive in $(0,1)$. Let $p = f/F = (\log F)' >0$ and $q = g/G = (\log G)' <0$. Moreover, we have $p' = f'/F- p^2$ and $q' = g/G -q^2$. Then the second equation, after dividing by... | 4 | https://mathoverflow.net/users/13972 | 267458 | 120,004 |
https://mathoverflow.net/questions/267450 | 7 | In [this 2005 paper](http://link.springer.com/article/10.1007/s00208-004-0577-3), Kimura introduces a notion of *finite dimensionality* for Chow motives, defined in terms of vanishing of high symmetric and wedge powers. Toward the end of his paper, he conjectures that the Chow motive $h(X)$ of any smooth variety $X$ is... | https://mathoverflow.net/users/64153 | Special cases of the Kimura-O’Sullivan conjecture, i.e., examples of finite dimensional motives? | Q1 + Q2: At present the Chow motives known to be finite dimensional are precisely those that are contained in the thick tensor subcategory generated by motives of abelian varieties. That is, the motives that can be obtained from motives of abelian varieties by tensorial operations, extensions, quotients and subobjects.... | 8 | https://mathoverflow.net/users/1310 | 267461 | 120,006 |
https://mathoverflow.net/questions/267452 | 5 | Let $X$ be a smooth complex manifold and
$\phi:\; X \mapsto Y$ a proper holomorphic
map which is birational ("birational contraction"),
and $Z= \phi^{-1}(y)$ its fiber in a point $y$.
The variety $Y$ is not assumed to be smooth.
In this case I think that $Z$ is Moishezon.
I would be very grateful for a reference or a... | https://mathoverflow.net/users/3377 | fibers of birational contraction for complex manifolds - are they Moishezon? | Up to taking a log resolution of $(X,Z)$ (which works in the analytic category), you may as well assume that $Z$ is a simple normal crossings divisor. Now I expect that the "usual" argument proves that every irreducible component $Z\_i$ of $Z$ is Moishezon. Iteratively do the following. First, blow up $y$ in $Y$ to obt... | 4 | https://mathoverflow.net/users/13265 | 267473 | 120,009 |
https://mathoverflow.net/questions/266837 | 0 | Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J\_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$.
**Q) Are there examples of ideals $J$ such that $l(J\_p)\leq l(J)-1$ for all $p\supset J$ such that $ht p=ht J+1$ and $J^n\neq J^{(n)}$ for... | https://mathoverflow.net/users/9485 | Analytic spread of localization of an ideal | Let $(R, \mathfrak{m}): = \mathbb{C}[[x,y,z]]$ a formal power series. Let $J = (x^2, xy, xz) = \mathfrak{m}(x)$. We can check that $\ell(J) = 3$, $J^{(n)} = (x^n)$. This ideal satisfies the requirements.
| 1 | https://mathoverflow.net/users/17901 | 267488 | 120,012 |
https://mathoverflow.net/questions/267484 | 5 | I remember seeing somewhere that whenever $f$ is a holomorphic function with radius of convergence at $z$, $0<R<\infty$ the following holds
$$\limsup\_{n\to \infty}(\max\_{|w-z|\leq \rho R} |S\_n(f,z)(w)|^{1/n})=\rho,$$
where $\rho\geq 1$ and $S\_n(f,z)(w)$ is the $n$-th partial sum of the Taylor series of $f$ with cen... | https://mathoverflow.net/users/108630 | Reference for result on partial sums of Taylor series | This has an elementary proof, using the formula for the radius of convergence and Cauchy's estimates. It suffices to treat the case $R=1$. We then know that the coefficients satisfy $|a\_n|\lesssim (1+\epsilon)^n$. Thus, if $|z|\le \rho$, then
$$
|S\_N(z)|\le \sum\_{n=0}^N |a\_n| \rho^n \lesssim ((1+\epsilon)\rho)^{N+1... | 4 | https://mathoverflow.net/users/48839 | 267490 | 120,014 |
https://mathoverflow.net/questions/267144 | 1 | Given a filtered space $(\Omega,\mathcal F,\mathbb F,\mathbb P)$ supporting a Brownian Motion $B$, where the filtration $\mathcal F$ is the augmented Brownian filtration, the Azema's martingale is defined by $M\_t=\mathbb E(B\_t|\mathcal G\_t)$, where $\mathcal G\_t=\sigma(sign(B\_s):s\leq t)$, completed over all $\mat... | https://mathoverflow.net/users/101188 | Azema's martingale and quadratic covariation | The Riemann sum converges to zero in probability since $M$ is quadratic pure jump and $B$ is continuous.
| 0 | https://mathoverflow.net/users/101188 | 267493 | 120,016 |
https://mathoverflow.net/questions/267475 | 2 | This question is inspired partly by this question [Any reference on Brownian Motion continuity](https://mathoverflow.net/questions/87735/any-reference-on-brownian-motion-continuity). In this post, the author asked if the following three axioms can define a Brownian motion without assuming the continuity axiom
"**4**... | https://mathoverflow.net/users/25437 | "Brownian motion" without assuming continuity of path at origin of state space | Yes, let $W$ be Brownian motion and let $V$ be the following modification:
$V\_t=W\_t$ except that we pick a number $s\in [0,1]$ according to the uniform distribution, independently of $W$, and let $V\_s=0$.
Then 1,2,3 are satisfied but the sample path of $V$ is *almost surely discontinuous* (at $s$).
To get almost... | 3 | https://mathoverflow.net/users/4600 | 267495 | 120,017 |
https://mathoverflow.net/questions/267464 | 15 | **Edit:** I revise the question based on the comment conversations
Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation.
We define a pseudo metric $d$ on $\mathcal{F}$ as follows:
$$d(G,H)= \inf \{Hd(\tilde {G}\_{n},\tilde{H}\_{n})\} $$
where $\inf$... | https://mathoverflow.net/users/36688 | The completion of the space of finite groups | i don't think it's a metric. Take a large prime $p$. By embedding $\mathbb Z/p\mathbb Z$ and $\mathbb Z/(p^2+p)\mathbb Z$ in the circle $S^1 \subseteq GL(2,\mathbb R)$, one sees that the distance between them is at most $O(1/p)$. By embedding $\mathbb Z/p \mathbb Z \times \mathbb Z/p \mathbb Z$ and $\mathbb Z/p \mathbb... | 12 | https://mathoverflow.net/users/18060 | 267498 | 120,020 |
https://mathoverflow.net/questions/267501 | 24 | If $X$ is a compact Hausdorff space, we can consider the Grothendieck ring of real vector bundles on $X$,
$\mathit{KO}^0(X)$, and this extends to a generalized cohomology theory represented by a ring spectrum
$\mathit{KO}$. Using complex vector bundles, we get another generalized cohomology theory, represented by
$\mat... | https://mathoverflow.net/users/97265 | Why not $\mathit{KSO}$, $\mathit{KSpin}$, etc.? | I am not sure if this is going to be a real answer to the question. However I believe these observations might be interesting.
Let me briefly sketch a way to describe a $G$-structure in (excessively) wide generality. Consider a fibration of spaces $\theta:X\to \coprod\_n BO\_n$. Then a $\theta$-structure on a vector ... | 18 | https://mathoverflow.net/users/43054 | 267504 | 120,023 |
https://mathoverflow.net/questions/266798 | 2 | What is an example of two bounded lattices $L, K$ such that there exist surjective lattice homomorphisms $f:L\to K$ and $g:K\to L$, but there are no injective lattice homomorphisms between $L, K$?
| https://mathoverflow.net/users/8628 | Bounded lattices with lattice surjections but no injections between them | Let $\bf n$ be the $n$-element antichain. Define $L\_{m,n}$ to be the lattice that is the ordinal sum ${\bf 1}+{\bf m}+{\bf 1}+{\bf n}+{\bf 1}+{\bf m}+{\bf 1}+{\bf n}+\cdots$, with $\omega$-many summands. Let $L\_{m,n}^\*$ be the bounded lattice obtained by adding a top element $1$ to $L\_{m,n}$ and letting the origina... | 3 | https://mathoverflow.net/users/75735 | 267514 | 120,025 |
https://mathoverflow.net/questions/267438 | 5 | Let $[0]\_q:=0$ and $[n]\_q:=\frac{1-q^n}{1-q}=1+q+\cdots+q^{n-1}$, for $n\geq1$.
>
> **Question.** Is there a closed formula (with proof) for the determinant of the matrix of $(i,j)$-entries
> $$[i+j\bmod n]\_q, \qquad i,j=1,2,\dots,n.$$
>
>
>
**Remark.** To bring in some context to the problem, this determi... | https://mathoverflow.net/users/66131 | Determinant of the "quantum" version of the group $\mathbb{Z}_n$ | Let me give a few details on Fedor's calculation. If we swap the columns with index $j$ and $n+1-j$ we get the matrix $M\_n(q)$ with $(i,j)$ entry equal to $[1+i-j \mod n]\_q$. Since we have swapped $\lfloor \frac{n}{2}\rfloor$ columns, the determinant of your original matrix is equal to
$$(-1)^{\lfloor \frac{n}{2}\rfl... | 4 | https://mathoverflow.net/users/2384 | 267515 | 120,026 |
https://mathoverflow.net/questions/267516 | 5 | Does there exist a finitely generated discrete amenable group $G$ that acts on a separable Hilbert space $\mathcal{H}$ by unitary transformations, and where (1) $\mathcal{H}$ has no finite dimensional $G$-invariant closed subspaces and (2) there does not exist a sequence $(v\_n)\_n$ of unit vectors in $\mathcal{H}$ suc... | https://mathoverflow.net/users/23661 | Can an amenable group have a weak mixing unitary representation without almost invariant vectors? | If $G$ is amenable, then every weakly mixing representation $\pi$ has almost invariant finite-dimensional subspaces. This means that $\pi \otimes \bar \pi$ has almost invariant vectors.
Results like this can be found in
M.E.B. Bekka. *Amenable unitary representations of locally compact groups*. Invent. Math. **100*... | 5 | https://mathoverflow.net/users/8176 | 267528 | 120,029 |
https://mathoverflow.net/questions/267422 | 9 | The question was asked by a Computer Scientist and is closely related to parallel computing. But it is clearly of algebraic nature, so I decided to post it here.
Let $X$ be a set and $\bar X$ be the union of all Cartesian powers $X^n$.
Let $f$ be a function from $\bar X$ to $X$. We say that $f$ is *inductive* if
the... | https://mathoverflow.net/users/nan | Inductive and reducible functions | I assume that $\bar X$ also contains $\varnothing$ as the Cartesian product of 0 copies of $X$.
Set $Y=\bar X$. If $y\_i=(x\_{i,1},\dots,x\_{i,k\_i})\in X^{k\_i}$ with $i=1,\dots,\ell$, set
$$
g(y\_1,\dots,y\_\ell)=(x\_{1,1},\dots,x\_{1,k\_1},x\_{2,1},\dots,x\_{2,k\_2},\dots)\in X^{k\_1+k\_2+\dots}\subset Y
$$
(so $... | 2 | https://mathoverflow.net/users/17581 | 267535 | 120,031 |
https://mathoverflow.net/questions/267541 | -4 | I just can't get this number out of my head. It's a number that has the decimal digits composed by all the prime numbers.
The first digit is not important, it can be 0,2357... or 1,2357...
Does any mathematician has studied this number? Is it transcendental? I guess I can prove it's not a normal number. Where can... | https://mathoverflow.net/users/108399 | Does the number 1,23571113... has been studied before? | The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value is approximately
0.235711131719232931374143… (sequence A033308 in the OEIS).
In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős in 1946 (... | 4 | https://mathoverflow.net/users/108399 | 267544 | 120,034 |
https://mathoverflow.net/questions/267529 | 4 | In the Wikipedia article [Diophantine set](https://en.wikipedia.org/wiki/Diophantine_set) there is a section entitled "Further applications" in regards to Matiyasevich's theorem and it states:
>
> Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are uns... | https://mathoverflow.net/users/102926 | Matiyasevich's theorem and Gödel's first incompleteness theorem | If you are writing a research paper on this, the usual practice in mathematics is that you do not need a reference for such an obvious result which is known to everyone in the field already. For things like this, there is often no paper that claims credit for the result - one of the roles of books is to summarize these... | 10 | https://mathoverflow.net/users/5442 | 267546 | 120,035 |
https://mathoverflow.net/questions/267545 | 3 | Let $G$ be a discrete, finitely generated group. Let $f\in \mathbb{C} G$ be given.
Consider $g\in G\setminus \operatorname{supp} f$ and let $\delta\_g$ denote the Dirac delta at $g$.
>
> Is it true that $\Vert f\Vert\le \Vert f+\delta\_g\Vert$?
>
>
>
The norms here are in $B(\ell\_2(G))$, as convolution oper... | https://mathoverflow.net/users/104535 | Norm inequality for convolution operators on groups | The operator norm of a convolution operator on $\mathbb Z$ is the supnorm of its Fourier transform.
Let $f(i) = -1$ if $i= -1 ,0,-1,-2$ and $0$ otherwise. Then the operator norm of convolution with $f$ is certainly $4$. If we add the delta function at $0$ the operator norm should be $ \max\_{z \in S^1} | z + z^2 + z^... | 6 | https://mathoverflow.net/users/18060 | 267549 | 120,036 |
https://mathoverflow.net/questions/267459 | 2 | $\require{AMScd}$If $\mathcal{X}$ is a category and $I$ a small category, the category of functors $\mathcal{X}^I$ inherits a (orthogonal) factorization system for each (orthogonal) factorization system on $\mathcal{X}$, defining the two classes objectwise.
It seems to me that I can define this factorization system "... | https://mathoverflow.net/users/7952 | Induced factorization system as a pullback in $\bf Cat$ | If by $\mathcal{A}^I$ you mean the non-full subcategory of $\mathcal{X}^I$ corresponding to the left class of the induced factorization system (which is not the functor category of $I$ into $\mathcal{A}$), then yes, it does fit into such a pullback square. This doesn't construct the whole factorization system however.
... | 1 | https://mathoverflow.net/users/49 | 267556 | 120,038 |
https://mathoverflow.net/questions/267321 | 2 | I am looking for a bound on the empirical Rademacher complexity of the following class:
$G=\left\{x \rightarrow \frac{h^T f(x)}{\|h\|\_2 \cdot \|f(x)\|\_2} : h\in R^d, f()=(f\_1(),\ldots,f\_d()), f\_j \in F \right\}$, where $F$ is some other function class.
$$\hat{R}\_N(G) = E\_\sigma \sup\_{h\in R^d, \forall j, f\_... | https://mathoverflow.net/users/61472 | Rademacher complexity of composition of functions | Assume that $h$ belongs to a set $H$, where each vector $v\in H$ satisfies
$\sum\_{i=1}^d |v\_i|\le1$. Then any expression of the form
$ h\cdot \bar f$, where $\bar f=(f\_1,\ldots,f\_d)\in F^d$ belongs to the *absolute convex hull* of $F$. Thus, under the above assumption on $H$, we have
$$
\frac1n\mathbb{E}\_\sigma \... | 2 | https://mathoverflow.net/users/12518 | 267558 | 120,040 |
https://mathoverflow.net/questions/267542 | 3 | We are given an arbitrary finite subset $P$ of the plane containing $N$ points. Let $Q$ be a subset of $P$ such that the pairwise distances $d(p,q)$ are unique for all $p,q\in Q$.
EDIT: How large can $Q$ be while remaining a strict subset of $P$? Is there a non-trivial lower bound for $Q$?
| https://mathoverflow.net/users/105925 | The size of a subset $Q$ of $P\subset\mathbb{R}^2$ that has all distinct pairwise distances | An earlier version of this answer had a simple induction proof that $|Q|$ is bounded below by a nonconstant function of $|P|$ for points in any dimension,
but I have since discovered that much stronger bounds are known.
For some of the history of the problem, and generalizations to volumes of higher-dimensional simpl... | 3 | https://mathoverflow.net/users/440 | 267561 | 120,041 |
https://mathoverflow.net/questions/267054 | 28 | Suppose we have a round-robin tournament (i.e., each player plays exactly one game with each other player) with $n$ players, who are all equally skillful except for one player, the *favorite*, whose probability of winning a game against any other player is some fixed value $p > 1/2$. Assume that all games are independe... | https://mathoverflow.net/users/3106 | For a round-robin tournament, what is the favorite's least favorite size? | I can show that $N(\epsilon)$ is equal to $\epsilon^{-2}$ up to a log factor on each side.
The strategy I'll use is to give an upper bound for $\pi(1/2+\epsilon,n)$. Optimizing it, we obtain an upper bound for $\pi(1/2+\epsilon,N(\epsilon))$. Then using lower bounds for $\pi(1/2+\epsilon,n)$ we can rule out certain v... | 11 | https://mathoverflow.net/users/18060 | 267563 | 120,042 |
https://mathoverflow.net/questions/267567 | 11 | Can Khovanov homology have arbitrarily large torsion?
That is, given $N\gg 0,$ does there exist $k>N$, a knot (diagram) $D$ and $i,j \in \mathbb{Z}$ such that $\operatorname{Kh}^{i,j}(D) = \mathbb{Z}/k\mathbb{Z}$?
| https://mathoverflow.net/users/104690 | Can Khovanov homology have arbitrarily large torsion? | [This paper](https://arxiv.org/abs/1701.04924) from earlier this year (Jan 18, to be precise) proves the existence of $\mathbb{Z}/n\mathbb{Z}$-torsion for $n\le 8$ and $\mathbb{Z}/2^s\mathbb{Z}$-torsion for $s\le23$. It also states at the beginning of Section 3.4:
>
> Until now, no knot or link with torsion larger ... | 17 | https://mathoverflow.net/users/13119 | 267571 | 120,044 |
https://mathoverflow.net/questions/267576 | 8 | Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \to B^n$ is induced by some $A \to B$.
>
> Assume that $C$ can be generated by $<n$ elements, does it follow $C=0$?
... | https://mathoverflow.net/users/2841 | Exact sequence of $n$th powers of abelian groups | By Ycor's arguments, it seems we can reduce to $p$-groups.
Assume $A$ and $B$ are finite $p$-groups. The $p$-rank of $A$ must equal the $p$-rank of $B$ or else the right exact sequence $(C/pC) \to (A/pA)^n\to (B/pB)^n \to 0$ would imply that the $p$-rank of $C$ is at least $n$ and hence that $C$ has at least $n$ gene... | 8 | https://mathoverflow.net/users/18060 | 267583 | 120,047 |
https://mathoverflow.net/questions/267586 | 15 | The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory that behaves well up to linear equivalence of divisors in the arithmetic case. In particular, we need some analytic data on... | https://mathoverflow.net/users/65980 | Meaning of the determinant of cohomology | It doesn't so much represent a dimension of a cohomology group as it does an Euler characteristic.
More precisely, it's based on Grothendieck's generalization of the Riemann-Roch theorem to families. Given a proper map $\pi: Y\to X$ and a line bundle $L$ on $Y$, we could of course expect Riemann-Roch or a generalize ... | 14 | https://mathoverflow.net/users/18060 | 267589 | 120,050 |
https://mathoverflow.net/questions/267600 | 2 | Could anyone give an insight on how to prove the following formula?
$$\sum\_{n=-\infty}^{+\infty}J\_{n}(\alpha)J\_{N+n}(\alpha)=\delta\_{N0} \, ,$$
where $N$ is an integer. I checked many references but failed to figure out the calculation method. By the way, I found this relation during some numerical calculation... | https://mathoverflow.net/users/108692 | Indefinite summation of multiplication of two Bessel functions | This is the case $y+z=0$ of the [addition formula](https://archive.org/stream/treatiseontheory00watsuoft#page/30/mode/1up) $J\_N(y+z)=\sum\_{n\in\mathbf Z}J\_n(y)J\_{N-n}(z)$, plus the fact that $J\_k(z)=\frac1{2\pi}\int\_0^{2\pi}\cos(k\theta-z\sin\theta)\,d\theta=J\_{-k}(-z)$.
| 1 | https://mathoverflow.net/users/19276 | 267605 | 120,059 |
https://mathoverflow.net/questions/267543 | 32 | Among the papers indexed by MathSciNet and Zentralblatt MATH,
I occasionally have seen papers which consist essentially only
of text copied from elsewhere without proper attribution and without
adding any significant value. I would be interested whether anyone
has an idea what the frequency of such papers among those i... | https://mathoverflow.net/users/28104 | Frequency of papers showing academic misconduct among the articles indexed by MathSciNet and Zentralblatt MATH | On behalf of zbMATH (which is certainly also the case for MathSciNet), we would very much appreciate a notification of such cases, if they have not yet been detected at the level of editors or reviewers. There is the general impression of our editors (which has been discussed with our MathSciNet colleagues who seem to ... | 49 | https://mathoverflow.net/users/100979 | 267609 | 120,060 |
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