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https://mathoverflow.net/questions/376681 | 14 | This sounds like a very silly question which should have have a negative answer but I don't see an argument. The precise question is this:
>
> Does there exist a covariant functor $ord$ from the category of sets and functions to the category of totally ordered sets and increasing functions such that every set $X$ i... | https://mathoverflow.net/users/21051 | Does there exist an ordering-functor? | **Conceptual answer.**
There can be no such functor. Let $C$ be any concrete category of finite sets and mappings such that the only automorphisms in $C$ are trivial. I claim there is no underlying set preserving functor $F$ from the category $\mathbf{FSet}$ of finite sets to $C$. The category of finite totally order... | 19 | https://mathoverflow.net/users/15934 | 376695 | 157,075 |
https://mathoverflow.net/questions/376664 | 7 | The well known [partition](https://en.wikipedia.org/wiki/Partition_(number_theory)) function $p(n)$ is defined as the number of ways to represent $n$ as the sum of natural numbers. An asymptotic formula for $p(n)$ is
$$p(n)\sim\frac{1}{4n\sqrt{3}}\exp\left(\pi\sqrt{\frac{2n}{3}}\right)$$
which was obtained by Ramanujan... | https://mathoverflow.net/users/nan | A generalization of partition function to the sums of squares | You also asked about the generating function. Write $r^k(n)$ for the number of partitions of $n$ with each part the $k$th power of a positive integer. That generating function is
$$\sum\_{n=0}^\infty r^k(n)q^n = \prod\_{m=1}^\infty \frac{1}{1-q^{m^k}}$$
since the $m$th factor on the right is a geometric series $(1+q^{m... | 4 | https://mathoverflow.net/users/14807 | 376706 | 157,078 |
https://mathoverflow.net/questions/376708 | 6 | Is it possible to have a non-split short exact sequences of vector bundles (on some smooth variety) $0\rightarrow V\_1 \rightarrow V\_2 \rightarrow V\_3 \rightarrow 0$. Such that $V\_2\cong V\_1\oplus V\_3$ as vector bundles?
| https://mathoverflow.net/users/127776 | Can non-split extension be isomorphic to the split one as objects | $\newcommand{\cO}{\mathcal{O}}$Consider exact sequence of trivial vector bundles $$0\to\cO\xrightarrow{\left(\begin{matrix}x \\ y\end{matrix}\right)}\cO\oplus\cO\xrightarrow{\left(\begin{matrix}y & -x\end{matrix}\right)}\cO\to 0$$ on $X=\mathbb{A}^2\_{x,y}\setminus\{0\}$. One checks easily that it is exact on stalks (b... | 10 | https://mathoverflow.net/users/39304 | 376715 | 157,081 |
https://mathoverflow.net/questions/376711 | 2 | So for me the definition the independence of two random variables $X,Y$ is intuitivly very clear.
But what I have never seen motivated is why the heck one would be interested in the covariance $$\operatorname{Cov}(X,Y):=\mathbb{E}\left((X-\mathbb{E}(X))(Y-\mathbb{E}(Y))\right).$$
Since independent variables are also ... | https://mathoverflow.net/users/117393 | What concept does covariance formalise? | $\DeclareMathOperator\Cov{Cov}\DeclareMathOperator\Var{Var}$I. **"[W]hy the heck one would be interested in the covariance"?**
At least for two reasons:
1. The correlation
$$\rho\_{X,Y}\mathrel{:=}\frac{\Cov(X,Y)}{\sqrt{\Var X}\sqrt{\Var Y}}$$
is the normalized covariance (assuming $\Var X\ne0$ and $\Var Y\ne0$), wit... | 12 | https://mathoverflow.net/users/36721 | 376720 | 157,083 |
https://mathoverflow.net/questions/376580 | 4 | Let $X,Y$ be completely regular Baire spaces. Is it true that every real valued separately continuous function on $X\times Y$ has a point of continuity?
| https://mathoverflow.net/users/168777 | Point of continuity of separately continuous functions | A counterexample to this problem (with $X$ Baire and $Y$ compact) was recently constructed by Mykhaylyuk and Pol in [this preprint](https://arxiv.org/pdf/1809.05799.pdf).
| 4 | https://mathoverflow.net/users/61536 | 376733 | 157,089 |
https://mathoverflow.net/questions/376680 | 0 | Given a set of many variables $S=\{x\_1,x\_2, ...., x\_i\}$, and any subset $S'$ of $S$, I need a function $f$ which maps $S'$ to a value $x$ and a function $f'$ which maps $x$ back to set $S'$.
I know my question can be solved with [Gödel\_numbering](https://en.wikipedia.org/wiki/G%C3%B6del_numbering), but it will c... | https://mathoverflow.net/users/168850 | Encoding numbers with relationship into one and back | This is inspired by answers in the other linked threads. You can let $f(S') = \sum\_{x\_j \in S'} 2^{j-1}$.
If you let your set only contain powers of 2, i.e. $S = \{ 1, 2, 4, 8, \ldots , 2^i\}$, then $f(S') = \sum\_{x\_j \in S'} x\_j$, the sum of all numbers in $S'$.
Note that this is the same as interpreting $S'$... | 0 | https://mathoverflow.net/users/82838 | 376734 | 157,090 |
https://mathoverflow.net/questions/376736 | 2 | Determinant modulo $2$ of biadjacency matrix of bipartite graphs provide mod $2$ information on number of perfect matchings on bipartite graphs providing polynomial complexity in bipartite situations.
Is there a similar trick for general graphs which is in polynomial complexity?
| https://mathoverflow.net/users/10035 | Mod $2$ information on perfect matchings in general graphs | One can use [Pfaffians](https://en.wikipedia.org/wiki/Pfaffian) of the adjacency matrix. It will look like
$\text{pf}(A) = \sum\_{\sigma} \text{sgn}(\sigma) \cdot (\text{product of n entries of A})$
over certain permutations where the product is 1 if and only those $n$ edges make a perfect matching. Hence, we sum $... | 3 | https://mathoverflow.net/users/51668 | 376739 | 157,092 |
https://mathoverflow.net/questions/376728 | 1 | I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form
$$\mu\_4 = \sum\_{i,j,k,l=1}^n w\_i w\_j w\_k w\_l \mu\_{ijkl}$$
where $\mu\_{ijkl}$ are the fourth-order co-moments of the $n$ random variables and $w\_i$ are the weights. The variables I assume to be iden... | https://mathoverflow.net/users/168376 | Can I prove that a polynomial representing the 4th moment of a weighted-sum of random variables is a sos? | By a well-known result due to Richter and Rogosinsky (see e.g. [Kemperman, Lemma 1, p. 69](https://link.springer.com/chapter/10.1007%2F978-3-642-46477-5_5)), there is a probability measure $\nu$ on a *finite* set $T\subset\mathbb R^n$ such that
$$\mu\_{ijkl}=\int\_{T}\nu(dt)t\_it\_jt\_kt\_l=\sum\_{t\in T}\nu(\{t\})t\_i... | 2 | https://mathoverflow.net/users/36721 | 376741 | 157,093 |
https://mathoverflow.net/questions/376754 | 0 | I am trying to derive the "[thin plate energy functional](https://en.wikipedia.org/wiki/Thin_plate_energy_functional)". Given a thin plate $z = z(x,y)$, how does one derive easily the energy functional
$$\iint\_{\mathbb{R}^2} \,\left[\left(\frac{\partial ^2z}{\partial x^2}\right)^2+2\left(\frac{\partial^2 z}{\partial... | https://mathoverflow.net/users/128758 | The derivation of thin plate spline interpolation energy function？ | It very much depends on what you consider a "derivation" and what you consider an "easy" one. You can start *assuming* that the energy density depends on $(\kappa\_1^2+\kappa\_2^2)g^{\frac 12}$, where $\kappa\_i$ are the principal curvatures, which can be considered quite natural from a physical point of view (in one d... | 1 | https://mathoverflow.net/users/167834 | 376757 | 157,098 |
https://mathoverflow.net/questions/376758 | 0 | We work over an algebraically closed field $k$, say of characteristic $0$, just in case, and we let $C$ be a smooth curve over $k$. First-order deformations of $C$ (or of any smooth variety for that matter) are captured by the cohomology group $H^1(C,\mathcal{T}\_C)$, where $\mathcal{T}\_C$ is the tangent sheaf of $C$.... | https://mathoverflow.net/users/168894 | Deformations of the Fermat curve | It might be helpful for you to distinguish embedded deformations of the plane curve $C\subset{{\mathbb P}\_k^2}$ from general deformations of the projective variety $C$. The embedded deformations are all of the form $\{x^n+y^n-z^n+f(x,y,z)=0\}\subset {{\mathbb P}\_k^2}$ for a general homogeneous polynomial $f$ of degre... | 3 | https://mathoverflow.net/users/6107 | 376764 | 157,100 |
https://mathoverflow.net/questions/376740 | 4 | I'm looking for an inverse system $(X\_\alpha)\_{\alpha < \omega\_1}$ of vector spaces (**EDIT:** over a *finite* field) such that, for some $\lambda \geq 2$ with $\lambda < \lambda^{\omega\_1}$ (I believe the case where $\lambda = \kappa^\omega$ for some $\kappa \geq 2$ is particularly interesting), the following cond... | https://mathoverflow.net/users/2362 | Example of an inverse system which suddenly "jumps" in size in a specific "controlled" way? | Here is my argument, which assumes $|2^\omega|<|2^{\omega\_1}|$ and uses an infinite base field. See Tim Campion's answer <https://mathoverflow.net/a/376790/164965> for the general case.
Take the base field to be $\mathbb Q,$ set $\lambda=2^\omega=\mathfrak c,$ and $X\_{\alpha}=\ell^\infty(\alpha)$: the bounded funct... | 3 | https://mathoverflow.net/users/164965 | 376773 | 157,101 |
https://mathoverflow.net/questions/376752 | 3 | Let $u\_k$ be a sequence of subharmonic functions on an open set $X$ and $\psi\_\delta$ a family of standard mollifiers with compact support. Hörmander claims in *The Analysis of Linear Partial Differential Operators Vol I*, Theorem 4.1.9(b) that if $u\_k$ converge as distributions to a subharmonic $u$ then $v\_j \* \p... | https://mathoverflow.net/users/123448 | If subharmonic functions converge weakly to a subharmonic limit, why do their smoothings converge uniformly on compact sets? | More detail on weak convergence of subharmonic functions, including a proof of this statement, can be found in his other book:
Hormander, Notions of convexity, Theorems 3.2.12 and 3.2.13.
| 2 | https://mathoverflow.net/users/25510 | 376776 | 157,102 |
https://mathoverflow.net/questions/376775 | 6 | The inverse Galois problem asks whether every finite group appears as the Galois group of a Galois extension of the rational numbers.
Is anything known about the anologous problem, where the extensions are not required to be Galois? In other words, for a finite group $G$, does there exist a finite field extension $K$... | https://mathoverflow.net/users/37368 | Inverse Galois problem for non-Galois extensions | The answer to this is positive. The first correct proof, it seems, was given in
Michael D. Fried. A note on automorphism groups of algebraic number
fields. Proc. Amer. Math. Soc., 80(3):386–388, 1980.
For a generalization to Hilbertian fields and some history see for example
F. Legrand and E. Paran. Automorphism ... | 8 | https://mathoverflow.net/users/50351 | 376780 | 157,103 |
https://mathoverflow.net/questions/376770 | 5 | $\DeclareMathOperator{\GL}{GL}$Let $G$ be a finitely presented group with generators $g\_1,\dotsc, g\_k$. Suppose we have a family of representations $\rho\_t:G\to \GL(n,\mathbb C)$ with $t\in [0,1]$ smoothly dependent on $t$. Suppose that for any $t\in [0,1]$ and $i\in\{1,\dotsc,k\}$, $\rho\_t(g\_i)$ is diagonalisable... | https://mathoverflow.net/users/13441 | A limit of conjugate representations in $\mathrm{GL}(n,\mathbb C)$ | It is proven in Lemma 1.25 of the book "Varieties of representations" by Lubotzky and Magid (Memoirs of AMS, vol. 336, 1985) that each $GL(n,{\mathbb C})$-conjugation orbit of a semisimple representation $\rho\in R\_n={\mathrm Hom}(\Gamma, GL(n, {\mathbb C}))$ is Zariski-closed in $R\_n$. If a representation is unitari... | 7 | https://mathoverflow.net/users/39654 | 376793 | 157,107 |
https://mathoverflow.net/questions/376760 | 3 | * Can we find an explicit example of a sequence of functions $f\_k \in H^1({\mathbf R}^3)$ such that, $f\_k \rightharpoonup f$ weakly converges in $H^1({\mathbf R}^3)$ and $f\_k \to f$ strongly converges in $L^6(R^3)$, but $f\_k$ does not strongly converge to $f$ in $H^1({\mathbf R}^3)$?
* What happens if one changes $... | https://mathoverflow.net/users/114101 | Explicit example $f_k \to f$ converging strongly in $L^6(R^3)$, but only weakly in $H^1(R^3)$ | First, note that the embedding $H^1(\mathbf{R}^3)\hookrightarrow L^p\_{\text{loc}}(\mathbf{R}^3)$ is compact only for $p<6$, and the "loc" is mandatory for this compactness to hold. I know that you did not wrote anything in contradiction with this, but your assumptions surprised me a bit.
Now, consider $f\_n:(x\_1,x\... | 7 | https://mathoverflow.net/users/27767 | 376803 | 157,113 |
https://mathoverflow.net/questions/376808 | 9 | The eigenfunctions of the Laplacian on $SL(2,\mathbb Z)\backslash \mathbb H$ are known to be given by three types: the constant function, the [real analytic Eisenstein series](https://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series) (which come in a continuous family), and the Maass cusp forms (which come in a di... | https://mathoverflow.net/users/168923 | Spectral decomposition of product of modular functions | I'm guessing you're a physicist by your unusual notation! The real-analytic Eisenstein series is usually denoted $E(z,s)$ (where $z = \tau$ in your notation), not $E\_s(\tau,\bar{\tau})$ (why does $\bar{\tau}$ appear?). I'll also let $f(z)$ denote a cusp form rather than $\nu\_i(\tau,\bar{\tau})$ (I've never seen this ... | 10 | https://mathoverflow.net/users/3803 | 376822 | 157,120 |
https://mathoverflow.net/questions/376782 | 3 | For $d \geq 3$ (degree) and $r \geq 3$ (radius), say that a $d$-regular (finite, simple, non-oriented) graph $G$ is $r$-almost-tree if it contains no cycle of length $\leq 2 r$: in other words, we want our graph to look locally like a $d$-regular tree, in the sense that its restriction to any ball of radius $r$ always ... | https://mathoverflow.net/users/118629 | Minimum size of regular graph with no short cycles | The problem of determining the smallest regular graphs with degree $k$ and girth $g$ is normally known as the cage problem.
It has a large literature which is nicely summarised in the Dynamic Cage Survey in the Electronic Journal of Combinatorics.
<https://www.combinatorics.org/ojs/index.php/eljc/article/download/D... | 5 | https://mathoverflow.net/users/1492 | 376825 | 157,121 |
https://mathoverflow.net/questions/376576 | 3 | I'm reading *Geometric Invariant Theory* by Mumford, and confuse about the Proposition 2.4 on P54.
It states that:
Let $G$ be a reductive group, act on an algebraic scheme. Then the action of $G$ on $X$ is proper if and only if for every non-trivial 1-PS $\lambda: \mathbb{G}\_m\to G$, the induced action of $\mathbb... | https://mathoverflow.net/users/153842 | Question on geometric invariant theory | I believe this is addressed on page 52, underneath the statement of Iwahori's theorem. He provides an argument for why Iwahori's result can be strengthened to G reductive by considering $G \rightarrow G'$ where $G
$ is reductive and $G'$ is the associated adjoint group.
| 5 | https://mathoverflow.net/users/119460 | 376836 | 157,128 |
https://mathoverflow.net/questions/191275 | 1 | Let $G$ be a topological group, let $K$ be a closed cocompact subgroup (i.e. the coset space $G/K$ is compact in the quotient topology) and let $g \in G$. Is there a sequence (edit: or net) of positive powers $g^{i\_n}$ of $g$ such that $g^{i\_n}K$ converges to $K$ in the coset space $G/K$?
If the answer is `no' in g... | https://mathoverflow.net/users/4053 | Powers in compact coset spaces | $\DeclareMathOperator{\eps}{\varepsilon}$No. I'll write $P$ instead of $K$, as $K$ often denotes a compact subgroup. Write $X=G/P$.
For $x\_0\in X$ such that $g\mapsto gx\_0$ induces a homeomorphism $G/P\to X$, the question is whether $(g^nx\_0)\_{n\ge 1}$ always accumulates at $x\_0$. This fails in the most classica... | 1 | https://mathoverflow.net/users/14094 | 376853 | 157,133 |
https://mathoverflow.net/questions/376855 | 7 | Let $G$ be a compact group and $u: G \to B(H)$ be a strongly continuous unitary representation on the Hilbert space $H$. Then is $u: G \to B(H)$ strictly continuous?
That is, give $B(H)$ the topology induced by the $\*$-isomorphism $M(B\_0(H))\cong B(H)$. Explicitely, a net $(x\_i)$ in $B(H)$ converges strictly to $x... | https://mathoverflow.net/users/nan | Unitary representation is strictly continuous | As you note, on bounded sets, the strict topology and the strong-$\ast$ topology agree on bounded sets. As the set of unitary operators *is* bounded, we can just work with the strong-$\ast$ topology. If $(u\_i)$ is a net of unitary operators converging strongly to $u$ a unitary, then for $\xi\in H$,
$$ \| u\_i^\ast(\... | 5 | https://mathoverflow.net/users/406 | 376862 | 157,137 |
https://mathoverflow.net/questions/376875 | 2 | Let $X$ be a path-connected manifold (or a CW complex).
Let $\pi\_1(X)$ be the fundamental group of $X$.
Let $\pi: \tilde X\longrightarrow X$ be a covering map.
For each $m\geq 0$, let $C\_m(\tilde X)$ be the real chain group generated by all the $m$-cells of $\tilde X$.
Then $C\_m(\tilde X)$ is a module over t... | https://mathoverflow.net/users/41075 | can the actions of fundamental groups annihilate homology? | There are finitely presented groups that do not have any non-trivial linear representations, so for these groups as fundamental group you are just asking whether the ordinary real homology of $X$ is trivial, which it usually won't be. As was pointed out in a comment the trivial group has this property, but there are pl... | 6 | https://mathoverflow.net/users/124004 | 376886 | 157,144 |
https://mathoverflow.net/questions/376878 | 1 | Let $X$ be an $n$-dimensional cell complex.
We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$.
Take the universal cover (or a general covering space) $\tilde X'$ of $X'$.
We have a covering map $\pi': \tilde X'\longrightarrow X'$.
................
Question. Whether or not can we ta... | https://mathoverflow.net/users/41075 | can we take skeletons of covering maps to give new covering maps? | Yes, first one should check that the restriction of a covering space is a covering space. This is true either by just checking the axioms or appealing to the fact that the pullback of a covering space is a covering space, and the restriction of a covering space is just the pullback of an inclusion into the base space.
... | 3 | https://mathoverflow.net/users/134512 | 376894 | 157,146 |
https://mathoverflow.net/questions/376846 | 5 | Assume I have an inductive system of short exact sequences of $C^{\ast}$-algebras (i.e., short exact sequences $0 \to A\_n \to B\_n \to C\_n \to 0$ together with transformations from the $n$-th to the $(n+1)$-st short exact sequence so that all squares commute). If I form now the colimit of the $C^{\ast}$-algebras, is ... | https://mathoverflow.net/users/13356 | Colimits of short exact sequences of C*-algebras | My notation
$$
i\_n:A\_n\to B\_n,
$$
$$
p\_n:B\_n\to C\_n,
$$
$$
i:\displaystyle \lim\_\to A\_n\to \displaystyle \lim\_\to B\_n,
$$
$$
p:\displaystyle \lim\_\to B\_n\to \displaystyle \lim\_\to C\_n,
$$
$$
\beta \_n:B\_n\to\displaystyle \lim\_\to B\_n.
$$
I suppose the only contentious point is to prove that... | 3 | https://mathoverflow.net/users/97532 | 376898 | 157,147 |
https://mathoverflow.net/questions/376656 | 11 | The choice principle $\text{AC}\_{\text{WO}}$ proves a large amount of cardinal arithmetic. It's well-known to imply DC, that successor cardinals are regular, and that for all $X$, there is $\lambda$ such that $\aleph(X)=\aleph^\*(X)=\lambda^+.$ Furthermore, we have the following:
>
> ($\text{ZF + AC}\_{\text{WO}}$... | https://mathoverflow.net/users/109573 | Does $\text{AC}_{\text{WO}}$ prove $\Theta \neq \aleph_{\omega+1}?$ | ($\text{ZF + AC}\_{\text{WO}}$) For any cardinals $\kappa\_1, \kappa\_2,$ there is $\lambda$ such that $\aleph(^{\kappa\_2}\kappa\_1)=\lambda^+$ and $\text{cf}(\lambda)>\kappa\_2.$
Pf: Let $\lambda$ be such that $\aleph(^{\kappa\_2}\kappa\_1)=\lambda^+,$ and fix a cofinal sequence $\langle\gamma\_{\xi}: \xi<\text{cf}... | 5 | https://mathoverflow.net/users/109573 | 376906 | 157,149 |
https://mathoverflow.net/questions/376774 | 2 | I am looking for the following questions:
(1) **True or false?** for every $p<q$, one may find a function $f\in L^1(\mathbb{R})$ such that $\hat{f}\in L^q (\mathbb{R})$ but $\hat{f}\notin L^p (\mathbb{R})$.
(2) **True or false?** There exists a function $f\in L^1(\mathbb{R})$ such that $\hat{f}\notin L^p (\mathbb{R... | https://mathoverflow.net/users/84390 | Two classic problems concerning Fourier transform of an integrable function | Here is a proof that (2) is true.
It suffices to find $f\in L^1(\mathbf{R})$ such that for any integer $n\in\mathbf{N}^\star$, one has $f^{\star n}\notin L^2(\mathbf{R})$, where
\begin{align\*}
f^{\star n}:=\stackrel{n\text{ times}}{\overbrace{f\star f\star \cdots \star f}.}
\end{align\*}
Indeed, using the formula $\... | 2 | https://mathoverflow.net/users/27767 | 376910 | 157,150 |
https://mathoverflow.net/questions/376912 | 2 | Is there any implementation available of an algorithm which solves in full generality the $S$-unit equation $x+y=1$ in a number field? It seems that Magma solves $ax+by=c$ but only in the algebraic integers, while Sage solves $x+y=1$ with $x,y$ $S$-integers, but only for $S$ the set of primes over a fixed rational prim... | https://mathoverflow.net/users/36370 | Software for $S$-unit equation | This SageMath implementation promises the full generality you are seeking:
[A robust implementation for solving the S-unit equation and several applications](https://arxiv.org/abs/1903.00977)
See also this [Phys.Org](https://phys.org/news/2019-04-centuries-old-mathematical-puzzles.html) announcement.
| 3 | https://mathoverflow.net/users/11260 | 376914 | 157,152 |
https://mathoverflow.net/questions/376925 | 0 | Let $X$ be a random variable with variance $\tau^2$ and $Y$ be another random variable such that $Y-X$ is independent of $X$ and has mean zero and variance $\sigma^2$. (One can think of $Y$ as a noisy observation of $X$.) It follows from the law of total variance that $\mathbb{E}[\operatorname{Var}(X|Y)]\leq\operatorna... | https://mathoverflow.net/users/28006 | Lower bound for reduced variance after conditioning | The LHS of the expression you wrote is the minimum mean square error (i.e., the expectation $\inf E(X-\hat X)^2$ where $\hat X$ is measurable on $Y$). On the other hand, the expression
you wrote ($\hat \sigma^2:=\sigma^2 \tau^2/(\sigma^2+\tau^2)$) is the error of the optimal linear estimator, so it always bounds from a... | 1 | https://mathoverflow.net/users/35520 | 376927 | 157,157 |
https://mathoverflow.net/questions/376820 | 10 | $\DeclareMathOperator\SL{SL}$This question came up in a class ["Total Positivity and Cluster Algebras" being taught by Chris Fraser](https://sites.google.com/site/cmfraser37/teaching/8680).
Let $N^+$ denote the space of uni-upper-triangular matrices in $\SL(n,\mathbb{R})$, and $N^+\_{\geq 0} \subseteq \SL\_{\geq 0}(n... | https://mathoverflow.net/users/25028 | Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices | The Catalan strata are unions of Bruhat strata, and the resulting map from permutations to Dyck paths is indeed given by taking the left-to-right maxima.
There is a way to parametrize totally nonnegative matrices by writing them as Lindström-Gessel-Viennot matrices for a certain weighted directed graph. This is expla... | 7 | https://mathoverflow.net/users/2384 | 376931 | 157,158 |
https://mathoverflow.net/questions/376949 | 3 | For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int\_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
$f=\check{f}$ almost everywhere? If the answer is NO, is it possible that the Lebesgue measure of $\{x: f(x)\neq\ch... | https://mathoverflow.net/users/84390 | On the Fourier inversion formula | *Note: I am not sure if I understand the word "converges" correctly.*
This is completely analogous to the similar question regarding convergence of Fourier series, which is classical.
Let $$g(x,r) = \int\_{-r}^r \hat f(\zeta) e^{2\pi i \zeta x} d\zeta$$ by "partial sums" of the inverse Fourier transform, and denote... | 9 | https://mathoverflow.net/users/108637 | 376955 | 157,167 |
https://mathoverflow.net/questions/376839 | 150 | In his talk, [The Future of Mathematics](https://www.youtube.com/watch?v=Dp-mQ3HxgDE), [Dr. Kevin Buzzard](https://en.wikipedia.org/wiki/Kevin_Buzzard) states that [Lean](https://en.wikipedia.org/wiki/Lean_(proof_assistant)) is the only existing proof assistant suitable for formalizing *all of math*. In the Q&A part of... | https://mathoverflow.net/users/30352 | What makes dependent type theory more suitable than set theory for proof assistants? | I apologize for writing a lengthy answer, but I get the feeling the discussions about foundations for formalized mathematics are often hindered by lack of information.
I have used proof assistants for a while now, and also worked on their design and implementation. While I will be quick to tell jokes about set theory... | 226 | https://mathoverflow.net/users/1176 | 376973 | 157,174 |
https://mathoverflow.net/questions/376970 | 8 | Chapter XI Theorem 3 from [here](https://math.sjtu.edu.cn/faculty/tyaglov/courses/linear%20algebra/The_book_add_2.pdf "Gantmacher: Theory of matrices") implicitly states that an invertible complex symmetric matrix always has a complex symmetric square root.
It's clear that a square root exists, by appealing to the Jo... | https://mathoverflow.net/users/75761 | Why does an invertible complex symmetric matrix always have a complex symmetric square root? | Higham, in *Functions of Matrices*, Theorem 1.12, shows that the Jordan form definition is equivalent to a definition based on Hermite interpolation. That shows that the square root of a matrix $A$ (if based on a branch of square root analytic at the eigenvalues of $A$) is a polynomial in $A$. Therefore, if $A$ is symm... | 15 | https://mathoverflow.net/users/9025 | 376980 | 157,179 |
https://mathoverflow.net/questions/376067 | 3 | Is there a generalization of Borel-Weil-Bott for partial flag varieties, i.e. homogeneous spaces of the form $G/P$ with $P$ parabolic and $G$ semisimple? If so, I would like a reference.
| https://mathoverflow.net/users/36720 | Borel–Weil–Bott for partial flag varieties | There are many places that give a complete answer to your question: One is a paper in the Annals of Math written by Kostant around the middle fifties, other more geometrical is due to Griffits-Schmid published in Acta Mathematica in the late sixties.
best regards
| 1 | https://mathoverflow.net/users/67162 | 376991 | 157,182 |
https://mathoverflow.net/questions/376998 | 4 | In [1, 5.6.3] Pedersen states without proof or reference that there are non-unital C\*-algebras whose Pedersen
ideal is the whole algebra.
* Does anyone know where can I find such an example?
* Is it possible to characterize algebras with this property?
---
[1] *Pedersen, Gert K.*, C\*-algebras and their automo... | https://mathoverflow.net/users/110570 | Is there a C*-algebra whose Pedersen ideal is not proper? | Examples include all non-unital algebraically simple $C^\ast$-algebras. By [Blackadar, Bruce E.; Cuntz, Joachim The structure of stable algebraically simple C∗-algebras. Amer. J. Math. 104 (1982), no. 4, 813–822.] a simple, stable $C^\ast$-algebra is algebraically simple if and only if it contains an infinite projectio... | 5 | https://mathoverflow.net/users/126109 | 377000 | 157,188 |
https://mathoverflow.net/questions/377001 | -1 | this is a problem from Durret's probability textbook.
Show that if $\varphi$ is a ch.f., then $Re\varphi$ and $|\varphi|^2$ are also ch.f.
I am wondering how to prove this. Actually I'm not even sure how to show something is ch.f. The only idea I have is to use the inversion formula to show that we can get the dist... | https://mathoverflow.net/users/169059 | how to prove that the real part and the modulus of a characteristic function is still a characterisitc function? | Although <https://math.stackexchange.com/> would be a better place to ask this question, let me answer it.
Since the characteristic function is basically the Fourier transform, let me explain it in terms of the Fourier transform $\varphi=\hat{f}$ and you can translate it to the language of the characteristic function... | 2 | https://mathoverflow.net/users/121665 | 377004 | 157,190 |
https://mathoverflow.net/questions/377002 | 3 | Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\gamma(0) = x, \gamma(1) = y$ and $\nabla\_{\gamma'(t)} \gamma'(t)=0$ for all $t$. This is a way to say that $\gamma$ is a ge... | https://mathoverflow.net/users/140013 | Almost geodesic on non complete manifolds | Start with the plane $\mathbb R^2$ and remove a slab, but keep a line going through the slab:
$$ Slab = \{(x, y) \in \mathbb R^2 : 0 < y < 1, x \neq 0\} $$
$$ M = \mathbb R^2 - Slab$$
```
y
--------o-------------
--------o-------------
--------o-------------
x
```
Note that $M\_1$ is connected but curves goi... | 4 | https://mathoverflow.net/users/46591 | 377006 | 157,191 |
https://mathoverflow.net/questions/377011 | 9 | $\newcommand\la\lambda\newcommand\w{\mathfrak w}\newcommand\R{\mathbb R}$Numerical calculations and other considerations ([The min of the mean of iid exponential variables](https://mathoverflow.net/questions/376819/the-min-of-the-mean-of-iid-exponential-variables/377007#377007)) suggest that
$$\int\_\R \frac{1-e^{it... | https://mathoverflow.net/users/36721 | An integral identity | I would close the contour in the upper half of the complex plane, the principal value picks up $i\pi$ times the residue$^\ast$ at $t=0$, which is $u/(1-u)$. There are no other poles.$^{\ast\ast}$
$^\ast$ $\frac{1-e^{i t u}}{e^{i t u}-i t-1}=\frac{u}{1-u}+{\cal O}(t^2).$
$^{\ast\ast}$ poles are at $t=i\tau$ with $e^... | 6 | https://mathoverflow.net/users/11260 | 377015 | 157,193 |
https://mathoverflow.net/questions/376795 | 3 | Fix a complete first order theory $T$ and a set of parameters $A$ in the monster model $\mathcal{U}$. Recall that an *$A$-invariant global type* is a type $p(x) \in S\_x(\mathcal{U})$ which is fixed by any automorphism of $\mathcal{U}$ which fixes $A$. An equivalent statement is that for every formula $\varphi(x,\bar{y... | https://mathoverflow.net/users/83901 | Existence of invariant types whose Morley sequences are all indiscernible sets | It's possible to lift the counterexample you described to a counterexample to Question 2 (and hence also to Question 1).
Let $E$ be an equivalence relation with infinitely many infinite classes. Let $\leq$ be preorder linearly ordering the set of $E$-classes. Paint each $E$-class by a copy of the counterexample you d... | 2 | https://mathoverflow.net/users/2126 | 377022 | 157,196 |
https://mathoverflow.net/questions/376399 | 1 | Let $\mathfrak{g}=\mathfrak{gl}\_3$ over $\mathbb{C}$ with positive roots
\begin{equation\*}
\Phi\_+=\{\alpha\_1=(1,-1,0),\alpha\_2=(1,0,-1),\alpha\_3=(0,1,-1)\}.
\end{equation\*}
Consider the morphism
\begin{align}
M((-2,1,1)) \oplus M((-1,-1,2) &\xrightarrow\phi M((0,-1,1))\\ (v\_{(-2,1,1)},v\_{(-1,-1,2)}) &\mapsto(y... | https://mathoverflow.net/users/135674 | Computing kernel in the category $\mathcal{O}$ | Here is a [less direct](https://mathoverflow.net/a/376625), but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M\_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously t... | 2 | https://mathoverflow.net/users/15292 | 377034 | 157,202 |
https://mathoverflow.net/questions/376984 | 5 | Let $A$ be an abelian category closed with respect to small coproducts (that is, and AB3 category). Which assumptions are sufficient to ensure the existence of an exact faithful functor from $A^{op}$ into abelian groups that respects products (i.e., it should send $A$-coproducts into the corresponding products)?
I su... | https://mathoverflow.net/users/2191 | Which abelian categories possess an exact faithful functor into abelian groups that respects products? | I find it less confusing to work directly with $A^{op}$ so let me do that; I'll rename it $C$. We have a complete abelian category $C$ (completeness is equivalent to being closed under small products) and we want to know when it admits an exact faithful functor $G : C \to \text{Ab}$ which respects products (equivalentl... | 6 | https://mathoverflow.net/users/290 | 377048 | 157,205 |
https://mathoverflow.net/questions/376985 | 2 | Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial derivatives is non-zero.
I am looking for $F$ which satisfies the following condition: There is a point in $P \in \math... | https://mathoverflow.net/users/84272 | Question about the implicit function theorem. an example of a homogeneous form for which its implicit function satisfies certain conditions | Here is a simple example: Take $F = w^3 +3 w u^2 -v^3$ on $\mathbb{R}^3$ with coordinates $(u,v,w)$. At the point $p=(u,v,w)=(1,0,0)$, we have that $F=0$ can be solved for $w$ as a function of $(u,v)$. Meanwhile, via implicit differentiation,
$$
w\_u = \frac{-2uw}{u^2+w^2}\quad\text{and}\quad w\_v = \frac{v^2}{u^2+w^2}... | 2 | https://mathoverflow.net/users/13972 | 377067 | 157,210 |
https://mathoverflow.net/questions/377073 | 1 | Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota\_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical inclusion. Suppose also that $f(\mathbb{R}^n)\cong \mathbb{R}^n\cong g(\mathbb{R}^n)$ via some $C^{\infty}$-diffeomorphism.
F... | https://mathoverflow.net/users/36886 | Extension of homeomorphisms | It sounds like you're looking for something like [the Klee trick](https://www.ams.org/journals/tran/1955-078-01/S0002-9947-1955-0069388-5/). If $K,K' \subset \mathbb{R}^n$ are compact and homeomorphic, it gives a construction of a self-homeomorphism $\phi$ of $\mathbb{R}^{2n}$ such that $\iota\_n^{2n}(K) = \phi(\iota\_... | 7 | https://mathoverflow.net/users/798 | 377076 | 157,212 |
https://mathoverflow.net/questions/376417 | 2 | $\newcommand\Cb{C^\text b}$Let $\Cb(\mathbb R)$ be the C\*-algebra formed by all bounded, continuous, complex valued functions on $\mathbb R$.
Consider the action $\tau $ of $\mathbb R$ on $\Cb(\mathbb R)$ given by
$$
\tau \_t(f)\mathclose|\_s = f(s-t), \quad \forall f\in \Cb(\mathbb R), \quad \forall s,t\in \mathbb R... | https://mathoverflow.net/users/110570 | Minimal components of the translation action on the Stone–Čech compactification | $\newcommand{\Cb}{C^{\text b}}$I think I have a negative answer to my own question. Consider the function $f(x)=\sin(1/x)$, defined for $x$ in $(0,+\infty )$, and let us denote the graph of $f$ by $G$.
Let $\alpha $ be the arc length parametrization of $G$ oriented in such a way that $\alpha (s)$ approaches the verti... | 0 | https://mathoverflow.net/users/110570 | 377082 | 157,213 |
https://mathoverflow.net/questions/377079 | 3 | A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K\_X-D$ is ample and the pair $(X,D)$ is klt.
Now, let $f:X\dashrightarrow Y$ birational map which is an isomorphism in codimension one between two projective normal and $\mathbb{Q... | https://mathoverflow.net/users/nan | Weak Fano varieties and small transformations | You need to know that $Y$ as at worst klt singularities. In this case $Y$ is a weak Fano variety with klt singularities and then it is log Fano.
Now, you are done since a small $\mathbb{Q}$-factorial transfomation of a log Fano variety is log Fano.
| 2 | https://mathoverflow.net/users/14514 | 377086 | 157,215 |
https://mathoverflow.net/questions/376847 | 2 | Let $F$ be a homogeneous from in $\mathbb{R}[x\_0, .., x\_n]$. Then $F$ defines a projective variety $X \subset \mathbb{P}\_{\mathbb{C}}^n$. Assume $X$ is smooth. In this case $F=0$ also defines a submanifold $M = \{ \mathbf{x} \in \mathbb{R}^{n+1} \backslash \{ \mathbf{0} \} : F(x\_0, .., x\_{n+1}) = 0 \}$ of $\mathbb... | https://mathoverflow.net/users/84272 | Algebraic geometric conditions on the variety $V(F)$ such that the manifold defined by $F$ has nonvanishing second fundamental form? | As we know, the projective hypersurface in $\mathbb{P}^n$ defined by a homogeneous polynomial equation
$$
F(x^0,\ldots,x^n)=0
$$
of degree $m$ is *nonsingular* if $x=0$ is the only solution to the equations
$$
0 = F = \partial\_0F = \partial\_1F = \cdots = \partial\_nF.
$$
Because $mF = x^0\,\partial\_0F + \cdots + x^n... | 1 | https://mathoverflow.net/users/13972 | 377099 | 157,219 |
https://mathoverflow.net/questions/377098 | 3 | I recently gave an undergraduate course on group theory (which is not entirely my field of expertise, so the following questions might have a well-known answer of which I am simply unaware). As I was explaining the concept of solvability, I digressed a little and told the class about the odd-order theorem, also known a... | https://mathoverflow.net/users/70751 | Does the sequence (Number of groups of even order $\le n$) / (Number of groups of order $\leq n$) converge? If not, what are its cluster points? | As mentioned in the comments, conjecturally almost all finite groups are $2$-step nilpotent $2$-groups, so conjecturally the answers to 1) and 3) are that the limits both exist and both equal $1$; that is, almost all finite groups have even order and almost all finite groups are solvable (even nilpotent). As numerical ... | 4 | https://mathoverflow.net/users/290 | 377100 | 157,220 |
https://mathoverflow.net/questions/365532 | 3 | I'm studying
>
> M. A. A. de Cataldo, L. Migliorini - *The Hard Lefschetz Theorem and the topology of semismall maps*, Ann. sci. École Norm. Sup., Serie 4 **35** (2002) 759-772.
>
>
>
The premises are the following.
Let $f:X\to Y$ be a proper holomorphic (non constant) map of irreducible, complex, projective... | https://mathoverflow.net/users/57030 | Weak Lefschetz theorem for Lef line bundles | It is based on certain vanishing property of $U= X\backslash Y$.
First you have a long exact sequence (a derived categorical version is given in the end)
$$H^k(X,Y;\mathbb{Q})\rightarrow H^k(X,\mathbb{Q}) \rightarrow H^k(Y,\mathbb{Q}) \rightarrow H^{k+1}(X,Y;\mathbb{Q}).$$
Note that $H^{k}(X,Y;\mathbb{Q})=H^{k}\_c(U,\m... | 1 | https://mathoverflow.net/users/133871 | 377114 | 157,225 |
https://mathoverflow.net/questions/377103 | 10 | Let $T\_k(x\_1,\ldots,x\_n)$ be the Todd polynomials, $e\_k(x\_1,\ldots,x\_n)$ the elementary symmetric polynomials and $p\_k(x\_1,\ldots, x\_n)$ the power sums of degree $k$.
We have the following generating formulas
\begin{align\*}
\sum\_{k\geq 0}T\_k(x\_1,\ldots,x\_n)t^k = \prod\_{i=1}^n\frac{tx\_i}{1-e^{-tx^i}}\,... | https://mathoverflow.net/users/109370 | Todd polynomials | We have
$$\log \sum\_{k \ge 0} T\_k t^k = \sum\_{i=1}^n \log \frac{x\_i t}{1 - e^{-x\_i t}}$$
so if we write
$$\log \frac{x\_i t}{1 - e^{-x\_i t}} = \log \sum\_{k \ge 0} B\_k^{+} x\_i^k \frac{t^k}{k!} = \sum\_{k \ge 1} b\_k x\_i^k \frac{t^k}{k!}$$
(using the sign conventions explained on [Wikipedia](https://en.... | 14 | https://mathoverflow.net/users/290 | 377115 | 157,226 |
https://mathoverflow.net/questions/377110 | 6 | $\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Let $B\_d$ and $S\_{d-1}$ denote, respectively, the closed unit ball and the unit sphere in $\R^d$. Let us say that a finite subset $F$ of $B\_d$ is maximal if the sum of all pairwise Euclidean distances between the points in the set $F$ is the largest possible given $n... | https://mathoverflow.net/users/36721 | Subsets of a ball/sphere with the largest sum of distances | Negative answers to some of those questions:
**Q1** Not always; for $n=8$ a square antiprism is better than a cube.
For example, in radius $\sqrt 3$, the cube with vertices $(\pm 1, \pm1, \pm1)$
gives $16(1 + 2\sqrt{2} + \sqrt{3}) = 88.9676+$ while changing
the $z=+1$ vertices to $(\pm\sqrt2, 0, 1)$ and $(0,\pm\sqrt2... | 6 | https://mathoverflow.net/users/14830 | 377121 | 157,227 |
https://mathoverflow.net/questions/377093 | -2 | At lesson, the teacher considers a flow $\Phi$ given by the solutions of the ode system for $t\in[0, T]$ and $x\in\mathbb R^d$,
$$
\begin{cases}
y'(s)=b(y(s), s),&s\leq T\\
y(t)=x
\end{cases},\label{1}\tag{\*}
$$
that is $\Phi(x, t, s)=y(s)$ solving \eqref{1}. He said that we will be mostly concerned with $\Phi(\cdot, ... | https://mathoverflow.net/users/160186 | Definition and properties of the inverse of the flow of an ODE | I think this question is better suited for math.stackexchange. But here's a short answer anyway:
The flow is a map $\Phi : \mathbb{R}\_+ \times \mathbb{R}^d \to \mathbb{R}^d$ and if you take an initial value $y\_0$ at time $t\_0 = 0$, then $t \mapsto \Phi(t, y\_0)$ is the trajectory of the solution to your ODE with i... | 3 | https://mathoverflow.net/users/102441 | 377132 | 157,233 |
https://mathoverflow.net/questions/377071 | 26 | In classical field theory, many fields and related objects are described as differential
forms. For example, in electromagnetism, the field $F := B - \mathrm dt\wedge E$ is a 2-form, and Maxwell's
equations ask that it be closed. Quantization imposes integrality constraints on $F$, which may concisely be
pacakged as as... | https://mathoverflow.net/users/97265 | In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot? | **Traditional approach.** Notice that what is considered in [[DMW00](http://arxiv.org/abs/hep-th/0005091); [DFM03](https://arxiv.org/abs/hep-th/0312069)] and elsewhere to quantize the C-field flux $G\_4$ is not just ordinary cohomology, but ordinary cohomology with bells and whistles added as need be:
Foremost there ... | 18 | https://mathoverflow.net/users/381 | 377154 | 157,240 |
https://mathoverflow.net/questions/377108 | -1 | Let $E$ and $M$ be smooth manifolds (of finite dimension, Hausdorff and second countable). Let $\pi:E\longrightarrow M$ be a surjective submersion such that:
1. $E\_p:=\pi^{-1}(p)$ is a real vector space isomorphic to $\mathbb R^n$, $\forall p \in M$.
2. The pointwise sum, multiplication by scalar and zero section ar... | https://mathoverflow.net/users/157138 | Local triviality condition in vector bundles | If you see $\Xi$ as a map from $U\times\mathbb R^n$ to $\pi^{-1}(U)$, then it is smooth, and by the inverse mapping theorem it is locally invertible around $(p,0)$ (because of the submersion condition and the condition on the $\sigma\_i(p)$). It means that for all $q$ close enough to $p$, the restriction $d\Xi\_{(q,0)}... | 2 | https://mathoverflow.net/users/129074 | 377160 | 157,243 |
https://mathoverflow.net/questions/376378 | 4 | It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:
>
> For every $f$-divergence and every stochastic kernel
> \begin{equation\*}
> \begin{split}
> K: \mathcal{X} \times... | https://mathoverflow.net/users/121501 | Information monotonicity of divergence => function of $f$-divergence | It is known that, in general, a monotonic divergence measure does not have to a monotonically-increasing function of an $f$-divergence. See discussion after Definition 1 (and footnote 3) in
* Polyanskiy and Verdú, Arimoto Channel Coding Converse and Rényi Divergence, *48th Annual Allerton Conference on Communication,... | 2 | https://mathoverflow.net/users/76565 | 377162 | 157,244 |
https://mathoverflow.net/questions/377163 | 5 | Suppose we take an almost complex structure on $\mathbb{T}^{6}$ with $c\_{1} \neq 0$ (there should be infinitely many homotopy classes satisfying this requirement). Now pull it back to the universal cover $\mathbb{R}^6$, giving an almost complex structure $J$, it should be invariant by the deck group action of $\mathbb... | https://mathoverflow.net/users/99732 | almost complex $\mathbb{Z}^{6}$-action | Here is a construction of a family of such examples that will work, but you will have to choose a particular map to get an explicit example.
Let's use coordinates $v\_1,v\_2,v\_3,v\_4,x,y$ (each periodic of period $2\pi$). Choose a smooth map $u=u(x,y):\mathbb{T}^2\to S^2$ that has nonzero degree. (This will ensure t... | 4 | https://mathoverflow.net/users/13972 | 377168 | 157,247 |
https://mathoverflow.net/questions/377171 | 0 | A random vector $X \in \mathbb{R}^n$ is isotropic if $\mathbb{E}XX^T = I\_n$. However isotropic random vectors don't have the property of isotropy. See [1](https://math.stackexchange.com/questions/2569824/if-x-is-isotropic-random-vector-then-is-the-centered-random-vector-x-ex-als?rq=1). So why are they called isotropic... | https://mathoverflow.net/users/169213 | Why are isotropic random vectors called isotropic if they aren't? | This is a contamination quite common in probability when properties of distributions are instead attributed to the associated random objects. Strictly speaking one should talk about isotropic (i.e., rotation invariant) measures or distributions rather than vectors.
Yet another more recent example of this contaminatio... | 1 | https://mathoverflow.net/users/8588 | 377183 | 157,255 |
https://mathoverflow.net/questions/377167 | 2 | I am reading a proof for the existence of a solution to the Local Cauchy problem of the non-linear Schrodinger equation
$$
i\partial\_t u+\Delta u +\epsilon u |u|^{2} = 0 \\
u(x,0)=u\_0(x)
$$
The structure of the proof is due to J. Ginibre and G. Velo (*On a class of nonlinear Schrödinger equations. I. The Cauchy pro... | https://mathoverflow.net/users/121404 | An inequality of spacetime Banach space for non-linear Schrodinger equation | It is sufficient to write
$$u^2\bar{u}-v^2\bar{v}=(u-v)u\bar{u}+(u-v)v\bar{u}+(\bar{u}-\bar{v})v^2$$
and then differentiate each term.
| 1 | https://mathoverflow.net/users/7294 | 377186 | 157,257 |
https://mathoverflow.net/questions/377109 | 4 | For a given $n$ is there a guaranteed way to construct any possible function from $\mathbb{Z}/n\mathbb{Z}$ to itself in terms of polynomials? Specifically, for $T = \mathbb{Z}/n\mathbb{Z}$ I'd like to use polynomials with $T$ coefficients to describe all functions $T \to T$.
My original hypothesis was that I could ex... | https://mathoverflow.net/users/169164 | Given the set of integers modulo $n$, can all functions from this set to itself be expressed as polynomials? | The 1995 paper *On polynomial functions from $\mathbb{Z}\_n$ to $\mathbb{Z}\_m$* in Discrete Mathematics, Vol. 137, proves the following strongly related result:
**Theorem:** *Every function $f:\mathbb{Z}\_n \rightarrow \mathbb{Z}\_m$ is a polynomial function if and only if $n$ is not greater than the least prime fac... | 9 | https://mathoverflow.net/users/17773 | 377187 | 157,258 |
https://mathoverflow.net/questions/377179 | 32 | I have a lower $n\times n$ triangular matrix called $A$ and I want to get $A^{-1}$ solved in $O(n^2)$. How can I do it?
I tried using a method called "forward substitution", but the inversion is solved in $O(n^3)$ for full $n\times n$ matrix.
| https://mathoverflow.net/users/169219 | Inverting lower triangular matrix in time $n^2$ | No such method is known at present.
If one could invert lower triangular $n \times n$ matrices in time $O(n^2)$
then one could multiply $N \times N$ matrices in time $O(N^2)$.
Indeed let $n=3N$ and apply the putative inversion algorithm to
the block matrix
$$
\left(
\begin{array}{ccc} I & 0 & 0 \cr B & I & 0 \cr 0 ... | 91 | https://mathoverflow.net/users/14830 | 377192 | 157,260 |
https://mathoverflow.net/questions/377193 | 3 | I was told: if $X$ is bdd below and $p$-complete spectra then $X^{tC\_q}$ vanishes for primes $q \not= p$.
I do not see how this holds.
---
I am aware from [I.2.9](https://arxiv.org/abs/1707.01799) that if $X$ is bdd. below, then $X^{tC\_q} \simeq \left(X^{\hat{}}\_q \right)^{tC\_P}$.
Here $\hat{}\_p$ denotes... | https://mathoverflow.net/users/97321 | Vanishing tate of a $p$-complete spectra | That is how you prove this.
Recall that $(-)^\wedge\_p \simeq L\_{\mathbb{S}/p}(-)$, where $L\_E(-)$ is the Bousfield localization with respect to $E$. Now, multiplication by $q$ mod $p$ is an equivalence, so multiplication by $q$ becomes a weak equivalence on $p$-complete spectra. Then $\mathbb{S}/q\wedge X\simeq X/... | 5 | https://mathoverflow.net/users/131196 | 377196 | 157,261 |
https://mathoverflow.net/questions/377172 | 0 | Today I was reading LMFDB (the L-functions and Modular Forms DataBase), and I came across something that confused me. When discussing degree 3 L functions on [this](https://www.lmfdb.org/L/degree3/) page, they assert that all the ones found so far have Euler products of the form
$$L(s)=\prod\_{p|N}\left(1-a\_np^{-s}+... | https://mathoverflow.net/users/159298 | Why does LMFDB refer to L functions having coefficients of type $a_p-a_{p^2}$ instead of just $a_{p^2}$? | This seems to be related to taking a power-series truncation in a reciprocal.
The $p$th Euler factor can be written as
$$\sum\_{k=0}^\infty {a\_{p^k}\over p^{ks}}$$
The reciprocal of this is in general a polynomial of degree less than or equal to the degree of the $L$-function. When the prime is good it is equal, a... | 1 | https://mathoverflow.net/users/169234 | 377198 | 157,262 |
https://mathoverflow.net/questions/264827 | 6 | The set of constructible numbers
<https://en.wikipedia.org/wiki/Constructible_number>
is the smallest field extension of $\mathbb{Q}$ that is closed under square root and complex conjugation. I am looking for an algorithm that decides if two constructible numbers are equal (or, what is the same, if a constructible... | https://mathoverflow.net/users/3816 | Algorithm to decide whether two constructible numbers are equal? | Although this can be done using the complicated algorithms for general algebraic numbers, there’s a much simpler recursive algorithm for constructible numbers that I implemented in the [Haskell `constructible` library](https://hackage.haskell.org/package/constructible).
A constructible field extension is either $\mat... | 7 | https://mathoverflow.net/users/68546 | 377201 | 157,263 |
https://mathoverflow.net/questions/377124 | 1 | Let $x,y,u,v$ be positive integers with $x,y$ coprime and $u,v$ coprime
( $xy,uv$ not necessarily coprime). Assume $x+y \ne u+v$.
How small the radical of $xy(x+y)uv(u+v)$ can be infinitely often?
Can we get $O(|(x+y)(u+v)|^{1-C})$ for $C>0$?
These are just two pairs of good $abc$ triples so we can get $C=0$
with... | https://mathoverflow.net/users/12481 | How small the radical of $xy(x+y)uv(u+v)$ can be infinitely often? | Here is a solution where the radical is $O(k^9)$ and $(x+y)(u+v)=O(k^{12})$
The idea is that $x,y,z=a^2,b^2,c^2$ for a Pythagorean triple and $u,v,u+v=A^2,B^2,C^2$ for another with $C=c^2.$ I used the most familiar type of triple (hypotenuse and long leg differ by $1$), there might be others that do better, or specia... | 4 | https://mathoverflow.net/users/8008 | 377206 | 157,264 |
https://mathoverflow.net/questions/376972 | 2 | I asked this question on Mathematics Stack Exchange some months ago but I got no answer.
Suppose one has an orientable compact surface $S$ of genus $g\ge 2$, $x\in S$, and $G=\pi(S,x)$ the fundamental group. There is a well-known description of the group with generators and relations as $$G=\langle a\_1,b\_1,\dots, a... | https://mathoverflow.net/users/158462 | A description of the fundamental class of the group cohomology of the fundamental group of an orientable surface of genus $g$ | For $1\le i\le g$, consider the following elements of the second degree of the bar complex:
\begin{align\*}
\alpha\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)\otimes a\_i\\
\beta\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_i\otimes b\_i\\
\gamma\_i &= \left(\prod\_{j < i} [a\_i,b\_i]\right)a\_ib\_i\otimes a\_i^{-... | 4 | https://mathoverflow.net/users/35687 | 377210 | 157,265 |
https://mathoverflow.net/questions/376268 | 2 | This question has been cross-posted from this [MSE question](https://math.stackexchange.com/q/3902896) and is an offshoot of this [other MSE question](https://math.stackexchange.com/q/3555053).
(Note that [MSE user mathlove](https://math.stackexchange.com/u/78967) has posted an [answer in MSE](https://math.stackexcha... | https://mathoverflow.net/users/10365 | On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number | Middle of page 6 of
<https://arxiv.org/pdf/1312.6001v10.pdf>
" we always have $0 < n−\lceil\sqrt{n^2−q^k}\rceil$ "
No, this requires that $q^k\ge 2n-1$,
an helpful assumption when the goal is to prove $q^k > n$.
| 2 | https://mathoverflow.net/users/71090 | 377216 | 157,267 |
https://mathoverflow.net/questions/377081 | 10 | Let $X$ be a smooth genus one curve over $k$. I don't call it elliptic curve because it will have no rational points.
By **index** of $X$ we mean the *smallest degree* of a closed point on $X$; equivalently by Riemann-Roch that's the same as the smallest positive degree of a divisor, or the greatest common divisor of... | https://mathoverflow.net/users/111491 | degree five genus one curves without rational points? | I'll address the case $d = 5$ over any number field, without recourse to Gross-Zagier formulas and Tate-Shafarevich groups. If $X$ has index 5, then it has order 5 in $H^1(k,E)$, hence comes from $H^1(k,E[5])$, where $E$ is the Jacobian. I assume the converse is not true (presumably some 5-torsion classes come from tor... | 10 | https://mathoverflow.net/users/949 | 377231 | 157,271 |
https://mathoverflow.net/questions/376577 | 3 | I'm trying to solve an optimal stopping problem which led me to an obstacle problem involving the following family of ODE's
$$(x^2+d)y'(x)-2xy(x) = 1.$$
For simplicity I first considered the case $d = 0$ before moving to the much more relevant case $d > 0$. This revealed a somewhat unexpected phenomenon to me: The ... | https://mathoverflow.net/users/78650 | (Non-) Convergence of solutions in a family of linear ODE's | Using the expansion for $\arctan(t)$ as $t\to +\infty$ in the form
$$\arctan(t) = \frac{\pi}{2}-\frac{1}{t}+\frac{1}{3t^3}+\cdots$$
we get, for fixed $x>0$ and positive $d\to 0$:
$$ f\_d(x) \sim \frac{\pi(x^2+d)}{4d^{3/2}} =: C\_d(x^2+d) $$
and further (that's where the magic cancellation happens) $$g\_d(x):=f\_d(x)-C\... | 2 | https://mathoverflow.net/users/24309 | 377248 | 157,274 |
https://mathoverflow.net/questions/377144 | 5 | Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d\_1,…,d\_r)$.
If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so that we can conclude that $f$ is induced by an action of the Automorphism group of $\mathbb{P}^n$, $\operatorname{P... | https://mathoverflow.net/users/169131 | Isomorphisms of complete intersections | This always holds for $\dim(X)\geq 3$. The point is that in this case the Picard group of $X$ is cyclic, generated by the line bundle $\mathscr{O}\_X(1):= \mathscr{O}\_{\mathbb{P}^n}(1)\_{|X}$; this is SGA 2, Exp. 12, Cor. 3.7. Therefore any isomorphism $f:X\rightarrow Y$ induces an isomorphism $f^\*\mathscr{O}\_Y(1)\c... | 7 | https://mathoverflow.net/users/40297 | 377249 | 157,275 |
https://mathoverflow.net/questions/377253 | 0 | How would one show that if $\omega$ is the vorticity associated to $\partial\_t u+u\cdot \nabla u -\nu \Delta u +\nabla p=0$ (with smooth, compactly supported initial data) and
$$\omega\in L^\infty([0,T],H^1(\mathbb{R}^3))\cap L^2([0,T],H^2(\mathbb{R}^3))$$
then $u$ is a classical, smooth solution of the equation? Here... | https://mathoverflow.net/users/160298 | Regularity in Navier Stokes from $L^2$ bound on vorticity | For instance, you could prove that $u$ belong to the same space as $\omega$ in your assumption : in that case $u$ is a strong solution and becomes instantaneously smooth. To prove that $u$ have this regularity (at least locally, which should be sufficient to prove regularity properties), you can rely on the Biot-Savart... | 2 | https://mathoverflow.net/users/27767 | 377263 | 157,278 |
https://mathoverflow.net/questions/377274 | 11 | $\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$**Question 1:** What is an example of a sequence $(X\_\alpha)\_{\alpha<\kappa}$ of abelian groups such that $\varprojlim^2\_{\alpha < \kappa} X\_\alpha \neq 0$?
Here $\varprojlim^2\_{\alpha<\kappa}$... | https://mathoverflow.net/users/2362 | Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$? | A great survey on this and some related topics is Osofsky's "[The subscript of $\aleph\_n$, projective dimension, and the vanishing of $\varprojlim^{(n)}$](https://projecteuclid.org/euclid.bams/1183535284)." As far as I am aware, this 1974 paper still describes the state of the art on the matter.
Osofsky (exposing ma... | 11 | https://mathoverflow.net/users/43000 | 377277 | 157,283 |
https://mathoverflow.net/questions/377088 | 1 | I am certainly going to make a mess of any serious algebraic terminology, so bear with me as I present my problem arising from a probability problem.
Consider the space of sequences of $n$ zero-one valued bits -- that is, $\mathbb{Z}\_2^n$. For any element $\omega = (\omega\_1,\omega\_2,\ldots,\omega\_n)$ we can of c... | https://mathoverflow.net/users/58551 | Polynomial form/Fourier transform of rational function over finite affine space | You want [Kravchuk polynomials](https://en.wikipedia.org/wiki/Kravchuk_polynomials), and the following identity
$$
\sum\_{\substack{(\omega\_{1},\ldots,\omega\_{n}) \in \{0,1\}^{n}\\ \omega\_{1}+...+\omega\_{n}=\ell}} \prod\_{j \in S} (-1)^{\omega\_{j}} = \sum\_{j=0}^{\ell} (-1)^{j} \binom{|S|}{j}\binom{n-|S|}{\ell-j}.... | 2 | https://mathoverflow.net/users/50901 | 377278 | 157,284 |
https://mathoverflow.net/questions/377234 | 3 | Perhaps surprisingly, we work in the language of second-order arithmetic. I was wondering if the strength of the following statement LP was known:
*An uncountable closed set in $\mathbb{R}$ has a limit point*.
For reference, the perfect set theorem (equivalent to ATR$\_0$) implies LP, while LP implies ACA$\_0$ (usi... | https://mathoverflow.net/users/33505 | What is the strength of the second-order statement 'an uncountable closed set in $\mathbb{R}$ has a limit point'? | I'm pretty sure it follows from ACA$\_0$. The point is that since you're not trying to prove the full perfect set property, you only need one iteration of the Cantor-Bendixson derivative.
Given a closed set $C$, two or three jumps can obtain the derivative $C'$. If this is empty, then you can index the elements of C ... | 4 | https://mathoverflow.net/users/32178 | 377280 | 157,286 |
https://mathoverflow.net/questions/377284 | 6 | As I do more number theory, and in particular analytic number theory, I keep hearing more about the Möbius function $\mu(n)$ and how it is supposedly "pseudorandom". The values of the Möbius function at $n$ are determined by a formula, so what does this mean? Also, why is it so important that the Möbius function behave... | https://mathoverflow.net/users/159298 | What's the deal with Möbius pseudorandomness? | As you point out in your question, there is no one good way to define pseudorandomness. The first approach is to try to solidify the condition that $\mu(n)$ is "equally likely" to be $\pm1$. We do this by using the property that for any $\epsilon>0$
\begin{equation}
\left|\sum\_{n<x}X\_n\right|=O\left(x^{1/2+\epsilon... | 7 | https://mathoverflow.net/users/159298 | 377285 | 157,290 |
https://mathoverflow.net/questions/377220 | 4 | *For any complexes of $R$-modules, $P$ and $M$, $\hom\_{\mathcal{C}(R)}(P,M)$ is the complexe defined by,*
$$\forall n \in \mathbb{Z}\ \ \ \hom\_{\mathcal{C}(R)}(P,M)\_n = \prod\_{i \in \mathbb{Z}} \hom\_R(P\_i, M\_{i+n})$$
*We say that a complexe $P$ is $\pi$-projective or K-projective, if for all quasi-isomorphisme... | https://mathoverflow.net/users/135591 | Show that $\hom_R(f, M)$ is a quasi-isomorphism if $f:P \to P'$ is a quasi-isomorphism of $K$-projectives complexes | What you call $\pi$-projective is what Spaltenstein calls $K$-projective (and this is the only term I've heard). Spaltenstein shows in his original paper that a quasi-isomorphism between $K$-projective complexes is a chain homotopy equivalence, which implies the result you want.
| 6 | https://mathoverflow.net/users/1310 | 377290 | 157,294 |
https://mathoverflow.net/questions/377266 | 6 | My question is very direct:
>
> What are the motivations for the name "jet"(subjet, superjet) in the context of viscosity solutions for second order fully nonlinear elliptic PDE?
>
>
>
The definition of which can be seen in Crandall, Ishii, Lions:
*Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis*, ... | https://mathoverflow.net/users/113406 | Motivations for the term "jet" in the context of viscosity solutions for fully nonlinear PDE | Strictly speaking, he answer is that there is no motivation for the name "jet" in the context of viscosity solutions for second order fully nonlinear elliptic PDE, because it was initially introduced in the more basic framework of differential calculus/geometry.
Still one can wonder about when and where it was introd... | 10 | https://mathoverflow.net/users/14094 | 377309 | 157,297 |
https://mathoverflow.net/questions/377298 | 1 | Suppose there exist a zero-mean Gaussian process $\mathbb{G} f\_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f\_u \mathbb{G} f\_v \big]$ when both $u$ and $v$ are known, where $\mathcal{S}^{p - 1}$ is the $p$-dimensional unit sphere. Now I want to know what exactly the in... | https://mathoverflow.net/users/153595 | The integral of a Gaussian process on a unit sphere | Let $\newcommand{\bG}{\mathbb{G}}$ $\newcommand{\bE}{\mathbb{E}}$
$$
X=\int\_S \bG f\_u du,\;\;k(u,v)=\bE( \bG f\_u \bG f\_v).
$$
Then $X$ is a mean zero Gaussian random variable so it suffices to find its variance $\bE(X^2)$. Note that
$$
X^2=\int\_{S\times S} \bG f\_u\bG f\_v dudv
$$
so
$$
\bE(X^2)= \int\_{S\times S}... | 2 | https://mathoverflow.net/users/20302 | 377311 | 157,298 |
https://mathoverflow.net/questions/377236 | 12 | Consider the following statement:
>
> $(\dagger)$ $\ $ There is an inner model $M$ such that $M \models \mathsf{GCH}+\square$ and for every countable $X \subseteq \mathrm{Ord}$, there is a countable $Y \in M$ such that $X \subseteq Y$.
>
>
>
When I say $M$ is an ``inner model'' I mean that $M$ is a class (defi... | https://mathoverflow.net/users/70618 | Getting a model of $\mathsf{ZFC}$ that fails to nicely cover an inner model | The consistency strength of the failure of $(\dagger)$ is an inaccessible cardinal.
Building on the comment of Mohammad, if $\omega\_2^V$ is a successor cardinal in $L$ then there is a set $X \subseteq \aleph\_1^V$ such that $L[X]$ computes $\aleph\_1, \aleph\_2$ correctly, which (assuming $0^\#$ does not exist) is e... | 10 | https://mathoverflow.net/users/41953 | 377325 | 157,300 |
https://mathoverflow.net/questions/377273 | 8 | In a combinatorial computation, I came across the following quantity:
Consider a finite meet semilattice $L$, that is, a finite poset which is closed under $\min$. Denote the least element of $L$ by $0$.
Now, define $Z := \{ S \subset L : \min S = 0 \}$. I want to compute the quantity
$$ \chi := \sum\_{S \in Z} (-1... | https://mathoverflow.net/users/169294 | Euler characteristic of the simplicial complex of sets of elements in a semilattice with non-zero meet | This is a special case of the crosscut theorem. See e.g. Corollary 3.9.4 of *Enumerative Combinatorics*, vol. 1, second ed. Let $L'$ be $L$ with a top element $\hat{1}$ adjoined. In Corollary 3.9.4 take $X$ to be all elements of $L'$ not equal to $1$. We get $\chi=-\mu\_{L'}(0,1)$, where $\mu\_{L'}$ is the Möbius funct... | 5 | https://mathoverflow.net/users/2807 | 377334 | 157,303 |
https://mathoverflow.net/questions/377251 | 10 | For a natural number $n$, let $c\_b(n)$ denote the number of digit transitions in the representation of $n$ in base $b$. By a digit transition, we mean a pair of successive, unequal digits: for instance, the decimal number 114633366 has 4 transitions, given by 14, 46, 63 and 36, hence $c\_{10}(114633366) = 4$.
Are th... | https://mathoverflow.net/users/161058 | Are there numbers whose binary and ternary representations simultaneously have few digit transitions? How frequent are those numbers? | [Edited because I had misread $c\_b(k)$ to be the number of non-zero digits in the base $b$ representation, rather than the number of digit transitions. The argument works for both variants. -T]
The claim is true, and one can in fact argue by purely Archimedean methods (with the only number theoretic input being the ... | 14 | https://mathoverflow.net/users/766 | 377347 | 157,307 |
https://mathoverflow.net/questions/377260 | 7 | Given two statistical manifolds, is there a notion of "isomorphic"? What are morphisms?
| https://mathoverflow.net/users/168590 | What is the correct notion of morphism between statistical manifolds? | Any smooth (resp. $C^1$) statistical manifold [can be embedded into](https://link.springer.com/article/10.1007/s00022-005-0030-0) the space of probability measures on a finite set.
A probability measure [can be viewed as](https://arxiv.org/abs/1406.6030) a weakly averaging affine measurable functional taking values i... | 5 | https://mathoverflow.net/users/92164 | 377356 | 157,310 |
https://mathoverflow.net/questions/377349 | 11 | $\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$.
For an integer $p\ge 0$, write $R\_p=S^p R$; then $R\_1=R$ and $\dim R\_p=p+1$.
Using Table 5 in the book of Onishchik and Vinberg, I computed that the represen... | https://mathoverflow.net/users/4149 | To describe an invariant trivector in dimension 8 geometrically | Here's another very nice (but still algebraic) interpretation that explains some of the geometry: Recall that $\operatorname{SL}(2,\mathbb{C})$ has a $2$-to-$1$ representation into $\operatorname{SL}(3,\mathbb{C})$ so that the Lie algebra splits as
$$
{\frak{sl}}(3,\mathbb{C}) = {\frak{sl}}(2,\mathbb{C})\oplus {\frak{m... | 12 | https://mathoverflow.net/users/13972 | 377358 | 157,311 |
https://mathoverflow.net/questions/377350 | 3 | Do the real-valued functions of bounded variation on $[0,1]$ belong to some Sobolev/Besov class?
What about a fractal, such as the Weierstrass function?
| https://mathoverflow.net/users/12518 | BV spaces and fractals -- are they Sobolev? Besov? | (Summary post of comments)
1. BV functions are bounded, and hence trivially in any $L^p$. (Special case of Sobolev/Besov spaces.)
2. The distributional derivative $f'$ is a signed measure, so $t \hat{f}(t)\in L^\infty$. Also $f$ is bounded, so $\hat{f} \in L^\infty$. So we can in fact conclude that $f\in W^{s,p}$ for... | 5 | https://mathoverflow.net/users/3948 | 377364 | 157,314 |
https://mathoverflow.net/questions/373465 | 6 | The classic example of a function that has a drastic cancelation when summed over divisors is $\mu(n)$, with complete cancellation for every number other than $1$. Another such function is the Liouville function $\lambda(n)$. Both of these functions have have the property that $\sum\_{n=1}^{\infty}\frac{f(n)}{n}=0$. Is... | https://mathoverflow.net/users/159298 | Does asymptotic behavior of $\left|\sum_{d|n}f(d)\right|$ imply asymptotic properties of $f(d)$? | Interestingly enough, it is actually enough to know
\begin{equation}
\frac{1}{N}\sum\_{n=1}^{N}\left|\sum\_{d|n}f(d)\right|=o(1)\tag{1}
\end{equation}
to deduce
\begin{equation}
\sum\_{n=1}^{\infty}\frac{f(n)}{n}=0\tag{2}
\end{equation}
to do this, we work instead under the change of variables $g(n):=\sum\_{d|n... | 3 | https://mathoverflow.net/users/159298 | 377367 | 157,316 |
https://mathoverflow.net/questions/377372 | 3 | Let $(V,V^{p,q},Q)$ be a polarized integral Hodge strucutre of weight $n$. I would like to understand the automorphism of this datum better. Specifically, I'm wondering if there are good conditions where we can show that the automorphism group of polarized integral Hodge structures is finite.
Since an automorphism of... | https://mathoverflow.net/users/152554 | Automorphism of integral Hodge structures | The polarization $Q$ gives rise to a positive definite form $Q'$. Automorphisms preserve $Q'$, so they lie in a compact group. On the other hand, automorphisms preserve the lattice, so they lie in a discrete subgroup of a compact group. This gives finiteness. In the case of a Riemann surface, the polarization comes fro... | 7 | https://mathoverflow.net/users/4144 | 377379 | 157,318 |
https://mathoverflow.net/questions/377380 | 0 | For $j\in\mathbb{N}$, consider continuous functions $f\_j:[0,1]\to\mathbb{\mathbb{R}^+}$ such that
$$\sup\_{t\in[0,1]}\sum\_jf\_j(t)<+\infty,$$
namely $f\_j(t)\in L\_t^{\infty}((0,1),l\_j^1(\mathbb{N}))$. I would like to understand whether the quantity
$$S\_f:=\sum\_{j,k\in\mathbb{N}}\int\_0^1f\_j(t)f\_k(t)dt$$
is fini... | https://mathoverflow.net/users/54552 | Finiteness of a bilinear combination | The answer is yes. Indeed,
$$M:=\sup\_{t\in[0,1]}\sum\_jf\_j(t)<\infty,$$
and hence
$$\begin{aligned}S\_f&=\sum\_{j,k\in\mathbb{N}}\int\_0^1f\_j(t)f\_k(t)\,dt \\
&=\int\_0^1\sum\_{j,k\in\mathbb{N}}f\_j(t)f\_k(t)\,dt \\
&=\int\_0^1\Big(\sum\_{j\in\mathbb{N}}f\_j(t)\Big)^2\,dt \\
&\le\int\_0^1 M^2\,dt=M^2<\infty,
\end{... | 1 | https://mathoverflow.net/users/36721 | 377382 | 157,319 |
https://mathoverflow.net/questions/377375 | 7 | In a [recent paper,](https://arxiv.org/pdf/1605.06794.pdf) Hiroshi Kihara induced a model structure on the category of diffeological spaces. He generates the classes of fibrations, cofibrations, and weak equivalences by constructing a functor $d:\Delta \to \mathcal{D}$ (where $\mathcal{D}$ is the category of diffeologi... | https://mathoverflow.net/users/75783 | Inducing a model structure using a cosimplicial object | A somewhat general statement along these lines would be as follows. Suppose $\mathcal{D}$ is a cartesian closed locally presentable category and $d : \Delta \to \mathcal{D}$ is a cosimplicial object with associated geometric realization adjunction $|{\cdot}| : \mathrm{sSet} \to \mathcal{D}$, satisfying the following pr... | 6 | https://mathoverflow.net/users/126667 | 377384 | 157,320 |
https://mathoverflow.net/questions/377353 | 10 | I have already asked this question on [stack exchange](https://math.stackexchange.com/questions/3918462/is-surjective-holomorphic-self-map-on-compact-complex-manifold-finite), but I didn’t get any answer.
Let $X$ be a compact connected complex manifold.
>
> Let $f:X \to X$ be a **surjective** holomorphic map. Is ... | https://mathoverflow.net/users/130742 | Is every surjective holomorphic self-map on a compact complex manifold finite-to-one? | Let me give a sketch of proof for Gromov's claim in the case where $X$ is Kähler. More precisely, let me prove the following ${}$
>
> **Proposition [G03, p.223].** Let $X$, $Y$ be two complex manifolds (not necessarily compact) of the same dimension and having the same even Betti numbers. If $X$ is Kähler, then eve... | 12 | https://mathoverflow.net/users/7460 | 377390 | 157,322 |
https://mathoverflow.net/questions/377393 | -3 | For any infinitely differentiable function $f: \mathbb{R}\to \mathbb{R}$ and positive integer $k\in\mathbb{N}$, let $f^{(k)}$ denote the $k$-th derivative of $f$.
For which $n\in\mathbb{N}$, $n>1$, is there a periodical function $f:\mathbb{R}\to \mathbb{R}$ with the property that $f^{(n)} = f$, but $f^{(k)} \neq f$ f... | https://mathoverflow.net/users/8628 | Periodical functions with $f^{(n)} = f$, but $f^{(k)} \neq f$ for $k\in \{1,\ldots,n-1\}$ | You are asking for solutions $f$ to the simple differentiable equation $$f^{(n)}=f$$ that make $f$ real-valued for real-values of $f$ and not of lower degree $k$th-derivative-wise. As is well-known, the solution is given by $$f(x)=\sum\_{j=0}^{n-1}c\_j\exp(\zeta^jx)\,,$$ for some constants $c\_j$ and $x\in\mathbb{R}$, ... | 3 | https://mathoverflow.net/users/166628 | 377396 | 157,325 |
https://mathoverflow.net/questions/377188 | 5 | I'm reading Frank Neumann's ["Algebraic Stacks and Moduli of
Vector Bundles"](https://impa.br/wp-content/uploads/2017/04/PM_36.pdf) and have some problems to understand
a construction from the proof of:
>
> **Theorem 2.67.** (page 81) The moduli stack $\mathcal{Bun}\_X^{n,d}$
> of vector bundles of rank
> n and deg... | https://mathoverflow.net/users/108274 | Construction of an atlas for the moduli stack $\mathcal{Bun}_X^{n,d}$ in F. Neumann's 'Algebraic Stacks and Moduli of Vector Bundles' | I don't know what is going on exactly (misprints?), but here are some ideas:
If you take a point of $q\in R\_m$ (i.e. $U=Spec(k)$) defined by a sequence
$0 \rightarrow G \rightarrow \mathcal{O}\_X^{P(m)}\rightarrow F \rightarrow 0$
then by (ii) we have $H^1(F)=R^1(pr\_2)\_{\*}F=0$.
If you apply $Hom(-,F)$ to th... | 2 | https://mathoverflow.net/users/70593 | 377402 | 157,326 |
https://mathoverflow.net/questions/377403 | 1 | Working with Slater's inequality (a companion of Jensen's inequality) I found this statement:
>
>
> >
> > Let $f(x)$ be a continuous, twice differentiable function, convex or concave and non constant on $(0,\infty)$ and increasing on $(\alpha,\infty)$ with $\alpha>0$ a constant. Now define:
> > $$g(x)=\frac{f(x)f... | https://mathoverflow.net/users/147649 | Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$ | Here is one counter example: $f(x)=\sqrt x$ gives a function $g(x)=2-x^{1/4}-\dfrac{2}{1+\sqrt{x}}$ without an [asymptote](https://en.wikipedia.org/wiki/Asymptote) for $x\rightarrow\infty$.
*Note: this is a counter example to the original conjecture in the OP, not to the updated conjecture.*
| 2 | https://mathoverflow.net/users/11260 | 377406 | 157,328 |
https://mathoverflow.net/questions/377412 | 7 | I am reading the paper "On fibering certain 3-manifolds" by John Stallings and I was hoping someone could help me through a certain detail. In particular, I am confused at the very end of the proof of Theorem 1 which is as follows:
Theorem (Stallings):
Let $M^3$ be a compact 3-manifold such that there is a finitely g... | https://mathoverflow.net/users/99414 | Difficulty with "On fibering certain 3-manifolds" by Stallings | I think about it like this. For convenience, I'll assume $M$ is closed.
Given a homomorphism $\phi:\pi\_1M\to\mathbb{Z}$, Stallings explains how to find an essential surface $S\subset M$ with $\pi\_1S\leq\ker\phi$, as you outline nicely in the question.
Now we construct the cyclic cover $M\_\phi$ of $M$ correspondi... | 8 | https://mathoverflow.net/users/1463 | 377415 | 157,330 |
https://mathoverflow.net/questions/377411 | 5 | Let $X$ be a manifold or a CW-complex.
Let
$\pi: \tilde X\longrightarrow X$
be a covering map.
Let $\pi\_1(X)$ be the fundamental group of $X$ and let $\rho: \pi\_1(X)\longrightarrow O(n)$ be an orthogonal representation.
Define the $\rho$-twisted chain complex of $\tilde X$ by
$C\_\*(\tilde X,\rho)=C\_\*(\... | https://mathoverflow.net/users/41075 | triviality of homology with local coefficients | Maybe you are looking for something more interesting, but you can take $X=S^1$, universal cover $\tilde X$, and $\rho: {\mathbb Z}\to O(n)$ such that the image group has no fixed unit vectors in $R^n$. Then $H\_\*(\tilde X,\rho)=0$ (which is a nice exercise to work out if you are new to this material). A more challengi... | 5 | https://mathoverflow.net/users/39654 | 377417 | 157,332 |
https://mathoverflow.net/questions/377387 | 4 | Consider some $f: [0,1)\times [0,1)\to \mathbb{R}$. I'm interested in conditions that guarantee that the following one-sided second partial derivatives at $(x,y)=(0,0)$ are symmetric:
$$
\partial\_x^+ \partial\_y^+ f(x,y)= \partial\_y^+ \partial\_x^+ f(x,y).
$$
(where as usual $\partial\_x^+$ and $\partial\_y^+$ indica... | https://mathoverflow.net/users/76565 | Symmetry of one-sided partial derivatives | One such condition is that $f$ be absolutely continuous in $[0,h)^2$ for some $h\in(0,1)$ -- so that
$$f(x,y)+f(0,0)-f(x,0)-f(0,y)=\int\_0^x du\,\int\_0^y dv\,g(u,v)$$
for some function $g$ integrable on $[0,h)^2$ and for all $(x,y)\in[0,h)^2$ -- with $g$ continuous on the set $([0,h)\times\{0\})\cup(\{0\}\times[0,h))\... | 5 | https://mathoverflow.net/users/36721 | 377425 | 157,333 |
https://mathoverflow.net/questions/377421 | 8 | Today's arXiv has a paper by Pierpaolo Vivo, [Index of a matrix, complex logarithms, and multidimensional Fresnel integrals](https://arxiv.org/abs/2011.12007), which asks the question whether it is possible to calculate the number $N(\lambda\_1,\lambda\_2)$ of eigenvalues of a real symmetric matrix $M$ that lie in the ... | https://mathoverflow.net/users/11260 | Counting eigenvalues without diagonalizing a matrix | Here is an efficient method.
First of all, I must quote that diagonalizing $M$ is not a **method**, because there is no explicit way to carry this out. It amounts to calculating the roots of a polynomial ! At best, one can do this in an approximate way.
Instead, I suggest to perform a preliminary step : put $M$ in ... | 17 | https://mathoverflow.net/users/8799 | 377428 | 157,334 |
https://mathoverflow.net/questions/377336 | 2 | This problem itself, admittedly, is not a research problem; but rather an intermediate step I've encountered in my research.
Let $(X\_i:1\le i\le N)$ be a multivariate normal random vector where i) each coordinate $X\_i$ is standard normal and ii) $\mathbb{E}[X\_iX\_j]=\rho$ for every $1\le i<j\le N$.
My question. ... | https://mathoverflow.net/users/127150 | On the probability of the multivariate normal with fixed pairwise correlations being coordinate-wise non-negative | E.g., [Ruben, formulas (102) and (102') on p. 220](https://www.jstor.org/stable/2333017?seq=1) has a recurrence for your probability
$$
p\_n:=P(X\_1\ge0,\dots,X\_n\ge 0).
$$
It is stated there, on p. 213: "For dimensionality greater than three (spherical tetrahedra, spherical pentahedra, etc.) the areas can no longer b... | 2 | https://mathoverflow.net/users/36721 | 377434 | 157,335 |
https://mathoverflow.net/questions/377435 | 4 | What is the consistency strength of a cardinal $\kappa$, such that there is some $j: V\prec M$ such that $M^{\lt j^\omega(\kappa)}\subseteq M$; in other words, for every cardinal $\lambda\lt\delta$, $M^\lambda\subseteq M$, where $\delta$ is the least fixed point of $j$. Let's call such a cardinal **almost $\omega$-huge... | https://mathoverflow.net/users/141402 | What is the consistency strength of almost $\omega$-huge cardinals? | Almost $\omega$-huge is equivalent to $\omega$-huge, so it is inconsistent with AC. Closure under $\kappa\_n$-sequences plus closure under $\omega$-sequences implies closure under $\delta$-sequences: given a $\delta$-sequence $\langle a\_\alpha : \alpha < \delta\rangle\subseteq M$, $s\_n = \langle a\_\alpha : \alpha < ... | 11 | https://mathoverflow.net/users/102684 | 377441 | 157,338 |
https://mathoverflow.net/questions/377271 | 7 | Say that a set $X$ is $\Pi^1\_1$-pseudofinite if every first-order sentence $\varphi$ with a model with underlying set $X$ has a finite model. The existence of infinite $\Pi^1\_1$-pseudofinite sets is consistent with $\mathsf{ZF}$, since indeed [every amorphous set is $\Pi^1\_1$-pseudofinite](https://math.stackexchange... | https://mathoverflow.net/users/8133 | Is the hereditary version of this weak finiteness notion nontrivial? | It is consistent that there are infinite hereditarily $\Pi\_1^1$-pseudofinite sets. I'll just say "pseudofinite" instead of "$\Pi\_1^1$-pseudofinite" for the rest of this post.
>
> **Theorem 1.** Let $N$ be a model of ZF-Foundation satisfying "pseudofinite violations of choice for pseudofinite sets": there is a pse... | 2 | https://mathoverflow.net/users/164965 | 377442 | 157,339 |
https://mathoverflow.net/questions/377061 | 20 | Title asks it: Does the Fourier expansion of the j-function have any prime coefficients?
A superabundance of congruences involving primes up to 13 rule out many candidates, but calculation suggests that primes $p>13$ occur as divisors at frequencies (about?) $1/p$.
But
$$c\_{71}=278775024890624328476718493296348769... | https://mathoverflow.net/users/10909 | Does the Fourier expansion of the j-function have any prime coefficients? | There are seven prime values (passing a BPSW test) of $c\_n$ with $n \le 2 \cdot 10^7$, at indices 457871, 685031, 1029071, 1101431, 9407831, 11769911, and 18437999.
For a writeup about the computations, source code, and the prime numbers themselves, see:
<https://github.com/fredrik-johansson/jfunction>
The first p... | 22 | https://mathoverflow.net/users/4854 | 377448 | 157,340 |
https://mathoverflow.net/questions/374748 | 4 | Let $A,B(a,b,c,d)\in\mathsf{GL}(4,\mathbb{Z})$ be given by $$A=\begin{pmatrix} I\_2 & \begin{pmatrix} 0&0\\0&1 \end{pmatrix} \\0& -I\_2\end{pmatrix},\quad B(a,b,c,d)=\begin{pmatrix} -2a-b & 2c& 0& c\\ -\frac{1+(2a+b)^2}{2c}&2a+b&d&a\\0&0&-b& \frac{1+b^2}{2d}\\ 0&0&-2d&b\end{pmatrix}$$ (which implies $a,b,c,d\in \mathbb... | https://mathoverflow.net/users/150901 | Finding isomorphism between $\mathbb{Z}^2\ltimes_{A,B} \mathbb{Z}^4$ and $\mathbb{Z}^2\ltimes_{A,C} \mathbb{Z}^4$ | Finally I've come with an answer to my question. I wasn't sure about answer because it changes a little bit the approach of the question but I'll do anyway.
The problem originally was about finding a classification (up to isomorphism) about the groups $G\_{A,B}$ where $A=P^{-1} \tilde{A} P$ and $B=P^{-1} \tilde{B} P$... | 1 | https://mathoverflow.net/users/150901 | 377450 | 157,341 |
https://mathoverflow.net/questions/377424 | 9 | Have you ever seen this matrix? Each row is obtained from the previous one by multiplying each element by the corresponding element of the next cyclic permutation of $(a\_1,\dots, a\_n)$:
$$\left(
\begin{array}{llllllll}
1 & 1 & 1 & \dots & 1 & 1 \\
a\_1 & a\_2 & a\_3 & \dots & a\_{n-1} & a\_{n} \\
a\_1 a\_2 & a\_2... | https://mathoverflow.net/users/169399 | Matrix obtained by recursive multiplication and a cyclic permutation | Denote the matrix $A$, and index all $a\_i$, and all rows and columns starting from $0$ for convenience. If, say, $a\_0 = 0$, then $\det A = (-1)^{\lfloor (n - 1) / 2 \rfloor} \prod\_{i = 1}^{n - 1} a\_i^i$ by substituting and computing the remaining upper triangular determinant. Let's further assume that all $a\_i$ ar... | 8 | https://mathoverflow.net/users/106512 | 377452 | 157,342 |
https://mathoverflow.net/questions/377451 | 2 | Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
**Definition:** A representation $v$ on the Hilbert space $H$ is an element $v\in M(B\_0(H)\otimes A)$ such that $(\text{id}\otimes \Delta)(v) = v\_{(12)}v\_{(13)}$. Here the subscripts wi... | https://mathoverflow.net/users/nan | Kernel of intertwiner is invariant (compact quantum groups) | Let $e$ be the orthogonal projection onto $\ker(x)$.
If the result is not true, then there is $\xi\otimes\eta \in H\_1\otimes K$ with
$$ (e\otimes 1) v\_1 (e\xi\otimes\eta) \ne v\_1 (e\xi\otimes\eta), $$
because the linear span of such vectors in dense in $H\_1\otimes K$. Here $K$ is some auxiliary Hilbert space such t... | 2 | https://mathoverflow.net/users/406 | 377455 | 157,344 |
https://mathoverflow.net/questions/377381 | 9 | This is well beyond my expertise, but I just learned some of the history behind
$u$-invariants of fields $F$,
where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution,
but $u(F)$-variable equations may not.
Here is [the Wikipedia explanation](https://en.wikipedia.org/wiki/U-invariant).
In partic... | https://mathoverflow.net/users/6094 | Standard conjecture on u-invariants? | For the classical $u$-invariant of fields of characteristic $\neq 2$, some known results are:
1. The $u$-invariant of formally real fields is $\infty$.
2. If $K$ is an algebraically closed field, its $u$-invariant is $1$. More generally, if $K$ does not have quadratic extensions its $u$-invariant is $1$.
3. There are... | 8 | https://mathoverflow.net/users/3903 | 377466 | 157,347 |
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