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https://mathoverflow.net/questions/339932 | 5 | Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty\text{Sh}(C, J)$. There is a
natural geometric morphism to $\infty\text{Grpd}$ whose left adjoint is
the constant sheaf functor $\Delta : \infty \text{Grpd} \to \infty\text{Sh}(C, J)$.
In the 1-topos case, ... | https://mathoverflow.net/users/112642 | Are constant $\infty$-sheaves constant on connected components? | *You need $U$ "contractible"*. In general $\Delta(S)$ is defined exactly as in the $1$-topos case: "take the constant presheaf valued at $S$ and sheafify it" (i.e. applies the left adjoint to the forget full functor from sheaf to presheaf).
This theorem, when one takes $U$ contractible, is indeed also true for $\inft... | 4 | https://mathoverflow.net/users/22131 | 339938 | 144,879 |
https://mathoverflow.net/questions/339865 | 2 | Consider the category of manifolds $\text{Man}$.
A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the category $\text{Man}$) for this $\mathcal{G}$, the stack of principal $\mathcal{G}$ bundles, denoted by $B\mathcal{G}$. Any... | https://mathoverflow.net/users/118688 | Stack associated to Groupoid object in category $\text{Sch}/S$ | For any algebraic stack $X$, there is a groupoid in schemes whose fppf stackification is equivalent to $X$. You can construct such a groupoid following the stacks project, by choosing a smooth presentation in algebraic spaces ([Lemma 04T5](https://stacks.math.columbia.edu/tag/04T5)), then choosing an étale presentation... | 2 | https://mathoverflow.net/users/121 | 339953 | 144,885 |
https://mathoverflow.net/questions/338527 | 6 | I am currently reading Ghys and Sergiescu's paper [Sur un groupe remarquable de difféomorphisms du cercle](http://perso.ens-lyon.fr/ghys/articles/grouperemarquable.pdf) (French only I'm afraid), but part of their proof of Corollary 3.4 (page 20 of the pdf) is opaque to me.
Some notation first though. Let $K = \operat... | https://mathoverflow.net/users/63965 | A sufficient condition for $\pi_1(\operatorname B\Gamma) = 0$ | Peter Greenberg's paper [Pseudogroups from group actions](https://www.jstor.org/stable/2374493) held the key.
There is a weak equivalence between $\mathrm{B}\Gamma$ and $(\mathrm{B}K\_0 \ast \mathrm{B}K\_0) \vee (\mathrm{B}K/\mathrm{B}K\_0)$, where $\ast$ is the join, $\vee$ is the usual wedge sum, and $\mathrm{B}K/\... | 1 | https://mathoverflow.net/users/63965 | 339963 | 144,889 |
https://mathoverflow.net/questions/339705 | 2 | Do you know of any PhD programs that are highly flexible? I'm interested in either math or computer science programs, as I have done research in both areas and feel confident to apply to both.
With high flexibility I mean that there should be freedom for the PhD student to choose:
1) who they work with (so that I ... | https://mathoverflow.net/users/43263 | Highly flexible PhD programs/school (with individual freedom) | This is not really an answer, but an extended comment.
I am repsonding to the following remarks of the OP:
* "I'd also be interested in names of PhD advisors in the domain of analysis/data science/topology that are known to be permissive and allowing their students to go abroead." [Quoted from the comments.]
* "The... | 3 | https://mathoverflow.net/users/102946 | 339973 | 144,891 |
https://mathoverflow.net/questions/339976 | 1 | $x\_1 + x\_2 + \dots + x\_n = 1, 0 \leq x\_i \leq 1$, and $(x\_1, x\_2, \dots, x\_n)$ evenly distributes on its restricted space, obviously which is a polygon on $n - 1$ dimension plane.
Let random variable $Z = \max(x\_1, x\_2, \dots, x\_n)$, what is the probability distribution function $F(m) = P(Z < m)$.
Obviou... | https://mathoverflow.net/users/138336 | What the probability of the max value of restricted random variable? | It is now a textbook fact that the joint distribution of your random variables $x\_1,\dots,x\_n$ is the same as that of $R\_1,\dots,R\_n$, where $R\_i:=H\_i/(H\_1+\dots+H\_n)$ and the $H\_i$'s are iid standard exponential random variables. [Moran, page 93](https://www.jstor.org/stable/2983572?seq=1#page_scan_tab_conten... | 1 | https://mathoverflow.net/users/36721 | 339979 | 144,894 |
https://mathoverflow.net/questions/339982 | 0 | If $\kappa$ is a cardinal and $X$ is a set, let $[X]^\kappa$ denote the collection of subsets of $X$ that have cardinality $\kappa$.
Let $\beta>\omega$ and $\beta \leq 2^{\omega}$. Is there ${\cal C}\subseteq [\mathbb{R}]^\beta$ such that every member of $[\mathbb{R}]^\omega$ is contained in exactly one member of ${\... | https://mathoverflow.net/users/8628 | Collection $\cal{C}$ of uncountable subsets of $\mathbb{R}$ such that every countable subset is contained in exactly one member of $\cal{C}$ | Suppose that every countably infinite subset of $\mathbb R$ is contained in exactly one member of $\mathcal C$, where $\mathcal C\subseteq\mathcal P(\mathbb R)$ and $\mathbb R\notin\mathcal C$. Let $A$ be a countably infinite subset of $\mathbb R$. Choose a set $S\in\mathcal C$ such that $A\subseteq S$, and choose an e... | 6 | https://mathoverflow.net/users/43266 | 339992 | 144,897 |
https://mathoverflow.net/questions/339989 | 3 | The $r^{th}$ moment of the divisor function, for $r\geq 1,$ is well known to obey
$$
\sum\_{n\leq x} \tau(n)^r\sim C\_r x (\log x)^{2^r-1}
$$.
where $C\_1=1.$ In a paper by Florian Luca and L. Toth, available at <https://arxiv.org/abs/1703.08785>, the constant $C\_r$ is also given.
What about general $r \in (1,\inf... | https://mathoverflow.net/users/17773 | Noninteger moments of the divisor function | One can derive a similar asymptotic formula (and even an asymptotic expansion with decreasing powers of $\log x$) for any $r\in\mathbb{C}$. See Ch. II.5 (The Selberg-Delange method) and II.6 (Two arithmetic applications) in Tenenbaum: Introduction to analytic and probabilistic number theory.
| 8 | https://mathoverflow.net/users/11919 | 339996 | 144,899 |
https://mathoverflow.net/questions/339991 | 2 | This is a question from [math.stackexchange](https://math.stackexchange.com/questions/3317035/borel-sigma-algebra-of-a-borel-subset), which was not answered for a month now. I don't feel comfortable to post it on mathoverflow, but I am somehow blind to see the mistake in the argumentations below.
Let $(X, \tau)$ be a... | https://mathoverflow.net/users/58682 | Borel $\sigma$-algebra of a Borel subset | The problem is that you have to take uncountable unions of sets of the form $[a,b) \times [c,d)$ to get every open set in the Sorgenfrey plane, so the $\sigma$-algebra generated by $[a,b) \times [c,d)$ is strictly smaller than the Borel $\sigma$-algebra.
Which is to say, in your notation, $\sigma(\tau\_e) \subseteq ... | 7 | https://mathoverflow.net/users/61785 | 340000 | 144,900 |
https://mathoverflow.net/questions/339878 | 5 | Given tall matrices $A$ and $Y$ and the following overdetermined linear system in square matrix $X$
$$AX=Y$$
is there an explicit formula for the least-squares solution if $X$ is constrained to be symmetric?
| https://mathoverflow.net/users/70012 | Symmetric linear least-squares solution | I assume that $A$ is onto, so that $H:=A^TA$ is positive definite. Minimizing $\|AX-Y\|\_F^2$ in Frobenius norm (the least square) among symmetric matrices $X$ yields the optimality condition that
$$\langle AS,AX-Y\rangle=0$$
for every symmetric $S$. This amounts to saying that $A^T(AX-Y)$ is skew-symmetric. In other w... | 6 | https://mathoverflow.net/users/8799 | 340001 | 144,901 |
https://mathoverflow.net/questions/339998 | 0 | Let $sFre\_{\mathbb{R}}$ *(resp. $Fre\_{\mathbb{R}}$)* denote the category of *(resp. separable)* Fr\'{e}chet spaces over $\mathbb{R}$ as objects, and bounded linear operators as morphisms.
Is this a topological concrete category over $\mathbb{R}-Vect$; in the sense of [this post](https://ncatlab.org/nlab/show/topol... | https://mathoverflow.net/users/36886 | Category of Frechet Spaces is Topological? | Posted as an answer at the OP‘s request: Since the category of Fréchet spaces doesn‘t admit products and sums in general, the answer would appear to be negative. This is analogous to the fact that the category of metric spaces is apparently not topological whereas that of uniform spaces is (for similar reasons).
| 4 | https://mathoverflow.net/users/131781 | 340009 | 144,902 |
https://mathoverflow.net/questions/339789 | 9 | Write $p(n)$ for the number of integer partitions of $n$. For $S \subseteq \{1, \ldots, n\}$, let $p\_S(n)$ be the number of partitions of $n$ with all parts in $S$. So $p(n) = p\_{\{1,\ldots,n\}}(n)$.
>
> Conjecture: Given positive integers $n$ and $k$ with $0 \le k \le p(n)$, there is an $S \subseteq \{1, \ldots... | https://mathoverflow.net/users/14807 | Every possible number of partitions by restricting parts? | The conjecture seems to hold.
For brevity, denote $[k,n]=\{k,k+1,\dots,n\}$ and $[n]=[1,n]$. Start with $S = [n]$.
**Algorithm:** Given $0\leq k\leq p(n)$, consider the numbers $1,2,\dots,n$ one by one, removing the number if the remaining set $S$ satisfies $p\_S(n)\geq k$. After all $n$ numbers are considered, we ... | 2 | https://mathoverflow.net/users/17581 | 340021 | 144,904 |
https://mathoverflow.net/questions/339994 | 2 | The problem is the following:
>
> Let $k$ be a field and let $N,G,Q: (\operatorname{Sch}/k)\_{\operatorname{fppf}} \longrightarrow \operatorname{Grps}$
> be fppf-sheaves from the big fppf-site of $\operatorname{Sch}/k$ to the
> category of groups. Assume that we have an exact sequence of
> fppf-sheaves $$ e\lon... | https://mathoverflow.net/users/138402 | fppf-extension of algebraic groups is an algebraic group | Here is a possible road to a solution (I am fairly sure that Milne had something more elementary in mind). Algebraic spaces satisfy fppf descent; hence $G$ is a group object is in the category of algebraic spaces of finite type over $k$. But it is well known that a group object is in the category of finitely generated ... | 3 | https://mathoverflow.net/users/4790 | 340022 | 144,905 |
https://mathoverflow.net/questions/340038 | 3 | Suppose I have $\|f\_n\|\_{2}^2=\int\_{S}f\_n(x)^2dx\rightarrow 0$ over a compact set $S\subset R^d$, and $\{f\_n\}$ is $1$-Lipschitz and $1$-smooth. What kind of extra condition can I add on $S$ so that $\|f\_n\|\_{\infty}\rightarrow 0$?
If $S$ is $R^d$, no extra condition is needed, since for any $|f(x)|>0$ we can... | https://mathoverflow.net/users/145524 | How does convergence in $\ell^2$ norm imply convergence in $\ell^\infty$-norm with Lipschitz conditions? | It suffices that every neighborhood of every point in $S$ have a nonzero Lebesgue measure, that is,
$$g\_r(x):=|S\cap B\_r(x)|>0\tag{1}
$$
for all $x\in S$ and all real $r>0$, where $B\_r(x):=\{y\in\mathbb R^d\colon\|y-x\|<r\}$, $\|\cdot\|$ is the Euclidean norm, and $|\cdot|$ is the Lebesgue measure.
Indeed, it ea... | 4 | https://mathoverflow.net/users/36721 | 340041 | 144,907 |
https://mathoverflow.net/questions/340028 | 3 | For Hilbert spaces $\mathcal{H}\_X$, $\mathcal{H}\_Y$ and $\mathcal{K}$, consider a linear map $f\colon \mathcal{K} \oplus \mathcal{H}\_X \to \mathcal{K} \oplus \mathcal{H}\_Y$ that is given as a matrix
$$f = \begin{pmatrix} A \colon \mathcal{K} \to \mathcal{K} & B\colon \mathcal{H}\_X \to \mathcal{K} \\ C\colon \mat... | https://mathoverflow.net/users/145510 | Is this sum of nonexpansive maps itself nonexpansive? | The answer to your conjecture is **yes**, and you are completely right that the result for isometries implies that result for nonexpansive mappings (which I will simply all *contractions* here).
This follows from Sz.-Nagy's dilation theorem for contractions on Hilbert spaces, which says the following:
**Theorem.** ... | 3 | https://mathoverflow.net/users/102946 | 340046 | 144,909 |
https://mathoverflow.net/questions/340027 | 26 | Does the Sobolev space $H^1(R^n)$ of weakly differentiable functions on a bounded domain in $R^n$ (or a more general Sobolev space) contain a continuous but nowhere differentiable function?
| https://mathoverflow.net/users/56920 | Do Sobolev spaces contain nowhere differentiable functions? | As Nate Eldredge pointed out, $W^{1,2}(\mathbb{R})$ functions are absolutely continuous on $\mathbb{R}$, and therefore differentiable a.e., and so the answer is no.
For $n\geq 2$, the answer is yes.
When n=2, this is a classical result of L. Cesari
*Cesari, Lamberto*, [**Sulle funzioni assolutamente continue in ... | 19 | https://mathoverflow.net/users/nan | 340048 | 144,911 |
https://mathoverflow.net/questions/340055 | 7 | Let $S$ be the surface of a convex body, polyhedral or smooth,
embedded in $\mathbb{R}^3$.
For a point $x \in S$, let $F(x)$ be the set of furthest points
from $x$, measured by shortest paths on the surface $S$.
Let $f(x)$ be the length of those shortest paths:
$|x y|$ for $y \in F(x)$.
It seems natural to hope that... | https://mathoverflow.net/users/6094 | Furthest distance half the diameter? | Denote the diameter by $d$ and distance by $|x-y|$. Then there
are $y,z$ such that $d=|y-z|$ and we have by triangle inequality for every $x$:
$$d=|y-z|\leq |y-x|+|x-z|\leq 2f(x),$$
so we obtain your inequality. Notice that I did not use convexity, or any other of your assumptions, only the
triangle inequality.
| 14 | https://mathoverflow.net/users/25510 | 340056 | 144,914 |
https://mathoverflow.net/questions/339970 | 9 | Let $X$ be a a discrete RV with $\mathbb{P}(X=k)<p$ for every $k$ (that's all we know). Taking the independent sum $S=X\_1+X\_2+\cdots+X\_n$, with each $X\_i$ distributed like $X$, what can we say about an upper bound for
$$
\max\_k \mathbb{P}(S=k) \text{ ?}
$$
If it helps, $n$ is large, but $n\ll \frac1p$. Is it true ... | https://mathoverflow.net/users/12487 | Bounding maximum probabilities in sum of i.i.d discrete RVs | The requested estimate follows from a theorem of Kesten [2] about concentration functions.
See in particular the inequality (1.6) on page 135 of [2]. Taking $L=\lambda<1$ in
this inequality gives the proposed inequality with some constant $c$. The constant $c$ needed depends on the constant in Rogozin's Theorem [1].
... | 3 | https://mathoverflow.net/users/7691 | 340061 | 144,917 |
https://mathoverflow.net/questions/340068 | 2 | Given a quiver algebra $A=KQ/I$ with acyclic $Q$. Then $A$ has finite global dimension, lets say $g$.
>
> Question: Is there for any $0 \leq i \leq g$ a simple module with projective dimension equal to $i$? Is this true when $Q$ is a linear oriented quiver of Dynkin type $A\_n$ ?
>
>
>
I guess the first questi... | https://mathoverflow.net/users/61949 | Projective dimensions of simple modules in acyclic quiver algebras | This is true by induction on the number of vertices.
Let $v$ be a source vertex of $Q$, and $B$ the quiver algebra obtained by removing $v$. Assume that $B$ has simples of all projective dimensions up to $\text{gldim} B$.
The simples for $B$ have the same projective dimensions as the corresponding simples for $A$,... | 4 | https://mathoverflow.net/users/22989 | 340070 | 144,919 |
https://mathoverflow.net/questions/340015 | 3 | In Jardine's book Local Homotopy Theory P.102, a simplicial presheaf $X$ on a site $C$ is said to satisfy descent if some local injective fibrant replacement $ j : X → Z $ is a sectionwise weak
equivalence.
The definition in Dugger Hollander Isaksen defines a presheaf $F$ to satisfy descent if
$F(X)$ is weak equiva... | https://mathoverflow.net/users/144294 | Descent in the injective model structure and descent for simplicial presheaves | According to DHI's official definition (Def 4.3), a *sectionwise fibrant* simplicial presheaf $F$ satisfies descent if $F(X) \simeq \text{holim}\, F(U\_{\bullet})$ for all hypercovers. If $F$ is not sectionwise fibrant then it satisfies descent if there's a sectionwise weak equivalence $F \to G$ such that $G$ is sectio... | 2 | https://mathoverflow.net/users/33143 | 340081 | 144,922 |
https://mathoverflow.net/questions/340036 | -1 | In $\mathbb{H}^n$, equipped with its hyperbolic metric of constant curvature $-1$, if we have two points $p,q$ on a common horosphere $\partial S$, then $$d\_{\mathbb{H}}(p,q) = 2\sinh^{-1} (d\_{\partial S} (p,q)/2)$$ where $d\_{\partial S}(p,q)$ is the shortest path from $p$ to $q$ that lies on the horosphere.
Is th... | https://mathoverflow.net/users/135446 | Horospherical distance in CAT($-1$) space | I suppose I'll answer my own question.
According to this paper: <https://projecteuclid.org/download/pdf_1/euclid.jdg/1214434219>
if $M$ is a Hadamard manifold with pinched sectional curvature $-b^2\leq K \leq -a^2$, and if $H$ is a horosphere and $h$ represents distance in $H$, then $$\frac{2}{a}\sinh (a/2) d(p,q)\le... | 0 | https://mathoverflow.net/users/135446 | 340087 | 144,923 |
https://mathoverflow.net/questions/340065 | 5 | This is cross-posted at [MSE](https://math.stackexchange.com/q/3347019).
I'm looking for a reference for the following result. It seems like it must be known, or follow quickly from something known, but I have not been able to find it in any of the textbooks I have.
Let $(\Omega, \mathcal F, \mathbb P)$ be a proba... | https://mathoverflow.net/users/117180 | What is this disintegration-like theorem? | The measure-theoretic formulation of conditional probability/expectation is cumbersome, so it is often easy to lose connection between intuitive facts and the formal demonstrations of them. I personally often have difficulty with arguing for seemingly obvious facts.
In this case, I think more than your theorem is tru... | 2 | https://mathoverflow.net/users/23297 | 340104 | 144,929 |
https://mathoverflow.net/questions/340085 | -2 | What is the difference between the two sets of the following Cauchy integral,
$$
\begin{split}
\int\_c t^k \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=4\pi i \zeta^k\\[8pt]
\int\_c \frac{1}{t^k} \cdot \frac{t+\zeta}{t-\zeta} \frac{dt}{t} &=0
\end{split}
$$
from G. N. SAVIN (1968), *[Stress Distribution around Holes](h... | https://mathoverflow.net/users/125078 | Cauchy integral and residue theorem | A modest proposal. Could it be that there is a typo or misunderstanding in the formulation? If $\zeta$ were inside, rather than on, the circle, the first pair of formulae would be correct—-a direct consequence of the Cauchy integral formula. Analogously for the second one if it were outside.
If $\zeta$ is really on t... | 0 | https://mathoverflow.net/users/131781 | 340119 | 144,933 |
https://mathoverflow.net/questions/340110 | 0 | This appears in the section 3.7 of the book *Compact Riemann Surfaces* by Jurgen Jost, right after Lemma 3.7.3. The exact words are
>
> Now let $v:\Sigma\_1\to\Sigma\_2$ be a Lipschitz continuous map. Cover $\Sigma\_1$ by coordinate neighborhoods. Choose $R\_0<1$ so that, for every $z\_0\in\Sigma\_1$, a disc of the... | https://mathoverflow.net/users/143284 | What is a Lipschitz continuous map between Riemann surfaces in Jost's book Compact Riemann Surfaces? | While changing the Riemannian metric will change the constants, as you observe, on a compact manifold it will not change the class of Lipschitz continuous function, defined as usual as the set of functions that do not increase distances between points by more than a constant factor. This is a common regularity hypothes... | 2 | https://mathoverflow.net/users/4961 | 340121 | 144,934 |
https://mathoverflow.net/questions/340132 | 1 | I've known few days ago the known as Steffensen's inequality, see the article *Steffensen's inequality* from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't know what are) in research of this inequality or generalizations. I wondered if one can to elaborate some exercise at u... | https://mathoverflow.net/users/142929 | Examples of Steffensen's inequality at undergraduated level studies | As pointed out by [Mitrinovich](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&cad=rja&uact=8&ved=2ahUKEwiE45zCrsHkAhVLXKwKHYyPDVcQFjAHegQIABAB&url=https%3A%2F%2Fwww.jstor.org%2Fstable%2Fpdf%2F43667360.pdf&usg=AOvVaw0-KYTTRVu5OtsosK7su5Z2), Steffensen himself used his inequality to derive, in particula... | 4 | https://mathoverflow.net/users/36721 | 340140 | 144,937 |
https://mathoverflow.net/questions/340117 | 4 | Babai, Kantor and Lubotzky proved in 1989 the following theorem ([Sciencedirect link to article](https://www.sciencedirect.com/science/article/pii/S0195669889800678)).
THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ of at most 7 generators for which the diameter o... | https://mathoverflow.net/users/97333 | Diameter of Cayley graphs of finite simple groups | There are two examples, $\mathrm{Alt}\_n$ and $\mathrm{PSL}\_2(q)$, in [this](https://pages.uoregon.edu/kantor/PAPERS/STOCdiameter.pdf) paper (p.861).
For $\mathrm{Alt}\_n$, the authors used 3 generators and achieved diameter at most $(1+o(1))4n\log n$.
For $\mathrm{PSL}\_2(q)$, an upper bound is $12\log\_4(q)$ (E... | 3 | https://mathoverflow.net/users/125498 | 340141 | 144,938 |
https://mathoverflow.net/questions/340112 | 1 | An inequality from the following paper
MIKOLAS, M., *Sur un probleme d'extremum et une extension de l'inegalite de Minkowski*, **Ann. Univ. Sci. Budapest Eotvos, Sect. Math.** 1 (1958),
101-106
is discussed in Mitrinovic's *Analytic Inequalities* book. The inequality is stated as below.
>
> Let $x\_{uv},$ $p,r... | https://mathoverflow.net/users/17773 | Conditions for Minkowski Style Vector Inequality to Hold | As suggested by @FrancoisZiegler, I looked at the original with my basic French and have concluded a few things.
**Erratum:** The theorem is wrongly stated in Mitrinovic, as well as the more recent Mitrinovic, Pecaric and Fink *Classical and New Inequalities in Analysis*, Chapter 8, p. 107. The correct statement from... | 1 | https://mathoverflow.net/users/17773 | 340174 | 144,946 |
https://mathoverflow.net/questions/340173 | 9 | It's known, for example in the answer to this question: [Is there a computable model of ZFC?](https://mathoverflow.net/questions/12426/is-there-a-computable-model-of-zfc) that ZFC has no computable model. My questions is: is there a model of ZFC for which the order relation on the class of ordinals is computable?
| https://mathoverflow.net/users/50128 | Computable models of the ordinal numbers | Yes, this can happen: if $M$ is a countable $\omega$-model of ZF whose well-founded part has ordertype $\omega\_1^{CK}$ (that is: has the shortest well-founded part possible for $\omega$-models), then $Ord^M$ as a linear order is just the **Harrison order**: $$\omega\_1^{CK}+(\omega\_1^{CK}\cdot\eta),$$ where $\eta$ is... | 16 | https://mathoverflow.net/users/8133 | 340175 | 144,947 |
https://mathoverflow.net/questions/336209 | 3 | Consider $(\mathbb{C}P^2,\omega\_{FS})$ where $\omega\_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int\_{L} \omega\_{FS} = \pi$.
Then the map
$$\begin{align}
i : (B(1),\omega\_0) &\to (\mathbb{C}P^2 \setminus L, \omega\_{FS}) \\
... | https://mathoverflow.net/users/92483 | Star-shaped domain in $\mathbb{C}P^2$ | One way to prove this is as follows.
First, from the assumption $B(1)\subset \mathbb C^2$ and the centre of $B(1)$ is $(0,0)$. Now we need the following two claims.
*Claim 1.* Any straight geodesic unit segment $I$ in $B(1)$ going through $(0,0)$ is sent by $i$ to a geodesic segment in $\mathbb CP^2$ through the po... | 2 | https://mathoverflow.net/users/943 | 340186 | 144,950 |
https://mathoverflow.net/questions/340195 | 3 | Consider the class of groups in the signature {\*}. Is the quasi-equational theory of that class axiomatized by the associative law and the left and right cancellative laws?
| https://mathoverflow.net/users/43439 | Is the quasi-equational theory of groups the same as cancellative semigroups? | No.
Every group satisfies
$$
(xy\approx x'y')\wedge (zy\approx z'y')\wedge (zw\approx z'w')\to (xw\approx x'w')
$$
but this quasi-identity is not derivable from associativity + cancellativity. You can find this in Maltsev's papers *Uber die Einbettug von assoziativen Systemen in Gruppen* parts I and II. In fact,... | 7 | https://mathoverflow.net/users/75735 | 340200 | 144,956 |
https://mathoverflow.net/questions/340193 | 2 | Given a Riemannian manifold $(\mathcal{M}, \{g\_x\}\_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d\_\mathcal{M}(x, y) \le 1 \}$?
1. Sample uniformly from $\{u \in \mathcal{T}\_x \mathcal{M} : \lVert u \rVert\_x \le 1\}$ (s... | https://mathoverflow.net/users/127367 | Uniform sampling on a Riemannian manifold via tangent space and exponential map | As Nate Eldredge pointed out in his comment, your two-step procedure will not in general simulate (a random element of $\mathcal M$ with) the uniform density $f=1/|B\_x|$ on $B\_x:=\{y\in\mathcal M\colon d\_{\mathcal M}(y,x)\le1\}$, where $|B\_x|$ is the volume of $B\_x$. However, this "two-step" density, say $h$, can ... | 3 | https://mathoverflow.net/users/36721 | 340209 | 144,960 |
https://mathoverflow.net/questions/340187 | 7 | $\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\abs}[1]{\lvert #1\rvert}\newcommand{\comptensor}{\mathbin{\hat{\otimes}}}$
Let $k$ be a complete non-archimedian field and let $X = \Sp(B)$ be a $k$-affinoid space. Let $V = \Sp(B') \subseteq X$ be an affinoid subdomain. It is well-known that the corresponding map $B \to B'$ i... | https://mathoverflow.net/users/112369 | Are maps corresponding to affinoid subdomains flat in the Banach sense? | This is not true. Assume that $X$ is the closed unit disc (given by $|T| \le 1$, with algebra $B$) and $V$ is a smaller disc (given by $|T| \le r$ for some $r \in (0,1)$, with algebra $B\_V$). Consider the annulus $W$ defined by $|T|=1$ with algebra $B\_W$. Then the restriction map $B \to B\_W$ is injective and admissi... | 6 | https://mathoverflow.net/users/4069 | 340210 | 144,961 |
https://mathoverflow.net/questions/340214 | 9 | *Background:* I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier today I was reading the book [1] by Malek, Necas, Rokyta, Ruzicka, and I have a question from the Subsection 1.2.8 ti... | https://mathoverflow.net/users/117762 | Is the space of Radon measures a Polish space or at least separable? | With respect to the norm topology, the space of Radon measures on a domain $\Omega$ is not separable. Indeed, for any two distinct points $x,y$ in $\Omega$, the Dirac measures $\delta\_x$ and $\delta\_y$ (where $\delta\_x(f)=f(x)$) satisfy $\|\delta\_x-\delta\_y\|=2$
since you can always find a compactly supported smoo... | 8 | https://mathoverflow.net/users/7691 | 340218 | 144,964 |
https://mathoverflow.net/questions/340228 | 3 | Let N be a set of n arbitrary points in $\mathbb{R}^2$ and let $k \in \{1,...,n\}$. Is it true that there always exists a disk $D$ containing exactly k points of N?
| https://mathoverflow.net/users/145642 | Existence of a disk containing a given number of points of a discrete set | Yes. From a generic point (in this case, in the complement of the ${n \choose 2}$ perpendicular bisectors of each pair), the distances to the members of $N$ are all
distinct, so for each $k$, disks of suitable radius centred at such a point will contain exactly $k$ of them.
| 8 | https://mathoverflow.net/users/13650 | 340229 | 144,969 |
https://mathoverflow.net/questions/340202 | 3 | For non-negative integers $k$ and $l$ let $p(k,l)$ denote the number of vector partitions of $(k,l)$. In other words, $p(k,l)$ is the number of ways of writing
$$
(k,l) = (k\_1,l\_1)+\dotsb + (k\_r,l\_r),
$$
where $(k\_i,l\_i)$ are non-zero vectors with non-negative integer coordinates, the order of the summands being ... | https://mathoverflow.net/users/9672 | Strict unimodality of bipartite partitions | I can prove that $p(k+1, l-1) < p(k,l)$ for all $k \ge l > 1$ except $k=l$ odd. Modifying the notation from the articles, let $p\_j(k,l)$ be the number of partitions of $(k,l)$ into exactly $j$ parts, so that $p(k,l) = \sum\_{j=1}^\infty p\_j(k,l)$. The argument below shows that $p\_2(k+1,l-1) < p\_2(k,l)$ except when ... | 3 | https://mathoverflow.net/users/14807 | 340231 | 144,970 |
https://mathoverflow.net/questions/340233 | 4 | I know that in the literature there are a lot of integrals involving the fractional part (and other floor and ceiling functions). For some of these integrals is provide its evaluation as a series or sum of series involving particular values of special functions when isn't possible to get the sum in closed-form.
I've ... | https://mathoverflow.net/users/142929 | Express $\int_0^{\pi/2}\{ \operatorname{gd}^{-1}(x)\}dx$ as series of special functions, with $\operatorname{gd}^{-1}(z)$ the inverse Gudermannian | The integral is twice [Catalan's constant](https://en.wikipedia.org/wiki/Catalan%27s_constant)
$$
G = L(1,\chi\_4) = 1 - \frac1{3^2} + \frac1{5^2} - \frac1{7^2} + \frac1{9^2} - + \cdots.
$$
This constant can be computed efficiently to high precision, even though no further "closed form" is known or expected.
I guesse... | 8 | https://mathoverflow.net/users/14830 | 340243 | 144,974 |
https://mathoverflow.net/questions/340234 | 2 | Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y\_1,y\_2)$ and $W$ takes values on $(w\_1,w\_2)$. The conditional probability density of $W$ given $Y$ is given by $f\_{W|Y}$, assumed to be continuous and twice differentiable.
Let $X$ be a continuous increasing function of random v... | https://mathoverflow.net/users/143140 | Sufficient condition for function of conditional probability density to be increasing | First here, some preliminary remarks: To avoid uninteresting technical complications, let us understand the term "increasing" in the non-strict sense of "non-decreasing".
Related to this is the remark that the conditions of continuity and differentiability are inessential in this context, because nonsmooth functions... | 4 | https://mathoverflow.net/users/36721 | 340244 | 144,975 |
https://mathoverflow.net/questions/340247 | 0 | Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions", vol. 1, pg 104) if $u$ is subharmonic on $D$, then there is a unique Borel measure $\mu$ such that for all compact $E$ in $D$ w... | https://mathoverflow.net/users/100746 | A question about Riesz decomposition theorem | 1) The formula is then meaningless: $h$ can be completely arbitrary (as there is no interior of $E$).
2) Yes, this is true. Consider a larger compact set $E'$ which is contained in $D$ and such that the interior of $E'$ contains $D$. Apply the decomposition theorem to $E'$ to get $$u(x) = \int\_{E'} K(x - \xi) d\mu(\... | 2 | https://mathoverflow.net/users/108637 | 340250 | 144,976 |
https://mathoverflow.net/questions/340252 | 0 | Consider the following 'wrong' question.
*Let $f(x) \in F[x]$ be an irreducible polynomial in a polynomial ring of a field $F$. Let $L$ be the splitting field of $f(x)$ over $F$. Assume that $L$ is a Galois extension over $F$. Let $\alpha \in L$ be a root of $f(x)$. Consider an intermediate field $L-K-F$. Let $g(x) \... | https://mathoverflow.net/users/46260 | Splitting field of an intermediate field | Even when $K/F$ is quadratic (hence galois and abelian), it can happen that $M$ is smaller than $L$.
For a straightforward counterexample, take a chain of groups $\{1\}<I<H<D\_8$ such that $I$ is not normal (cf. [subgroups of D8](https://groupprops.subwiki.org/wiki/Subgroup_structure_of_dihedral_group:D8)). Let $L/F$... | 1 | https://mathoverflow.net/users/11919 | 340259 | 144,978 |
https://mathoverflow.net/questions/339988 | 6 | $\DeclareMathOperator\SO{SO}$Let $F$ be a finite extension of $\mathbb{Q}\_p$, and let $\mathfrak{o}$ denote the ring of integers, with maximal ideal $\mathfrak{p}$. Let $G\_l$ denote the finite group $\SO\_3(\mathfrak{o}/\mathfrak{p}^l)$.
>
> **Question:** Is there a formula for the number of irreducible represent... | https://mathoverflow.net/users/105652 | Number of irreducible representations of $SO_3(\mathfrak{o}/\mathfrak{p}^l)$ | As far as I know, there is currently no such explicit formula in the literature. In fact, even for $\mathrm{SO}\_3(\mathbb{F}\_q)$ (i.e., the case $l=1$), I have not seen a neat table of the dimensions and multiplicities of all irreps, although this could certainly be obtained from the work of Lusztig, [**Irreducible r... | 3 | https://mathoverflow.net/users/2381 | 340262 | 144,980 |
https://mathoverflow.net/questions/340127 | 1 | **There is a Uniform Hopf Inequality as follow:**
Let $\Omega \subset \mathbb{R}^n$, $n \geq 1$ denote a smoothly bounded domain. Also let $\rho(x)=\mathrm{dist}(x,\partial \Omega)$, the distance function from $\partial \Omega$. Assume that $f ≥ 0$ belongs to $L^∞(Ω)$ and let $u$ denote the solution of
$$
\begin{case... | https://mathoverflow.net/users/76453 | Uniform Hopf Inequality | It seems that you are looking for so-called **lower heat kernel bounds**.
* For the heat equation the inequality you are looking for can, for instance, be found in Theorem 1.1 of "Zhang: The Boundary Behavior of Heat Kernels of Dirichlet Laplacians (Journal of Differential Equations, 2002)".
* For the fractional Lapl... | 2 | https://mathoverflow.net/users/102946 | 340265 | 144,981 |
https://mathoverflow.net/questions/340277 | 5 | Let $\alpha$ be an algebraic number and $G$ be a connected $\mathbb Q(\alpha)$-algebraic group, and $d\in\mathbb Z^+$.
We fix a faithful representation of $G$ in some $GL\_n$ and identify $G$ with its image under this representation. For a subring $R$ of $\mathbb Q(α)$, $G(R):=G\cap GL\_n(R)$.
Does $G(\mathbb Z[d\a... | https://mathoverflow.net/users/142244 | For algebraic $\alpha$, does $G(\mathbb Z[d\alpha])$ have a finite index in $G(\mathbb Z[\alpha])$? | The answer is yes for $\alpha$ an integer and no for $\alpha$ a general algebraic number.
To see the problem with algebraic numbers, take $\alpha=1/2, d=2$. Then almost any algebraic group, in particular $\mathbb G\_m$ or $\mathbb G\_a$, will not have its $\mathbb Z$ points a finite-index subgroup of its $\mathbb Z[1... | 6 | https://mathoverflow.net/users/18060 | 340285 | 144,984 |
https://mathoverflow.net/questions/340274 | 1 | Define a *pro-perfect set* $S$ to be a finite set of positive integers satisfying the following three properties:
1. $1\in S$.
2. $\displaystyle\sum\_{n\in S}n^{-1}\in S$
3. There exists a unique permutation $\pi$ of $S$ of order $2$ such that $$\frac{1}{\vert S\vert}\sum\_{n\in S}n\pi(n)\in S.$$
**Question 1**: Is... | https://mathoverflow.net/users/13625 | A possible axiomatic characterization of the set of divisors of a perfect number | Try this.
$$
S = \{1,2,3,12,18,36\}
$$
then
$$
\sum\_{k \in S}\frac{1}{k} = 2 \in S
$$
and
$$
\frac{1}{6}\sum\_{k \in S} k \pi(k) = 36 \in S
$$
when $\pi$ is the permutation of $S$ that reverses the order, while
$$
\frac{1}{6}\sum\_{k \in S} k \pi(k) > 36
$$
for any other permutation $\pi$.
Uniqueness of the minimi... | 7 | https://mathoverflow.net/users/454 | 340286 | 144,985 |
https://mathoverflow.net/questions/340267 | 7 | Let's work over the complex numbers $\mathbb{C}$. Let $g\geq3$ be an integer. Let $\mathcal{M}\_g$ be moduli stack of smooth genus $g$ curves. Let $M\_g$ be the corresponding coarse moduli scheme. They share an open subscheme $M\_g^\circ$ parametrizing automorphism-free smooth genus $g$ curves.
What is the fundament... | https://mathoverflow.net/users/nan | Fundamental group of $M_g^\circ$ | Except in a few trivial cases, the locus of curves which have an extra automorphism will have codimension greater than one in $\mathcal M\_g$. When that happens, the fundamental group of $\mathcal M\_g$ must equal the fundamental group of $\mathcal M\_g^{\circ}$. (To see this, one can pass to the $3$-torsion cover of $... | 11 | https://mathoverflow.net/users/18060 | 340287 | 144,986 |
https://mathoverflow.net/questions/340275 | 21 | I am a little confused about what I think must be either a standard theorem or a standard counterexample in model theory, and I am hoping that the MathOverflow model theorists can help sort me out about which way it goes.
My situation is that I have a chain of submodels, which is not necessarily an elementary chain,
... | https://mathoverflow.net/users/1946 | Is the union of a chain of elementary embeddings elementary? | First I'll give a counterexample to the natural generalization, adapted from my earlier comment:
Let $M\_n = (\mathbb{N},<)$ for all $n$, and let $N\_n = (\mathbb{N}\sqcup \frac{1}{n!}\mathbb{Z},<)$, where all elements of $\frac{1}{n!}\mathbb{Z}$ are greater than all elements of $\mathbb{N}$ in the order. Note that ... | 19 | https://mathoverflow.net/users/2126 | 340298 | 144,993 |
https://mathoverflow.net/questions/338061 | 3 | As tell us the Wikipedia section dedicated to [*Odd perfect numbers*](https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers) (please, see also the related references if you need it), any perfect number has the form $$n=q^\alpha m^2$$
where the integer $\alpha\geq 1$ satisfies $\alpha\equiv 1\text{ mod }4$, t... | https://mathoverflow.net/users/142929 | Bounds for the number of prime numbers less than the Euler's factor, the radical and the greatest prime factor, respectively, of an odd perfect number | Since we have good asymptotics for $\pi(n)$ by the prime number theorem (and can get good explicit bounds on that from Rosser and Schoenfeld's work as well as later work such as that by Dusart) this question is essentially the same as asking for interesting upper and lower bounds on $q^a$, $\mathrm{rad}(n)$ and $\mathr... | 4 | https://mathoverflow.net/users/127690 | 340302 | 144,995 |
https://mathoverflow.net/questions/339796 | 21 | The [Hilbert–Pólya conjecture](https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture) is the name given to the idea that the "reason" or "explanation" for the collinearity of the non-trivial zeros of the Riemann zeta function $\zeta(s)$ is that they are the spectrum of some self-adjoint operator. The empi... | https://mathoverflow.net/users/3106 | Is the Hilbert–Pólya intuition vindicated in the function field case? | In the function field setting, the most natural spectral explanation for the Riemann hypothesis might be expressing the eigenvalues of Frobenius as the eigenvalues of a unitary operator on a finite-dimensional vector space (times $\sqrt{q}$). This would be related to the Hilbert-Polya picture by a logarithm.
There ar... | 6 | https://mathoverflow.net/users/18060 | 341302 | 144,996 |
https://mathoverflow.net/questions/341301 | 3 | Let $p$ be a prime and $\xi\_n$ a nth primitive root of unity. What is the ring of integers of $\mathbb{Q}\_p(\xi\_n)$? Is it $\mathbb{Z}\_p[\xi\_n]$?
| https://mathoverflow.net/users/143426 | ring of integers of a cyclotomic extension of $\mathbb{Q}_p$ | Yes. This follows e.g. from Propositions 16 and 17 in Section IV.4 of Serre: Local fields.
Indeed, let $n=p^mn'$, where $(n',p)=1$. By Proposition 17, $\mathbb{Q}\_p(\zeta\_{p^m})$ has ring of integers $\mathbb{Z}\_p[\zeta\_{p^m}]$. Then, applying Proposition 16 to the local field $\mathbb{Q}\_p(\zeta\_{p^m})$, we se... | 3 | https://mathoverflow.net/users/11919 | 341303 | 144,997 |
https://mathoverflow.net/questions/340002 | 0 | In this ocassion we consider the followgin series that involve ${n\brace k}$ the Stirling number of the second kind and $(n)\_k$ the Pochhammer symbols. I've known from an informative point of view that in the literature was explored an example versus the definition of irrational absolutely abnormal numbers (for exampl... | https://mathoverflow.net/users/142929 | The series $\sum_{n=1}^\infty {2n\brace n}^{-{2n\brace n}}$ and $\sum_{n=1}^\infty (2n)_{n}^{-(2n)_{n}}$ in the context of normal numbers | While it is very likely both numbers are absolutely normal, simply by appealing to the idea that there's no obvious reason why they should be abnormal, current proof techniques are very far from being able to prove the normality of such numbers in a given base, let alone in all bases simultaneously.
The closest thing... | 1 | https://mathoverflow.net/users/38622 | 341305 | 144,998 |
https://mathoverflow.net/questions/340300 | 2 | Given a semisimple complex Lie algebra $\frak{g}$ of rank $r$, with Chevally generators $E\_i,F\_i,K\_i$. Let $V$ be a finite dimensional representation of $\mathfrak{g}$ such that each weight space of $V$ is $1$-dimensional. Let $(i\_1,\dots,i\_k)$ be an ordered set of elements of $\{1,\dots,r\}$ (allowing repeats), a... | https://mathoverflow.net/users/126606 | Semisimple Lie algebra modules with $1$-dimensional weight spaces | Let me flesh out the answer suggested in the comments. Any complex simple Lie algebra can be obtained as a complexification of a (split) Lie algebra $\mathfrak{g}\_{\mathbb{Q}}$ defined over the field of rational numbers. The irreducible finite dimensional representations of $\mathfrak{g}$ are complexifications of repr... | 2 | https://mathoverflow.net/users/6818 | 341311 | 145,002 |
https://mathoverflow.net/questions/341332 | 2 | I would like to measure the similarity between a pair of weighted tree graphs. According to this [post](https://mathoverflow.net/a/38316/145438), this can be done by regarding the trees as metric spaces and then applying the Gromov-Hausdorff distance.
Given two metric spaces, $(A, d\_{A})$ and $(B, d\_{B})$, the Gro... | https://mathoverflow.net/users/145438 | Gromov-Hausdorff distance between weighted tree graphs | I would direct you to Theorem 7.3.25 of the book "A Course in Metric Geometry" by Burago-Burago-Ivanov.
Roughly speaking, the Gromov-Hausdorff distance between two compact metric spaces can be computed by looking for the infimum of all $\epsilon>0$ such that there is a correspondence between $X$ and $Y$ that changes ... | 2 | https://mathoverflow.net/users/145709 | 341335 | 145,008 |
https://mathoverflow.net/questions/341334 | 9 | I spoke with a computer scientist a few weeks who told me that in computer science there is something called a "vision paper", which is a paper that does not contain concrete results, but rather outlines a more general vision of what one either wants to do, or how a field may develop etc. A *future vision* in any case ... | https://mathoverflow.net/users/43263 | Is there something like a vision paper in mathematics? | The "mother of all vision papers", the [Langlands program,](https://en.wikipedia.org/wiki/Langlands_program) was simply a [handwritten letter;](http://publications.ias.edu/sites/default/files/handwritten-ltw.pdf) so I would not worry too much about "where to post it", it's the content that will determine the impact.
... | 5 | https://mathoverflow.net/users/11260 | 341339 | 145,010 |
https://mathoverflow.net/questions/341325 | 1 | All examples about geometric flow equations given in Wikipedia's [Geometric flow](https://en.wikipedia.org/wiki/Geometric_flow) article are first order in time derivative. Would it make sense to have a geometric flow equation which was second order in time derivative and is there examples of those? If we replace the fi... | https://mathoverflow.net/users/141581 | Geometric flow equations which are second order in time derivative | Geometric flows are modeled on fundamental flow equations, namely the heat and wave equations. The heat equation, which uses only a first order derivative in time, diffuses the function, so when $t \rightarrow \infty$, the solution not only converges to a limit, usually zero, but its derivatives all converge to zero. O... | 3 | https://mathoverflow.net/users/613 | 341342 | 145,012 |
https://mathoverflow.net/questions/341345 | 0 | Suppose we have a field with a total order which verifies
If $x,y\geq 0$, then $x+y\geq 0$ (here we relax the compatibility property for the addition).
If $x,y\geq 0$, then $xy\geq 0$.
Suppose also that every upper bounded nonempty set has a supremum.
We don't assume that $x\leq 0, -x\leq 0$ implies $x=0$.
I... | https://mathoverflow.net/users/111691 | Axioms for the real numbers | The answer is no. E.g., let $R$ be the field of all real numbers with the strict total order $\prec$ defined as follows: for any $x$ and $y$ in $R$ such that $x<y$ (where $<$ is the usual order on $R$), let $x\prec y$ if $y\ge0$, and $y\prec x$ if $y<0$. Then all your conditions on the total order will hold, but $(R,\p... | 4 | https://mathoverflow.net/users/36721 | 341349 | 145,015 |
https://mathoverflow.net/questions/338089 | 1 | Denote by $Q\_n$ the $n$-dimensional hypercube. A vertex of $Q\_n$ is represented by a vector of $n$ $\{0,1\}$-bits. An edge corresponding to two vertices whose vectors differ in one coordinate is represented by a bit vector with a $\*$ in that coordinate. A $d$-dimensional subcube $D$ is represented by a bit vector wi... | https://mathoverflow.net/users/73577 | Mapping of subcubes of a $(d+k)$-hypercube onto subcubes of a $d$-hypercube | Take any maximal face $a$ in the pre-image of $(\*,\*,\*)$ (i.e., of the 3-dimensional face). W.l.o.g., it has the form
$a=(\underbrace{\*,\dots,\*}\_k,0,0,\dots,0)$. Denote by $a(p,q)$ the face obtained from $a$ by replacing the $p$th star with $q$ (so $1\leq p\leq k$ and $q\in\{0,1\}$).
We say that a bit is *deter... | 2 | https://mathoverflow.net/users/17581 | 341364 | 145,020 |
https://mathoverflow.net/questions/341366 | 3 | Let $H$ be a Hilbert space. For given operators $a$ and $b$ on $H$, how can we find all solutions of the equation $xb=a$?
| https://mathoverflow.net/users/84390 | For given operators $a$ and $b$, solve the equation $xb=a$ | I will assume that you want $x$ to be a bounded operator on $H$. Then, a necessary condition for such an $x$ to exist, is that there must be some constant $C>0$ satisfying $\Vert a(\eta)\Vert \leq C\Vert b(\eta)\Vert$ for all $\eta\in H$.
Conversely, suppose that $a,b\in B(H)$ and $C>0$ satisfy these conditions. Let ... | 5 | https://mathoverflow.net/users/763 | 341370 | 145,023 |
https://mathoverflow.net/questions/341371 | 1 | So I am stuck at this situation . Suppose $m:B\_2(H\_1)\times B\_2(H\_2)\to \mathbb C$ be bilinear form given by $m(S,T)=\left<T,\phi(S)\right>$, where $\phi: B\_2(H\_1)\to B\_2(H\_2)$ be a bounded linear map and $B\_2(H)$ denotes the space of Hilbert Schmidt operators on $H$ with inner product $\left<S,T\right>=Tr(T^\... | https://mathoverflow.net/users/145729 | interchanging limits and summation | No, of course the sum and limit aren't interchangeable. E.g. take $H\_1 = H\_2 = l^2$, $\phi = {\rm id}$, $T = S =$ orthogonal projection onto the first coordinate, and $\tilde{T}\_j = \tilde{S}\_j =$ the rank 1 operator taking $e\_j$ to $e\_1$. (Note that the index $i$ does not come into the problem, it is a separate ... | 3 | https://mathoverflow.net/users/23141 | 341376 | 145,026 |
https://mathoverflow.net/questions/341381 | -1 | For a project I'm creating a program that must analyse a database of images to define average ratios for certain parts of the face (i.e. distance between eyes, distance from nose to chin etc.). I've ran into the problem where I can find distances between these points, however each subject (image) is of differing length... | https://mathoverflow.net/users/145738 | Calculating Average Ratios Of Human Faces From Images Of Differing Lengths To The Camera | Was way overthinking this, by dividing the two extreme points to get a factor and then dividing each ratio by this factor I can get ratios proportional to the head size.
| -1 | https://mathoverflow.net/users/145738 | 341384 | 145,029 |
https://mathoverflow.net/questions/341383 | 1 | Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed?
Sorry if this question is trivial. I am not familiar with theory of linear partial differential operators. This is a result I may use as a "black box".
Thank you in advance!
| https://mathoverflow.net/users/114334 | Non-closed range space of Laplace operators? | The range of the Laplace operator $-\Delta: L^2(\mathbb{R^d}) \supseteq H^2(\mathbb{R^d}) \to L^2(\mathbb{R^d})$ is **not** closed (for any dimension $d \ge 1$).
To see this, one can for instance use the following observations:
* $-\Delta$ has empty point spectrum, so $0$ is not an eigenvalue of $-\Delta$.
* $0$ is... | 3 | https://mathoverflow.net/users/102946 | 341391 | 145,032 |
https://mathoverflow.net/questions/341372 | 3 | Let $G$ be a finite cyclic group of order $p$. The tate cohomology groups $\hat{H}^\*(G, \mathbb{F}\_p)$ are defined using a complete resolution of $\mathbb{F}$ as $\mathbb{F}\_pG$-module.
There is a cup product on $\hat{H}^\*(G,\mathbb{F\_p})$, so I want to find the structure of this ring.
In this MO question
[... | https://mathoverflow.net/users/116775 | Cup product in Tate Cohomology Ring | In Chapter XII.7 of the book
*Cartan, Henri; Eilenberg, Samuel*, Homological algebra, Princeton Mathematical Series. 19. Princeton, New Jersey: Princeton University Press xv, 390 p. (1956). [ZBL0075.24305](https://zbmath.org/?q=an:0075.24305).
(starting on page 250) they calculate the cup products $\hat{H}^\*(G;A)... | 4 | https://mathoverflow.net/users/8103 | 341400 | 145,033 |
https://mathoverflow.net/questions/340180 | 5 | Let $A=\{a\_1<a\_2<a\_3<\dots< a\_k\}\subset\{1,2,\dots,N\}$ be a set of integers. Let $r\_A(n)=\#\{(a\_i,a\_j):a\_i+a\_j=n\}$ be the number of representations of $n$ as a sum of two elements from $A$. In typical parlance, $A$ is a Sidon set (or $B\_2$ set) if $r\_A(n)\le 2$ for all $n$. It is known that the maximum si... | https://mathoverflow.net/users/38622 | Eccentricity in the number of representations for sets too large to be Sidon sets | There exists a set $A$ on $\{1,...,N\}$ where $ |A|\geq\frac{2}{\sqrt{3}}\sqrt{N}$ and $E(A)=o(N)$.
Let $B$ be a Sidon set on $\{1,...,n\}$. Let $C = \{3n+1-b | b\in B\}$. Let $A=B\cup C$.
Suppose $a<b\leq c<d$ and $a+d=c+b$, where $a,b,c,d$ are elements of $A$. Now I will analyze the possibilities of $a,b,c,d$.
... | 4 | https://mathoverflow.net/users/125498 | 341404 | 145,034 |
https://mathoverflow.net/questions/340091 | 6 | Let $M$ be a manifold with boundary $\partial M$. Suppose that $M$ is equipped with some structure for which a notion of volume for chains can be defined. For example, if $M$ is triangulated, then the volume of a simplicial chain is just the number of cells with non-zero coefficients in the formal sum (let's work over ... | https://mathoverflow.net/users/104498 | Is the volume of relative cycles at least the systole of the manifold? | Here is a counterexample. Let $T$ be the $2$-torus obtained by identifying the opposite sides of the square on $\mathbb R^2$ with vertices $(\pm 1, \pm 1)$. Let $B\_{1-\varepsilon}$ be the disk $x^2+y^2<1-\varepsilon$ and let $M$ be the complement in $T$, $M=T/B\_{1-\varepsilon}$. It is clear that the systole $sys\_1(M... | 2 | https://mathoverflow.net/users/943 | 341410 | 145,035 |
https://mathoverflow.net/questions/50557 | 71 | The background is as follows: I have been whittling away at my commutative algebra notes (or, rather at commutative algebra itself, I suppose) recently for the occasion of a course I will be teaching soon.
I just inserted the statement of the theorem that the ring $\overline{\mathbb{Z}}$ of all algebraic integers is ... | https://mathoverflow.net/users/1149 | Is there a "purely algebraic" proof of the finiteness of the class number? | Yes, there exist purely algebraic conditions on a Dedekind domain which hold for all rings of integers in global fields and which imply that the class group is finite.
For a finite quotient domain $A$ (i.e., all non-trivial quotients are finite rings), a non-zero ideal $I\subseteq A$ and a non-zero $x\in A$, let $N\_... | 46 | https://mathoverflow.net/users/2381 | 341418 | 145,039 |
https://mathoverflow.net/questions/340257 | 0 | Let $M = \mathbb{Z}[\phi] \setminus \{0\}$ be the multiplicative monoid of the ring $\mathbb{Z}[\phi]$ with $\phi = \frac{1+\sqrt{5}}{2}$ the golden ratio.
We define the equivalence relationship $x\sim y$ iff $x =uy$ for some unit.
Is there a good description of the quotient $M/\sim$?
My initial thought was to do $a... | https://mathoverflow.net/users/94076 | Multiplicative monoid of ring modulo units | For general (commutative) rings $R$, we can construct $M = R \setminus \{0\}$ and $\sim$ as you did. It's not too hard to see (as GH mentioned) that in this general situation, $a \sim b$ if and only if they generate the same ideal, so $M/\sim$ will be isomorphic to the monoid of nonzero principal ideals of $R$.
Again... | 1 | https://mathoverflow.net/users/141571 | 341445 | 145,049 |
https://mathoverflow.net/questions/337664 | 6 | Consider some oriented surface $S$ with fundamental group $\pi\_1(S)$. The group cohomology of $\pi\_1(S)$ with coefficients in $\mathbb{R}$ is isomorphic to the de Rham cohomology of $S$. In degree 2, integration gives a map $\int\_S\colon H^2\_{dR}(S;\mathbb{R})\to \mathbb{R}$.
In terms of group cohomology, with c... | https://mathoverflow.net/users/124800 | Analogue of integration for group cohomology | We always have a pairing $H\_k(X, \mathbb{R}) \otimes H^k(X, \mathbb{R}) \to \mathbb{R}$ between chains and cochains. If $X$ is a closed oriented $n$-manifold, the orientation equips it with a class $[X] \in H\_n(X, \mathbb{R})$ called the *fundamental class*, and pairing this class with elements in $H^n(X, \mathbb{R})... | 3 | https://mathoverflow.net/users/290 | 341454 | 145,050 |
https://mathoverflow.net/questions/339952 | 2 | Let $G$ be a tripartite graph with partite sets $A,B,C$. The graphs $A\cup B$, $B\cup C$ and $C\cup A$ are each bipartite. Let the maximum degree of the graph be $\Delta$.
Now, we know that the List edge coloring conjecture is true for bipartite graphs, that is, the edge choosability is same as edge chromatic number ... | https://mathoverflow.net/users/100231 | List coloring of tripartite graph | The procedure, as such, may not give a proper coloring for a given graph. Consider the graph having $4$ vertices, $1,2,3,4$ given by the adjacency list $1-4,2-4,1-3,2-3,3-4$. Here the vertices $1,2$ form a part $A$, and vertices $3$ and $4$ belong to different parts, say $B$ and $C$ respectively. We know that $3$ is th... | 1 | https://mathoverflow.net/users/100231 | 341456 | 145,052 |
https://mathoverflow.net/questions/341395 | 7 | This is a follow-up to [this question](https://mathoverflow.net/questions/101067/are-all-irreducible-supercuspidal-representation-induced-from-compact-mod-center) by [Marc Palm](https://mathoverflow.net/users/10400/marc-palm) asked 7 years ago:
>
> Let $K$ be a finite extension of $\mathbb{Q}\_p$, and $G$ a re... | https://mathoverflow.net/users/105652 | Are all cuspidals induced? | **Question 1.** Yes indeed.
a) There are new results for classical groups and their inner forms (works of Shaun Stevens, Daniel Skodlerack, ...). In particular Skodlerack proved that in the case of "quaternionic forms" of classical groups, in residue characteristic not $2$, any irreducible supercuspidal representati... | 9 | https://mathoverflow.net/users/4767 | 341460 | 145,054 |
https://mathoverflow.net/questions/341450 | 2 | I need a reference for a theorem that states: Let $D$ be a domain of $\mathbb{R}^{m}$ and let $K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. Let $u$ be a subharmonic function on $D$. Then there is a unique Borel measure $\mu$ such that for all compact $E$ in $D$ we have
$$u(x)=\int\_{\partial E}u(\zeta)d\omeg... | https://mathoverflow.net/users/100746 | Reference for a theorem on subharmonic functions | Perhaps I misunderstood the question: you are right, there are two expressions available for $u$.
1. We have
$$ u(x) = \int\_{\partial E} u(z) \omega\_E^x(dz) - \int\_E G\_E(x, y) \mu(dy) ,$$
where $\omega^E\_x$ is the harmonic measure, $G\_E$ is the Green's function, and $\mu = -\Delta u$ in the sense of distributio... | 2 | https://mathoverflow.net/users/108637 | 341461 | 145,055 |
https://mathoverflow.net/questions/341435 | 1 | Let $a, b \in \mathbb R^n$ and $f, g \in L^1 [0,1]$. Assume for all $h \in AC[0,1 ]$ (space of absolutely continuous functions) following integral equality holds
$$ \int\_{0}^{1} \langle f(t) , h(t) \rangle \; dt + \int\_{0}^{1} \langle g(t) , h' (t) \rangle \; dt + \langle h(0) , a \rangle + \langle h(1) , b \rangl... | https://mathoverflow.net/users/108824 | Converting an integral equation into a differential equation | This is elementary but let's state it. First one only considers smooth test functions $h$ with compact support (so that the last two terms disappear). If $F(x):=\int\_0^xf(t)dt$, integrating by parts the first term gives $\int\_0^1\langle (F-g), h'\rangle dt=0$ for all these $h$, whence $F-g$ is a constant, that is, $g... | 1 | https://mathoverflow.net/users/6101 | 341462 | 145,056 |
https://mathoverflow.net/questions/341449 | 1 | Let $A:=\{ u \in H^k(0,2\pi): u^{(j)}(0)=u^{(j)}(2\pi) \mbox{ for } j=0,1,\ldots, k-1\}$, where $H^{k}(0,2\pi)\subseteq L^2(0, 2 \pi)$ is the Sobolev space of order $k$ on $(0, 2 \pi)$. Can we say that $u \in A$ iff
$$\sum\_{n=-\infty}^\infty (1+n^2)^k |\hat{u}(n)|^2<\infty?$$
In the above series $\hat{u}(n)$ are the ... | https://mathoverflow.net/users/143016 | Characterization of a subset of the Sobolev space $H^k(0,2\pi)$ in terms of Fourier series | This is indeed true, and let me illustrate this in the special case $k=1$.
We are given a function $u\in H^1(0, 2\pi)$ such that $u(0)=u(2\pi)$. First we note that this boundary condition indeed makes sense because by Sobolev embedding $u$ is continuous. We also know that $u$ has a (at least distributional) derivativ... | 0 | https://mathoverflow.net/users/37103 | 341468 | 145,060 |
https://mathoverflow.net/questions/338565 | 2 | In the paper [Anti-de Sitter space, squashed and stretched](https://arxiv.org/abs/gr-qc/0509076) Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up [Berger spheres](https://en.wikipedia.org/wiki/Berger%27s_sphere), it seems what is meant with "squashing" in the case of $S^... | https://mathoverflow.net/users/142501 | Explanation for "Squashing" and "Stretching" (Lorentzian Analogue of Berger Spheres) | The subgroup $H$ is not a normal subgroup (or *invariant subgroup*, in physicist's language), so the quotient $B(\beta)=G/H$ is not naturally a quotient group, only a quotient space. The condition $\alpha^2+\beta^2=1$ is not necessary. Indeed any vector with a nonzero component of $z\_4$ will do. Better: any one-dimens... | 2 | https://mathoverflow.net/users/13268 | 341469 | 145,061 |
https://mathoverflow.net/questions/341472 | 1 | Why is it true that if
$f(x):\mathbb{R}\longrightarrow\mathbb{R}$ is a positive, even function, decreasing for x>0, then it can be written as a convex linear combination of $\frac{1}{2h}\chi\_{[-h,h]}(x)$?
Thank you, I'm struggling with this a lot!
| https://mathoverflow.net/users/145757 | Even function and linear combination | Suppose that $f\colon\mathbb{R}\longrightarrow\mathbb{R}$ is a positive even function, which is decreasing and left-continuous on $[0,\infty)$, and such that $f(x)\to0$ as $x\to\infty$. Then for real $t\ge0$
\begin{multline}
f(t)=\int\_{[t,\infty)}-df(u)
=\int\_{[0,\infty)}-df(u)1\_{t\le u} \\
=\int\_{(0,\infty)}-df(... | 0 | https://mathoverflow.net/users/36721 | 341483 | 145,067 |
https://mathoverflow.net/questions/341408 | 4 | Let $\{X\_i\}\_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable *infinite-dimensional* Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
* (Dense $G\_{\delta}$) $X\_i$ is a dense $G\_{\delta}$ subset of $X$ not containing $0$,
* (Almost Contains a Linear Subspace) For each $i$, there exist... | https://mathoverflow.net/users/145751 | Haar-null union of dense subsets | In the Frechet space $X:=\mathbb R^\omega$ consider the dense linear subspace $$L\_0:=\{(x\_n)\_{n\in\omega}\in\mathbb R^\omega:|\{n\in\omega:x\_n\ne0\}|<\omega\}.$$
Fix a countable base $\{V\_n\}\_{n\in\omega}$ of the topology of the space $L\_0$ and in each set $V\_n$ choose a point $x\_n$, which is not contained i... | 3 | https://mathoverflow.net/users/61536 | 341497 | 145,072 |
https://mathoverflow.net/questions/341397 | 8 | By strict equivalence, I mean a monoidal equivalence whose underlying monoidal functors are strict, and here I am looking for two monoidal categories which are not strictly equivalent.
| https://mathoverflow.net/users/82023 | Is there an example of two strict monoidal categories which are (monoidally) equivalent, but not strictly? | Here is a very universal/abstract example:
(The following is some very classical machinery, but I'm not very familiar with the literature on this so I'm not sure which reference should be quoted)
Let $\textbf{Mon}$ be the $2$-category of monoidal categories and (pseudo) monoidal functor between and $\textbf{SMon}$ ... | 3 | https://mathoverflow.net/users/22131 | 341498 | 145,073 |
https://mathoverflow.net/questions/341505 | 6 | I am reading through Alan Baker's *Transcendental Number Theory* (don't worry if you don't know the book or the subject - this question is pretty much self-contained). Lemma 1 of Chapter 3 states an upper bound for $\nu(x;k)$, defined to be the least common multiple of the integers $x+1,\ldots,x+k$. The claim is that, ... | https://mathoverflow.net/users/50139 | Upper bound on least common multiple of consecutive integers | **1.** Assume first that $k\leq c''$. Then
$$\nu(x;k)\leq(x+k)^k\leq\left(\frac{c''}{k}\right)^{2k}(x+k)^{2k}=\left(\frac{c''(x+k)}{k}\right)^{2k}.$$
So the range $k\leq c''$ is fine as long as $c$ is chosen to satisfy $c\geq c''$.
**2.** Now assume that $x\leq c''k$. Then for some absolute constants $C$ and $C'$,
$... | 7 | https://mathoverflow.net/users/11919 | 341512 | 145,077 |
https://mathoverflow.net/questions/341506 | 1 | **The Setup:** Suppose $\Omega$ is a bounded, open, connected, simply connected subset of $\mathbb{R}^2$ with smooth boundary. Suppose that I am given a function $\Phi:\mathbb{R}^2\to\mathbb{R}$ and two continuous functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying
$$\begin{cases}\Delta\Phi=f(\Phi),& \text{ if }(x,y)\i... | https://mathoverflow.net/users/105925 | "Combining" two differential equations into one | When $\Phi(\Omega)$ and $\Phi(\Omega^c)$ overlap, $h$ does not necessarily exist. Take for example $\Omega = B\_1$ and $\Phi = (1-|x|^2)^2$. Then we may take $f(s) = 8(1-2\sqrt{|s|})$ and $g(s) = 8(1+2\sqrt{|s|})$. Since $f$ and $g$ disagree on $\Phi(B\_1) \cap \Phi\left(\overline{B\_1}^c\right) = (0,\,1]$, the functio... | 2 | https://mathoverflow.net/users/16659 | 341519 | 145,078 |
https://mathoverflow.net/questions/341518 | 0 | I am not sure if this is the right place to post this question, please point me to the correct forum if I posted in a wrong place.
I have an optimization problem like this
```
Min sum(yi)
st sum on j(wij) <= cyi for all i in N
sum on i(xij) = 1 for all j in N
wij = wjxij for all i, j in N
Lij = Ljxij ... | https://mathoverflow.net/users/145822 | How do I solve this integer programming problem with non convex constraints? | The nonlinear constraint
$$(L\_{ij} - D\_{ik})(L\_{ik} - D\_{ij}) \le 0$$
is a disjunction:
$$\left(L\_{ij} - D\_{ik} \ge 0 \wedge L\_{ik} - D\_{ij} \le 0\right) \bigvee \left(L\_{ij} - D\_{ik} \le 0 \wedge L\_{ik} - D\_{ij} \ge 0\right).$$
Introduce a binary variable $z\_{ijk}$ that enforces at least one side of the d... | 2 | https://mathoverflow.net/users/141766 | 341523 | 145,081 |
https://mathoverflow.net/questions/341457 | 3 | Let
$$V = \left\{ {v \in {H^2}(0,1):{v\_x}\left( 0 \right) - v\left( 0 \right) = {v\_x}\left( 1 \right) + v\left( 1 \right) = 0} \right\}.$$
**Question:** Is $V$ dense in ${H^1}\left( {0,1} \right)$?
I know ${C^\infty }\left( \mathbb{R} \right)$ is dense in ${H^1}\left( {0,1} \right)$. But I can't prove that $V$ i... | https://mathoverflow.net/users/135807 | Dense set in Sobolev space ${H^1}\left( {0,1} \right)$ | The space $V$ is dense in $H^1(0,1)$. Here are three (more or less) different proofs:
**Proof 1 (the pedestrian way):**
We set $H^1 := H^1(0,1)$ and $H^2 := H^2(0,1)$.
Fix $h \in H^1$. Since $H^2(0,1)$ is dense in $H^1(0,1)$, there exists a sequence $(f\_n)\_{n \in \mathbb{N}} \subseteq H^2(0,1)$ that converges t... | 6 | https://mathoverflow.net/users/102946 | 341525 | 145,083 |
https://mathoverflow.net/questions/341465 | 3 | Let $X$ be a complex algebraic curve, assumed to be connected, smooth and complete. Let $f: X \rightarrow X$ be a surjective morphism. Define a *backward complete set for $f$* as a subset $S$ of $X$ such that $f^{-1}(S) \subset S$ (I am not sure if it is the standard terminology).
>
> If $f$ has infinitely many fin... | https://mathoverflow.net/users/9317 | A question on dynamics on complex algebraic curves | For a map $f:\mathbb P^1\to\mathbb P^1$ of degree $d\ge2$, there are three cases: (1) There are no finite backward invariant sets. (2) There is one such set consisting of a single point.Moving that point to infinity, we have $f(x)\in\mathbb{C}[x]$ is a polynomial of degree $d$. (3) There are two such points. Moving the... | 5 | https://mathoverflow.net/users/11926 | 341533 | 145,086 |
https://mathoverflow.net/questions/341394 | 4 | I was wondering whether there is a characterization of perfect DG modules over a DG algebra as there is one for modules over a ring. Namely, an object in $D(R)$, where $R$ is a ring, is perfect if and only if it is isomorphic to a bounded complex of finitely generated projective modules. Is there any similar characteri... | https://mathoverflow.net/users/91572 | Perfect DG modules | Well by definition a DG module $X$ is perfect iff it is a direct summand of a module which is a finite iterated extension of free modules (it's then a nontrivial result that this property is the same as being compact). If $Y$ is a dg module and $Y'$ is an extension of $Y$ by $R[n],$ this extension is classified by an e... | 3 | https://mathoverflow.net/users/7108 | 341538 | 145,090 |
https://mathoverflow.net/questions/339069 | 5 | While analyzing a variational problem, I came to the following question:
>
> Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving **immersion**. Denote by $\omega\_f :\mathbb D^n \to \mathbb{R}^n$ the unique ha... | https://mathoverflow.net/users/46290 | Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible? | It seems that the answer is negative for dimension $n=2$ . I am not sure if higher dimensions can be reduced to the $2D$ case.
*Here is the argument for $n=2$:*
Suppose that there exist $f\_k \in C^{\infty}(\mathbb D^2, \mathbb{R}^2)$ such that $d\omega\_{f\_k} \in \text{GL}^+$ everywhere and $f\_k \to f$ in $W^{1,... | 0 | https://mathoverflow.net/users/46290 | 341544 | 145,093 |
https://mathoverflow.net/questions/341546 | 1 | Let $G$ be a torsion free group and $\alpha$ be a non-zero element in its complex group algebra. Assume that $\mathfrak A$ is the Banach sub-algebra of $\ell^1(G)$ generated by $\alpha$. Is it possible to extend a non-zero representation of $\mathfrak A$ (on a Hilbert space) to all of $\ell^1(G)$? What is the situation... | https://mathoverflow.net/users/84700 | Is it possible to extend this homomorphism? | No. Take $\alpha=g$, a group element, and consider a non-trivial one dimensional representation of the cyclic group generated by $g$. If $G$ has no abelian quotient then you're doomed.
| 4 | https://mathoverflow.net/users/89334 | 341547 | 145,095 |
https://mathoverflow.net/questions/340067 | 7 | Is there a (possibly hypergeometric-type) explicit evaluation of the
continued fraction
$$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$
Even the special case $d=0$, $a=1$ would be interesting. Note that
$$\tanh^{-1}(z)=\dfrac{z}{1-\dfrac{1^2z^2}{3-\dfrac{2^2z^2}{5-\ddots}}}$$
but that does no... | https://mathoverflow.net/users/81776 | Evaluation of hypergeometric type continued fraction | This is found in [1] $\S 82$, Satz 5. It covers the case where the numerator $a\_n$ is polynomial of degree $2$ in $n$ and the denominator $b\_n$ is degree $1$.
If I plugged in correctly, we get for your continued fraction: Let
$a, c, d$ be complex numbers satisfying: $c \ne 0, a \ne 0, a^2 \ne 4c$, and
$(a^2-4c)/a^... | 5 | https://mathoverflow.net/users/454 | 341566 | 145,101 |
https://mathoverflow.net/questions/341554 | 0 | Suppose I have an LP formulation as such:
$\min\ \ \sum\limits\_{i,j,t}\ w\_{ij}x\_{ijt} (\frac{t-r\_j}{p\_{ij}}+0.5)$
$\sum\limits\_{i,t}\frac{x\_{ijt}}{p\_{ij}}=1\,\forall\ j$
$\sum\limits\_{j}x\_{ijt}\leq 1\,\forall \ i,t$
$x\_{ijt}\geq 0\ \forall\ i,j,t\geq r\_j$
For understanding the context of its formu... | https://mathoverflow.net/users/67082 | Finding dual of a scheduling LP formulation | Two alternative approaches:
1. Rewrite the primal problem in standard form by replacing equality with two inequalities and multiplying both sides of $\le$ inequalities by $-1$ to reverse the sense to $\ge$.
2. For equality constraints in the primal, use free variables in the dual, and for $\le$ constraints in the pri... | 1 | https://mathoverflow.net/users/141766 | 341572 | 145,104 |
https://mathoverflow.net/questions/341446 | 1 | **Setting** Let $G=(V,E)$ be an undirected graph. A *walk* $\pi$ in $G$ of length $k$ is a sequence of $k+1$ vertices $v\_1,\ldots,v\_{k+1}$ such that for each $i\in[1,k]$,
$\{v\_i,v\_{i+1}\}\in E$. Let $H=(W,F)$ be another undirected graph having the same number of vertices as $G$, i.e., $|V|=|W|=n$.
If for each $k$... | https://mathoverflow.net/users/9839 | Characterisation of walk-equivalent digraphs | This should work: $G$ is given by $A\_G=\begin{bmatrix}0 & 1 &0 & 0\\0& 0 &1 &1 \\1 &0 &0 &0\\0 &0 &0 &0 \end{bmatrix}$ and $H$ given by $A\_H=\begin{bmatrix}0 & 1 &0 & 0\\0& 0 &1 &0 \\0 &0 &0 &1\\1 &0 &0 &0 \end{bmatrix}$. Both have $4$ walks, of any length.
Consider matrix $Q=\begin{bmatrix}a & b &c & d\\e& f &g &h ... | 2 | https://mathoverflow.net/users/145855 | 341573 | 145,105 |
https://mathoverflow.net/questions/341434 | 1 | I have been looking for litterature on results obtained by deep neural networks to find dense (and quite possibly non-lattice, perhaps even non-periodic) sphere packings, but I have not been too successful. In fact, I am only aware of [a 2002 paper](https://www.researchgate.net/profile/David_Cornforth/publication/22093... | https://mathoverflow.net/users/50912 | Prospects for deep learning of non-lattice sphere packings | There’s definitely a lot of potential for finding great packings using computers. I don’t believe the known sphere packings up through 24 dimensions are all optimal, and a clever heuristic algorithm could plausibly beat some of them. (Dimensions 19 and 21 might be the lowest-hanging fruit.) I don’t think this would be ... | 7 | https://mathoverflow.net/users/4720 | 341577 | 145,106 |
https://mathoverflow.net/questions/341398 | 1 | Let $A\subset B$ Be affine domains over a field of characteristic zero, say k. We know that the integral closure of $A$ in any finite extension of $Q(A)$ is a finite $A$ module. My question is why the integral closure of $A$ in $B$ is a finite $A$ module?
I have tried to show that the integral closure is again an aff... | https://mathoverflow.net/users/145745 | Integral closure of affine domains | $Q(B)$ is a finitely generated field extension of $Q(A)$ and so any intermediate field is a finitely generated field extension of $Q(A)$ (see below); in particular the algebraic closure of $Q(A)$ in $Q(B)$ is a finite extension of $Q(A)$ and so the integral closure of $A$ in the algebraic closure of $Q(A)$ in $Q(B)$ is... | 0 | https://mathoverflow.net/users/59248 | 341579 | 145,107 |
https://mathoverflow.net/questions/341549 | 1 | **Edit:** According to essential comment of YCore I revise the question.
Let $A$ be a finite dimensional graded algebra which is a unital, super commutative and associative algebra. Is there a Lie group $G$ whose differential graded algebra of all $G$-left invariant differential forms be isomorphic to $A$? This is a ... | https://mathoverflow.net/users/36688 | Is a finite dimensional graded algebra isomorphic to the equivariant de Rham complex of a Lie group? | No. A commutative differential graded algebra A is isomorphic
to the Chevalley-Eilenberg algebra of a finite-dimensional Lie algebra L
if and only if its underlying graded algebra is the exterior
algebra on A\_1, which must be finite-dimensional.
In this case we have L=(A\_1)\* and the Lie bracket on L is the dual of t... | 3 | https://mathoverflow.net/users/402 | 341584 | 145,110 |
https://mathoverflow.net/questions/341565 | 3 | This question is motivated by the work presented in article 358 of Gauss' *Disquisitiones Arithmeticae*. For the sake of completeness, let me say something about the background and present the question at the end. Any comment or correction is appreciated.
Let $n$ be an odd prime, and suppose further that $n=3m+1$ for... | https://mathoverflow.net/users/100553 | The local zeta-functions of some cubic plane curves | We have $3a+3b+3c+1 = n$ so $M + n = 9a+1$. Probably you mean $9a+3$ is the number of solutions of the equation, and not $a$, as each nonzero number has three cube roots. Except shouldn't it be $9a+6$ as there are also the solutions with $x$ or $y$ zero? The constant term will not affect whether the function is rationa... | 1 | https://mathoverflow.net/users/18060 | 341609 | 145,118 |
https://mathoverflow.net/questions/341558 | 2 | Let $0<\alpha\leq\frac{1}{2}$ a fixed real number. I wondered if it is possible to evaluate the sequence of definite integrals $$\int\_0^1\left(\sum\_{k=0}^n (f(x))^k\right)^{\alpha}dx\tag{1}$$
for some example of a continuous function and positive $f(x)>0$ for all $0<x<1$, and $f(x)\neq\text{constant}$.
>
> **Ques... | https://mathoverflow.net/users/142929 | Example of evaluation of $\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx$, for some choice of $f(x)$ satisfying certain requirements | If you allow for special functions, you of course allow for quite some beasts. Generally, your expression is
$$
I = \int\_{0}^{1} dx \left( \sum\_{k=0}^{n} (f(x))^k \right)^{\alpha }
= \int\_{0}^{1} dx \left( \frac{1-(f(x))^{n+1} }{1-f(x)} \right)^{\alpha }
$$
Let's choose $f(x)=x^2 $ and $\alpha =1/2$, and substitute ... | 2 | https://mathoverflow.net/users/134299 | 341610 | 145,119 |
https://mathoverflow.net/questions/341557 | -1 | I want to coin a theory that can speak about big sets like some of those present in NF, but at the same time comprehend over small collections of them as it is the case in ZFC. Is this known to be inconsistent. In particular I have the following system in my mind.
Have all axioms of Zermelo restricted to well founded... | https://mathoverflow.net/users/95347 | Can we have a theory that define small (ZFC set sizes) collections of big sets? | The Schema of Equivalence classes is inconsistent with the existence of an empty set. To see this let R be the equivalence relation defined by xRy iff ((x is not in x) and (y is not in y)) or x=y. By the
Schema of Equivalence classes {| 0} exists where 0 is empty.
| 8 | https://mathoverflow.net/users/133981 | 341615 | 145,121 |
https://mathoverflow.net/questions/341622 | 2 | I wondered, inspired in a result from [1] (**Proposition 17**) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture,
$$\sum\_{\substack{\text{primes }p\leq x\\\text{such that }p+2\text{ is prime}}}\frac{\log^m p}{p}$$
as $x\to\infty$, where $m\geq 1$ denote... | https://mathoverflow.net/users/142929 | On $\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{(\log p)^m}{p}$, on assumption of the first Hardy–Littlewood conjecture | Using integration by parts, it follows from the [first Hardy-Littlewood conjecture](https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture) that
$$\sum\_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{\log p}{p}\sim 2C\_2\log\log x,$$
and
$$\sum\_{\substack{p\leq x\\p,p+2\text{ twin prim... | 6 | https://mathoverflow.net/users/11919 | 341624 | 145,124 |
https://mathoverflow.net/questions/312507 | 17 | This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. Actually I know such a proof (from Skopets - Zharov book), but it uses the third dimension which has to be justified its... | https://mathoverflow.net/users/4312 | Why are the medians of a triangle concurrent? In absolute geometry | There is such a proof given by Hjelmslev; it is based on a clever application of central symmetries (point-reflections). You can find it on pages 102-104 of
Ф. Бахман, Построение геометрии на основе понятия симметрии. (Russian) [The development of geometry based on the concept of symmetry]
Translated from the German... | 7 | https://mathoverflow.net/users/21684 | 341640 | 145,126 |
https://mathoverflow.net/questions/341642 | 2 | Is second-order logic *with standard semantics* necessary to categorically characterise the natural number structure?
One can prove that any two models of Dedekind-Peano arithmetic are isomorphic (categoricity theorem), and, by invoking the isomorphism theorem, that one model of Dedekind-Peano arithmetic satisfies A ... | https://mathoverflow.net/users/145888 | Is second-order logic *with standard semantics* necessary to categorically characterise the natural number structure? | As you suspect, **you cannot do this**.
Here's one general fact which in particular kills this question (which is just an instance *compactness for Henkin semantics*): suppose $\Sigma$ is any language and $T$ is any second-order $\Sigma$-theory which has some infinite Henkin model. Then $T$ has Henkin models of arbit... | 2 | https://mathoverflow.net/users/8133 | 341660 | 145,134 |
https://mathoverflow.net/questions/341656 | 2 | Let $X$ be a smooth irreducible subvariety of an algebraic group over a field, assume $X$ is invariant under $n$-th power map for every integer $n$ ($n=0$ means the identity element is in $X$). Must $X$ be a subgroup?
The motivation for this question is the intuition that if a cone in a linear space is smooth, then i... | https://mathoverflow.net/users/102104 | Smooth irreducible subvarieties in an algebraic group that are stable under power maps | This is false in nonabelian unipotent groups. For these groups, the exponential map is algebraic, and an isomorphism. The image of any linear subspace under this map will be smooth, irreducible, and invariant under the $n$th power map. But it will not be a subgroup unless the subspace is closed under the Lie bracket. F... | 11 | https://mathoverflow.net/users/18060 | 341662 | 145,136 |
https://mathoverflow.net/questions/341600 | 2 | Do you know an example of an unbounded closed operator $T:D(T)\subseteq X \to X$, defined in a complex Banach space $X$, such that $\mathbb{C} \setminus\sigma(T)$ is unbounded and the equality
\begin{equation}\tag{1}
\lim\_{|z| \to \infty} \| R(z,T)\|=0
\end{equation}
holds?, where $R(z,T):=(T-zI)^{-1}$ for $z \notin \... | https://mathoverflow.net/users/142048 | Example of an unbounded closed operator $T$ such that $\lim_{|z| \to \infty} \| R(z,T)\|=0$ | As already noted by several users in this comments, something seems to be a bit odd with the question due to the behaviour of the resolvent close to the spectrum. Here are a few details about what is true and what is not:
**1)** If $(z\_n)$ is a sequence in $\mathbb{C} \setminus \sigma(T)$ such that $\|R(z\_n,T)\| \t... | 1 | https://mathoverflow.net/users/102946 | 341667 | 145,137 |
https://mathoverflow.net/questions/341659 | 2 | By lemma 4.9 in [Dugger-Hollander-Isaksen](https://arxiv.org/pdf/math/0205027.pdf), a hypercover is defined as an augmented simplicial object $U\_\bullet\to X$ in the category of simplicial presheaves such that each $U\_n$ is a coproduct of representables and each $U\_n\to (cosk\_{n-1} U\_\bullet)\_n$ is a local epimor... | https://mathoverflow.net/users/124163 | Definition of hypercover for simplicial presheaves and hypercovering in $\infty$-topos | The condition that $U\_n$ being a coproduct of representables is *not essential* for the definition of hypercover of simplicial presheaves. In fact, in Jardine's *Local Homotopy Theory*, a hypercover is just *defined* to be a local trivial fibration (more general class than DHI's), which one can prove to be equivalent ... | 4 | https://mathoverflow.net/users/42571 | 341670 | 145,138 |
https://mathoverflow.net/questions/339340 | 4 | I just read about the definitions about torsor of sheaf of groups and get a bit confused.
How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a similar weak equivalence $\Omega B G\simeq G$ for $G$ a sheaf of group?
Why is there an equivalence $\Omega BTors(G)\s... | https://mathoverflow.net/users/124163 | $G$-torsor for topological space compared to that for sheaf of groups | The notion of $G$-torsor for a topological space is the same as the sheaf-theoretic definition, once you are familiar with the language of site.
Your desired weak equivalences exist, in which the looping need (right) derived (w.r.t. the local model structure), they follow from Jardine's *Local Homotopy Theory*, Propo... | 2 | https://mathoverflow.net/users/42571 | 341671 | 145,139 |
https://mathoverflow.net/questions/341666 | 4 | I have an integral over a subspace of $\mathbb{R}^n \times \mathbb{R}^n$ with an integrand of the form
$$\exp\left(-\frac{1}{2}\left[||u^2|| + \langle u, v \rangle + ||v||^2\right]\right)$$
The subspace is exactly the space for which $u\_{i} = u\_{n-i}$ (assume $n$ is even). In other words, $v$ is a true $n$-dimensiona... | https://mathoverflow.net/users/134361 | Degenerate Gaussian Integral | $\newcommand{\R}{\mathbb{R}}$
Let $U:=\{u\in\R^n\colon u\_i=u\_{n-i}\ \forall i\}$ be your $n/2$-dimensional subspace. I am assuming that your integral is with respect to the product of the Lebesgue measures on $U$ and $\R^n$, and I will denote those measures by $du$ and $dv$, respectively. So, if $\cdot$ denotes the d... | 6 | https://mathoverflow.net/users/36721 | 341672 | 145,140 |
https://mathoverflow.net/questions/341663 | 1 | For some reason, I'm having difficulties proving something that is intuitively simple.
Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of $x$, i.e., it is distributed as, $X^{truncated}\sim X | X \in[-b,b]$ for some $b>0$, and $x$ is distributed symmetrical... | https://mathoverflow.net/users/145894 | Comparing noisy truncated RV with noisy regular RV | $\newcommand{\R}{\mathbb{R}}$
Let $Y:=X^{truncated}$ and $Z:=\Lambda$. Then, by rescaling, without loss of generality $Z\sim N(0,1)$.
We have to show that $P(Y+Z>a)\le P(X+Z>a)$ for all $a>0$.
Here we need to assume that $Z$ is independent of $X$ and $Y$ (I gather this assumption is missing in your question, and wi... | 0 | https://mathoverflow.net/users/36721 | 341676 | 145,142 |
https://mathoverflow.net/questions/331675 | 5 | Is there a group with a presentation
$\left< X \mid r\_i, i \in \mathbb{N} \right>$ (where $X$ is finite) with
$\left< X \mid r\_i, i \in A \right>$ is amenable
if and only if $A\subset \mathbb{N}$ is infinite.
| https://mathoverflow.net/users/7307 | Amenable groups with special presentations | An almost yes.
For $n,m\ge 2$ consider the group
$$H=H(n,m)=\langle t,x,y\mid txt^{-1}=x^n,\;t^{-1}yt=y^m\rangle$$
Remark: $H$ is a semidirect product $\mathbf{Z}\ltimes(\mathbf{Z}[1/n]\ast\mathbf{Z}[1/m])$, where the positive generator of $\mathbf{Z}$ acts by multiplication by $n$ on the first factor, and by $1/... | 5 | https://mathoverflow.net/users/14094 | 341677 | 145,143 |
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