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https://mathoverflow.net/questions/283832 | 4 | Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters](https://en.wikipedia.org/wiki/Ultrafilter) on $\omega$. The *Rudin-Keisler preorder* on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq\_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$
It i... | https://mathoverflow.net/users/8628 | Cardinality of maximal chains in the poset of ultrafilters with Rudin-Keisler ordering | Yes. I claim that such a maximal chain must always have cardinality $\frak{c}^{+}$.
First of all, take notice that for each ultrafilter $U$, there are at most $\frak{c}$ many Rudin-Kiesler equivalence classes below $U$ simply because there are at most continuumly many functions $f:\omega\rightarrow\omega$ and every ... | 6 | https://mathoverflow.net/users/22277 | 283899 | 125,533 |
https://mathoverflow.net/questions/283915 | 6 | Suppose $g\ge 0$ and $n\ge 0$ are integers. We have the space $\overline{\mathcal M}\_{g,n}$ of stable curves of arithmetic genus $g$ with $n$ marked points. The topology on this space is the one described in the paper "Compactness Results in SFT" (by Bourgeois et al) starting from page 811.
The notes "Holomorphic Cu... | https://mathoverflow.net/users/110236 | Chart for Deligne-Mumford Compactification | For a careful discussion of Deligne-Mumford compactification of the moduli space of curves from a complex-analytic point of view, see the following paper (which also has a good bibliography of earlier work from this perspective):
J. Hubbard, S. Koch,
An analytic construction of the Deligne-Mumford compactification of... | 6 | https://mathoverflow.net/users/317 | 283916 | 125,539 |
https://mathoverflow.net/questions/283913 | -1 | It is quite well-known that $\Pi\_2^0$-Indescribability is the same as Strong Inaccessibility and $\Pi\_n^0$-Indescribability for every $n>2$.
It is also quite simple to show that $\Pi\_0^0$-Indescribability is equivalent to nonemptiness (i.e. $\kappa>0$).
However, this gives some relations for almost every $\Pi\_n... | https://mathoverflow.net/users/111429 | What is known about $\Pi_1^0$-Indescribable cardinals? | A $\Pi\_1$ statement $\varphi$ is absolute downwards between transitive structures: if $(V\_\alpha; \in, A)\models\varphi$ and $\beta<\alpha$, then $(V\_\beta;\in, A\cap V\_\beta)\models\varphi$. So $\Pi^0\_1$-indescribability, like $\Pi^0\_0$-indescribability, is trivial.
| 5 | https://mathoverflow.net/users/8133 | 283919 | 125,541 |
https://mathoverflow.net/questions/283581 | 1 | Let $R$ be a ring. An endomorphism $\alpha:R\to R$ is said to be right central reflexive if for all $a,b\in R,$ $aRb=(0)\implies bR\alpha (a)\subset Z(R)$, where $Z(R)$ denotes the centre of the ring.
The ring $R$ is called right $\alpha$-central reflexive if $\alpha$ is a right central endomorphism for $R$.
Let ... | https://mathoverflow.net/users/41137 | Regarding central reflexive rings | Here's a simple example.
Fix a domain $k$. Consider the (associative unital) ring $R=k\langle x,y\rangle/\langle\!xy=0\rangle\!\rangle$. Let $t$ be its endomorphism $x\mapsto 0$, $y\mapsto y$.
0) A basis of $R$ is given by the $y^nx^m$, $n,m\ge 0$.
1) I claim that $R$ right central reflexive, and even satisfies $... | 2 | https://mathoverflow.net/users/14094 | 283931 | 125,543 |
https://mathoverflow.net/questions/283930 | 5 | We say that a space $(X,\tau)$ has the *fixed point property (FPP)* if for every continuous map $f:X\to X$ there is $x\in X$ with $f(x) = x$.
What is an example of a space $X$ with FPP such that $X^2$ (with the product topology) does not have FPP?
| https://mathoverflow.net/users/8628 | Squaring a space with the fixed-point property | The topological space
$$
X = \{ (x,\sin \left( \frac{\pi}{1-x} \right)) \ | \ 0 \leq x < 1 \} \cup {(1,1)} \subseteq \mathbb{R}^2
$$
has the FPP but its square does not. See Example $2$ p.$977$ of [E. H. Connell, Properties of Fixed Point Spaces](https://www.jstor.org/stable/pdf/2033633.pdf?refreqid=excelsior%3A7894cfc... | 2 | https://mathoverflow.net/users/21724 | 283932 | 125,544 |
https://mathoverflow.net/questions/283914 | 2 | This is a short question:
In the symmetric monoidal category of chain complexes (over a field if necessary), does the endomorphism operad carries a Hopf structure, i.o.w. can it be considered as a Hopf operad?
The answer is probably no, but just to be sure
| https://mathoverflow.net/users/21965 | Is there a Hopf structure on the dg-endomorphism operad? | No. Given any Hopf operad $\mathcal{O}$, its arity one part $\mathcal{O}(1)$ is a bialgebra. In particular, you would get that $\mathrm{End}\_V(1) = \mathrm{End}(V)$ is a bialgebra.
Suppose that for a vector space $V$, $\mathrm{End}(V)$ is a bialgebra. Then the counit is an algebra map $\epsilon\colon \mathrm{End}(V)... | 5 | https://mathoverflow.net/users/18512 | 283946 | 125,549 |
https://mathoverflow.net/questions/283757 | 5 | Let $G$ be a profinite group. Let $\Lambda$ be a discrete valuation ring over $\mathbb{Z}\_p$. Let $A\_0,A\_1$ be artinian ring over $\Lambda$. Let $\alpha : A\_1 \rightarrow A\_0$ is a small map that is $\mathfrak{m}\_{A\_1}.ker(\alpha) = 0$. Assume that a representation $\rho : G \rightarrow GL\_n(A\_0)$ is given. I ... | https://mathoverflow.net/users/58056 | Obstructions to lift of representations | Something like what you want is true, though it needs to be formulated slightly differently. Let $k$ be the residue field of $A\_0$, and let $\overline{\rho}$ denote the ``residual'' representation $\rho \otimes\_{A\_0} k$.
Let $I$ denote the kernel of the map from $A\_1$ to $A\_0$; then $I$ is a $k$-vector space. We... | 3 | https://mathoverflow.net/users/116208 | 283947 | 125,550 |
https://mathoverflow.net/questions/283934 | 1 | We are reading John Roe's book *Lectures on Coarse Geometry*. We come across a question in P27 line 9:
Suppose $X$ is a paracompact and locally compact Hausdorff space, $\bar{X}$ is a compactification of $X$, how to use Urysohn's Lemma to choose continuous functions $f,g:\bar{X}\times\bar{X}\to \mathbb{R}^+$ such that... | https://mathoverflow.net/users/102003 | How to choose a continuous function which vanishes **only** on the closed set | 1. The function $f:\bar X\times \bar X\to\mathbb R$ exists if and only if the compactification $\bar X$ of $X$ is metrizable, which implies that the locally compact space $X$ is metrizable and separable.
This follows from a well-known metrizability theorem saying that a compact topological space $K$ is metrizable if ... | 3 | https://mathoverflow.net/users/61536 | 283948 | 125,551 |
https://mathoverflow.net/questions/275960 | 7 | Consider an $n$-dimensional convex polytope with $k$ vertices. In the worst case the number of faces is exponential in $n$ and $k$. Consider a $2$-dimensional plane which intersects this polytope, i.e., it intersects only a subset of all faces. Can I bound the number of such intersected faces? In the worst case, will t... | https://mathoverflow.net/users/112492 | Section of an $n$-dimensional convex polytope by $2$-dimensional plane | I found a paper wich shows that in the worst case it still can be exponential:
"Shadows and slices of polytopes, Nina Amenta et al., 1996"
<https://dl.acm.org/citation.cfm?id=237228>
| 2 | https://mathoverflow.net/users/112492 | 283960 | 125,555 |
https://mathoverflow.net/questions/283959 | 5 | For $f:[0,1]\to\mathbb{R}$, let $f'\_{app}(x)$ denote the approximate derivative (that is, the derivative calculated along some set with density $1$ at $x$, if such a thing exists). Assume that $f'\_{app}$ exists and is continuous on $[0,1]$. Does this imply that $f'$ exists on $[0,1]$ (of course immediately $f'$ would... | https://mathoverflow.net/users/35158 | If $f:[0,1]\to\mathbb{R}$ has continuous approximate deriv., is it $C^1[0,1]$? | Yes. Let $g$ be an antiderivative of $f'\_{app}$, then $(f-g)'\_{app}=0$ , this implies that $h:=f-g=\text{const}$.
Sketch of the proof of this fact. It suffices to prove that $h(x)=h(0)$ a.e. Indeed, if $h(a)\ne 0$, than this would imply $h'\_{app}(a)=\infty$. We may fix $c>0$ and $\rho\in (0,1)$ and prove that the ... | 6 | https://mathoverflow.net/users/4312 | 283961 | 125,556 |
https://mathoverflow.net/questions/283940 | 1 | There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If not, is there a calculable obstruction? Are there some natural classes of triangulated categories for which it is surje... | https://mathoverflow.net/users/41291 | Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group | One obstruction is that if all Hom-sets are finite dimensional vector spaces, and for all objects $X$ and $Y$, $\text{Hom}(X,Y[i])=0$ for all but finitely many $i$, then any self-equivalence must preserve the bilinear form
$$\langle [X],[Y]\rangle =\sum\_i(-1)^i\dim\text{Hom}(X,Y[i])$$
on the Grothendieck group.
| 5 | https://mathoverflow.net/users/22989 | 283969 | 125,560 |
https://mathoverflow.net/questions/283965 | 2 | Suppose we have an algebraic group over an algebraic closed field of prime characteristic, for example we may think of a subgroup of $GL\_n(\bar{\mathbb{F}}\_q)$ (where $\bar{\mathbb{F}}\_q$ is the algebraic closure of the field with $q$ elements) (that is the case I'm interested).
My question is what can be said, o... | https://mathoverflow.net/users/47300 | Representations of Algebraic Groups | To say anything meaningful here, you should start with a *connected* algebraic group. Finite groups for example are technically affine algebraic groups, but their representations over $\mathbb{C}$ have a literature of their own.
The key reference, following years of partial exploration, involves connected *semisimple... | 6 | https://mathoverflow.net/users/4231 | 283974 | 125,563 |
https://mathoverflow.net/questions/283950 | 1 | There is an well known algorithm given by E.M Luks for bounded graph isomorphism. There are three important steps of the algorithm two of them are given below:
1. If the group action on vertex set of input graph is transitive (single orbit) then solve the isomorphism problem in each coset:
$$ISO(G,X,Y) = ISO(K\_i\s... | https://mathoverflow.net/users/111831 | Some doubts on the algorithm of graph isomorphism of bounded degree graph | 1. We know that each $ISO$ is either a coset or an empty set. Suppose that non-empty answers to children $ISO$'s are cosets $g\_i H\_i$. Take $g\_1$ as a representative for the resulting coset, then the subgroup $H$ corresponding to the resulting coset is induced by $g\_1^{-1}g\_i h\_i$, where $h\_i$ ranges over genera... | 4 | https://mathoverflow.net/users/106512 | 283983 | 125,568 |
https://mathoverflow.net/questions/283453 | 10 | Both volume computation and lattice point enumeration of convex polyhedron are $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.
1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?
2. ... | https://mathoverflow.net/users/10035 | Approximate volume computation and lattice point enumeration - hardness | Below the line I address the question after "Update" about the relationship between volume and the number of lattice points. Since OP asks about conditions under which we can count the number of lattice points in $K$, let me point to a recent [chapter](https://www.csun.edu/~ctoth/Handbook/chap7.pdf) by Barvinok, which ... | 6 | https://mathoverflow.net/users/35733 | 283988 | 125,570 |
https://mathoverflow.net/questions/283989 | 4 | Let be $G, H$ groups and $\Phi(G), \Phi(G)$ their Frattini groups. I'm looking for a conterexample that shows that in general $\Phi(G \times H)= \Phi(G) \times \Phi(H)$ doesn't hold, therefore that there exist $G, H$ such that $\Phi(G) \times \Phi(H)\not \subset \Phi(G \times H)$ (inclusion in other direction is indeed... | https://mathoverflow.net/users/108274 | Product of Frattini Groups | The inclusion can indeed be proper. Indeed, there exist simple groups $S$ with no maximal subgroup. So $\Phi(S)=S$. But the diagonal in $S\times S$ is a maximal subgroup, and in particular contains $\Phi(S\times S)$. Since $\Phi(S\times S)$ is normal, it follows that $\Phi(S\times S)=\{1\}$, which is properly contained... | 10 | https://mathoverflow.net/users/14094 | 283991 | 125,571 |
https://mathoverflow.net/questions/284000 | 3 | What is a good book on topological rings and modules?
I'm interested in topological rings and modules typically endowed with non-linear topologies, e.g.. non-linearly topologized normed rings.
I have some references at my disposal, but I would like to ask for some here, just in case.
Thanks.
*Current references... | https://mathoverflow.net/users/nan | References on topological rings | Here are 3 references that haven't been mentioned yet. I am not sure if the latter two would be of any use to you, but probably are worth a look.
* Arnautov, Glavatsky, Mikhalev - Introduction to the Theory of Topological Rings and Modules (1995)
* Prolla - Topics in Functional Analysis over Valued Division Rings (19... | 5 | https://mathoverflow.net/users/1849 | 284002 | 125,576 |
https://mathoverflow.net/questions/283942 | 2 | *Cross-Posted from [Math Stackexchange](https://math.stackexchange.com/questions/2464439/upper-bound-lemma-implies-the-ergodic-theorem-for-random-walks-on-groups).*
>
> **Ergodic theorem.** A random walk on a finite group $G$ driven by a probability $\nu\in M\_p(G)$ is *ergodic* if $\operatorname{supp}(\nu)$ is not... | https://mathoverflow.net/users/35482 | Upper bound lemma implies the ergodic theorem for random walks on groups? | Here is a proof of necessity.
First suppose $\nu$ is concentrated on a proper subgroup $S$. Then consider the module $\mathbb C[G/S]$. It is not the trivial module and contains the trivial module with multiplicity $1$. Therefore, it contains a non-trivial irreducible constituent $\rho\_a$. By Frobenius reciprocity, ... | 2 | https://mathoverflow.net/users/15934 | 284003 | 125,577 |
https://mathoverflow.net/questions/272362 | 5 | The classical mean value result of Murty and Murty (1991) and Bump, Friedberg, and Hoffstein (1990) on derivatives of modular form L-functions $L(s,f)$ proves (roughly speaking) the existence of infinitely many imaginary quadratic fields $K$ for which the Heegner hypothesis holds, ie., every prime dividing the level of... | https://mathoverflow.net/users/48554 | The Heegner hypothesis for a mean value result of Murty-Murty/Bump-Friedberg-Hoffstein | Let $K/\mathbb{Q}$ be a number field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}\_2(\mathbb{A}\_K)$, where $\mathbb{A}\_K$ is the ring of adeles over $K$. Consider the standard $L$-function $L(s,\pi)$ associated to $\pi$. The proof of Theorem B (part 2) of Friedberg and Hoffstein (Nonvanishi... | 5 | https://mathoverflow.net/users/111215 | 284008 | 125,579 |
https://mathoverflow.net/questions/283970 | 5 | Who was the first to prove this theorem and is there an "official" name for it?
Let $\phi:X\rightarrow Y$ be a map of H-spaces that are also CW-complexes. Assume $\phi$ induces isomorphisms on homology groups $H\_{\*}(-,\mathbb{Z})$. Then $\phi$ is a homotopy equivalence.
I learned about this theorem from my under... | https://mathoverflow.net/users/70751 | Attribution of theorem saying that inducing isomorphism on homology implies homotopy equivalence between H spaces that are CW complexes | The result for simply-connected spaces is usually attributed to J.H.C. Whitehead, in particular Theorem 14 in
* J.H.C Whitehead. Combinatorial homotopy II. Bull. Amer. Math. Soc. 55, (1949). 453–496.
The result was generalized to nilpotent spaces in
* E. Dror. A generalization of the Whitehead theorem. In: Lectu... | 7 | https://mathoverflow.net/users/50846 | 284013 | 125,581 |
https://mathoverflow.net/questions/283998 | 2 | I am dealing with Riemannian immersions and I am stuck on the following: Given a totally geodesic immerserd surface $S$ on a compact riemannian manifold $M$ with metric $g$, is there another metric $\tilde g \neq g$ on $M$ such that $S$ is isometric immersed (not necessarely totally geodesically) on $(M,\tilde g)$? In ... | https://mathoverflow.net/users/94097 | Preserving connections immersions | Suppose that $S$ is a pair of disjoint Euclidean 2-spheres, say of radius 1 and radius 2. Isometrically immerse to the 3-sphere $M$ as totally geodesic spheres of radius 1, intersecting along a totally geodesic curve $C$ lying on each of the 2-spheres. The preimage of $C$ in $S$ is a pair of circles of different length... | 2 | https://mathoverflow.net/users/13268 | 284014 | 125,582 |
https://mathoverflow.net/questions/283955 | 0 | Assume we have two p.p. simple abelian surfaces $(A\_i,D\_i)$, i=1,2, over $\mathbb{C}$ with the following commutative diagram:
$\require{AMScd}
\begin{CD}
A\_1 @>{birational}>> A\_2\\
@V{2:1}VV @VV{2:1}V \\
A\_1/(-1) @>{birational}>> A\_2/(-1)\\
@V{\iota\_1}VV @VV{\iota\_2}V\\
\mathbb{P}^3 @>{birational}>> \mathbb{P... | https://mathoverflow.net/users/70593 | Behaviour of (principal) polarizations of (singular) surfaces under birational maps | @ulrich points out that the OP might be asking a question different than the one I answered, and I completely agree. So let me explain what the comment above actually proves.
Let $(A\_1,\mathcal{L}\_1)$ and $(A\_2,\mathcal{L}\_2)$ be Abelian varieties of dimension $g$ together with a specified ample invertible sheaf.... | 2 | https://mathoverflow.net/users/13265 | 284018 | 125,583 |
https://mathoverflow.net/questions/284034 | 2 | Suppose $\pi:X \rightarrow \Delta$ is a fibration and $\pi^{-1}(0)$ is the only singular fiber, and let's also assume that all the fibers are complex projective varieties. Then we have a fibration between smooth manifolds,
\begin{equation}
\pi: X \setminus X\_0 \rightarrow \Delta^\*
\end{equation}
For every smooth fibe... | https://mathoverflow.net/users/87910 | Monodromy of a one parameter fibration | No, $\pi:X\to\Delta$ could be a family of cubic surfaces and then the monodromy will be generated by an element $w$ of the Weyl group $W(E\_6)$.
If the total space $X$ is smooth and the closed fiber $X\_0$ has a singularity of type $E\_6$ then $w$ is a Coxeter element. (This is a result of Demazure but I don't have the... | 2 | https://mathoverflow.net/users/8726 | 284036 | 125,587 |
https://mathoverflow.net/questions/252416 | 3 | A totally order-disconnected space (TOD) is a tuple $(P, \leq, \tau)$ where $(P, \leq)$ is a poset and $(P,\tau)$ is a topological space such that for $x\not\leq y$ in $P$ there is a clopen down-set that contains $y$, but not $x$. If $P, Q$ are TODs then a map $f:P\to Q$ is a *morphism* if $f$ is continuous and order-p... | https://mathoverflow.net/users/8628 | Compactification of order-disconnected spaces | It seems that the answer to this problem is affirmative:
Take a well-behaved pospace $P$. To show that $\beta\_2(P)$ is well-behaved, take any clopen upper set $U\subset \beta\_2(P)$. We should prove that its lower set ${\downarrow}U$ is clopen in $\beta\_2(P)$.
For this consider the upper clopen set $U\cap P$ in ... | 1 | https://mathoverflow.net/users/61536 | 284046 | 125,590 |
https://mathoverflow.net/questions/283597 | 12 | Let $\mathcal{L}\_{\kappa \lambda}$ denote the infinitary logic that allows conjunction of less than $\kappa$-many formulas and simultaneous quantification of less than $\lambda$-many variables. It is well-known that
>
> if a sentence $\varphi \in \mathcal{L}\_{\omega\_1 \omega}$ has
> arbitrarily large models, t... | https://mathoverflow.net/users/33039 | Models with few types in infinitary logics | Here is a partial answer: consistently, the generalization can fail for all uncountable $\kappa$. Namely:
Suppose $\mathbb{V} = \mathbb{L}$ and let $\kappa$ be any uncountable cardinal. Let $\mathcal{L}$ be the language consisting of the binary relation $\epsilon$ and constant symbols $(c\_\alpha: \alpha < \kappa)$.... | 6 | https://mathoverflow.net/users/26705 | 284053 | 125,595 |
https://mathoverflow.net/questions/284037 | 2 | Let $\mathcal{X}\rightarrow B$ be a family of projective varities ($B$ is DVR say) whose generic fibre is smooth, but the closed fibre is divisor with normal crossing singulairty. Is there some special cases known where the limiting Hodge structure on the closed fibre is pure? I am asking is there a known sufficient co... | https://mathoverflow.net/users/nan | Limiting Hodge structure | Sure. Let $H$ denote the limit MHS in degree $i$ associated to $\mathcal{X}\to B$, where I will assume that it's semistable. $H$ is isomorphic as a vector space to $H^i(X\_t)$ with $t\not=0$. Let $N$ denote the logarithm of monodromy which is unipotent by semi stability.
The weight filtration of $H$ is determined by th... | 2 | https://mathoverflow.net/users/4144 | 284056 | 125,597 |
https://mathoverflow.net/questions/284059 | 5 | Let $A$ be a finite subset of the group $H$. I am interested in sets with the property that
(1)$\qquad\qquad |\{ab\ \colon\ (a,b)\in A\times A\}| = |A|^{2}$.
Thus $A$ has property (1) if the product set $A^{2}$ is as large as possible.
Is it true that any infinite group has arbitrarily large finite sets satisfyin... | https://mathoverflow.net/users/90186 | Maximum-sized product sets in infinite groups | Indeed we even have, in a infinite group: for every finite subset $S$ and $n$ there exist a finite subset $F$ of cardinal $n$ such that the multiplication is injective on $S\times F$. Indeed, first find $F'=\{x\_1,\dots,x\_{n-1}\}$ of cardinal $n-1$ (by induction). Choose $x\_n$ distinct from $s^{-1}tx\_i$ for all $s,t... | 6 | https://mathoverflow.net/users/14094 | 284066 | 125,599 |
https://mathoverflow.net/questions/283997 | 5 | For a seperated scheme of finite type $X$ over $\mathbf{C}$, let $H\_\*(X)$ denote its Borel-Moore homology, which is defined by
$$
H\_k(X) = R^{-k}\Gamma(X, \omega\_X)
$$
where $\omega\in D\_c(X, \mathbf{C})$ is a dualising object in the derived category of constructible $\mathbf{C}$-sheaves on $X$. It is immediate fr... | https://mathoverflow.net/users/38052 | Base change for Borel-Moore homology | $\require{AMScd}$
After the struggle of a whole day due to my unfamiliarity with the category theory, I found the answer.
The commutativity is due essentially to the following facts:
1. Given any morphism of schemes $f: X \to Y$ we have the adjunction $(f^\*, f\_\*, \epsilon\_f, \eta\_f)$:
$$
f^\*: D\_c(Y, \mathbf{... | 3 | https://mathoverflow.net/users/38052 | 284076 | 125,604 |
https://mathoverflow.net/questions/263549 | 3 | The formula
$$
\small\sum\_{n=-\infty}^\infty (bq^n,p/aq^n;p)\_\infty
z^n q^{n(n-1)/2}=\frac{(-z,-q/z;q)\_\infty}{\ln\frac{1}{q}}\int\limits\_0^\infty\frac{\left(bt/z,pz/at;p\right)\_\infty}{\left(-t,-q/t;q\right)\_\infty}\frac{dt}{t},\quad |p|<|q|,\tag{1}
$$
or alternatively in more symmetric form
$$
\sum\_{n=-\infty... | https://mathoverflow.net/users/82588 | Does $\sum_{n=-\infty}^\infty (bq^n,p/aq^n;p)_\infty z^n q^{n(n-1)/2}$ have a closed form? | It turns out that the answer is quite easy.
Take $p=q^2$,$~z=1$,$~b=q^2a$, then
$$
\sum\_{n=-\infty}^\infty (bq^n,p/aq^n;p)\_\infty
z^n q^{n(n-1)/2}=2 \left(q a,q/a;q^2\right){}\_{\infty }\sum \_{n=-\infty}^\infty \frac{\left(-1/a\right)^n q^{n^2+n}}{1-a q^{2 n+1}}.
$$
The sum on the RHS is proportional to Appell-Le... | 1 | https://mathoverflow.net/users/82588 | 284097 | 125,609 |
https://mathoverflow.net/questions/284098 | 10 | How can one construct surfaces in $\mathbb R^3$ of constant negative Gaussian curvature containing a line in $\mathbb R^3$? (this question is inspired by [this MSE post](https://math.stackexchange.com/q/2478117/72694)).
| https://mathoverflow.net/users/28128 | Constant Gaussian curvature surfaces in 3-space containing lines | Given any point $p$ on a surface $S$ of Gauss curvature -1,
there exists an open neighborhood $U\subset S$ and $p$-centered
coordinates $(x,y):U\to\mathbb{R}^2$, whose image is
a domain $R = (x,y)(U)\subset\mathbb{R}^2$
and a function $u:R\to (0,\pi/2)$ such that the first
and second fundamental forms of the surface ... | 17 | https://mathoverflow.net/users/13972 | 284100 | 125,610 |
https://mathoverflow.net/questions/284099 | 4 | A group $G$ is *equationally Noetherian* if every system of group equations with coefficients from $G$ is equivalent to finite sub-system over $G$. It seems that this property must be invariant under geometric equivalences like quasi-isometry or at least bi-Lipschitz equivalence. I have no proof and no counterexample. ... | https://mathoverflow.net/users/44949 | Is being equationally Noetherian a quasi-isometry invariant? | No. Yes. $(\*)$ This is **not** a QI-invariant, not bilipschitz invariant, and not even (unlabeled) Cayley graph invariant.
Indeed, consider two finite groups $F\_1,F\_2$ of the same order: then $F\_1\wr\mathbf{Z}$ and $F\_2\wr\mathbf{Z}$ have isomorphic Cayley graphs (namely with respect to the generating subset $F\... | 5 | https://mathoverflow.net/users/14094 | 284109 | 125,613 |
https://mathoverflow.net/questions/284103 | 1 | I ask this question because of the apparent incoherence of the value of following integral:
$$I=\int\_{0}^{1} \int\_{0}^{\infty} \left|\sum\_{n=1}^{\infty} f(nx) e^{2 i \pi n y} \right|^2 dx dy$$
Where $f(x)$ is such that $\int\_{0}^{\infty} f(x)dx=0$, with near zero $f(x)=ax+o(x)$ and $f(x)$ exponentially decreasin... | https://mathoverflow.net/users/38290 | Incoherence of Fubini therorem with integral on Fourier series | The problem is not with Fubini's theorem (or more precisely, Tonelli's theorem), which is valid for any non-negative function measurable in the product sigma algebra. The error instead lies in the claim "the integral on $x$ beeing [sic] continuous in $y$". In fact it appears that the integral blows up like $1/y$ as $y ... | 5 | https://mathoverflow.net/users/766 | 284117 | 125,615 |
https://mathoverflow.net/questions/284074 | 10 | In some books, mainly about PDEs, I read that any open set can be approximated by sub-domain with smooth boundary (not just piecewise smooth). In 2 dimensional case, this seemly to be quite trivial: for any subdomain, use small open balls to cover its boundary and then mollify the connection parts. But in the higher di... | https://mathoverflow.net/users/114035 | Can we approximate any open set by sub-domains with smooth boundary? | By a well-known theorem of Whitney, any closed subset of $R^n$ coincides with the zero set of a $C^\infty$ function:
[Whitney, Hassler: Analytic extensions of differentiable functions defined in closed sets, Trans. AMS 36 (1934), 63--89.](http://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/S0002-99... | 9 | https://mathoverflow.net/users/68969 | 284118 | 125,616 |
https://mathoverflow.net/questions/284107 | 14 | Suppose I have two maps of topological spaces, $f:X\rightarrow B$ and $g:Y\rightarrow B$, such that $f$ induces a homology isomorphism and $g$ is a fibration and $B$ is connected. Is it true that the natural map $X\times\_{B}Y\rightarrow Y$ induces an isomorphism in homology?
| https://mathoverflow.net/users/103287 | Pullback and homology | This is not necessarily true. For example, there is a space $X$ constructed by attaching a 3-dimensional cell to $S^1 \vee S^2$, which serves as a standard counterexample to several questions. The map $X \to S^1$ produces isomorphisms $\pi\_1(X) \to \pi\_1(S^1)$ and $H\_\* X \to H\_\* S^1$, but $\pi\_2 X \to \pi\_2 S^1... | 20 | https://mathoverflow.net/users/360 | 284122 | 125,617 |
https://mathoverflow.net/questions/284079 | 6 | Suppose we work in an algebraically closed field. Then, do the Waring rank (symmetric tensor rank) and tensor rank of a symmetric tensor coincide in general? Recall that tensor rank is rank with respect to the Segre variety and Waring rank is rank with respect to the Veronese variety.
| https://mathoverflow.net/users/98093 | Waring rank vs tensor rank of symmetric tensors? | A counterexample in $\mathbb{C}$ is given in [A counterexample to Comon's conjecture](https://arxiv.org/abs/1705.08740):
>
> We present an example of a symmetric tensor of size 800×800×800 which
> can be written a sum of 903 simple tensors with complex entries but
> not as a sum of 903 symmetric simple tensors.
>... | 10 | https://mathoverflow.net/users/11260 | 284134 | 125,623 |
https://mathoverflow.net/questions/284141 | 1 | This question is a bit of a follow up to [*this*](https://mathoverflow.net/questions/283965/representations-of-algebraic-groups) question.
Let us consider the finite field $\mathbb{F}\_q$ and its algebraic closure $\mathbb{F}$, viewed as an additive abelian group. Its group of linear characters, ${\rm Hom}(\mathbb{F}... | https://mathoverflow.net/users/47300 | Linear characters of algebraic closed fields | $\newcommand{\IF}{\mathbb{F}}\newcommand{\IC}{\mathbb{C}}$
For simplicity I'm going to assume $q=p$.
Since all elements in $\overline{\IF\_p}$ have order $p$, any group homomorphism $\overline{\IF\_p}\to\IC^\times$ has either image contained in the group of $p$-th roots of unity $\mu\_p$. In other words we can equiva... | 3 | https://mathoverflow.net/users/3041 | 284158 | 125,628 |
https://mathoverflow.net/questions/284131 | 3 | In general, my problem can be formulated as follows: Let $X$ be a random variable with value in $\mathbb R^2$, and let $G:\mathbb R^2 \times \mathbb R\rightarrow \mathbb R$ be a function which is continuous in the first argument and measurable in the second(i.e., a Caratheodory function). Assume the partial maximizatio... | https://mathoverflow.net/users/101188 | measurable selection and values of optimization problem | Undery our assumptions, the argmax-correspondence need not even be nonempty-valued. But one can find almost-selections of an almost-argmax-correspondence, and that is enough.
Let $\epsilon>0$. Define the function $h$ by $h(x)=\sup\_yG(x,y)$. By assumption, $h$ is measurable. Therefore, the set
$$B=\big\{(x,y)\mid G(... | 2 | https://mathoverflow.net/users/35357 | 284159 | 125,629 |
https://mathoverflow.net/questions/284153 | 4 | Let $T\_n$ be a random tree on $n$ labelled vertices chosen equiprobably among all $n^{n - 2}$ trees, and $I(T)$ be the number of distinct independent sets of a tree $T$. I'm interested in the average number $\mathrm{E}I(T\_n)$ of independent sets of a random tree, at least asymyptotically.
Here are the values for a ... | https://mathoverflow.net/users/106512 | Number of independent sets of a random tree | The exact average number of independent sets in a random labelled tree of $n$ vertices is
$$ E\_n = \sum\_{k=0}^{n-1} \binom{n}{k} n^{1-k} (n-k)^{k-1}. $$
To prove this, use the Matrix Tree Theorem (or alternatively Ilya's method) to show that the number of labelled trees with a specified independent set of size $k$ ... | 6 | https://mathoverflow.net/users/9025 | 284163 | 125,631 |
https://mathoverflow.net/questions/284105 | 2 | Please feel free to delete or move it to somewhere. I just need a confirmation or a reference.
Let $D\_r(\mathbb{R}^l,S^n)=F(\mathbb{R}^l,r)\_+\wedge\_{\Sigma\_r}(S^n)^{\wedge r}$ be the $r$-th stable piece in $\Omega^lS^{n+l}$ obtained from Snaith splitting. Is $l(n+r-1)-(r-2)$ the best upper bound on the dimension ... | https://mathoverflow.net/users/51223 | Homological dimension of configurations spaces | Yep, that is the top class, when $r$ is a power of 2, in degree $(2^t-1)(l-1)+ 2^tn$. This also happens to be the top dimension of a cell complex homotopy equivalent to $D\_{2^t}(\mathbb R^l, S^n)$: the generalization to all $l$ of the Fox-Neuwirth cell structure shows that there is an equivariant cell complex of dimen... | 5 | https://mathoverflow.net/users/102519 | 284167 | 125,633 |
https://mathoverflow.net/questions/284135 | 1 | This question is related to my previous post: [New edge coloring problem in graph theory](https://mathoverflow.net/q/283886/90655).
*Added:* Let $G$ be a simple graph. Consider the following edge coloring:
1. We are allowed to use repetitive colors on some edges incident to a vertex such that the result does not c... | https://mathoverflow.net/users/90655 | Edges of every simple graph can be colored with at most $s+1$ color | It might be less confusing if you just claimed/asked if $\lfloor \frac{\Delta+3}2\rfloor$ colors suffice. However it is not correct. The $10$ edges of the complete graph $K\_5$ can not be $3$ colored without creating a monochromatic $3$ edge path (which might be closed, i.e. a triangle.)
Here is a sketch which seems ... | 2 | https://mathoverflow.net/users/8008 | 284175 | 125,634 |
https://mathoverflow.net/questions/284054 | 9 | Let $n \geq 7$ and $V = \mathbb{C}^n$ be the standard representation for $S\_{n+1}$, the symmetric group of cardinal $(n+1)!$
Let $k$ be an integer such that $2 \leq k \leq n$. Is it true or false that $\bigwedge^k V$ does not appear as an irreducible sub-representation of $\mathrm{Sym}^k V$? I am looking for a refe... | https://mathoverflow.net/users/37214 | alternating and symmetric powers of the standard representation of the symmetric group | Let $W = \mathbb C^n$, with the obvious representation of $S\_n$.
**Lemma** Let $\lambda$ be a partition of $n$. The multiplicity of the irreducible representation $V\_\lambda$ inside $W^{\otimes k}$ is given by the number of ways of obtaining the partition $\lambda$ by starting with the partition $[n]$, and then al... | 11 | https://mathoverflow.net/users/1310 | 284176 | 125,635 |
https://mathoverflow.net/questions/284173 | -1 | The Riemann Sphere is another imagination of the complex plane. In that imagination infinity is represented by the top pole of the sphere. Therefore it is imagined as only one single point.
But when we (in our lessons) deal with real numbers only, we image there is a +∞ and a -∞ (for example when talking about limits... | https://mathoverflow.net/users/116325 | Why don't we have a Riemann Circle with Infinity as one Point? | If you allow the Riemann sphere then in it you could say you have the real line with one point at infinity.
However in Calculus $\infty$ and $-\infty$ are not really numbers or places. It is a convenient notation because some of what it makes you think you can do is correct, but you have to check.
The way to read... | 0 | https://mathoverflow.net/users/8008 | 284178 | 125,636 |
https://mathoverflow.net/questions/284174 | -1 | Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes\_A k \le 1$? If not, is there a known upper-bound for $\dim M \otimes\_A k$?
| https://mathoverflow.net/users/58203 | On rank one torsion-free modules over local rings | $B=k[x\_1,\dots,x\_n]$,
$\mathfrak m=(x\_1,\dots,x\_n)\vartriangleleft\_{\rm max} B$,
$A=B\_{\mathfrak m}$,
$M=\mathfrak m\_{\mathfrak m}$.
Then $K:=\mathrm{Frac} A\simeq k(x\_1,\dots,x\_n)\simeq M\otimes\_A K$, so $M$ has rank one, but $\dim\_k M\otimes k=n$.
| 3 | https://mathoverflow.net/users/10076 | 284180 | 125,637 |
https://mathoverflow.net/questions/284172 | 2 | I remember I read those problems some place, but I cannot find it. Does anyone have any idea where I can find it?
1. If $X$ is a Banach space, then $(L^1(a,b;X))^\*\cong L^\infty(a,b;X^\*)$?
2. $X, Y$ are both Banach spaces, if $X$ is compactly embedded into $Y$, then do we have $L^p(a,b;X)$ is embedded in $L^p(a,b;Y... | https://mathoverflow.net/users/82561 | Some questions on parabolic function spaces | 1. This is true if and only if $X$ has the Radon-Nikodym property, see Diestel & Uhl: [Vector measures](http://bookstore.ams.org/surv-15), Chapter IV.1, Theorem 1.
2. This would be very useful indeed, but unfortunately, no. The standard reference to get started is probably the paper by Jacques Simon, [Compact sets in t... | 4 | https://mathoverflow.net/users/85906 | 284182 | 125,638 |
https://mathoverflow.net/questions/284188 | 10 | Assume that $G$ is a Lie group and at the same time it admits a symplectic structure.
>
> Does $G$ necessarily admit a symplectic structure such that the right multiplication preserves the symplectic structure?
>
>
>
| https://mathoverflow.net/users/36688 | Symplectic Lie groups | Let $H$ be the universal covering of $\mathrm{SL}\_2(\mathbf{R})\times\mathbf{R}$. Then $H$ is diffeomorphic to $\mathbf{R}^4$ and hence has a symplectic structure (as a manifold). However, every Lie group with a right-invariant symplectic structure is solvable (see Baues-Cortès ([arXiv link](http://arxiv.org/abs/1307.... | 12 | https://mathoverflow.net/users/14094 | 284193 | 125,640 |
https://mathoverflow.net/questions/284186 | 2 | Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map.
How to compute $l(R/m^{[p^e]})?.$
I know the answer is $p^{ed}$ but I do not know how to prove it.
| https://mathoverflow.net/users/9485 | Length of a module and Frobenius map | This is false. The noetherian local ring $R =
\mathbb{F}\_3[[X,Y]]/(Y^2 - X^3)$ has dimension one, and if $x,y$ are the images of $X,Y$ in $R$ then the sequence
$$
R \supseteq (x,y) \supseteq (x^2,y) \supseteq (y) \supseteq (y^2) = \mathfrak{m}^{[3]}
$$
shows that $R / \mathfrak{m}^{[3]}$ does not have $R$-length $3$.... | 3 | https://mathoverflow.net/users/21724 | 284199 | 125,642 |
https://mathoverflow.net/questions/216272 | 72 | This question has been crossposted from [Math.SE](https://math.stackexchange.com/q/1413071/98602) in the hopes that it reaches a larger audience here.
$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is dif... | https://mathoverflow.net/users/40804 | Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure? | The $m$-fold connected sum $m\# {\mathbb{CP}}^{2n}$ admits an almost complex structure if and only if $m$ is odd, as we show in our recent [preprint](https://arxiv.org/pdf/1710.05316.pdf). (By the way, thanks to Mike for this interesting question, which motivated us to write the paper!)
Here's a brief summary of the ... | 40 | https://mathoverflow.net/users/20999 | 284200 | 125,643 |
https://mathoverflow.net/questions/284150 | 3 | We define the upper density $\rho (G)$ of a finite group $G$ as the ratio of the number of finite groups of order $<n$ which contain a subgroup isomorphic to $G$ to the number of groups of order $<n$, in the limit as $n \rightarrow \infty$ (if such a limit exists, but we can always take limsup/liminf).
For example, ... | https://mathoverflow.net/users/103594 | On the Upper Density of $C_2$ in finite groups | Pyber showed that the number of groups of order $n$ is $\leq n^{\frac{2}{27}\nu(n)^3+C\nu(n)^{3/2}}$, where $\nu$ is the highest power of a prime dividing $n$ and $C$ is an absolute constant. On the other hand, Sims showed that for $n$ a power of $p$, this upper bound is achieved. Since for odd $n$ we have $\nu(n)\leq\... | 10 | https://mathoverflow.net/users/37555 | 284205 | 125,644 |
https://mathoverflow.net/questions/284183 | 5 | For $A, B \subseteq \omega$ we set $A \leq\_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a **down-set** of $B$. It is easy to see that $\leq\_{\text{inj}}$ is an ordering relation on ${\cal P}(\omega)$. (This is different from the lexicographic ordering di... | https://mathoverflow.net/users/8628 | A different ordering on ${\cal P}(\omega)$ | Yes there is a surjective order-preserving map from $\leq\_{inj}$ to $\subseteq$, no in the reverse direction.
If you restrict the $\leq\_{inj}$ order to the family of all infinite subsets of $\omega$ the result is isomorphic to the set of all functions from $\omega$ to $\omega$ with the coordinatewise order ($f \leq... | 5 | https://mathoverflow.net/users/23141 | 284209 | 125,647 |
https://mathoverflow.net/questions/247946 | 1 | Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL\_2(Z)$ and let $\lambda\_f(n)$ be the $n$-th normalized Fourier coefficient of $f.$ Can someone provide me with an effective estimate for this infinite product
$$\prod\_{\lambda\_f(p)=0} \left(1-\frac{1}{p+1}\right).$$
Thanks in a... | https://mathoverflow.net/users/76102 | Effective estimate for this infinite product over Hecke eigenvalues | In general, this product is hard to work with, and good upper/lower bounds require a bit of work. I'll also assume the simplest case of $k=12$, in which case $\lambda\_f(n)$ is the famous Ramanujan tau function.
Since the density of primes $p$ for which $\lambda\_f(p)=0$ is zero (a result first proved by Serre his pa... | 3 | https://mathoverflow.net/users/111215 | 284214 | 125,651 |
https://mathoverflow.net/questions/284202 | 1 | A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(X,\tau )$ is strongly nowhere dense if for any nonempty open set $U\in \tau$, there exists nonempty open set $V \subset U... | https://mathoverflow.net/users/114580 | Concerning a strongly nowhere dense subset | I add another answer in light of your comment to the answer above. Here I construct a generalized topological space (which is not a topological space) along with a non-empty strongly nowhere dense subset.
Let $X = \{0,1,2,3\}$ and let $$\tau = \big\{\{0,1\}, \{1,2\}, \{0,1,2\}, X\big\}.$$
Note that $X$ is the only ... | 1 | https://mathoverflow.net/users/8628 | 284216 | 125,652 |
https://mathoverflow.net/questions/284169 | 8 | Due to the recent spate of detections of gravitational waves by LIGO, my amateurish interest in the mathematics of general relativity has been revived.
---
The wave-forms of the detected gravitational waves are said to have been compared with predicted wave-forms produced by a combination of post-Newtonian approx... | https://mathoverflow.net/users/50614 | Some Mathematical Questions on Gravitational Waves and Numerical Relativity | Question 3:
You should not think of the singularity (corresponding to a black hole) as moving in space-time. [It is not](https://en.wikipedia.org/wiki/Penrose_diagram). So you are asking the wrong question if your motivation is gravitational waves.
The answer to the question you did ask however is "yes", see the [... | 2 | https://mathoverflow.net/users/3948 | 284219 | 125,654 |
https://mathoverflow.net/questions/284194 | 4 | This is a cross-post from [MSE](https://math.stackexchange.com/questions/2473216/do-trivial-homotopy-groups-imply-existence-of-boundary-preserving-homotopies).
Let $N$ be a smooth $d$-dimensional connected orientable manifold which have the following property:
>
> For every smooth $d$-dimensional manifold $M$ wit... | https://mathoverflow.net/users/46290 | Do trivial homotopy groups imply existence of boundary preserving homotopies? | Let me first talk only about continuous maps. Your question becomes equivalent to asking whether every map from $\partial (M\times I) \to N$ (for $I$ the interval) can be extended to $M\times I$. This follows from the following two facts.
1) $\partial (M\times I) \to M\times I$ is cofibration.
2) If $i: A \to B$ i... | 4 | https://mathoverflow.net/users/2039 | 284222 | 125,657 |
https://mathoverflow.net/questions/284189 | 1 | Let $G$ be a hyperbolic group, and $\mu$ a (nonsymmetric) probability measure on $G$ whose support generates $G$ as a semigroup.
Let $\nu$ be the associated harmonic ($\mu$ stationary) on $\partial G$.
Let $\hat{\mu}$ be the reflected random walk given by $\hat{\nu}(g)=\mu(g^{-1})$.
Let $\hat{\nu}$ be the $\hat{\mu}$ h... | https://mathoverflow.net/users/12395 | Absolute continuity of harmonic measure for a random walk and its reflection | In general there is no reason for the coincidence of the measure classes of the harmonic measures of the original and of the reflected random walks. It they do coincide, then this indicates that the random walk in question must be in a certain respect "special'', and it is very easy to construct counterexamples, alread... | 2 | https://mathoverflow.net/users/8588 | 284227 | 125,658 |
https://mathoverflow.net/questions/284230 | 60 | Here's a familiar conversation:
**Me:** Do you think Conjecture A and Conjecture B are equivalent?
**Friend:** Yes, because I think they're both true.
**Me:** [eye roll] You know what I mean...
Does there exist a rigorous notion of what I mean? Perhaps something about the existence of a proof of $p \Leftrightar... | https://mathoverflow.net/users/29873 | What does it mean to suspect that two conjectures are logically equivalent? | First of all, in practice when we say "Conjecture A is equivalent to Conjecture B," what we mean is "We *have a proof* that Conjecture A is true iff Conjecture B is true." We can have such a proof without having a proof of either conjecture, so this is a meaningful situation. Of course, it will (hopefully) *later becom... | 88 | https://mathoverflow.net/users/8133 | 284231 | 125,660 |
https://mathoverflow.net/questions/284261 | 4 | Let $O$ be an order in a number field (So $O$ is an one dimensional noetherian domain), I am curious about multiplication of norm function in $O$.
For every nonzero ideal $I \subseteq O$, let $N(I)=\#O/I$. The norm function need not to be multiplicative, for example take $O=\Bbb Z[2i], I=(2,2i),$ then $N(I^n)=2^{2n-1... | https://mathoverflow.net/users/102104 | Norm in an order of a number field and Hilbert polynomial | This is not true. An easier type of ring to think about is a subring of $k[x]$, e.g. $R = k[x^3,x^4]$; we will deal with the number ring case later. For this type of ring, your question basically asks that $\dim R/I^n$ equals $an - b$ for some constants $a, b$. For $R = k[x^3,x^4]$, we get:
* $I = (x^3, x^4)$, so $\d... | 7 | https://mathoverflow.net/users/82179 | 284263 | 125,672 |
https://mathoverflow.net/questions/284124 | 5 | This is a follow-up to [this question:](https://mathoverflow.net/questions/283987/the-largest-piece-of-circumference/283999#283999) Given $n$ random chords of a circle, what is the distribution of the number of intersections? Random is defined by "endpoints uniform on the circle".
**Update** Numerical experiments ind... | https://mathoverflow.net/users/11142 | The distribution of the number of chord intersections | The number of crossings in a chord diagram depends only on the order that the ends of the chords appear on the circle. Also, for any chord diagram every pairing of the $2n$ endpoints is equally likely. Therefore, it suffices to take any $2n$ distinct points on the circle and pair them at random.
This is a widely stud... | 1 | https://mathoverflow.net/users/9025 | 284274 | 125,677 |
https://mathoverflow.net/questions/284276 | 2 | Is the following true?
Let $X$ and $Y$ be separable Banach spaces and consider their dual spaces $X^\*$ and $Y^\*$ equipped with weak\* topology. Suppose that a linear map $T:X^\*\to Y^\*$ is sequentially continuous. Then, is it true that $T$ is continuous?
I suspect this may be true using the following two facts:
... | https://mathoverflow.net/users/116379 | Sequential Continuity in dual spaces of separable Banach Spaces | This is true. To show it, in the following I will use $\langle \mbox{-}, \mbox{-} \rangle$ for the pairing between a space and its dual (with the vectors from the space on the left, the dual on the right).
For each $y \in Y$, if we define $f(y) = \langle y, \mbox{-}\rangle \circ T$, then $f(y)$ is a sequentially cont... | 4 | https://mathoverflow.net/users/61785 | 284283 | 125,679 |
https://mathoverflow.net/questions/284281 | 7 | I always had the impression that there was a duality (i.e. a contravariant equivalence of categories) between Banach spaces and certain notion of pointed compact convex set (something like algebras for a certain Giri monad acting on the category of compact topological spaces, maybe with additional conditions) given by:... | https://mathoverflow.net/users/22131 | Duality between Banach spaces and compact convex spaces | In my opinion, the most natural expression of this symmetric duality is that between Banach spaces and Waelbroeck spaces (the latter being Banach spaces with a suitable compact topology on the unit ball). This goes back to work of Lucien Waelbroeck and Henri Buchwalter. For a systematic treatment, I would refer to the ... | 6 | https://mathoverflow.net/users/115244 | 284285 | 125,681 |
https://mathoverflow.net/questions/284292 | 30 | Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the quasi circle on $\mathbb R^2$ is an example. Yet I've not written down the (dirty) proof. But in this case all its homot... | https://mathoverflow.net/users/33064 | Does there exist any non-contractible manifold with fixed point property? | Take the space $\mathbb{CP}^2$. Its cohomology ring is given by $\mathbb{Z}[a]/a^3$, where $a$ has degree $2$. A map $f:\mathbb{CP}^2\rightarrow \mathbb{CP}^2$ induces a map on the second cohomology group with $f^\*(a)=k a$ with $k\in \mathbb{Z}$. From this you can compute the action on (co)homology on the other degree... | 49 | https://mathoverflow.net/users/12156 | 284296 | 125,685 |
https://mathoverflow.net/questions/284094 | 0 | **Let R be a commutative ring.$r\in R$.Let M and N are R-modules such that $rM=0$ and there is a short exact sequence $0\rightarrow N\xrightarrow r N\rightarrow N/rN \rightarrow 0$.**
if X is arbitrary R-module,$X\xrightarrow r X$ is R module morphism by multiplication,then we can get the induced map $Hom\_R(M,r):Hom... | https://mathoverflow.net/users/106580 | How to prove this by extension group | Let $\,\,E: 0 \to N \xrightarrow{i} W \xrightarrow{\varepsilon} M \to 0$ be a short exact sequence and let $r \in R$.
The short exact sequence induced from $E$ by the homomorphism $\mu: N \to N,\,n \mapsto r\cdot n$ is by definition$$\,\,E': 0 \to N \to X \to M \to 0$$
where $X= W \oplus N\,/\,\{(-i(n),rn)\mid n \i... | 1 | https://mathoverflow.net/users/18571 | 284301 | 125,687 |
https://mathoverflow.net/questions/283715 | 5 | Mobius transformations map circles to circles.
Wiki says 'Möbius transformations can be more generally defined in spaces of dimension n>2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere'.
>
> Is there an explicit way to given such transformations in $n$-dimenstions?
>
>
... | https://mathoverflow.net/users/10035 | Explicit generalizations of Mobius transformations? | Yes, and, indeed, these examples special cases of the situation that $G$ is a simple real Lie group, $K$ is a maximal compact subgroup, and we look at the action of $G$ on the left on the Riemannian symmetric space $G/K$.
For the so-called classical groups $G$, the spaces $G/K$ are model-able in terms of matrices, ve... | 6 | https://mathoverflow.net/users/15629 | 284320 | 125,692 |
https://mathoverflow.net/questions/284319 | 3 | I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem.
A preliminary result is stated in various books (Stanislaw lojasiewicz's *Intro to Complex Analytic Geometry* page 239 lemma 3, Grauert-Remmert's *Coherent Analytic Sh... | https://mathoverflow.net/users/105628 | Understanding Remmert-Stein extension theorem | I am posting my comment as an answer. I assume that you are taking the union of $\text{Zero}(y-ax)$ as $a$ ranges over an infinite set of integers. The set $V$ is not a complex analytic subvariety of $D\setminus N.$ The set $V$ accumulates on the line $\text{Zero}(x).$
| 3 | https://mathoverflow.net/users/13265 | 284334 | 125,698 |
https://mathoverflow.net/questions/186791 | 6 | I am not sure that this question even makes sense, which I suppose is part of the questions itself.
In any case, I attended a talk recently wherin there was some discussion about a "tropical Teichmuller space", as for example in [this](http://arxiv.org/pdf/1207.2443v2.pdf) paper, associated to the tropical moduli sp... | https://mathoverflow.net/users/22991 | Hodge Bundles on Tropical Spaces | There is a preprint [Towards a tropical Hodge bundle](https://arxiv.org/abs/1701.04385) by Bo Lin and Martin Ulirsch, which discusses the Hodge bundle over the moduli space of genus $g$ tropical curves.
| 4 | https://mathoverflow.net/users/81295 | 284337 | 125,699 |
https://mathoverflow.net/questions/284340 | 2 | For $A, B \subseteq \omega$ we set $A \leq\_{\text{inj}} B$ if there is an injective and order-preserving map $f:\omega\to \omega$ , such that $f(A)$ is a down-set of $B$. It is easy to see that $\leq\_{\text{inj}}$ is an ordering relation on ${\cal P}(\omega)$.
Is $({\cal P}(\omega), \leq\_{\text{inj}})$ a lattice?... | https://mathoverflow.net/users/8628 | Is $({\cal P}(\omega), \leq_{\text{inj}})$ a distributive lattice? | Let
$$A=\{a\_0<a\_0+a\_1<a\_0+a\_1+a\_2<\dots\},\qquad B=\{b\_0<b\_0+b\_1<\dots\},$$
so that the $a\_i$ and $b\_i$ are the gaps in $A$ and $B$.
Then $A\le\_{\mathrm{inj}}B$ iff $a\_i\le b\_i$ for each $i$. Thus $(\mathcal P(\omega),\le\_{\mathrm{inj}})$ is indeed a distributive lattice, isomorphic to $\omega^\omega$ wi... | 3 | https://mathoverflow.net/users/4600 | 284341 | 125,700 |
https://mathoverflow.net/questions/284345 | 5 | let $x\in (0,1)$, and $k$ be postive intgers,such $k\ge 2$,
>
> I conjecture following inequality maybe hold?
> $$x\cdot \dfrac{(1-x)^{k-1}}{(k+1)^{k-2}}+\dfrac{(1-2x)^k}{k^k}\le \dfrac{1}{(k+2)^{k-1}}$$ creat by wang yong xi
>
>
>
This is my attempt
when $k=2$,then inequality can be written as
$$x\cdot (1-x... | https://mathoverflow.net/users/38620 | Prove this conjecture inequality $x\cdot \frac{(1-x)^{k-1}}{(k+1)^{k-2}}+\frac{(1-2x)^k}{k^k}\le \frac{1}{(k+2)^{k-1}}$ | Consider only $k>2$. Denote $$f(x)=x\cdot \frac{(1-x)^{k-1}}{(k+1)^{k-2}}+\frac{(1-2x)^k}{k^k},$$
we need to prove that $f(x)\leqslant f(\frac1{k+2})$ for $x\in [0,1]$. We have $f'(\frac1{k+2})=0$ and $$f'(0)=(k+1)^{2-k}-2k^{1-k}=k^{2-k}\left(\left(1+\frac1k\right)^{2-k}-\frac2k\right)>0$$
by Bernoulli inequality $(1+x... | 8 | https://mathoverflow.net/users/4312 | 284361 | 125,706 |
https://mathoverflow.net/questions/284363 | 5 | Suppose I have $n$ *samples* $(x\_i, f(x\_i))\_{i=1}^n$ from an *unknown* function $f$. **I need to approximate (estimate) the derivative $f'(x^\*)$ at some new test point $x^\*$**, that is not necessarily one of the $x\_i$. I am assuming nothing about the $x\_i$: They can be regularly or irregularly sampled, have larg... | https://mathoverflow.net/users/99132 | Automatic vs numerical differentiation of a function known from samples | Automatic differentiation needs the structure of the function ( computation graph, or preferably a straight line program).
In your case, I am not sure how numeric differentiation helps to get a reliable result. If your parameter space is high-dimensional, you are completely screwed. If not, you can interpolate the f... | 8 | https://mathoverflow.net/users/11142 | 284372 | 125,707 |
https://mathoverflow.net/questions/284368 | 14 | If I have a system of linear equations, $A x = c$, with $A$ an $n\times n$ complex matrix, it is relatively easy to see that the set of matrices $A$ for which there is no (complex) solution has measure zero, as this is the set of matrices such that $\det(A) = 0$.
Can something similar be said for systems of *quadrati... | https://mathoverflow.net/users/84108 | Do almost all systems of quadratic equations have solutions? | Yes. The magic words are "elimination theory" and "resultant". In essence, the system has a solution unless some determinant (the iterated resultant) vanishes.
| 16 | https://mathoverflow.net/users/11142 | 284373 | 125,708 |
https://mathoverflow.net/questions/284228 | 1 | A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(X,\tau )$ is strongly nowhere dense if for any nonempty open set $U\in \tau$, there exists nonempty open set $V \subset U... | https://mathoverflow.net/users/114580 | Existence of a weak Baire space which is not Baire space | It is easy to see that every finite generalized topological space is a weak Baire space. Now, consider $X=\{ a,b,c \}$. Clearly, $\tau =\{ \{ a,b\} , \{ a,c\} , \{ b,c\} , X, \emptyset \}$ is a generalized topology on $X$. Then $X$ is weak Baire space, but it is not Baire space. Because $\{ a,b \} =\{ a\} ... | 1 | https://mathoverflow.net/users/114476 | 284380 | 125,710 |
https://mathoverflow.net/questions/284393 | 1 | In all the literature I have read, etale cohomology is defined for smooth varieties. Suppose $X/\mathbb{Q}$ is a singular variety over $\mathbb{Q}$, does there exist etale cohomology $H\_{\text{et}}^\*(X\_{\bar{\mathbb{Q}}},\mathbb{Q}\_{\ell})$ defined for it? If $H\_{\text{et}}^\*(X\_{\bar{\mathbb{Q}}},\mathbb{Q}\_{\e... | https://mathoverflow.net/users/87910 | Etale cohomology of singular varieties? | Just to answer the last question, the key statement is that for a proper variety over $\mathbb F\_q$, the number of $\mathbb F\_{q^n}$ points is equal to the alternating sum of the traces of $\operatorname{Frob}\_{q^n} = \operatorname{Frob}\_q^n$ on the etale cohomology. (Or for a general variety, if you take etale coh... | 12 | https://mathoverflow.net/users/18060 | 284396 | 125,715 |
https://mathoverflow.net/questions/284377 | 12 | Given two elements $A,B \in \text{SL}(2, \mathbb{F}\_{2^n})$, is there a (computationally inexpensive) test one could perform to check whether together they generate the entire group?
| https://mathoverflow.net/users/116435 | Testing whether two elements of $\text{SL}(2, \mathbb{F}_{2^n})$ generate the entire group | The answer is **yes**, it is *easy to check* whether the ordered pair $(A, B)$ generates $\text{SL}\_2(\mathbb{F}\_q)$ for $q$ the power of a prime number. Indeed, there exist *simple criteria* according to Daryl McCullough and Marcus Wanderley, see [1, Section 11]. (*Italic* means that I am quoting the authors).
> ... | 13 | https://mathoverflow.net/users/84349 | 284403 | 125,718 |
https://mathoverflow.net/questions/284398 | 14 | The Plünnecke–Ruzsa Inequality states that for a finite subset $A$ of an abelian group $G$ with small doubling $|A+A|\le K|A|$, the iterated sum and difference
sets are also small:
$|tA-sA| \le K^{t+s} |A|$.
It seems natural to expect that the optimal exponent on $K$ should actually be $K^{t+s-1}$, since we're thin... | https://mathoverflow.net/users/70060 | Optimality of the Plünnecke-Ruzsa Inequality | No. See for instance Exercise 2.3.5 of
*Tao, Terence; Vu, Van H.*, Additive combinatorics, Cambridge Studies in Advanced Mathematics 105. Cambridge: Cambridge University Press (ISBN 978-0-521-13656-3/pbk). xviii, 512 p. (2010). [ZBL1179.11002](https://zbmath.org/?q=an:1179.11002).
See also a number of papers of Ru... | 19 | https://mathoverflow.net/users/766 | 284407 | 125,719 |
https://mathoverflow.net/questions/283504 | 7 | I have heard that differential equations on $\mathbb{S}^1$ of the form
\begin{equation} \hspace{40mm} \dot{\theta}(t) \ = \ A\sin(\theta(t)) + g(t) \hspace{4mm} \mathrm{mod} \ 2\pi, \hspace{40mm} (1) \end{equation}
where $A$ is a constant and $g$ is a continuous $1$-periodic function, cannot admit locally asymptoticall... | https://mathoverflow.net/users/15570 | Can a periodically additively perturbed sinusoidal vector field on the circle have a stable periodic orbit of higher least period? | The claim is **true**. (As proved [here](https://arxiv.org/pdf/1505.06975.pdf), it is quite unique to sinusoidal vector fields. The same reference also mentions, in its introduction, existing applications of equation (1) with $g$ taking the particular form mentioned above.)
In short (as in Theorem 1 of [here](https:/... | 4 | https://mathoverflow.net/users/15570 | 284416 | 125,722 |
https://mathoverflow.net/questions/284401 | 4 | Let $G$ be a connected simple graph with two distinct edges $e,f \in E(G)$. Choose a random spanning tree $T\subseteq G$, my question is whether there are any known upper bound for the following
\begin{equation} \frac{\mathbb{P}(e\in E(T))\mathbb{P}(f\in E(T))}{\mathbb{P}(e,f\in E(T)) } \end{equation}
for all possible... | https://mathoverflow.net/users/116446 | Conditional probability that a random spanning tree contains the edge e | There is no constant upper bound, as shown by the following example. Take two vertices $v, u$ and connect them with $n \geq 2$ edge-disjoint paths of two edges. This graph has $n 2^{n - 1}$ spanning trees. The number of spanning trees containing any fixed edge $e$ is $(n + 1)2^{n - 2}$. However, for any $v-u$ path $e, ... | 6 | https://mathoverflow.net/users/106512 | 284421 | 125,724 |
https://mathoverflow.net/questions/140791 | 11 | Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a Shimura variety to a sub-Shimura variety. However I formulate the question for the special case I am interested in.
L... | https://mathoverflow.net/users/1310 | A frustrating cohomology class on the moduli of abelian surfaces | The answer to the main question above is indeed positive: the restriction map is nonzero for all $k \geq 2$. This was proved in Section 5 of my paper *Tautological rings of spaces of pointed genus two curves of compact type.* Compos. Math. 152 (2016), no. 7, 1398–1420.
The idea of the calculation is that these are Ei... | 5 | https://mathoverflow.net/users/1310 | 284429 | 125,728 |
https://mathoverflow.net/questions/284433 | 25 | I learned from a colleague that if one sums translates of the Gaussian density $f(x)=(2\pi)^{-1/2}e^{-x^2/2}$ translated by the integers (i.e. one considers $F(x)=\sum\_{n\in\mathbb Z}f(x+n)$), the resulting function is *remarkably* constant: that is, the function differs from its average value by less than one part in... | https://mathoverflow.net/users/11054 | Sum of Gaussian pdfs | First of all this has nothing to do with the inflection point of $e^{-\alpha x^2}$. According to Poisson summation formula (see Whittaker, Watson, Modern analysis, chapter 21.51)
$$
\sum\_{n=-\infty}^\infty e^{-\alpha (x-n)^2}=2{\sqrt{\frac{\pi}{\alpha}}}\left(1+2\sum\_{n=1}^\infty e^{-\frac{\pi^2}{\alpha}n^2}\cos2\pi ... | 34 | https://mathoverflow.net/users/82588 | 284439 | 125,730 |
https://mathoverflow.net/questions/284448 | 2 | In an introductory lecture note by T. Saito,
<http://www.ms.u-tokyo.ac.jp/~t-saito/pp/GR2.pdf>
he said that the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ could be seen as the fundamental group of the set set of primes $2,3,5,... \infty$. The local systems over $2,3,5,... \infty$ could be seen ... | https://mathoverflow.net/users/87910 | $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ as the fundamental group of $2,3,5,... \infty$? | Be aware that this is a somewhat vague comment by Saito which can be interpreted in numerous ways. I would not take what he has written too seriously.
I would guess that he means that
$$\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) = \lim\_U \,\pi\_1(U)$$
where the limit is over all open subsets $U \subset \mathrm{Spec} ... | 3 | https://mathoverflow.net/users/5101 | 284451 | 125,732 |
https://mathoverflow.net/questions/284432 | 3 | Let $\kappa$ be an infinite cardinal. Then we call $F\subseteq{\cal P}(\kappa)$ a *[Fano](https://en.wikipedia.org/wiki/Fano_plane) plane* on $\kappa$ if
1. $\bigcup F = \kappa$; $|F|=\kappa$; and $|a| = \kappa$ for all $|a|\in F$,
2. if $a\neq b\in F$ then $|a\cap b|=1$, and
3. if $x\neq y\in\kappa$ there is $a\in F... | https://mathoverflow.net/users/8628 | Infinite Fano planes | Let $K$ be a field of cardinality $\kappa$. Fix a bijection between $\kappa$ and the projective plane over $K$. Then let $F$ be (the image under this bijection of) the set of lines in the projective plane.
| 4 | https://mathoverflow.net/users/114758 | 284453 | 125,733 |
https://mathoverflow.net/questions/284225 | 3 | Let $C$ be a smooth projective curve over $\mathbb{C}$ with automorphism group $G$. When is it possible to construct an analytic family of smooth projective curves $\mathcal{C} \longrightarrow T$ (e.g. $T = \mathbb{C}$) such that $\text{Aut }C\_t \cong G$ for (almost) every $t \in T$ ($C\_t$ is the fiber over $t \in T$... | https://mathoverflow.net/users/38282 | Family of smooth projective curves with given automorphism group | You might also want to have a look at a book about Hurwitz stacks by Bertin-Romagny called "[Champs de Hurwitz](https://perso.univ-rennes1.fr/matthieu.romagny/articles/champs_de_Hurwitz.pdf)". There they construct spaces $\mathcal{H}\_{g,G,\xi}$ of smooth projective curves $C$ with an action of a finite group $G$ and s... | 2 | https://mathoverflow.net/users/69630 | 284456 | 125,734 |
https://mathoverflow.net/questions/284455 | 1 | I am in an ideal situation but I would appreciate a hint. First here is the scenario.
Let $\mathcal{J}$ be an the abelian variety obtained from the Jacobian of a genus $2$ curve $\mathcal{H}/k$ which has one $k$-rational point $\infty$ . Consider $\Theta\subset\mathcal{A}$ given by the the image of the Abel-Jacobi ma... | https://mathoverflow.net/users/91023 | Intersection number of divisors with its pull back and its push forward | There is a very general projection formula that's valid for any proper morphism $f:X\to Y$ and any cycles $x$ and $y$ in the Chow groups of $X$ and $Y$ respectively:
$$ f\_\*(x\cdot f^\*y) = f\_\*(x)\cdot y. $$
In your case, the intersections are 0-cycles, so taking the degree of both sides gives the formula that you w... | 4 | https://mathoverflow.net/users/11926 | 284459 | 125,736 |
https://mathoverflow.net/questions/284445 | 7 | Let $G$ be a group and $M$ a $G$-module. The basic definitions:
* $H^0(G, M)$ will be the set of $G$-fixed points in $M$.
* $Z^1(G, M)$ is the group of $1$-cocycles, i.e. the maps $f: G \rightarrow M$ such that $f(gg') = f(g) + g f(g')$ for all $g, g' \in M$.
* $B^1(G, M)$ is the group of $1$-coboundaries, ie. maps $... | https://mathoverflow.net/users/38068 | Is there an explicit description of the corestriction map $H^1(H, M) \rightarrow H^1(G, M)$? | I believe it is the following. Let $f$ be a cocyle for $H$. Take a set of representatives $X$ of $G/H$ in $G$. Then $\operatorname{cor}(f)(g) = \sum\_{x \in X} y\cdot f(y^{-1}gx)$ where $y\in X$ is the unique representative such that $gxH=yH$. Then $\operatorname{cor}(f)$ is a cocycle whose class is the well-defined co... | 7 | https://mathoverflow.net/users/5015 | 284461 | 125,737 |
https://mathoverflow.net/questions/283787 | 6 | Consider a completely bounded unital map $\Phi: \mathbf M\_h(\mathbb C) \to \mathbf M\_k(\mathbb C)$. Suppose that $\Phi$ has right-inverse $\Psi$ which is completely positive. Is the operator norm of $\Phi$ stable under tensor products with other spaces — *i.e.*, is $$\lVert \Phi \rVert \stackrel?= \lVert \Phi \rVert\... | https://mathoverflow.net/users/3723 | Completely bounded norm for unital maps with completely positive sections | Unfortunately, the answer is no.
Suppose $\Phi : M\_4 \rightarrow M\_2$ and $\Psi : M\_2 \rightarrow M\_4$ are given by
$$\Phi\left(\left[\begin{array}{cc} A & B\\ C& D\end{array}\right]\right) = A + 10B^T \ \ \textrm{and} \ \ \Psi(A) = \left[\begin{array}{cc} A&0\\ 0&A\end{array}\right] $$
where $A,B,C,D\in M\_2$. ... | 7 | https://mathoverflow.net/users/76593 | 284466 | 125,738 |
https://mathoverflow.net/questions/284462 | 5 | This question is from [here](https://math.stackexchange.com/questions/2490828/exponential-derivative-of-delta-distribution). I'm asking it here as well to increase the number of people who see it and might be able to help.
The question is, what is the result of the following integral for integer $n$ and real $x$?
$... | https://mathoverflow.net/users/37039 | Exponential derivative of delta distribution? | Note that
$$e^{a\partial/\partial x}f(x)=f(x+a)$$
is the translation operator, so your exponent of the delta function gives $2\pi \delta(x-in)$, which is indeed consistent with
$$\int\_{-\infty}^\infty e^{izy}dy=2\pi\delta(z)$$
for $z=x-in$.
---
All of this is purely formal, but there have been attempts to p... | 1 | https://mathoverflow.net/users/11260 | 284468 | 125,739 |
https://mathoverflow.net/questions/284444 | 7 | For a positive integer $n$ put $\omega(n)$ for the number of distinct prime divisors of $n$. It is a well-known theorem of Erdős and Kac that the probability distribution for the quantity
$\displaystyle \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}}$
is the standard normal distribution. In other words, we have... | https://mathoverflow.net/users/10898 | Density of integers with many prime factors | This is answered in:
*Mehrdad, Behzad; Zhu, Lingjiong*, [**Moderate and large deviations for the Erdős-Kac theorem**](http://dx.doi.org/10.1093/qmath/hav035), [ZBL06553541](https://zbmath.org/?q=an:06553541).
(can be found on [arxiv.org](https://arxiv.org/abs/1311.6180))
The paper also has an excellent bibliography... | 5 | https://mathoverflow.net/users/11142 | 284470 | 125,740 |
https://mathoverflow.net/questions/284449 | 4 | In Harry Gonshor's *An Introduction to the Theory of Surreal Numbers*, on page 50, Gonshor points to a method for intuitively guessing what the square root of the countable infinity is in his construction -- I have a generalization of this intuitive guess method that I am curious about.
For some context, in Gonshor'... | https://mathoverflow.net/users/92164 | Roots of $\omega$, larger $\gamma$-numbers | Chapter 3 of ONAG (see parts "$\omega$-map" and "Sign-expansions and normal forms") contains a way to obtain $\omega^x$ for any number $x$ (including transfinite cardinals). **UPD (replacing a false statement)**: according to the reference, for $x$ having sign expansion $x\_0 x\_1 \ldots$ (indexed by ordinals up to som... | 4 | https://mathoverflow.net/users/106512 | 284471 | 125,741 |
https://mathoverflow.net/questions/284388 | 2 | In the paper
*The volume of a small geodesic ball in a Riemannian manifold*, Michigan Mathematical Journal **20** (1973), doi: [10.1307/mmj/1029001150](http://dx.doi.org/10.1307/mmj/1029001150)
A. Gray proved the following result:
>
> Let $M$ be a Riemannian manifold of dimension $n$ with positive scalar Ricci... | https://mathoverflow.net/users/7460 | Volume of large balls in embedded submanifolds | Since volume is an intrinsic geometric invariant, the fact that $M$ is a submanifold is irrelevant. If you assume that Ricci curvature is nonnegative, then the inequality follows from the [Bishop-Gromov inequality](http://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality) (in fact, just the Bishop inequality).
A... | 2 | https://mathoverflow.net/users/613 | 284478 | 125,746 |
https://mathoverflow.net/questions/284472 | 7 | Let $f : \mathbb R \to \mathbb R$ be a function such that $f \in L^1[-a,a] , \forall a \in (0,\infty)$ and $g(x) : = \lim \_{r\to \infty} \dfrac 1r \int\_{x-r}^{x+r} f(t) dt$ exists in $\mathbb R$ for every $x \in \mathbb R$ . Then is it true that $g$ is an affine map i.e. $g(x)-g(0)=x(g(1)- g (0)) , \forall x \in \mat... | https://mathoverflow.net/users/nan | $f$ locally (Lebesgue) integrable function on real line, $g(x):= \lim _{r\to \infty} \frac 1r \int_{x-r}^{x+r} f(t) dt$ exists for every real $x$ | The answer is yes.
Let $\int\_a^b:=\int\_a^b f(t) dt$, with $\int\_b^a:=-\int\_a^b$ (if $a<b$). We have
\begin{equation}
\int\_{x-r}^{x+r}\sim rg(x)\tag{\*}
\end{equation}
as $r\to\infty$, where $A\sim B$ is understood as $A-B=o(r)$, and $x,y,\dots$ are any real numbers. Hence, substituting $s:=r-x$, we have
\beg... | 4 | https://mathoverflow.net/users/36721 | 284482 | 125,748 |
https://mathoverflow.net/questions/268709 | 2 | Ultimately this is about how primes jump. I will abstract the situation somewhat as there may be related applications which do not spring to my mind.
I want to find small spoilers to Hall's Marriage Theorem for a specific situation. As the index i increases, I will be processing a sequence S\_i of subsets of Y, and I... | https://mathoverflow.net/users/3402 | A faster way to spoil an injection? | Bipartite matching.
I am using a refactored version of an algorithm of Hopcroft and Karp found at
<https://en.wikipedia.org/wiki/Hopcroft>–Karp\_algorithm . I found that instead of reintializing the Pairu and Pairv arrays for each run, I can modify them from the previous run to take advantage of previous work. I may... | 0 | https://mathoverflow.net/users/3402 | 284493 | 125,754 |
https://mathoverflow.net/questions/284498 | 1 | Let $G$ be a semisimple algebraic group over the complex numbers and we fix a maximal torus $T$. Let $w\in W$ be an element in the Weyl group, and let $T^{w}$ be the elements in $T$ that are fixed by $w$. Is $T^{w}$ still a torus? Is the Lie algebra of $T^{w}$ going to be $t^{w}$, the fixed points of $w$ acting on the ... | https://mathoverflow.net/users/58689 | Weyl group action on maximal tori | The first statement amounts to determining whether $T^w$ is connected. In the simplest case $G={\rm SL}\_2$, $T$ may be identified with the multiplicative group ${\rm G}\_m=\Bbb{C}^{\times}$, the Weyl group is cyclic of order 2 and the simple reflection $s$ acts by $x\mapsto x^{-1}$. Therefore, its fixed point set is $... | 7 | https://mathoverflow.net/users/5740 | 284514 | 125,760 |
https://mathoverflow.net/questions/284507 | 0 | It is about deformation theory on algebraic surfaces.
If there are two singular points on an algebraic surfaces, is it possible that two singular points collapse to a new point as the surface deforms?
Also, if it is possible, I am wondering that the new point might be nonsingular.
| https://mathoverflow.net/users/75699 | Can two singular points collapse to a new singular point? | Yes, it is possible, as shown by the following simple example.
Think of a double cover $S$ of $\mathbb{P}^2$ branched on two smooth conics intersecting transversally: it has four singularities of type $A\_1$ (locally, they are of the form $x^2+y^2+z^2=0$).
When you deform the branch locus to two smooth conics tha... | 5 | https://mathoverflow.net/users/7460 | 284518 | 125,762 |
https://mathoverflow.net/questions/284525 | 2 | I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. A coupling is a joint distribution of $A,B$ with marginal distributions $A,B$. I know there are several ways to couple any two variables, but I am placing a restriction on the coupling $(A',B')$ (if such a coupling exists) of ... | https://mathoverflow.net/users/95756 | Reference Request for Couplings with Conditions | I think the Hall Marriage Theorem will give you necessary and sufficient conditions for the existence of a coupling:
Write $i\sim j$ if $i/\gcd(i,j)$ is prime or $i=j$.
Define functions:
\begin{align\*}
a\colon\mathcal P(B)\to\mathcal P(A);
&\quad a(T)=\{i\in A\colon \exists j\in T\text{ with }i\sim j\}\text{; and}\\
... | 5 | https://mathoverflow.net/users/11054 | 284538 | 125,766 |
https://mathoverflow.net/questions/284512 | 0 | Suppose that the function $f$ is defined by
$f(z) = \sum\_{j=0}^\infty a\_{2j} z^{2j}$ where $a\_{2j} \ge 0, z \in \mathbb{C}$. My questions are the following:
1. First I want to check this point: if we have a bound$f(z) \le e^{z^2}$ for real $z$, then does this imply that $f$ is entire? I think $f$ should be entire... | https://mathoverflow.net/users/54494 | Bounds for the coefficients of the even entire function with positive coefficients | The answer to the question 1 is yes, and this is called Pringsheim's theorem:
For a series with non-negative coefficients the singular point which is closest to
the origin lies on the positive ray. Thus, if the series converges for all positive $z$, it converges everywhere and the function is entire. (Your estimate
$f(... | 4 | https://mathoverflow.net/users/25510 | 284540 | 125,768 |
https://mathoverflow.net/questions/284491 | 2 | Let us define an abstract "cycle complex" as the following combinatorial object: it is $(V, C)$, where $V$ is a set of $n$ nodes, $C$ is a set of $c$ cyclically ordered subsets of $V$, each of size exactly $s$, and it is assumed that two sets in $C$ intersect on at most one node. Let us define the genus of $(V, C)$ as ... | https://mathoverflow.net/users/25121 | Maximum genus of an abstract "cycle complex" | Let's first consider the special case that each node appears in exactly two sets in $C$. (Nodes that only appear in one set are irrelevant and can be deleted without changing the possible embeddings, and I will say something about nodes that appear in more than 2 sets later).
Then we can define the "dual" of a cycle ... | 1 | https://mathoverflow.net/users/353 | 284541 | 125,769 |
https://mathoverflow.net/questions/284508 | 2 | Let $R\_\alpha$ be the $\alpha$-th infinite regular ordinal. This question assumes AC, so the following is true:
$$R\_\alpha=\left\{
\begin{array}{ll}
\omega\_\alpha & \alpha=\kappa+n\;\mathrm{where}\;n<\omega\land\kappa\;\mathrm{is}\;\mathrm{wk. inaccessible}\;\mathrm{or}\;0 \\
\omega\_{\alpha+1} & \mathrm{otherwi... | https://mathoverflow.net/users/115951 | Elementary extensions of infinitary languages | $\alpha$-supercorrect cardinals are consistent, for any $\alpha$.
Work in a mild second-order set theory, namely Gödel–Bernays set theory along with the assertion that there is a $\mathcal L\_{R\_\alpha,R\_\alpha}$-truth predicate. (This is defined similarly to the usual Tarskian definition of a truth predicate, with... | 5 | https://mathoverflow.net/users/64676 | 284554 | 125,774 |
https://mathoverflow.net/questions/284543 | 7 | Let $p(x) = \sum\_{n=1}^N e^{2 \pi i a\_n x}$ be a trigonometric polynomial, where $a\_n$ are distinct positive integers. There is a classical trick which (using Hölder's inequality) allows to give a lower bound for the $L^1$ norm of $p$ in terms of the $L^4$ norm of $p$. One obtains
$$
\|p\|\_1 \geq \frac{\|p\|\_2^{3}... | https://mathoverflow.net/users/46852 | $L^1$ and $L^4$ norms of trigonometric polynomials | No this need not be the case. Take $f(x) = \sum\_{n=1}^{N/2} e(nx)$ and $g(x) = \sum\_{k=N/2}^N e(2^kx)$. Then the $L^4$ norm of $f$ is big -- of size $N^{\frac 34}$ -- and its $L^1$ norm is very small -- of size $\log N$. On the other hand the $L^4$ norm of $g$ is small -- of size $\sqrt{N}$ -- and its $L^1$ norm is c... | 12 | https://mathoverflow.net/users/38624 | 284561 | 125,777 |
https://mathoverflow.net/questions/284536 | 8 | I'm looking for references on the development of (some of) Category theory without the axiom of choice. One possible axiom system (that, to me, seems the natural setting) is ZF + there are arbitrarily large inaccessible cardinals.
I found a link to [these](http://cs.nyu.edu/pipermail/fom/2008-March/012783.html) notes... | https://mathoverflow.net/users/95265 | Category theory without axiom of choice | As was pointed out in a comment (that should have been an answer), Makkai's work on anafunctors gives a way to replace the notion of "functor" by a more general notion that doesn't require the axiom of choice in order to prove, for instance, that every fully faithful essentially surjective functor is an equivalence (in... | 11 | https://mathoverflow.net/users/49 | 284563 | 125,779 |
https://mathoverflow.net/questions/284559 | 2 | I have a field $L$ and two field automorphisms $f, g: L \to L$ such that $f \circ g = g \circ f$. Under what conditions can they be extended to automorphisms of an extension $K \mid L$ that still commute? In particular they can certainly be extended to automorphisms of the algebraic clousure $K = \overline L$, but can ... | https://mathoverflow.net/users/17353 | Extending commuting automorphisms of a field | **Reformulation in terms of group cohomology of Galois groups.** Assume that $f$ and $g$ both have finite order dividing the positive integer $n.$ The the fixed subfield $E=L^{f,g}$ of $f$ and $g$ gives a finite, Galois extension $L/E.$ The two commuting elements $f$ and $g$ define a surjective group homomorphism, $$\r... | 2 | https://mathoverflow.net/users/13265 | 284567 | 125,780 |
https://mathoverflow.net/questions/244029 | 3 | It is known that the orientable genus of union of two (disjoint) graphs is the sum of their genus. So, it is natural to ask
*What can be said about the non-orientable genus of union of two (disjoint) graphs?*
Note that upper and lower bounds are known for the non-orientable genus of $k$-amalgams of two graphs with... | https://mathoverflow.net/users/40723 | Non-orientable genus of union of graphs | The non-orientable genus $\tilde{\rm{g}}$ is not additive: consider the union of $K\_5$ and $K\_7$. We have $\tilde{\rm{g}}(K\_5)=1$, $\tilde{\rm{g}}(K\_7)=3$ and $\tilde{\rm{g}}(K\_5\cup K\_7)=3$: embed $K\_5$ on the projective plane, $K\_7$ on the torus, and take the connected sum of these two surfaces.
| 2 | https://mathoverflow.net/users/24076 | 284574 | 125,783 |
https://mathoverflow.net/questions/245246 | 4 | This question is a continuation of the following two questions:
[Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion.](https://mathoverflow.net/questions/133445/discriminants-of-indefinite-integral-binary-quadratic-forms-admitting-3-or-6-tor)
[On certain solutions of a quadratic for... | https://mathoverflow.net/users/10898 | On the automorphism group of binary quadratic forms | Here is a necessary and sufficient condition for $G^{-}(f)$ to be non-empty, taken from [1, Exercise 6.21]:
>
>
> >
> > Let $S = \begin{pmatrix} a & b/2 \\ b/2 & c \end{pmatrix}$ with $a,b$ and $c \in \mathbb{Z}$. Then the following are equivalent:
> >
> >
> >
>
>
> $(1)$ There exists $A \in \text{GL}\_2(\ma... | 3 | https://mathoverflow.net/users/84349 | 284581 | 125,785 |
https://mathoverflow.net/questions/284590 | 3 | Let $I$ be a set, $\mathcal{U}$ be an ultrafilter on $I$ and $1\leq p<\infty$. Let $X\_{i}=L\_{p}(\mu\_{i})(i\in I)$, where $\mu\_{i}$ is a probability measure for each $i\in I$. Relying on standard results from Banach lattice theory, we can prove that the ultraproduct $(X\_{i})\_{\mathcal{U}}$ of $(X\_{i})\_{i\in I}$ ... | https://mathoverflow.net/users/41619 | A question on ultraproducts of $L_{p}(\mu)$-spaces | As for the first question the answer is **no** even if $\mu$ is assumed to be $\sigma$-finite as by a simple change of measure you may reduce the problem to the case where $\mu$ is finite. Let us fix a finite measure $\mu$.
When $p=1$ we may use the following reasoning. The space $L\_1(\mu)$ is [weakly compactly gene... | 5 | https://mathoverflow.net/users/15129 | 284609 | 125,795 |
https://mathoverflow.net/questions/284594 | 0 | The question is simply: let $|S| = n,$ how big can a subset $I$ of $2^S$ be such that for any $A, B \in I,$ $A\not\subset B$ (so, this is an independent set in the graph whose vertices are subsets, and edges correspond to inclusion in some direction). It is clear that $|I| \geq \binom{n} {\lfloor \frac{n}{2} \rfloor},$... | https://mathoverflow.net/users/11142 | Independent sets of subsets | To address the updated question, a family of subsets of $[n]$ is called *$k$-Sperner* if it does not contain a chain of length $k+1$. By taking all sets whose size lies in the middle $k$ values of $[n]$, there exist $k$-Sperner families who size is the sum of the $k$ middle bịnomial coefficients. Erdős proved that this... | 5 | https://mathoverflow.net/users/2233 | 284616 | 125,797 |
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