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https://mathoverflow.net/questions/373104 | 3 | It can be found that there are the following [bordism group](https://en.wikipedia.org/wiki/Cobordism) $\Omega\_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds:
$$
\Omega\_0^{SO} = \mathbb{Z} , \quad \Omega\_1^{SO} = 0.
$$
$$
\Omega\_0^{Spin} = \mathbb{Z} , \quad \Omega\_1^{Spin} = \mathb... | https://mathoverflow.net/users/106497 | Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$ | $\newcommand{\Z}{\mathbb Z}$Part 2 is correct: for any $G$-structure, $\Omega\_0^G$ is either isomorphic to $\Z$ or
$\Z/2$. Part 3 isn't correct, and I'll give a counterexample.
First, part 2: let $\rho\colon G\to O$ be a $G$-structure. $\Omega\_0^G$ is isomorphic to $\pi\_0(MG)$, where $MG$ is
the Thom spectrum of t... | 7 | https://mathoverflow.net/users/97265 | 373108 | 155,841 |
https://mathoverflow.net/questions/373007 | 8 | I heard that the Rips complexes associated to the Cayley graphs of hyperbolic groups are contractible for a sufficiently large radius. Is the converse true? Namely, if a group is non-hyperbolic, then is its Rips complex never ``asymptotically" contractible?
For example, we can ask if the non-hyperbolic group $\mathbb... | https://mathoverflow.net/users/156792 | Contractible Rips complex from non-hyperbolic group | Another source of Cayley graphs with contractible Rips complexes comes from Helly graphs.
**Proposition:** *Rips complexes of uniformly locally finite Helly graphs are contractible.*
See Lemma 5.28 and Theorem 4.2(v) from the preprint [arXiv:2002.06895](https://arxiv.org/abs/2002.06895).
One construction of Helly... | 8 | https://mathoverflow.net/users/122026 | 373114 | 155,843 |
https://mathoverflow.net/questions/373118 | 12 | The famous Kirchhoff's Matrix-Tree theorem counts the number of spanning trees of a connected graph, that is, the number of bases of its cycle matroid. But it appeals to vertices, that's why I do not see how to generalize it to general matroids. Also the graphs with isomorphic cycle matroids may have quite different La... | https://mathoverflow.net/users/4312 | Is there Matrix-Tree theorem for counting the bases of a connected matroid? | A broader class of matroids for which you have a Matrix Tree theorem are the [regular matroids](https://en.wikipedia.org/wiki/Regular_matroid) (those representable over every field): see, e.g., <https://arxiv.org/abs/1404.3876>.
**EDIT**: Let me actually try to give a very simple explanation of what's going on here.
... | 12 | https://mathoverflow.net/users/25028 | 373123 | 155,845 |
https://mathoverflow.net/questions/373132 | 2 | I look for a reference of the following implication
Let $ X $ be a compact complex manifold,
If : 1) $ \chi (O\_X) \neq0 $
2) the Universal covering does not contain compact subvariety
So $ K\_X $ is big .
We know that $ K\_X $ is big $\implies$ $ K\_X $ is nef, when can we have the equivalent?
| https://mathoverflow.net/users/148120 | A big line bundle in complex compact manifold | At least in the projective setting the following holds true (this is taken from J. Kollár "Shafarevich maps and automorphic forms", Proposition 13.14.2).
**Proposition.** Let $X$ be a smooth projective variety. If $K\_X$ is nef but not big, and $X$ has generically large fundamental group, then $\chi(\mathcal O\_X)=0$... | 5 | https://mathoverflow.net/users/9871 | 373142 | 155,851 |
https://mathoverflow.net/questions/373135 | 3 | Let $T$ be a self-adjoint operator on a Hilbert space $\mathcal{H}$, with spectrum $\sigma(T)$. For any $x,y\in \mathcal{H}$, denote by $\mu\_{xy}$ the spectral measure of $T$ with respect to $x$ and $y$, that is the unique Borel measure on $\sigma(T)$ such that
$$ \langle x,f(T)y\rangle = \int\_{\sigma(T)} f(\lambda... | https://mathoverflow.net/users/127070 | Spectrum of a self-adjoint operator and spectral measures | Yes, it's true.
I prefer to work with positive measures, so I only deal with $x=y$ (the $\mu\_{x,y}$ have to reason to be positive otherwise). This is not problematic, as the spectrum $\sigma(T)$ is also the closure of $\cup\_x \mathrm{Supp}(\mu\_{x,x})$. So we have to show that the support of $\mu\_{x,x}$ is contain... | 5 | https://mathoverflow.net/users/10265 | 373143 | 155,852 |
https://mathoverflow.net/questions/372649 | 2 | Let $M$ be a free right $R$-module. When $M\_R\cong R\_R^n$ with $n\in \mathbb{Z}\_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M\_R)$ is isomorphic to $\mathbb{M}\_n(R)$. We also know that $\mathbb{M}\_{n}(R)$ is generated as a ring by its idempotents, when $n\geq 2$. Is $E$ generated by its idempot... | https://mathoverflow.net/users/165991 | Endomorphism rings of infinitely generated free modules generated by idempotents? | Yes it's true: every element can be written as $tu+vw+x+y-4z$ with each of $t,\dots, z$ idempotent.
More generally, this holds for an arbitrary module $M$ that is isomorphic to $N\times N$ for some module $N$.
Indeed, in this setting, every endomorphism of $M$ can be written as block matrix $\begin{pmatrix} A & B\\... | 3 | https://mathoverflow.net/users/14094 | 373155 | 155,855 |
https://mathoverflow.net/questions/373163 | 4 | Let $\sigma$ denote the sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$, let $x,y \in \mathbb{R}$. Given the following two conditions: $|\sigma(-x) - \sigma(y)| < \epsilon$ and $x - y > c > 0,$ where $\epsilon$ can be regarded as a small positive number and $c$ as a large positive number.
**Revised question:** can ... | https://mathoverflow.net/users/106253 | Inequality involving sigmoid function | Such a function $f$ does not exist.
Suppose the contrary. Let $t:=\epsilon\downarrow0$ and let $c$ go to $\infty$ fast enough so that $c\ge\ln\frac1t$ and $c\ge2f(t)+2\ln\frac1t$. Let then $x=c$ and $y=\ln t$. Then eventually $y\ge-x$, $\sigma(y)\ge\sigma(-x)$, $|\sigma(-x)-\sigma(y)|=\sigma(y)-\sigma(-x)<\sigma(y)=\... | 5 | https://mathoverflow.net/users/36721 | 373171 | 155,857 |
https://mathoverflow.net/questions/373164 | 5 | Let $(M, J)$ be a Fano projective manifold. Can $(M, -J)$ be general type?
For complex curves and surfaces Kodaira dimension is diffeomorphism invariant so this cannot happen.
| https://mathoverflow.net/users/nan | Fano manifold becoming general type upon conjugation | No. $(M,J)$ and $(M,-J)$ have conjugate pluri-canonical rings, hence have same Kodaira dimension.
*Proof.* Take a section $\mu$ of $K\_{(M, J)}^{\otimes n}$, then $\bar \mu$ is a holomorphic section of $K\_{(M,-J)}^{\otimes n}$. And vice versa.
| 8 | https://mathoverflow.net/users/943 | 373172 | 155,858 |
https://mathoverflow.net/questions/373156 | 10 | Let $\pi:E\to X$ be a complex vector bundle\*, and $f:E\to E$ a bundle isomorphism.
Consider the mapping torus
$$E(f) := \frac{E\times [0,1]}{E \times \{0\}\sim\_f E \times \{1\}}$$
where the identification is the obvious one: $(x,0)\sim\_f (f(x),1)$.
$E(f)$ is also a complex vector bundle over $ X\times \mathbb{S}^1... | https://mathoverflow.net/users/158806 | Chern classes of a mapping torus vector bundle in terms of the construction data | In the case that $E$ is trivial, there is a "universal" example of the construction you describe, which is the vector bundle on $S^1\times U(n)$ formed the "canonical" automorphism (each point acts on the fiber over it). For a general $X$ you take the pullback along the map to $S^1\times U(n)$ defined by the given auto... | 6 | https://mathoverflow.net/users/163893 | 373189 | 155,861 |
https://mathoverflow.net/questions/373185 | 4 | Let $(M, J)$ be a complex projective manifold. Can $(M, -J)$ have different Chern/Hodge numbers?
| https://mathoverflow.net/users/nan | Chern/Hodge numbers of the conjugate complex manifold | These are all the same.
As for Hodge numbers, you can choose a Kahler metric $g$ on $(M,J)$, and it will also be Kahler for $(M,-J)$. Now we know that $h^{p,q}$ is the dimension of the space of harmonic $(p,q)$-forms. A harmonic $(p,q)$ form for $(M,g,J)$ gives you a harmonic $(q,p)$ form on $(M,g,-J)$. And since $h^... | 11 | https://mathoverflow.net/users/943 | 373192 | 155,862 |
https://mathoverflow.net/questions/373180 | 4 | One can easily show that if $p$ is a prime that does not divide $a$, then $a^{p(p-1)}\equiv 1 \pmod{p^2}$.
However, my question is: If instead of $p$ being a prime, it were a pseudoprime to the base $a$, would the result still be valid? Moreover, in general, for what (if any) composite numbers $p$ would the above res... | https://mathoverflow.net/users/166340 | On composite divisors of certain terms in the extended Lucas sequences | In base $2$ the solutions are [A306259](https://oeis.org/A306259):
Composite numbers k such that $2^{(k(k-1))} \equiv 1 \pmod{ k^2}$
From a comment: It contains all Fermat pseudoprimes to base 2,
[A001567](https://oeis.org/A001567).
The sequence starts:
```
21, 105, 165, 205, 231, 273, 301, 341, 385, 465
```
... | 2 | https://mathoverflow.net/users/12481 | 373193 | 155,863 |
https://mathoverflow.net/questions/373197 | 7 | Do you know where I can find proof of equivalence Baire Category Theorem and DC (Axiom of Dependent Choice)? It is well known fact but I can't find appropriate literature with the proof.
| https://mathoverflow.net/users/143974 | BCT equivalent to DC | You can find it, amongst other places in my write up:
>
> [Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice.](http://karagila.org/wp-content/uploads/2016/10/axiom-of-choice-in-analysis.pdf)
>
>
>
If you need a source to cite, my money is on Handbook of Analysis and it... | 16 | https://mathoverflow.net/users/7206 | 373200 | 155,866 |
https://mathoverflow.net/questions/373191 | 14 | Many interesting C\*-algebras can be realized as **convolution algebras** over a groupoid, an idea introduced in 1980 by Jean Renault ([this](https://ncatlab.org/nlab/show/category+algebra) entry in nLab provides plenty of context to the general approach of attaching an algebra to a groupoid).
Perhaps due to my incom... | https://mathoverflow.net/users/15293 | Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra? | The main obstruction to this kind of duality is not so much that not every $C^\*$-algebra is a convolution algebra (though, at least if we don't use twisted convolution algebra, there are known obstruction as mentioned in the comment), but rather that the construction that attach a convolution $C^\*$-algebra to a group... | 17 | https://mathoverflow.net/users/22131 | 373206 | 155,869 |
https://mathoverflow.net/questions/373204 | 2 | What is the Kahler cone of $\mathbb{C}P^1 \times \mathbb{C}P^n$ blown-up along a co-dimension two subvariety of the form $pt \times H$ where $H \subset \mathbb{C}P^n$ is a hyperplane?
| https://mathoverflow.net/users/149600 | Kahler cone of blow up of $\mathbb{C}P^1 \times \mathbb{C}P^n$ | The Kahler classes are the following: $a \mathbb CP^n+b H\times \mathbb CP^1-c E$ where $E$ is the exceptional divisor, and $a, b>c>0$. This can be proven using the fact that manifold is toric.
Indeed, the moment image of $\mathbb CP^1\times \mathbb CP^n$ is a segment times an $n$-simplex, and to get your manifold on... | 3 | https://mathoverflow.net/users/943 | 373207 | 155,870 |
https://mathoverflow.net/questions/373208 | 6 | I want to find reference of Waldspurger's paper referred at *"Sur les coefficients de Fourier des formes modulaires de poids demi-entier"* J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (available [here at J. Voight's web page](https://math.dartmouth.edu/%7Ejvoight/notes/Waldspurger.pdf)).
The name of ref. is [W]... | https://mathoverflow.net/users/166376 | Reference of J.L. Waldspurger's paper on Shimura correspondence | The full reference is
Jean-Loup Waldspurger, "Sur les coefficients de Fourier des formes modulaires de poids demi-entier", (French) Journal de Mathématiques Pures et Appliquées, IX Séries, 60, 375-484 (1981), [MR0646366](http://www.ams.org/mathscinet-getitem?mr=MR0646366), [Zbl 0431.10015](https://zbmath.org/?q=an%3A... | 7 | https://mathoverflow.net/users/113756 | 373211 | 155,872 |
https://mathoverflow.net/questions/373179 | 2 | I am trying to find a mathematical relationship between the size of a tree (or - in other terms - the cardinality of set or permutations) for a set of elements which are subject to precedence constraints.
To illustrate the problem, I'll provide a few examples. Suppose our set is {a, b, c, d} and our precedence constr... | https://mathoverflow.net/users/166341 | Number of permutations of a set given arbitrary precedence constraints | If the precedence constraints are a disjoint union of rooted trees, as in your examples, then there is an explicit formula due to Knuth, *The Art of Computer Programming*, vol. 3, 1973, p. 70. Let $n$ be the size of the set $S$. For each $x\in S$, let $\nu(x)$ denote one more than the number of elements that come after... | 8 | https://mathoverflow.net/users/2807 | 373212 | 155,873 |
https://mathoverflow.net/questions/373210 | 4 | I am confused with what seems to be a standard notation in analytic number theory and I'd appreciate any clarification. I am interested in the zero density estimates, for example link.springer.com/article/10.1007/BF01403187 . In this paper and in many other sources I have seen, $N\_{\chi}(\alpha, T)$ is defined to be t... | https://mathoverflow.net/users/84272 | Question about the notation $N_{\chi}(\alpha, T)$, the number of zeroes of the $L(s, \chi)$ in a rectangle | Zeros are always counted with multiplicity, both in $N\_\chi(\alpha,T)$ and in sums over zeros. This becomes clear when you look at how this quantity is estimated. Note also that the multiplicity of each zero $s$ of $L(s,\chi)$ is small, namely $O(\log q(2+|s|))$ by [Jensen's formula](https://en.wikipedia.org/wiki/Jens... | 7 | https://mathoverflow.net/users/11919 | 373215 | 155,875 |
https://mathoverflow.net/questions/370042 | 7 | I apologize in advance if this sounds vague but I am trying to find directions as to what to look for.
All the sets in this problem are finite.
Suppose we have two functions $f\_1\colon X\_1\times Y\_1\to X\_1$ and $f\_2\colon X\_2\times Y\_2\to X\_2$.
**Problem**. Decide whether there exist two surjective mappings... | https://mathoverflow.net/users/164236 | Search algorithms with mappings/functions/sets as variables | You can solve the problem via integer linear programming as follows. Let binary decision variables $P(x\_2,x\_1)$ and $Q(y\_2,y\_1)$ indicate whether $p(x\_2)=x\_1$ and $q(y\_2)=y\_1$, respectively. The constraints are:
\begin{align}
\sum\_{x\_1 \in X\_1} P(x\_2,x\_1) &= 1 &&\text{for $x\_2 \in X\_2$} \tag1 \\
\sum\_{y... | 0 | https://mathoverflow.net/users/141766 | 373216 | 155,876 |
https://mathoverflow.net/questions/373228 | -1 | I consider the following set
$$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$
Is it possible to identify the closure of $A$ in the reals?
| https://mathoverflow.net/users/150564 | Limiting points of elementary set | Set $x=n/m$, rational dense in the reals, so the closure of $A$ is the image
of the function $f(x)=3x/(2(x^2+x+1))$, so the interval $[-3/2,1/2]$.
| 4 | https://mathoverflow.net/users/81776 | 373230 | 155,880 |
https://mathoverflow.net/questions/373079 | 17 | Lately there has been a lot of progress on the foundations of $(\infty,2)$-categories (for example, all currently-known models for them [were shown](https://arxiv.org/abs/1911.01905) to be equivalent and finally we have a [construction](https://arxiv.org/abs/2003.11757) of the Gray tensor product).
While I'm aware th... | https://mathoverflow.net/users/130058 | $(\infty,2)$-categories: current applications and future prospects | Topological field theory (TFT) is a major client of higher-dimensional category theory. For $(\infty,
2)$-categories specifically, this specializes to two-dimensional TFT. One significant research area in this field
is taking physics ideas, making them mathematically rigorous, and then using the resulting mathematical ... | 16 | https://mathoverflow.net/users/97265 | 373233 | 155,881 |
https://mathoverflow.net/questions/371959 | 9 | I'm interested in numerical methods on $\mathrm{SO}(3)$ manifold, and working on a particular problem using the exponential coordinates:
$$
R(u) := \exp(u\_\times)
$$
with $u\in \mathbb{R}^3$ and where $u\_\times \in \mathfrak{so}(3)$ is the cross-product matrix of vector $u$.
The directional derivative of $R(u)$ in ... | https://mathoverflow.net/users/164738 | What is the Lipschitz constant of the differential of the matrix exponential $\mathfrak{so}(3)\to \mathrm{SO}(3)$ | Found the proof! It's done using the integral definition of $T$:
$$
T(v) = \int\_0^1 R(su) ds = \lim\_{n\rightarrow \infty} \frac{1}{n}\sum\_{i=1}^n R\left(\tfrac{i}{n}v\right)
$$
So for any vectors $X$ and $Y$:
\begin{align\*}
&\biggl|\left[\mathrm{D}\_v \left(R(v)X\right)\right]Y - \left[\mathrm{D}\_u \left(R(u)X\rig... | 2 | https://mathoverflow.net/users/164738 | 373244 | 155,887 |
https://mathoverflow.net/questions/373145 | 3 | Let $a(\cdot), b(\cdot)$ be non-negative multiplicative functions supported on square-free integers (that is, $a(p^k) = b(p^k) = 0$ for all primes $p$ and $k \geq 2$). Consider the summatory functions
$$\displaystyle A(x) = \sum\_{n \leq x} a(n), B(x) = \sum\_{n \leq x} b(n).$$
Suppose that $A(x), B(x)$ satisfy asy... | https://mathoverflow.net/users/10898 | Estimating a sum of the shape $\sum_{n \leq x} a(n) b(n)$ | If for every $q>1 $ the function $b(n)^q$ has average $O\_q(1)$ then by H"{o}lder one can get $$ \sum\_{n<x} \lambda^{\omega(n)} b(n) \ll\_\epsilon x (\log x)^{\lambda-1+\epsilon}$$ for every fixed $\epsilon >0$. Apart from the $(\log x)^\epsilon$ term, this is close to the best one can hope in general.
If you know t... | 4 | https://mathoverflow.net/users/9232 | 373250 | 155,889 |
https://mathoverflow.net/questions/373248 | 2 | In Khinchin's book, "Continued Fractions," he considers the question, given an irrational, $\alpha$, and a real number, $\beta$, how to find integral $x$ and $y$ such that
$$\alpha x - y \approx \beta$$
to a given level of accuracy.
He then says that Chebyshev "obtained the first basic results connected with it, and ... | https://mathoverflow.net/users/166394 | Continued fractions, Chebyshev and non-homogenous approximation | Our question is a very special case of the general problem of restricted simultaneous Diophantine approximation. Thus, I believe that the best place to start is to consult the paper of Schmidt: *Two questions in Diophantine approximation*, Monatsh. Math. 82, 237-245. Moreover, you also should consult the paper of P. Th... | 2 | https://mathoverflow.net/users/164119 | 373262 | 155,894 |
https://mathoverflow.net/questions/373199 | 10 | One of the most famous application of number theory is the RSA cryptosystem, which essentially initiated asymmetric cryptography.
I wonder if there are applications of number theory also in **symmetric** cryptography.
Thank you in advance for any comment / reference.
**NOTE:** Since RSA is based on [Euler's theor... | https://mathoverflow.net/users/164852 | Number theory in symmetric cryptography | Here are a few interesting examples of *symmetric* primitives whose claimed security is/was based on number-theoretic problems:
1. From the 1980s: the famous [Blum-Blum-Shub deterministic random bit generator](https://en.wikipedia.org/wiki/Blum_Blum_Shub) is a classic example. Let $N = pq$ be the product of two large... | 11 | https://mathoverflow.net/users/156215 | 373263 | 155,895 |
https://mathoverflow.net/questions/371089 | 8 | Do there exist non-algebraic Kähler threefolds with abelian $\pi\_1$ of arbitrarily large rank?
| https://mathoverflow.net/users/nan | Non-algebraic Kähler threefolds with abelian $\pi_1$ of arbitrarily large rank | Let's construct such a Kahler $3$-fold $X$. It will be obtained as an elliptic fibration over a projective surface $S$ with abelian fundamental group $\mathbb Z^{2g}$.
**Construction.** Recall first that the space of principally polarised Abelian varieties of dimension $g$ has dimension $g(g+1)/2$. Let us consider th... | 0 | https://mathoverflow.net/users/943 | 373272 | 155,897 |
https://mathoverflow.net/questions/373264 | 4 | I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$
It is fairly straightforward to obtain
$$ \frac{d^n f}{dx^n} = \frac{P\_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $$
where $P\_n(x)$ is given by the recursive relationship $P\_0(x) = 1$ and
$$P\_{n+1}(x) = P\_n'(x) \cdot x \cdot (x^2... | https://mathoverflow.net/users/41452 | Higher-order derivatives of $(e^x + e^{-x})^{-1}$ | Using the tried-and-true method of calculating small examples and plugging them into the OEIS, one finds that the $P\_n(x)$ are, up to sign, known as MacMahon polynomials, and their coefficients are given by [Eulerian numbers of type B](http://oeis.org/A060187). The OEIS also has a separate entry for the [maximal coeff... | 20 | https://mathoverflow.net/users/3106 | 373275 | 155,898 |
https://mathoverflow.net/questions/373274 | 9 | Narrow question: Did Publ. Math. Institute Hung. (Publications of the Mathematical Institute of the Hungarian Academy of Sciences) change its name? I am finalizing the bibliography of an article in progress that refers to an article that supposedly appeared in Publ. Math. Institute Hung. (see e.g. [this MathSE question... | https://mathoverflow.net/users/3621 | Does Publ. Math. Institute Hung. have a new name? | According to Mathscinet, the name of this journal was
A Magyar Tudományos Akadémia. Matematikai Kutató Intézetének Közleményei.
Abbreviation: Magyar Tud. Akad. Mat. Kutató Int. Közl.
It changed the name in 1977 and now is called Alkalmazott Matematikai Lapok.
But the paper you refer to is listed under the old name.... | 9 | https://mathoverflow.net/users/25510 | 373276 | 155,899 |
https://mathoverflow.net/questions/373254 | 9 | Suppose I have a Heegaard splitting of a closed oriented irreducible 3-manifold $M$, defined by the Heegaard diagram $(\Sigma\_{g},\{\alpha\_{1},\dots,\alpha\_{g}\},\{\beta\_{1},\dots,\beta\_{g}\})$. Are there any obvious sufficient or necessary conditions for the attaching curves for when $M$ is toroidal (or atoroidal... | https://mathoverflow.net/users/149240 | Toroidal Heegaard splittings | In Hempel's "[3-manifolds as viewed from the curve complex](https://arxiv.org/abs/math/9712220)," one of the main theorems is a necessary criteria for being toroidal. In particular, he shows that if a 3-manifold is toroidal then all of its Heegaard splittings, $\Sigma$, have $d(\Sigma) \leq 2$, where $d$ is the Hempel ... | 7 | https://mathoverflow.net/users/84721 | 373286 | 155,902 |
https://mathoverflow.net/questions/373285 | 8 | Let $\phi : M \to M$ be a diffeomorphism. Is there a metric $g$ on $M$ and a diffeomorphism $\psi$ isotopic to $\phi$ so that $\psi$ is an isometry with respect to $g$? I'm guessing the answer is no, since there are manifolds such that all metrics admit no nontrivial isometries and maybe some of these manifolds have a ... | https://mathoverflow.net/users/99414 | Realizing mapping classes as isometries? | As you suspect, in general, no.
For example, if $M$ is compact, and $\psi:M\to M$ fixes a metric $g$ on $M$, then the closure of $\{\psi^k\ |\ k\in\mathbb{Z}\ \}$ is a compact abelian subgroup of $\mathrm{Isom}(M,g)$, and, hence, its identity component is a torus, so $\psi$ must be isotopic in $\mathrm{Isom}(M,g)$ to... | 12 | https://mathoverflow.net/users/13972 | 373288 | 155,903 |
https://mathoverflow.net/questions/373271 | 3 | $\DeclareMathOperator{\Sub}{\operatorname{Sub}}$ Let $G$ be a profinite group and consider the space $\Sub(G)$ of closed subgroups of $G$ equipped with the profinite topology. That is, we have $G = \underleftarrow\lim(G\_i)$ for finite groups $G\_i$, and we construct $\Sub(G)$ as $\lim(\Sub(G\_i))$. The space $\Sub(G)$... | https://mathoverflow.net/users/166413 | Spaces of closed subgroups of a profinite group up to conjugacy | Just to restate the question concisely:
>
> Let $G$ be a second-countable profinite group. If $G$ has uncountably many subgroups, does it have uncountably many closed subgroups modulo conjugacy?
>
>
>
The answer is no. A counterexample is the $p$-adic group $\mathrm{SL}\_2(\mathbf{Z}\_p)$ for $p$ prime.
It c... | 4 | https://mathoverflow.net/users/14094 | 373290 | 155,905 |
https://mathoverflow.net/questions/373301 | 11 | Let $M$ and $N$ be $3$-manifolds obtained by zero-surgery on (left-handed) trefoil and figure-eight knot respectively.
What is the easy way to prove that $M$ and $N$ are not homeomorphic?
Note: When they are knot homology spheres (they are both homology $S^1 \times S^2$'s), I cannot use the classical invariants.
| https://mathoverflow.net/users/nan | $0$-surgeries on trefoil and figure-eight | If you're happy bringing in heavy machinery then you could compute some sort of Floer homology, like the 'hat' version of Heegaard Floer homology: this has rank 2 for $S^3\_0(3\_1)$ and rank 4 for $S^3\_0(4\_1)$, so they're different.
On the other hand, in cases like this where you have a very specific pair of 3-mani... | 12 | https://mathoverflow.net/users/428 | 373306 | 155,911 |
https://mathoverflow.net/questions/373296 | -2 | *Note: This question aims to be a generalization of [Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$?](https://mathoverflow.net/q/371972/165539) and [Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? -- Part 2](https://mathoverflow.net/q... | https://mathoverflow.net/users/157462 | Is there a function $f$ that is a finite sum of functions with finite products of the inputs of $f$ as inputs with this property? | There's no such family of functions for $\epsilon = 0$, and even for $\epsilon\_n$ depending on $n$ decaying at a rate to be determined later. I don't know what happens if $\epsilon > 0$ is independent of $n$.
For a multiset of indices $I = \{ i\_1, \dots i\_n \}$ write $a\_I = \prod\_{i \in I} a\_i$ for the correspo... | 3 | https://mathoverflow.net/users/290 | 373314 | 155,914 |
https://mathoverflow.net/questions/373282 | 3 | We have seen so many norms we need for PDE. For example, for elliptic PDE, we require a continuous version of $C^k$, i.e. $C^{k,\alpha}$. Roughly speaking, under appropriate norm, we could capture the topological information we want. But a question (maybe too vague), how can we know what kind of norm we want in PDE?How... | https://mathoverflow.net/users/46341 | Discovery of norm in PDE | One striking example, though perhaps considered "antique" by now, is Levi's 1906 use of a (true/correct!) minimum principle in Hilbert spaces, and forming what is now called $W^{1,2}$, to prove a (true/correct!) version of "Dirichlet's principle" (which, in effect, had been asserted for certain concrete Banach spaces, ... | 5 | https://mathoverflow.net/users/15629 | 373316 | 155,916 |
https://mathoverflow.net/questions/373313 | 3 | Given a manifold $M$, we can always embed it in some Euclidian space (general position theorem). Hence we can define the minimal embedding space of $M$ to be the smallest euclidean space that we can embed $M$ in. My question is, will this depend on the category of $M$ (piece-wise linear or smooth)? I am not an expert i... | https://mathoverflow.net/users/103418 | minimal embedding space of a manifold in smooth and PL case | Yes, it depends on the category of manifolds you are considering.
For example, by Corollary 1.4 of [Hsiang-Levine-Szczarba](https://www.sciencedirect.com/science/article/pii/0040938365900418), the 16-sphere with non-standard smooth structure does not admit a smooth embedding into $\mathbb{R}^{19}$, and hence also not... | 10 | https://mathoverflow.net/users/798 | 373319 | 155,918 |
https://mathoverflow.net/questions/373321 | 11 | Is every matrix $A \in \mathrm{SL}\_n(\mathbb C)$ a product of four unipotent matrices?
I have verified that this is true if $n = 2$, and I believe I have came across this result before. However, I cannot find a reference to this.
| https://mathoverflow.net/users/88670 | Is every $A \in \mathrm{SL}_n(\mathbb C)$ a product of four unipotent matrices? | In response to Qiaochu's question in the comments, Fong and Sourour prove in their paper [The group generated by unipotent operators](https://www.ams.org/journals/proc/1986-097-03/S0002-9939-1986-0840628-0/) that every element of $\mathrm{SL}\_n(\mathbb C)$ is a product of three unipotent matrices.
Edit: Sourour prov... | 15 | https://mathoverflow.net/users/2384 | 373330 | 155,923 |
https://mathoverflow.net/questions/373318 | 3 | Let $X$ be a projective manifold and $\Delta$ a divisor with simple normal crossings. Consider $X$ as the compactification of a quasi-projective variety $X\_0$ with boundary $\Delta$, i.e. $X\_0 = X \backslash \Delta$. Suppose that $(X,\Delta)$ is of log general type, i.e. $K\_X+D$ is big.
A theorem of Cadorel [Cad16... | https://mathoverflow.net/users/105103 | Curvature of varieties of log general type | No, this is not true, even for $\Delta=\emptyset$. If $X$ admits a Kähler metric with negative holomorphic bisectional curvature, then so do all its subvarieties; in particular, all its subvarieties are of general type.
However, many varieties of general type admit subvarieties with non-maximal Kodaira dimension (blo... | 6 | https://mathoverflow.net/users/5659 | 373331 | 155,924 |
https://mathoverflow.net/questions/373337 | 1 | I need this [reference](http://actamath.com/EN/abstract/abstract1243.shtml), but I couldn't find it online as a PDF. Any help please?
J. Sun, X, Zhang, *The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations*, Acta Math. Sinica (in Chinese) **48** (2005... | https://mathoverflow.net/users/102228 | Find this reference or an alternative where I can find this result | It is online and freely accessible, but the web site of the journal is somewhat crippled. This site does not work at all for me
<http://actamath.com/EN/abstract/abstract1243.shtml#>
but I managed to download it from
<http://actamath.com/EN/volumn/volumn_1555.shtml#>
(follow the pdf link and if needed rename the... | 3 | https://mathoverflow.net/users/11260 | 373339 | 155,930 |
https://mathoverflow.net/questions/373346 | 16 | For a Grothendieck topos $\mathcal{E}$, are the following assertions equivalent?
$(i)$ $\mathcal{E}$ is localic.
$(ii)$ The diagonal geometric morphism $\mathcal{E} \to \mathcal{E} \times \mathcal{E}$ is an embedding. (Here $\mathcal{E} \times \mathcal{E}$ is the product topos, not the product category.)
$(iii)... | https://mathoverflow.net/users/166281 | Toposes with only preorders of points | $(i) \Leftrightarrow (ii)$ is true and is Proposition C.2.4.14 in Peter Johnstone's Sketches of an elephant. More generally he shows that a bounded geometric morphism $f: \mathcal{E} \to \mathcal{S}$ is localic if and only if $\mathcal{E} \to \mathcal{E} \times\_{\mathcal{S}} \mathcal{E}$ is an embedding.
$(ii)$ and ... | 18 | https://mathoverflow.net/users/22131 | 373349 | 155,932 |
https://mathoverflow.net/questions/373343 | 9 | Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert\_{\mathrm{F}}=\sqrt{\sum\_{i,j} A\_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert\_{\mathrm{op}}=\sup\_{x \not = 0} \frac{\left\Vert Ax\right\Vert\_2}{\left\Vert x \right\Vert\_2}$ where the the norm in the numerator and denominator are the... | https://mathoverflow.net/users/166470 | When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold? | The term "operator Lipschitz function" is definitely not reserved to the Hilbert-Schmidt norm. On the opposite, I would say that it is mostly used for the operator norm (but not only, see for example <https://arxiv.org/abs/0904.4095> ). In particular, the survey that you are citing is using the operator norm.
It is k... | 9 | https://mathoverflow.net/users/10265 | 373358 | 155,936 |
https://mathoverflow.net/questions/373369 | 6 | Let $M$ be a connected closed complex manifold. Assume it has an antiholomorphic involution. Must it have an antiholomorphic involution with a fixed point?
| https://mathoverflow.net/users/nan | Antiholomorphic involution with a fixed point | No. There exist both non-algebraic and projective counterexamples.
1 *Non-algebraic example.* Take a flat Euclidean torus $T^4=M$ and let $Z$ be its twistor space. It has an antiholomorphic involution without fixed points which is central symmetry in all the fibres. I claim that $Z$ doesn't have an anti-holomorphic i... | 10 | https://mathoverflow.net/users/943 | 373375 | 155,941 |
https://mathoverflow.net/questions/373384 | 3 | Let $f(z\_1,z\_2,\ldots,z\_n)$ be a function on $\mathbf{C}^n$ such that for all $i$, the restriction
$$
[z\_i\mapsto f(z\_1,z\_2,\ldots,z\_n)]
$$
is a "rational function".
(**added:** to be precise here one should allow $(z\_2,z\_3,\ldots,z\_n)$ to avoid a closed exceptional variety $E\subseteq\mathbf{C}^{n-1}$, now... | https://mathoverflow.net/users/11765 | On a variation of Hartogs' separate analyticity theorem | Let us prove the desired result for $n=2$. We have
$$f(x,y)=\frac{\sum\_{i=0}^m a\_i(y)x^i}{\sum\_{i=0}^k b\_i(y)x^i}=r\_x(y),\tag{1}$$
where the $a\_i$'s and $b\_i$'s are some functions and, for each $x$, $r\_x$ is a rational function. We want to show that $f$ is a rational function. Without loss of generality (wlog),... | 3 | https://mathoverflow.net/users/36721 | 373390 | 155,945 |
https://mathoverflow.net/questions/373368 | 3 | In this [book](https://link.springer.com/book/10.1007/978-3-0348-5727-7) (proof of $4.1.3.$ Lemma. exactly), one can find this passage, that I tried to rephrase here:
>
> Let $f:I\times E\rightarrow E$ a [Pettis integrable](https://en.wikipedia.org/wiki/Pettis_integral) function, where $I:=[0,T]\subset \mathbb{R}$,... | https://mathoverflow.net/users/102228 | Family of Pettis integrals functions "uniformly approximated" by sums | As @Jochen commented, the result isn't true as originally stated. The book demands that the functions $(s\mapsto f(s,y(s)) : y\in \Omega)$ are equicontinuous, i.e. that for all $\varepsilon>0$, there exists $\delta>0$ such that for all $y\in \Omega$ and $u,v\leq t$,
$$|u-v|<\delta \implies \lVert f(u,y(u))-f(v,y(v))\rV... | 2 | https://mathoverflow.net/users/138576 | 373392 | 155,947 |
https://mathoverflow.net/questions/372518 | 19 | I am quite interested in moduli spaces for Rings and Ideals, a letter from Deligne to Bhargava is cited in Melanie Wood's thesis *Moduli spaces for Rings and Ideals* ([pdf](https://math.berkeley.edu/%7Emmwood/Publications/WoodthesisFinal.pdf)), studying the minimal free resolution of $n$ points in $\mathbb P^{n−2}$. It... | https://mathoverflow.net/users/165896 | Deligne's letter to Bhargava from March 2004 | The letter is [here](http://people.math.harvard.edu/%7Emmwood/DeligneToBhargava2004.pdf). Thanks to Will Sawin for alerting me to this request.
| 26 | https://mathoverflow.net/users/166496 | 373412 | 155,952 |
https://mathoverflow.net/questions/373397 | 1 | Let $c,d\in\mathbb{N},\varepsilon>0$ and $p$ be a prime. Question: is it true that for all $\varepsilon>0$, if $p$ is sufficiently large depending on $c,d$ and $\varepsilon$, then for any varieties $X,Y\subseteq \mathbb{F}\_{p}^{d}$ of "complexity" at most $c$, either $\vert X\cap Y\vert< \varepsilon\vert X\vert$ or $X... | https://mathoverflow.net/users/166521 | Counting the number of points in a variety over a finite field | This is true if $X$ is geometrically irreducible by the [Lang-Weil bound](https://terrytao.wordpress.com/2012/08/31/the-lang-weil-bound/), which gives us that the size of $|X \cap Y|$ is $(c(X \cap Y) + O\_c(p^{-1/2})) p^{\dim (X \cap Y)|}$ where $c(X \cap Y)$ is the number of top-dimensional components of $X \cap Y$, ... | 3 | https://mathoverflow.net/users/290 | 373427 | 155,956 |
https://mathoverflow.net/questions/373407 | 3 | Let $\mathcal J$ be an ideal sheaf on a (Noetherian) $Y$-scheme $X$, and let $\mathcal I$ be the unique primary ideal in a primary decomposition $\mathcal J$ corresponding to a minimal associated prime $x \in X$ with the closure $Z$. In this case $\mathcal O\_X/\mathcal I$ puts a canonical scheme structure on $Z$. If $... | https://mathoverflow.net/users/2234 | flatness and reduction | Your current question as stated is a little weaker than the linked original question. So I will answer both negatively by giving an example of $I$ a prime ideal and $J$ is $I$-primary inside a polynomial ring $R$ over the complex number $k$.
The point is the so-called [miracle flatness](https://en.wikipedia.org/wiki/... | 4 | https://mathoverflow.net/users/2083 | 373434 | 155,957 |
https://mathoverflow.net/questions/373426 | 1 | Is every regular local ring $R$ a filtered colimit of regular local rings which are essentially of finite type over $\mathbb{Z}$ (i.e. localizations of finitely generated rings)?
For comparison, Popescu's theorem says that under the stronger assumption that $\mathrm{Spec}\,R\to\mathrm{Spec}\,\mathbb{Z}$ is a [regular... | https://mathoverflow.net/users/86006 | Is every regular local ring a filtered colimit of essentially finitely generated regular local rings? | It seems it was proved recently by Popescu himself, see [On a question of Swan (with an appendix by Kęstutis Česnavičius)](http://content.algebraicgeometry.nl/2019-6/2019-6-030.pdf).
| 3 | https://mathoverflow.net/users/15505 | 373435 | 155,958 |
https://mathoverflow.net/questions/373170 | 8 | Consider the following: Suppose that $K$ is a perfect field, $V$ and $W$ are integral $K$-varieties, $V \to W$ is a dominant morphism, and the function field of $V$ is a separable extension of the function field of $W$. Then there is a dense open subvariety $U$ of $V$ such that $U \to W$ is smooth.
I would like a ref... | https://mathoverflow.net/users/152899 | Reference request for generic smoothness | Well, thanks everyone, but in the end I found a good reference. I will post it here as this might be useful for someone else.
The following is (a special case of) Corollary 5.4.3 in Mumford and Oda's Algebraic Geometry II: Suppose that $V$ and $W$ are integral $K$-varieties, $W$ is regular, and $f : V \to W$ is a dom... | 1 | https://mathoverflow.net/users/152899 | 373438 | 155,959 |
https://mathoverflow.net/questions/373425 | 2 | By coincidence I noticed that the following two matrices yield the same eigenvalues
\begin{pmatrix} A & B \\ B^\* & A \end{pmatrix} and \begin{pmatrix} 0& A+b1\_{\mathbb C^{2 \times 2}} \\ A+b^\* 1\_{\mathbb C^{2 \times 2}} & 0 \end{pmatrix}
where $A = \begin{pmatrix} 0 & a \\ a^\* & 0 \end{pmatrix}$ and $B=\begin{pm... | https://mathoverflow.net/users/119875 | Two equivalent matrices? | Define the unitary matrix
$$U=\left(
\begin{array}{cccc}
i e^{i \pi /4} & 0 & 0 & 0 \\
0 & 0 & 0 & -e^{-i \pi /4} \\
0 & 0 & i e^{i \pi /4} & 0 \\
0 & -e^{-i \pi /4} & 0 & 0 \\
\end{array}
\right),$$
then
$$U\begin{pmatrix} A & B \\ B^\* & A \end{pmatrix}U^{-1}=\begin{pmatrix} 0 & A+b\mathbb{1} \\ A+b^\ast\mathbb{1... | 3 | https://mathoverflow.net/users/11260 | 373440 | 155,960 |
https://mathoverflow.net/questions/373430 | 3 | Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have
$$t \mapsto f(t,x) \text{ is measurable}$$
$$x \mapsto f(t,x) \text{ is continuous}.$$
My question is: given $w \in L^1(0,T;X)$, is $t \mapsto f(t,w(t))$ measurable ... | https://mathoverflow.net/users/58845 | Measurability of superposition operator with non-separable Banach space | [Copying here the content of the comments, for the question not to appear as unanswered]
If by $L^1(0,T;X)$ you mean (as it is standard) the space of Bochner-measurable functions, then by definition any $w \in L^1(0,T;X)$ takes values in a separable subspace of $X$, so the general case follows from the separable case... | 5 | https://mathoverflow.net/users/10265 | 373442 | 155,961 |
https://mathoverflow.net/questions/373441 | 49 | I received an email today about the award of the [2020 Nobel Prize in Physics](https://www.aip.org/science-news/nobel2020) to **Roger Penrose**, **Reinhard Genzel** and **Andrea Ghez**. [Roger Penrose](https://en.wikipedia.org/wiki/Roger_Penrose) receives one-half of the prize "for the discovery that black hole formati... | https://mathoverflow.net/users/51189 | What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize? | It seems (as mentioned by Sam Hopkins above) that the **Singularity Theorem** is the official reason for the Nobel Award.
But that is by no means the only (and perhaps not even the most important) contribution of Sir Roger Penrose to mathematical physics ( not to mention his works as a geometer and his research on ti... | 34 | https://mathoverflow.net/users/15293 | 373446 | 155,962 |
https://mathoverflow.net/questions/373255 | 3 | While investigating non-periodic RNG's (random number generators) for irrational numbers, I came up with a version that actually produces pseudo-random words consisting of $N$ bits, where $N$ is typically a large prime number. Here I explain my RNG. My question is whether it suffers from the same problems as [Xorshift]... | https://mathoverflow.net/users/140356 | Question about a new pseudo-random number generator | Floating point division varies across platforms especially if the language and the system supports hardware accelerated floating point arithmetic. It is risky to use it in an encryption algorithm standard. You can find a better more secure method if you use a fixed size seed starting at digit m of square root of 2 over... | 1 | https://mathoverflow.net/users/166490 | 373458 | 155,965 |
https://mathoverflow.net/questions/363292 | 19 | For any groupoid, it's groupoid cardinality is the sum of the reciprocals of the automorphism groups over the isomorphism classes. Let us consider the category of vector spaces over a finite field $\mathbb F\_q$ with only invertible morphisms allowed.
Then, the size of the automorphism groups are $g\_n= \prod\_{i=1}^... | https://mathoverflow.net/users/58001 | What is the groupoid cardinality of the category of vector spaces over a finite field? | Upon substituting $x=\frac{1}{q}$ we obtain
$$\sum\_{n\geq 0}\frac{1}{|\mathrm{GL}\_n(\mathbb F\_q)|}=\sum\_{n\geq 0}\frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}$$
and this evaluates to the product $\prod\_{i\geq 1}\frac{1}{(1-x^{5i-4})(1-x^{5i-1})}$ by the first [Rogers-Ramanujan identity](https://en.wikipedia.org/wiki/R... | 16 | https://mathoverflow.net/users/2384 | 373461 | 155,967 |
https://mathoverflow.net/questions/373469 | 3 | Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Can we find $l>0$ such that $$\bigcap\_{-l\leq t \leq l}t+E$$ is a set of positive Lebesgue measure?
Notation: $t+E=\{t+e|e\in E\}$
| https://mathoverflow.net/users/166207 | To show a set is a set of positive Lebesgue measure in $ \mathbb{R}$ | No. Every set $E$ without interior points (e.g. the complements of the rationals) has the property that $$\bigcap\_{|t|<\varepsilon}(t+E)=\emptyset$$ for every $\varepsilon>0$. Indeed, for every $x\in E$, there is $t$ with $|t|<\varepsilon$ and $x-t\notin E$, hence $x\notin\bigcap\_{|t|<\varepsilon}(t+E)$.
| 7 | https://mathoverflow.net/users/165275 | 373473 | 155,969 |
https://mathoverflow.net/questions/373468 | 2 | Let's consider the usual Hilbert transform $H$ defined as
$$Hf = P.V. (\frac{1}{x}\*f).$$
A well-known unique continuation principle states that if $Hf = f =0$ on some **interval** $I$, then $f \equiv 0$. My question is whether the argument is still true if we replace the interval $I$ with a **point** $x\_0$. More spec... | https://mathoverflow.net/users/114951 | Unique continuation of the Hilbert transform | No. Let $$u(z) = \exp(-(-iz)^{1/2}-(-iz)^{-1/2})$$ for $z$ in the closed upper complex half-plane, with the principal branch of the complex power. Then $u$ is a bounded holomorphic function in the open half-plane, continuous up to the boundary, and vanishing sufficiently fast at complex infinity. Thus, the Hilbert tran... | 3 | https://mathoverflow.net/users/108637 | 373479 | 155,971 |
https://mathoverflow.net/questions/373472 | 0 | Let $A, B$ be $\mathbb{C}$-algebras, which are also integral domains. Suppose there is an injective ring homomorphism $f:A \to B$. Assume further than $f$ is a finite morphism in the sense that $f$ induces a finite $A$-module structure on $B$. Let $M$ be a finitely generated $A$-module. Let $m \in M$ such that there ex... | https://mathoverflow.net/users/45397 | Naive question on tensor product | A trivial counterexample is $m=m'=0$. Perhaps more interesting is a situation where $b$ does not belong to the image of $A$, but some multiple or power of $b$ does. Say $M = B = \mathbb{C}[x]$ and $A = \mathbb{C}[x^2,x^3]$. Take $m' = x^3$ and $b = x$. Then
$$ m' \otimes b = x^3 \otimes x = x \otimes x^3 = x^4 \otime... | 5 | https://mathoverflow.net/users/88133 | 373486 | 155,973 |
https://mathoverflow.net/questions/373452 | 4 | There are well-known techniques for inverting convolutions over the whole or half real line with Fourier and Laplace transformations, but on the face of it they can't be applied to an integral equation of the form:
$$\int\_{-c}^c f(x-t)\rho(t) dt = g(x)\\ -c \le x \le c$$
where the unknown function to be determined... | https://mathoverflow.net/users/23829 | Inverting convolutions over finite intervals | A modification of the Wiener-Hopf method for this type of problems is described in [Convolution equations on finite intervals and factorization of matrix functions](https://link.springer.com/article/10.1007%2FBF01202095) and in [Finite interval convolution operators with transmission property.](https://idp.springer.com... | 3 | https://mathoverflow.net/users/11260 | 373491 | 155,977 |
https://mathoverflow.net/questions/373460 | 1 | Let $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m $. I have the following two problems:
P.1.
\begin{equation}
\underset{x\in\mathbb{R}^n}{\text{minimize}} \| Ax-b \|\_1 \\
\text{s.t. } \| x \|\_{\infty} \leq 1
\end{equation}
P.2.
\begin{equation}
\underset{x\in\mathbb{R}^n}{\text{minimize}} \| x \|\_1... | https://mathoverflow.net/users/166574 | How to minimize l1-norm constrained by "infinity norm" | This work like this: The $\infty$-norm constraints are straigtforward. In the first problem you write
$$
-1 \leq x\_i \leq 1
$$
or, more explicitely
$$
x\_i\leq 1\\-x\_i\leq 1.
$$
One could even just write $x\leq 1$ and $-x\leq 1$ with the all-ones vector.
In the second problem you get
$$
Ax\leq 1+b\\
-Ax\leq 1-b.
$$
F... | 2 | https://mathoverflow.net/users/9652 | 373496 | 155,978 |
https://mathoverflow.net/questions/373500 | 12 | For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?
For example it is true for $S^2$.
| https://mathoverflow.net/users/nan | Unique almost complex structure up to diffeomorphism | You were lucky to find the only possible example. If you take any manifold of dimension $\ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrable outside of it. Since complex $n$-balls have more than one holomorphic structure, we are done. And all surfaces apart ... | 19 | https://mathoverflow.net/users/943 | 373501 | 155,979 |
https://mathoverflow.net/questions/373503 | 17 | For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general information about the map(s) $\Phi\_\*$ induced on the homotopy groups of these spaces. I’m mainly interested in the simply c... | https://mathoverflow.net/users/3460 | Homotopy groups of Diff(X) and Homeo(X) | No, the statement about the kernel and cokernel being finite is not true.
For a closed $d$-manifold, $d \neq 4$, smoothing theory identifies the homotopy fibre of
$$B\mathrm{Diff}(M) \longrightarrow B\mathrm{Homeo}(M)$$
with (certain path components of) the space of sections of a bundle
$$Top(d)/O(d) \longrightarrow ... | 16 | https://mathoverflow.net/users/318 | 373506 | 155,981 |
https://mathoverflow.net/questions/373481 | 9 | I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance generating its topology, and endow the set of self-homeomorphisms of $S$ with the distance max(uniform distance between two ... | https://mathoverflow.net/users/58307 | Group of surface homeomorphisms is locally path-connected | This is a particular case of Corollary 1.1 of Edwards, Robert D.; Kirby, Robion C. Deformations of spaces of imbeddings. Ann. of Math. (2) 93 (1971), 63--88. MR0283802, which says that the group of homeomorphisms of any compact manifold is locally contractible.
| 12 | https://mathoverflow.net/users/798 | 373508 | 155,982 |
https://mathoverflow.net/questions/373504 | 6 | Bernhard Böhmler (who is also on MO) and myself had the following idea:
Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides the order of $G$.
Let $A=kG$ be the group algebra of $G$ and $M$ the direct sum of indecomposable all trivial source m... | https://mathoverflow.net/users/61949 | Endomorphism ring of trivial source modules for abelian p-groups | Representations of $B$ (or at least an equivalent category) are studied in the literature under the name of "cohomological Mackey functors".
Theorem 1.1 of
*Bouc, Serge; Stancu, Radu; Webb, Peter*, [**On the projective dimensions of Mackey functors**](http://dx.doi.org/10.1007/s10468-017-9695-y), Algebr. Represent.... | 9 | https://mathoverflow.net/users/22989 | 373509 | 155,983 |
https://mathoverflow.net/questions/372992 | 4 | Let $H$ be a 3-partite 3-uniform hypergraph with minimum vertex cover number $\tau(H)$ (i.e. $\tau(H)=\min\{|Q|: Q\subseteq V(H), e\cap Q\neq \emptyset \text{ for all } e\in E(H)\}$).
**Question:** Is $\tau(H)$ at most 3 times the *matching width* of $H$?
Given a matching $M$ in $H$, let $\rho(M)$ be the minimum si... | https://mathoverflow.net/users/17798 | Relationship between minimum vertex cover and matching width | Your suspicion is correct. The following hypergraph $H$ provides a negative answer to your question. Let $V=\{0,1,\dots, 11\}$. Then $V=V\_0\cup V\_1\cup V\_2$, where $V\_0=\{0,1,2,3\}$, $V\_1=\{4,5,6,7\}$, and $V\_2=\{8,9,10,11\}$. Let $E(H)$ is a family of all three-element subsets $e$ of $V$, such that $|e\cap V\_i|... | 3 | https://mathoverflow.net/users/43954 | 373536 | 155,991 |
https://mathoverflow.net/questions/373420 | 2 | I asked this [question in MathStackExchange](https://math.stackexchange.com/questions/3842894/a-problem-about-an-unramified-prime-in-a-galois-extension), but I didn't receive any answer.
Let $K/\mathbb{Q}$ be a Galois extension of degree $n$, and denote its ring of integers by $\mathcal{O}\_K$. Let $\mathfrak{p}$ be ... | https://mathoverflow.net/users/166540 | A problem about an unramified prime in a Galois extension | A set of representatives for $\mathcal{O}$ modulo $\mathfrak{p}$ is given by
$$S:=\{a\_0+a\_1x+\dotsb+a\_{f-1}x^{f-1}\ :\ a\_0,a\_1,\dotsc,a\_{f-1}\in\{0,1,\dotsc,p-1\}\}.$$
As $P(x)$ lies in $\mathfrak{p}\setminus\mathfrak{p}^2$, a set of representatives for $\mathfrak{p}$ modulo $\mathfrak{p}^2$ is given by
$$S\cdot ... | 1 | https://mathoverflow.net/users/11919 | 373541 | 155,993 |
https://mathoverflow.net/questions/373535 | 11 | I understand that for any nonempty set $S$ of characteristics, there exists a PID $R$ such that the set of characteristics of residue fields of $R$ (i.e. quotients by of $R$ by maximal ideals -- I'm not including the residue field at the generic point. Thanks to Steven Landsburg for pointing out this terminological amb... | https://mathoverflow.net/users/2362 | Example of a PID with a residue field of finite characteristic and a residue field of characteristic 0? | You can take the ring of fractions $\frac{a}{b}$ with $a,b \in \mathbb Z[x]$, where $b$ is nonzero mod $p$ and nonzero mod $px-1$.
Given any polynomial $a$, we can remove all factors of $p$ and remove all factors of $px-1$, obtaining a polynomial that is nonzero mod $p$ and nonzero mod $px-1$. So every polynomial is ... | 21 | https://mathoverflow.net/users/18060 | 373559 | 156,002 |
https://mathoverflow.net/questions/373538 | 14 | Let $M$ be a smooth orientable compact connected (with boundary) manifold of dimension $4$. In addition $M$ is assumed to be aspherical and acyclic.
>
> **Question:** is there a "classification" of such manifolds? Or can they be classified in any effective way?
>
>
>
| https://mathoverflow.net/users/17895 | Very particular kind of 4-manifolds. Classification | There are plenty of such manifolds, but as Danny indicates in his answer, there is not a known classification.
Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi\_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skel... | 18 | https://mathoverflow.net/users/1345 | 373570 | 156,008 |
https://mathoverflow.net/questions/373585 | 6 | Let $C$ be a pointed $\infty$-category which admits finite limits.
---
Let $Sp(C)$ denote the $\infty$ category of *spectrum objects*. One way to define, i.e. [1.4.2.24](https://www.math.ias.edu/%7Elurie/papers/HA.pdf), is by taking the homotopy limit in $Cat\_\infty$, the $\infty$-category of categories.
$$Sp(C)... | https://mathoverflow.net/users/139900 | When does the loop functor $\Omega^\infty:Sp(C) \rightarrow C$ commute with filtered colimits? | The result is true, more generally, if you take a class of diagrams $\mathcal K$ and the $\infty$-category $\widehat{Cat\_\infty}^\mathcal K$ of $\infty$-categories that have all $\mathcal K$-indexed colimits, and functors between them that preserve those, then the forgetful functor $\widehat{Cat\_\infty}^\mathcal K\to... | 5 | https://mathoverflow.net/users/102343 | 373588 | 156,012 |
https://mathoverflow.net/questions/373582 | -1 | This question comes from some reasoning I made myself about a "joke block chain" where every new block is labeled with a triplet **<S, P, N>** where where *S = sum of the N transactions so far* and *P = product of the N transactions*.
So let's say we start with:
```
<5, 5, 1> => [5]
<- include a transaction of 3
<... | https://mathoverflow.net/users/166641 | Are <sum, product, N> triplets unique and hard to solve? | If I understand the question, then the triple $(30,840,3)$ could come from $6+10+14=30$, $6\times10\times14=840$ or from $7+8+15=30$, $7\times8\times15=840$.
| 6 | https://mathoverflow.net/users/158000 | 373589 | 156,013 |
https://mathoverflow.net/questions/250487 | 3 | Is there any characterization of the non-compact connected Lie groups that possess faithful finite-dimensional unitary representations?
| https://mathoverflow.net/users/50457 | Faithful finite-dimensional unitary representations | **Proposition.** *Equivalences ($G$ connected Lie group):*
* (i) $G$ has a faithful finite-dimensional continuous unitary representation;
* (ii) $G$ is locally isomorphic to some compact Lie group;
* (iii) $G$ is direct product of some Euclidean group (=$\mathbf{R}^d$ for some $d$) with a compact Lie group.
$\bulle... | 3 | https://mathoverflow.net/users/14094 | 373591 | 156,015 |
https://mathoverflow.net/questions/373475 | 2 | Let $e\_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l\_2(\mathbb{N})$.
Let $h(n) = J\_2(n)$ be the second Jordan totient function.
Define:
$$\phi(n) = \frac{1}{n} \sum\_{d|n}\sqrt{h(d)} e\_d$$.
Then we have:
$$ \left < \phi(a),\phi(b) \right > = \frac{\gcd(a,b)^2}{ab}=:k(a,b)$$
The vectors $\... | https://mathoverflow.net/users/165920 | A geometric approach to the odd perfect number problem? | We have for all $n$:
$$|\hat{\phi}(n)|^2 = |\sum\_{d|n} \phi(d)|^2 = \left < \sum\_{d|n} \phi(d),\sum\_{d|n} \phi(d)\right >$$
$$= \sum\_{d|n} |\phi(d)|^2 + 2 \sum\_{d\_1 < d\_2,d\_1,d\_2|n} \left < \phi(d\_1),\phi(d\_2)\right >$$
$$= \tau(n) + 2 \sum\_{d\_1 < d\_2} \frac{\gcd(d\_1,d\_2)^2}{d\_1 d\_2}$$
$$\ge \tau(n)... | 2 | https://mathoverflow.net/users/165920 | 373592 | 156,016 |
https://mathoverflow.net/questions/373515 | 4 | If I have a smooth positive scalar function $h$ defined on a 2-dimensional manifold $M$, then $h:M\rightarrow (0, \infty)$, where the metric of $M$ is $g=\frac{dx^2+dy^2}{y^2}$.
$h$ must satisfy the following $|\nabla h|^2=\frac{(h+1)^2}{2}$.
Considering that the gradient of a smooth function on manifold is $\nabla... | https://mathoverflow.net/users/111304 | Gradient of a function defined on a Riemannian-manifold | The requirement that $h$ be positive coupled with the assumption that the metric on $M$ be complete implies that there is no solution. It doesn't really matter what the metric is as long as it's complete. Here is why:
The equation $|\nabla h|^2 = \tfrac12 (h+1)^2$ implies that, if we set $f = \log (h+1)$, then we hav... | 7 | https://mathoverflow.net/users/13972 | 373596 | 156,019 |
https://mathoverflow.net/questions/373523 | 4 | Let $F$ a Fréchet space.
This means that $F$ is a complete Hausdorff topological space whose topology can be generated by an increasing family of seminorms $\{ p\_{n} \}\_{n \in \mathbb{N}}$.
Let's denote by $F\_{n}$ the completion of of $F$ with respect to $p\_{n}$.
Now, $F$ is *nuclear* if the family $\{ p\_{n} \}\... | https://mathoverflow.net/users/99745 | If $F$ is a countably normed, nuclear Fréchet space, can I then find a fundamental system which exhibits both of these properties at once? | **Prolog.**
As the arguments below are somewhat technical and probably not too interesting for many readers, I would like to point out that such problems can be quite subtle. The crucial problem (which could also be interesting for Banach spacers) is that the unique extension of a linear *injection* between normed spac... | 6 | https://mathoverflow.net/users/21051 | 373598 | 156,021 |
https://mathoverflow.net/questions/373580 | 5 | Fix an integer $i\geq 3$ and a finite abelian group $G$.
Is there a connected closed Kähler manifold $M$ such that $H^i(M, \mathbb{Z})\approx \mathbb{Z}^n\oplus G$ for some integer $n\geq 0$?
| https://mathoverflow.net/users/nan | Arbitrary torsion in cohomology of Kähler manifolds | The answer is positive and can be deduced from Proposition 15 of "Sur la topologie des varietes algebriques en characteristique p" by Serre. According to this proposition for any finite group $G$ there exists a complete intersection $X$ on which $G$ is acting freely. Set $Y=X/G$. Then $\pi\_1(Y)=G$. Let now $G$ be your... | 9 | https://mathoverflow.net/users/943 | 373599 | 156,022 |
https://mathoverflow.net/questions/359792 | 9 | We refer to the book *Tensor categories* by Etingof-Gelaki-Nikshych-Ostrik ([MR3242743](https://mathscinet.ams.org/mathscinet-getitem?mr=3242743)) for the notion of (unitary) fusion category. Two fusion categories are *Grothendieck equivalent* if they have the same fusion ring.
**Question**: Is there a fusion catego... | https://mathoverflow.net/users/34538 | Is there a fusion category not Grothendieck equivalent to a unitary one? | Yes, according to Andrew Schopieray. He just provided a categorifiable fusion ring, of rank 6 and multiplicity 2, without pseudounitary categorification (so without unitary categorification), in the following preprint called *Non-pseudounitary fusion*.
<https://arxiv.org/abs/2010.02958>
| 7 | https://mathoverflow.net/users/34538 | 373601 | 156,023 |
https://mathoverflow.net/questions/373553 | 4 | I'd like to know how to show $$\min\_{\Vert x\Vert\_2=1=\Vert y\Vert\_2}\left(\sum\_{k=1}^nx\_ky\_k\right)^2-\sum\_{k=1}^nx\_k^2y\_k^2\geq -1/2.$$
The inequality is discussed in a previous post [Minimum of squared sum minus sum of squares](https://mathoverflow.net/questions/206615/minimum-of-squared-sum-minus-sum-of-... | https://mathoverflow.net/users/166627 | Minimising the squared sum minus the sum of squares | It should oftentimes be the case that, analyzing a "thoughtless" Lagrange multiplier solution, one finds a more elegant, "clever" solution. At least, this is the case here. Analyzing the previous Lagrange multiplier solution, one can obtain the following.
We need to show that
$$\sum x\_j^2 y\_j^2\le1/2+\Big(\sum x\_j... | 6 | https://mathoverflow.net/users/36721 | 373604 | 156,024 |
https://mathoverflow.net/questions/373381 | 5 | In his intro to ( Skolem 1923a), Van Heijenoort (From Frege to Godel, p. 509) describes Skolem as giving “an alternative to the axiomatic approach” to proving a first-order formula. This is referring to the effective procedure Skolem gives for checking whether or not a first-order formula *U* has a solution of level n.... | https://mathoverflow.net/users/116705 | Skolem's method for checking truth-value assignments - a "cut-free proof procedure" for first-order logic? | The answer to the question can be found in Section I (especially p.11) of [this source](https://arxiv.org/abs/1904.10540) (it is a newly typeset version of Joseph E. Quinsey's 1980 Oxford doctoral thesis *Some Problems in Logic: APPLICATIONS OF KRIPKE’S NOTION OF FULFILMENT*).
| 5 | https://mathoverflow.net/users/9269 | 373614 | 156,028 |
https://mathoverflow.net/questions/373621 | 7 | A Betti sequence is a map $\mathbb{Z}\_{\geq 0}\to \mathbb{Z}\_{\geq 0}$.
A Betti sequence $b$ is realizable if there is a connected closed Kähler manifold $M$ such that $b(k)=b\_k(M)$.
A Hodge diamond is a map $\mathbb{Z}\_{\geq 0}\times \mathbb{Z}\_{\geq 0}\to \mathbb{Z}\_{\geq 0}$. To any Hodge diamond $h$ we as... | https://mathoverflow.net/users/nan | Inverse Hodge and inverse Betti problems for Kähler manifolds | The Hodge diamond \begin{array}{ccccc}&&1&&\\&0&&0&\\a&&1&&a\\&0&&0&\\&&1&&\end{array} is naively realisable.
Suppose $M$ is a compact Kähler surface with the given Hodge diamond with $a \geq 2$. As $h^{2,0}(M) > 1$, the Kodaira dimension of $M$ is either $1$ or $2$. Note that $$c\_1(M)^2 = 2\chi(M) + 3\sigma(M) = 2(... | 11 | https://mathoverflow.net/users/21564 | 373631 | 156,032 |
https://mathoverflow.net/questions/373609 | 4 | I'd like to know how to prove
$$\min\_{\Vert x\Vert\_2=1=\Vert y\Vert\_2}\left|\sum\_{k=1}^nx^\*\_ky\_k\right|^2-\sum\_{k=1}^n|x\_k|^2|y\_k|^2\geq -1/2$$
for $x,y\in\mathbb{C}^n$ with $\Vert x\Vert\_2=\Vert y\Vert\_2=1$.
This is a generalisation from $\mathbb{R}^n$ to $\mathbb{C}^n$ of the inequality which was prov... | https://mathoverflow.net/users/166627 | Generalisation of the squared sum minus the sum of squares inequality | Replacing $x\_k^\*$ by $x\_k$ and noting that $|x\_k^\*|=|x\_k|$, we see that the problem is to show that
$$|s|^2-\sum\_{k=1}^n|x\_k|^2|y\_k|^2\ge-1/2$$
for $x,y\in\mathbb{C}^n$ with $\|x\|\_2=\|y\|\_2=1$, where
$$s:=\sum\_{k=1}^n v\_k,\quad v\_k:=x\_ky\_k.$$
For any fixed values of the $|x\_j|$'s and $|y\_k|$'s (so ... | 3 | https://mathoverflow.net/users/36721 | 373634 | 156,033 |
https://mathoverflow.net/questions/373623 | 9 | Suppose $A$ and $B$ are $E\_{\infty}$ rings, then $\mathrm{Mod}(A)$ and $\mathrm{Mod}(B)$ are $E\_{\infty}$ monoidal categories (left modules over those rings). We can ask about $E\_n$ colimit-preserving morphisms between these two categories. How do we characterize them? If $n=\infty$ we just have maps from $A$ to $B$... | https://mathoverflow.net/users/136287 | How to characterize $E_n$ morphisms from $\mathrm{Mod}(A)$ to $\mathrm{Mod}(B)$? | By Corollary HA.4.8.5.20, the functor from $\mathbb{E}\_{n+1}$-algebras to $\mathbb{E}\_n$-monoidal categories and colimit-preserving, $\mathbb{E}\_n$-monoidal functors is fully faithful. (Notice that the condition of being "linear over Sp" is automatic from exactness). So, indeed, you'd get just $\mathbb{E}\_{n+1}$-al... | 9 | https://mathoverflow.net/users/6936 | 373644 | 156,035 |
https://mathoverflow.net/questions/373646 | 5 | I am looking for a function with the following property:
Let $v\_1,v\_2$ be two linearly independent vectors in $\mathbb{R}^2.$
I am given a smooth function $g:(0,1) \rightarrow (0,\infty).$
I am trying to understand if there exists a smooth (non-constant) function $f:(0,1) \times \mathbb R^2 \rightarrow \mathbb ... | https://mathoverflow.net/users/119875 | Does such a function exist? | Define $f(t,x\mathbf v\_1+y\mathbf v\_2)=e^{2\pi i(x/g(t)+yg(t))}$.
Then $f(t,\mathbf x+j\mathbf v\_1)=e^{2\pi ij/g(t)}f(t,\mathbf x)$ and similarly $f(t,\mathbf y+k\mathbf v\_2)=e^{2\pi ikg(t)}f(t,\mathbf x)$.
Suppose $t$ is such that $g(t)=\frac nm$ (in lowest terms). Then $f(t,\mathbf x+j\mathbf v\_1)=e^{2\pi ijm/... | 6 | https://mathoverflow.net/users/11054 | 373658 | 156,039 |
https://mathoverflow.net/questions/373514 | 2 | Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.
Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-module $kG^{reg}$.
Then $M$ corresponds to a right ideal of $kG$.
**Question:**
>
> Is there a MAGMA command / pr... | https://mathoverflow.net/users/12826 | MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra | I am not sure if I understand completely what you are trying to do, but I get the impression that the heart of the problem is that you are given $kH$-module homomorphism $M \to N$, and you want to compute the induce homomorphism $M\_H^G \to N\_H^G$. I think the following code does that.
```
InducedHom := function(ph... | 3 | https://mathoverflow.net/users/35840 | 373671 | 156,040 |
https://mathoverflow.net/questions/373677 | 1 | Let $X\_1, ..., X\_n \quad i.i.d \sim U[a,b]$ Then $Z\_i$ defined as:
$$
Z\_i = \frac{X\_{(i)}- X\_{(1)}}{X\_{(n)} - X\_{(1)}}, \quad i = \overline{2,n-1},
$$
where $X\_{(k)}$ is the $k$-th order statistic.
I wonder if there is a simple way to find its distribution. I am confused because $X\_{(k)}$ are not independen... | https://mathoverflow.net/users/157203 | How can one calculate distribution of ratio of differences of order statistics of uniform distribution? | As long as $a<b$, the distribution of $Z:=Z\_i$ does not depend on $a,b$. This follows because for $Y\_k:=(X\_k-a)/(b-a)$ we have $Y\_k\overset{iid}\sim U[0,1]$ and
$Z\_i=\dfrac{Y\_{(i)}-Y\_{(1)}}{Y\_{(n)}-Y\_{(1)}}$. So, without loss of generality, $a=0$ and $b=1$.
The pdf of $Z$ can be found using the transformatio... | 0 | https://mathoverflow.net/users/36721 | 373687 | 156,047 |
https://mathoverflow.net/questions/373365 | 4 | We say that a simple, undirected graph $G=(V,E)$ is *separating* if for all $x\neq y\in V$ there are $e\_x,e\_y\in E$ such that $x\in e\_x$ and $y\in e\_y$, and $e\_x\cap e\_y = \varnothing$. We say $G$ is *minimally separating* if it is separating and for all $E'\subseteq E$ with $E'\neq E$ we have that $(V,E')$ is no... | https://mathoverflow.net/users/8628 | Minimally separating graphs | Yes, every separating graph has a spanning subgraph which is minimally separating. The proof uses the same idea as the [Banakh–Petrov theorem](https://mathoverflow.net/questions/324504/graphs-with-minimum-degree-deltag-lt-aleph-0).
Let $G=(V,E)$ be a separating graph. I will write $N(x)$ and $d(x)=|N(x)|$ for the nei... | 3 | https://mathoverflow.net/users/43266 | 373691 | 156,049 |
https://mathoverflow.net/questions/373656 | 3 | The following is a purely combinatorial problem that I came across in the course of research in non-classical logic. It sounds to me like the kind of question that someone may very well have considered at some point, but not being a very combinatorially minded person myself, I have not managed to find it in the literat... | https://mathoverflow.net/users/145176 | Transversals and almost transversals of a finite family of sets | Here is a family of counterexamples with arbitrarily large $l$ in the case $m=n=2$:
$$T\_1 = {\*}111111\cdots1$$
$$T\_2 = 0{\*}11111\cdots1$$
$$T\_3 = 00{\*}1111\cdots1$$
$$T\_4 = 000{\*}111\cdots1$$
$$T\_5 = 0000{\*}11\cdots1$$
$$\cdots$$
$$T\_{l-1} = 00000\cdots0{\*}1$$
$$T\_{l} = 00000\cdots00{\*}$$
i.e. $T\_i$ ... | 3 | https://mathoverflow.net/users/160416 | 373699 | 156,050 |
https://mathoverflow.net/questions/373633 | 7 | I first asked this on StackExchange, but no dice; so apologies in advance if this question really belongs there.
Suppose a functor $F \colon \mathcal{C} \to \mathcal{D}$ between two model categories (i) sends cofibrant objects to cofibrant objects and (ii) sends weak equivalences between cofibrant objects to weak equ... | https://mathoverflow.net/users/156259 | Terminology: "left homotopical"? | Thanks to Zhen Lin and Denis-Charles Cisinski for their comments, as well as "jgon" who commented on the Stack Exchange post. It seems there is no standard terminology for such a functor.
There is, however, a related notion. A functor that preserves weak equivalences between cofibrant objects is said to be *left defo... | 1 | https://mathoverflow.net/users/156259 | 373702 | 156,051 |
https://mathoverflow.net/questions/373695 | 5 | Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let also $V\_1, V\_2, V\_3$ finite-dimensional simple modules over $\mathcal{H}$ and $Q$ be a simple quotient of $V\_1\otimes V\_2\otimes V\_3$. Is it possible to show that one of the following statements is true? Is there any counterexample?
i) $Q$ is a quotient... | https://mathoverflow.net/users/137269 | Simple quotients of a triple tensor product | Both of these statements are true (at least if $H$ is semisimple). It suffices to prove the first one. By hypothesis there is a nonzero map $V\_1 \otimes V\_2 \otimes V\_3 \to Q$. It dualizes to a nonzero map $V\_1 \otimes V\_2 \to Q \otimes V\_3^{\ast}$ (I don't know if I need to distinguish between left and right dua... | 7 | https://mathoverflow.net/users/290 | 373707 | 156,052 |
https://mathoverflow.net/questions/373711 | 7 | If $N$ is a normal subgroup of a group $G$ such that $G/N= \mathbb{Z}$. Suppose that the classifying space of $G$ is a finite CW-complex of dimension $n$. Does it follow that the classifying space of $N$ is a finite CW-complex of dimension $n-1$ ?
| https://mathoverflow.net/users/136909 | Dimension of classifying space of a group | No. Baumslag-Solitar groups of type $(1, n)$, which are semidirect products $\Bbb Z \ltimes \Bbb Z[1/n]$ have finite two-dimensional classifying spaces, but $\Bbb Z[1/n]$ clearly cannot have one-dimensional classifying space.
| 9 | https://mathoverflow.net/users/81055 | 373713 | 156,055 |
https://mathoverflow.net/questions/373704 | 6 | **Question 1:** Let $\mathcal A$ be an abelian group. Does there exist an inverse system $(A^n)\_{n \in \mathbb N} = (\cdots \to A^n \to A^{n-1} \to \cdots \to A^0)$ such that $\varprojlim^1 A^\bullet \cong \mathcal A$? If not, can we characterize the abelian groups which are $\varprojlim^1$ groups or at least say anyt... | https://mathoverflow.net/users/2362 | Which abelian groups are $\varprojlim^1$ groups? | Abelian group $A$ is cotorsion if $\rm{Ext}(F, A) = 0$ for every flat $F$, or, equivalently, $\rm{Ext}(\Bbb Q, A) = 0$
Every $\varprojlim^1$ of an inverse system of abelian group is cotorsion, and, conversely, every cotorsion group is a $\varprojlim^1$. Proof can be found in *Warfield, Huber. On the values of the fun... | 14 | https://mathoverflow.net/users/81055 | 373715 | 156,056 |
https://mathoverflow.net/questions/373682 | 6 | Let $X$ be a Banach space (over $\mathbb C$), and let $\mathcal L(X)$ be its algebra of bounded linear operators.
Let $U\subset \mathbb C^N$ be an open subset, and $f:U\to \mathcal L(X)$ a function that is locally bounded (with respect to the operator norm on $\mathcal L(X)$), and holomorphic when $\mathcal L(X)$ is ... | https://mathoverflow.net/users/5690 | holomorphy in infinite dimensions (holomorphic families of operators) | In addition to the information given by user bathalf15320, I think that a bit more information on the Banach space case could be useful:
Here is a very general theorem about vector valued functions:
**Theorem 1.** Let $Y$ be a complex Banach space and let $f: U \to Y$ be locallly bounded. Let $W \subseteq Y'$ be a ... | 3 | https://mathoverflow.net/users/102946 | 373718 | 156,057 |
https://mathoverflow.net/questions/373685 | 7 | By the cobordism hypothesis, there is an $O(2)$-action on the maximal subgroupoid $\hat{\mathcal{C}}$ of the subcategory of fully dualizable objects in a bicategory $\mathcal{C}$. The $SO(2)$-part of this action can equivalently be described by a natural transformation $id\_{\hat{\mathcal{C}}} \to id\_{\hat{\mathcal{C}... | https://mathoverflow.net/users/122457 | An explicit expression for the naturality of the Serre automorphism in the bicategory of algebras | We will use the fact that $M$ is invertible. Let ${}\_BN\_A$ be an inverse to $M$. Thus we have isomorphisms
$${}\_AM \otimes\_B N\_A \cong {}\_AA\_A$$
and
$${}\_BN \otimes\_A M\_B \cong {}\_BB\_B$$
If we make this data part of an adjoint equivalence (as we should, and as I will assume) then the construction I am about... | 4 | https://mathoverflow.net/users/184 | 373719 | 156,058 |
https://mathoverflow.net/questions/373697 | 6 | I'm trying to get a grasp of Barwick's model for genuine $G$-spectra, that is, spectral Mackey functors [1](https://www.maths.ed.ac.uk/%7Ecbarwick/papers/mack1.pdf). There's a classical formula about induction, that should be easy to prove, that I was trying to prove in this model; but I failed, and it's worse than tha... | https://mathoverflow.net/users/102343 | An induction formula for spectral Mackey functors, and a fake proof | $\newcommand{\Hom}{\mathrm{Hom}} \newcommand{\res}{\mathrm{res}}$
Ah, well, I found the mistake (at a surprising time: I'm more tired now than I was when I looked for it earlier) : $A^{eff}(H)\to A^{eff}(G)$ given by $G\times\_H-$ preserves pullbacks, not products ! In particular, in my computation for $\Hom\_H(\res\_H... | 4 | https://mathoverflow.net/users/102343 | 373720 | 156,059 |
https://mathoverflow.net/questions/373690 | 9 | Let $(M^{n+k},g)$ be a Riemannian manifold. Call a surface $\Sigma^n \subset M$ *calibrated* if there is a closed $n$-form $\omega$ defined on a neighbourhood $U \subset M$ of $\Sigma$ so that $\omega \lvert \Sigma = \mathrm{vol}\_\Sigma$ and for any $p \in U$ and $n$-tuples $(X\_1,\dots,X\_n) \in T\_p M$ of orthonorma... | https://mathoverflow.net/users/103792 | Non-calibrated area-minimising surface | Actually, a better example along the lines Otis suggests would be the geodesic $\mathbb{RP}^1\subset\mathbb{RP}^2$. Of course, $\mathbb{RP}^1$ is orientable and it is homologically mass-minimizing, but it can't be calibrated on any open set $U\subset\mathbb{RP}^2$ containing $\mathbb{RP}^1$ because twice it is not even... | 8 | https://mathoverflow.net/users/13972 | 373740 | 156,063 |
https://mathoverflow.net/questions/373727 | 8 | Let $(M^n,g)$ be a complete Riemannian manifold with bounded geometry, that is, it has bounded curvature and positive injectivity radius. Given two disjoint smoothly embedded homotopically trivial closed curves $\gamma\_1$ and $\gamma\_2$, we consider the problem of minimizing annulus $\Sigma$ with $\partial \Sigma=\ga... | https://mathoverflow.net/users/105900 | Plateau's Problem from an annulus | Such an annulus need not exist. For example, consider two circles in $\mathbb{R}^3$ defined by $x^2+y^2 = 1$ and $z = \pm R$. If $R$ is sufficiently large, then there cannot be a minimizing annulus (or, indeed, any minimizing connected surface) with these two circles as boundary.
The reason is the following: First, o... | 14 | https://mathoverflow.net/users/13972 | 373741 | 156,064 |
https://mathoverflow.net/questions/373739 | 29 | I am chemist and ask for apologies for all my mathematical inabilities when asking this question in advance, but after quite a bit of searching I found that this problem could be "open" or at least hard enough to find addressed in the literature and also advanced enough that it's possibly suitable to be asked here.
I... | https://mathoverflow.net/users/83999 | What determines the maximal dimension of the irreps of a (finite) group? | Your question touches on many issues in group representation theory, and I can only give a few general remarks which may point you in interesting directions for further reading.
As to your question regarding the maximal real irreducible representation of a finite group, there is an interesting connection with the Fro... | 19 | https://mathoverflow.net/users/14450 | 373748 | 156,065 |
https://mathoverflow.net/questions/373664 | 2 | I'm doing some research in Control theory, and a stumbled with this problem. Any help is appreciated.
**QUESTION**
Let $P\_1,\dots,P\_m$ be $m$ symmetric positive definite $n\times n$ matrices with $m<n$ and real entries. I'm looking for necessary conditions for the existence of nontrivial real coefficients $\alpha\_... | https://mathoverflow.net/users/166253 | Necessary conditions for existence of linear combination of these matrices to be singular | Eigenvalues are continuous functions of the matrix entries if this is expressed carefully. Consider $H(c) = cP\_1+P\_2+\cdots+P\_m$. When $c$ is large and negative, the eigenvalues of $H(c)$ are all negative. When $c$ is positive, $H(c)$ is psd and the eigenvalues are all positive. So there is some $c$ such that $H(c)$... | 2 | https://mathoverflow.net/users/9025 | 373760 | 156,068 |
https://mathoverflow.net/questions/373749 | 0 | I have a jar containing `n` numbered marbles, where `1...x` marbles are red and marbles `x+1...n` are black, and I want to remove them one by one. However, red marbles are larger and more likely to be grabbed: at any time, each red marble is `k` times more likely to be selected than each black marble, regardless of the... | https://mathoverflow.net/users/166773 | What is the most likely sequence? | This is to note that I answered the question in the comments.
| 2 | https://mathoverflow.net/users/297 | 373761 | 156,069 |
https://mathoverflow.net/questions/373661 | 3 | I started to read this preprint: <https://arxiv.org/abs/2010.03696>
In it, the author states that $\sum\_{n\leq x}\mu\_{k}(n)=\zeta(k)^{-1}x+O(x^{1/k})$ and that under RH, the exponent in the error term becomes $\frac{1}{k+1}$ (where $\mu\_{k}$ is the indicator of $k$-free numbers).
What would an exponent of the fo... | https://mathoverflow.net/users/13625 | Error term for the summatory function of $k$-free numbers indicator and RH | The Dirichlet series of the indicator function of $k$-free numbers is $\zeta(s)/\zeta(ks)$. Hence any exponent less than $1/k$ in the error term implies a quasi-Riemann Hypothesis. More precisely, if the number of $k$-free numbers is $x/\zeta(k)+O(x^c)$, then $s=1$ is the only pole of $\zeta(s)/\zeta(ks)$ in the half-p... | 4 | https://mathoverflow.net/users/11919 | 373770 | 156,071 |
https://mathoverflow.net/questions/373728 | 3 | In 1980, C. Pomerance, J. Selfridge, and S. S. Wagstaff defined a *pseudoprime to the base a* to be any composite odd $n$ such that $n \mid a^{n-1} - 1$.
More recently, in 2013, S. S. Wagstaff referred to such numbers as ``Fermat pseudoprimes.''
Are either of the following known to be true?---
(1) Every Lucas seq... | https://mathoverflow.net/users/166753 | On pseudoprimes to the base $a$ (Fermat pseudoprimes) | (1) is true.
Let $p$ be a prime such that $p\nmid (a-1)a$ and $\frac{a^p-1}{a-1}$ is composite. Then $\frac{a^p-1}{a-1}$ is a base-$a$ pseudoprime.
Also, if $q$ is a Carmichael number comprime to $(a-1)a$, then both $q$ and $\frac{a^q-1}{a-1}$ are base-$a$ pseudoprimes.
| 3 | https://mathoverflow.net/users/7076 | 373780 | 156,076 |
https://mathoverflow.net/questions/368638 | 5 | We denote by $B\_{p}^s(\mathbb{T}) := B\_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$.
For $p>0$ fixed and $f \in \mathcal{S}'(\mathbb{T})$ a generalized function, set
\begin{equation}
s\_p(f) = \sup \{s \in \mathbb{R} , \ f ... | https://mathoverflow.net/users/39261 | Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$? | The answer to my question is actually **no**: there exists generalized functions such that $s\_p(f)$ is not of the proposed form. Stéphane Jaffard discusses the possible forms of the functions $s\_p(f)$, denoted by $\eta(p)$, in his paper [*On the Frisch-Parisi Conjecture*](https://www.sciencedirect.com/science/article... | 0 | https://mathoverflow.net/users/39261 | 373789 | 156,080 |
https://mathoverflow.net/questions/373737 | 4 | The prime counting function $\pi(x)$ is defined as
\begin{equation}
\pi(x)=\sum\_{p\leq x}1
\end{equation}
where $p$ runs over primes.
I have seen many bounds for $\pi(x)$ such as
\begin{equation}
\frac{x}{\log x}\left(1+\frac{1}{2\log x}\right)<\pi(x)<\frac{x}{\log x}\left(1+\frac{3}{2\log x}\right)
\end{equatio... | https://mathoverflow.net/users/166729 | Bounds for prime counting function | The following explicit version of the Prime Number Theorem was proved by [Trudgian](https://link.springer.com/article/10.1007%2Fs11139-014-9656-6):
$$ |\pi(x)-\mathrm{li}(x)|<x e^{-0.39\sqrt{\ln x}},\qquad x\geq 229.\tag{$\ast$}$$
In fact Trudgian's Theorem 2 is somewhat stronger than $(\ast)$, and with Mathematica it ... | 14 | https://mathoverflow.net/users/11919 | 373790 | 156,081 |
https://mathoverflow.net/questions/373762 | 2 | Say, we're given smooth functions $f\_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying
1. $\Delta f\_n\ge 0$ (subharmonic)
2. $f\_n\ge 0$
3. $\int\_\Omega f\_n=I>0$ for all $n\in\mathbb{N}$
4. ${f\_n}\_{|\partial\Omega}=n$
Then, say $B\subset\subset \Omega$. Can we conclud... | https://mathoverflow.net/users/166785 | Positive subharmonic functions with constant integral blowing up at boundary | Let $\Omega$ be the unit ball, $B$ some smaller concentric ball, and
$u\_n(x)=1$ for $|x|\leq 1-1/n$ and $u\_n(x)=n(n-1)|x|+2n-n^2$ for $1-1/n\leq|x|\leq 1$.
Then your conditions 1,2,4 are satisfied exactly, and 3 is satisfied approximately
(integrals tend to a positive constant), so a slight modification will give you... | 2 | https://mathoverflow.net/users/25510 | 373798 | 156,083 |
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