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https://mathoverflow.net/questions/355864 | 9 | Let $G$ be a connected Lie group. Recall that the topological group $G^\delta$ is $G$ endowed with the *discrete* topology. The inclusion $G^\delta \to G$ induces a map between the classifying spaces $\eta: BG^\delta\to BG$.
**Question 1**
Let $\eta^\*:H^\*(BG,\mathbb{Z})\to H^\*(BG^\delta,\mathbb{Z})$ be the indu... | https://mathoverflow.net/users/99042 | About the cohomology of $BG^\delta$. Making a Lie group discrete | I will only attempt to answer your first question. The reason there is no contradiction is that it is **not** true for arbitrary spaces that $H^{\ast}(X;\mathbb Q) = H^{\ast}(X;\mathbb Z) \otimes \mathbb Q$.
For instance, take $X = B\mathbb Q$. Then $H^2(X;\mathbb Z) = \text{Ext}(\mathbb Q,\mathbb Z)$ is a $\mathbb Q... | 13 | https://mathoverflow.net/users/14233 | 355870 | 150,186 |
https://mathoverflow.net/questions/354236 | 20 | A closed, oriented $d$-manifold $M$ is said to *dominate* another such manifold $N$ if there exists a map $M \to N$ of non-zero degree. (This notion should not be confused with the unrelated concept of *homotopical domination* that Wall's finiteness obstruction relates to.)
This relation turns $d$-manifolds into a po... | https://mathoverflow.net/users/14233 | Can every manifold be dominated by a parallelizable one? | I think this follows from the stable Hurewicz map $\pi\_n^s(N) \to H\_n(N;\mathbb{Z})$ being an isomorphism modulo torsion. If we set $n = \dim(M)$ then some positive multiple of the fundamental class of $M$ is hit. Under the Pontryagin-Thom isomorphism this gives a stably framed $M$ and a continuous $f: M \to N$ such ... | 11 | https://mathoverflow.net/users/154199 | 355876 | 150,187 |
https://mathoverflow.net/questions/355874 | 1 | If $A\_G$ is the adjacency matrix of a k-regular graph, let $B = J+xA\_G$, where J is the matrix whose elements are all 1s and $x\in R$ is a scalar. If $\lambda\_1\geq\lambda\_2\geq \dots \geq \lambda\_n$ are eigenvalues of $A\_G$, how do we prove that $\underset{x\in R}{\min} \lambda\_{max}(J+xA\_G) = \frac{n\lambda\_... | https://mathoverflow.net/users/153397 | Eigenvalues of adjacency matrix of a k-regular graph | If $G$ is regular, then $J$ and $A\_G$ are simultaneously diagonalizable (i.e. they have a common set of eigenvectors).
That is, the eigenvalues of $xA\_G$ and $J$ (to the same eigenvectors) just add up to the eigenvalues of $B=J+xA\_G$.
Note that the spectrum of $J$ is $\{0^{n-1}, n^1\}$, and that the eigenvalue $n$... | 8 | https://mathoverflow.net/users/108884 | 355880 | 150,190 |
https://mathoverflow.net/questions/355858 | 1 | In Munkres Theorem 20.4 it is shown that the (relative) uniform topology induced by:
$$
d(x,y)\triangleq \sup\_{n \in \mathbb{N}} d(x\_n,y\_n)
$$
is strictly finer than the product topology on $\prod\_{n \in \mathbb{N}} \mathbb{R}$ and its relative topology on $\ell^{\infty}(\mathbb{R})$.
Now, for each positive inte... | https://mathoverflow.net/users/36886 | Comparison of product topology and colimit topology in sequence spaces | As user131781 noted the correct answer depends on the topology of $K\_n$. If it is the norm topology the result is a special case of the following more general case:
Let $K$ be a metric space with topology $\cal{O}$ and $K\_n$, $n \in \mathbb{N}$ a sequence of subsets with $K\_n \subset K^o\_{n+1}$ ($X^0$ the interior ... | 1 | https://mathoverflow.net/users/100904 | 355889 | 150,191 |
https://mathoverflow.net/questions/355251 | 4 | A correspondence $\_{N} H\_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras was proved in [Pe].
>
> **Theorem ([Pe,Theorem 3.2.])** Let $N$ be a factor and $N\_0 \subset N$ weakly-$\ast$ den... | https://mathoverflow.net/users/12604 | Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations | For separable von Neumann algebras the Haagerup property for $N$ is equivalent to the existence of a real closable derivation $\delta$ such that $\delta^\*\delta$ has compact resolvents in $\mathcal B(L^2(N))$. This is true even in the non-tracial case. See Theorem 7.7 in:
Martijn Caspers, Adam Skalski, The Haagerup... | 2 | https://mathoverflow.net/users/6460 | 355890 | 150,192 |
https://mathoverflow.net/questions/355884 | 10 | I can prove the following result without too much trouble:
Let $f: X \to S$ be a proper flat morphism of finite presentation with $S$ irreducible having generic point $\xi \in S$. Then the following are equivalent,
(1) the generic fiber $X\_\xi \to \mathrm{Spec}{\kappa(\xi)}$ is smooth
(2) there exists a smooth f... | https://mathoverflow.net/users/154157 | Smoothness of the generic fiber implies smoothness over a dense open of the target? | As Will Sawin said in his comment, this is true using Chevalley's theorem. A sketch goes as follows.
Suppose that $X\_\xi \to \mathrm{Spec}(\kappa(\xi))$ is smooth. Let $N \subset X$ be the nonsmooth locus. For a morphism of finite presentation, the smooth locus is retrocompact open so $N$ is constructible. Thus by C... | 9 | https://mathoverflow.net/users/154157 | 355893 | 150,193 |
https://mathoverflow.net/questions/355840 | 5 | Let $W \subset V$ be quadratic spaces over a number field $F$.
Let $G\_n=SO(V)$ and $G\_m=SO(W)$ and we consider $G\_m$ as a subgoup of $G\_n$ via a diagonal embedding.
Let $f$ be an automorphic form of $G\_n$ and $g$ an automorphic form of $G\_m$.
I am wondering whether the function $h$ on $G\_m$ defined by $h(k... | https://mathoverflow.net/users/29422 | Restriction of product of automorphic forms | This is a special case of restricting automorphic forms on a larger group $G$ to a smaller (sub)group $H$, of course. So far as I know, orthogonal groups are not special in this regard, although, yes, there are some obvious natural maps among them.
Certainly if the map $H\to G$ is a $k$-morphism (with groundfield $k$... | 4 | https://mathoverflow.net/users/15629 | 355894 | 150,194 |
https://mathoverflow.net/questions/355896 | 1 | The distribution of functions of random variables is well-studied for various different and general cases, but I didn't find much result for the reverse.
>
> Suppose that $X\_1, X\_2$ are (probably independent) random variables and we have $X\_1\sim f(x\_1)$ and $X\_2\sim g(x\_2)$. Construct a new distribution for ... | https://mathoverflow.net/users/97167 | Random variable corresponding to sum of density functions | I assume $X\_1 \sim f(x\_1)$ means that that the distribution of $X\_1$ has density function $f$.
Note first that your $\kappa$ can only be $2$, otherwise the integral of $p$ will not equal $1$. Then $p$ is the density of a [mixture](https://en.wikipedia.org/wiki/Mixture_distribution) of $X\_1, X\_2$; it corresponds ... | 1 | https://mathoverflow.net/users/4832 | 355906 | 150,197 |
https://mathoverflow.net/questions/355910 | 1 |
>
> I'm trying to eigendecompose the following matrix $A$, i.e. to find $Q$ and $\Lambda$ such that
> $$
> A = \begin{bmatrix}
> -\alpha & \alpha & -\gamma^{-1} & 0\\
> \beta & -\beta & 0 & -\gamma^{-1}\\
> -1 & 0 & \alpha & -\beta\\
> 0 & -1 & -\alpha & \beta
> \end{bmatrix}=Q\Lambda Q^{-1}
> $$
> where $\al... | https://mathoverflow.net/users/136012 | Spectral decomposition of a $4\times4$ real nonsymmetric matrix with unknown elements | I don't know how explicit or simple you want the expressions for $\Lambda$ and $Q$ to be, but here is a description. Using your notations, we write $A$ as
$\begin{bmatrix}
B&-\gamma^{-1}{\rm{I}}\_2\\
-{\rm{I}}\_2&-B^{\rm{T}}
\end{bmatrix}$
where
$B=\begin{bmatrix}
-\alpha&\alpha\\
\beta&-\beta
\end{bmatrix}$.
Let $\l... | 2 | https://mathoverflow.net/users/128556 | 355922 | 150,199 |
https://mathoverflow.net/questions/355929 | 2 | Is there a derivation of an uncertainty principle or uncertainty-type principle from the symplectic non-squeezing theorem?
| https://mathoverflow.net/users/154158 | Derivation of an uncertainty principle from the symplectic non-squeezing theorem | Yes. see:The symplectic camel and phase space quantization, by Maurice De Gosson.
Journal of Physics A: Mathematical and General, Volume 34, Number 47
| 3 | https://mathoverflow.net/users/30684 | 355933 | 150,202 |
https://mathoverflow.net/questions/355928 | 5 | What is an explanation for what the theory of symplectic rigidity is and what kind of questions it can answer? I was led to this after reading about the symplectic non-squeezing theorem of Gromov.
| https://mathoverflow.net/users/154158 | What is symplectic rigidity? | **Rigidity**, as used throughout mathematics and not just symplectic geometry, indicates that some structure attached to an object captures more data than one would "expect" from the underlying object itself. There are a number of helpful examples already in [this Wikipedia entry](https://en.wikipedia.org/wiki/Rigidity... | 12 | https://mathoverflow.net/users/66405 | 355934 | 150,203 |
https://mathoverflow.net/questions/355936 | 13 | 1. Is it possible there are examples of where classical logic proves a theorem that provably is false within constructivism? Is so what are some examples?
2. What are some examples of most contrasting theorems provable in these two logics that does not fall in 1.?
| https://mathoverflow.net/users/136553 | Contrasting theorems in classical logic and constructivism | This is a very natural question, but as it happens one needs some more background to give a natural answer (is my humble opinion).
For clarity let me give a summary first indication:
As to your **Question 1.** : This is commonly thought of as not being possible in strict terms, because '*constructive mathematics*'... | 12 | https://mathoverflow.net/users/101577 | 355943 | 150,205 |
https://mathoverflow.net/questions/355913 | 0 | Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the logarithm integral $x$ , I computed $\int\_{0}^{1} (\pi(x)-Li (x)) $ I [have got](https://www.wolframalpha.com/input/?i=integral... | https://mathoverflow.net/users/51189 | What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution? | The integral does not converge. See <https://academic.oup.com/blms/article-abstract/31/4/424/277640?redirectedFrom=fulltext>.
On the other hand, a proof that this integral is less than $K.t^{\alpha}$ almost everywhere (i.e. on a set of the form $(t\_{0},\infty)\setminus J$ where $J$ has Lebesgue measure $0$) for some... | 2 | https://mathoverflow.net/users/13625 | 355945 | 150,206 |
https://mathoverflow.net/questions/355947 | 5 | Let us assume we have a square matrix $A$ whose entries are sampled from a standard Gaussian distribution of mean $0$. Do we have any information about the distribution of its eigenvalues?
Particularly, I'm aware that there are different results on symmetric gaussian matrices (or, the Gaussian orthogonal ensemble of... | https://mathoverflow.net/users/93775 | Eigenvalues and eigenvectors of Gaussian random matrices | This is the Ginibre ensemble, see [Eigenvalue statistics of the real Ginibre ensemble](https://arxiv.org/abs/0706.2020) for the eigenvalue distribution. For an $N\times N$ matrix with $N\gg 1$ there are on average $\sqrt{2N/\pi}$ eigenvalues on the real axis, uniformly in the interval $(-\sqrt N,\sqrt N$). The rest of ... | 8 | https://mathoverflow.net/users/11260 | 355952 | 150,207 |
https://mathoverflow.net/questions/355970 | 0 | Suppose we order the rational numbers using the diagonal method (used to prove they are countable) using an $n\times n$ grid. Now suppose we count the distinct rational numbers (those points on the grid where gcd(numerator, denominator) = 1. Denote this by $S\_n$.
Does $\lim\_{n\to\infty} n^2/|S\_n|$ exist, and if so... | https://mathoverflow.net/users/155247 | does the ratio of the count of rational numbers on an $n\times n$ grid to $n^2$, converge as $n$ tends to infinity | The limit you are looking for is the *reciprocal* of the probability that two random positive integers are coprime. This probability (with the proper intepretation) is the density of square-free numbers:
$$\prod\_{\text{$p$ is prime}}\left(1-\frac{1}{p^2}\right)=\left(\sum\_{n=1}^\infty \frac{1}{n^2}\right)^{-1}=\frac{... | 4 | https://mathoverflow.net/users/11919 | 355972 | 150,214 |
https://mathoverflow.net/questions/355966 | 0 | Here’s an easy one, I hope:
Suppose $\tau$ is a stopping time and $(M\_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M\_\tau]= \mathbb{E}[M\_0]$.
Will it also be the case that $\mathbb{E}[M\_\sigma]= \mathbb{E}[M\_0]$ for an arbitrary random time $\si... | https://mathoverflow.net/users/155244 | Martingale optional stopping before a stopping time | No. The problem is that $\tau-1$ is not a "stopping time" for the martingale.
Example. Martingale $(X\_n), n=0,1,2$ is the standard random walk on $\mathbb Z$. Our sample space is $[0,1)$ with Lebesgue measure.
\begin{align}
X\_0(\omega) &= 0,\qquad \omega \in [0,1).
\\
X\_1(\omega) &= \begin{cases}
1,\quad & \omega ... | 0 | https://mathoverflow.net/users/454 | 355975 | 150,217 |
https://mathoverflow.net/questions/355911 | 6 | Let us consider a directed graph $\Gamma=(V,E,s,t)$ ($V$ set of vertices, $E$ set of edges, $s,t: E \rightarrow V$ are the "source" and "target" maps).
Assume that $\Gamma$ is **bi-regular**, that is there are two integers $d\_1 \geq 1$ and $d\_2 \geq 1$ such that $|s^{-1}(v)|=d\_1$ and $|t^{-1}(v)|=d\_2$ for every $... | https://mathoverflow.net/users/9317 | Random walks on infinite directed regular graphs | Here is a counterexample:
Let $G\_1$ be the digraph with vertex set $\mathbb N$, two loops at $0$, an edge from $0$ to $1$, and for every $i \geq 1$ an edge from $i$ to $(i+1)$ and two parallel edges from $i$ to $(i-1)$. Let $G\_2$ be any countable digraph in which every vertex has $2$ outgoing edges and $4$ incomin... | 5 | https://mathoverflow.net/users/97426 | 355980 | 150,218 |
https://mathoverflow.net/questions/355279 | 5 | Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see [here](https://cds.cern.ch/record/1383342/files/978-3-642-14090-7_BookBackMatter.pdf), the functional derivative in Physics is defined in terms of Taylor expansions. Let me elaborate.
**[The Physicist po... | https://mathoverflow.net/users/150264 | Functional derivatives on Banach spaces | **Premise**: almost (if not) all derivations below are kept at a *formal* level, i.e. (apart from the notes, almost) *no discussion of the hypotheses needed to make the result rigorous are given*. This is because the question asks to show a way to extend a particular notion of derivative of functionals in order to incl... | 2 | https://mathoverflow.net/users/113756 | 355984 | 150,220 |
https://mathoverflow.net/questions/355977 | 0 | Let $A$ be a commutative C\*-algebra.
**I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.**
---
Notes and reminders:
A C\*-algebra is a Banach algebra over $\mathbb{C}$ with involution $\*$ s.t. $\|x^\*x\| =... | https://mathoverflow.net/users/122931 | Separability of an algebra is equivalent to separability of its spectrum | It's not true.
For simplicity, suppose $A$ is unital, so that its spectrum is compact Hausdorff.
If $X$ is a compact Hausdorff space and $C(X)$ is separable, then you can show that $X$ is second countable. (Let $\{f\_k\}$ be a countable dense subset of $C(X)$, and $\{U\_n\}$ a countable basis of open sets in $\math... | 4 | https://mathoverflow.net/users/4832 | 355986 | 150,221 |
https://mathoverflow.net/questions/289658 | 3 | **Note:** A pointer to a reference, or a yes/no answer with a 1-2 sentence incomplete/non-rigorous justification would suffice for answers. I am just curious about whether the result is true; it is fairly unambiguous that I presently lack the means to understand it adequately. **/Note**
In 1992, Studeny gave a proof ... | https://mathoverflow.net/users/93694 | Is there a complete countable axiomatization of conditional independence? (Graphoids) |
>
> Studeny says in the abstract of his 1992 paper that:
>
>
>
> >
> > However, under the assumption that CIRs [conditional-independence relations] are grasped the existence of a countable characterization of CIRs is shown.
> >
> >
> >
>
>
> Since Studeny seems to be calling a "complete finite axiomatization... | 5 | https://mathoverflow.net/users/144619 | 355994 | 150,224 |
https://mathoverflow.net/questions/355758 | 5 | I asked this question on [Mathematics Stackexchange](https://math.stackexchange.com/questions/3589223/u-t-aufu-where-a-is-the-infinitesimal-generator-of-c-0-semigroup), but got no answer.
In Pavel's book: Nonlinear Evolution Operators and Semigroups - Applications to Partial Differential Equations, we have the follo... | https://mathoverflow.net/users/115618 | $u_t=Au+F(u)$ where $A$ is the infinitesimal generator of $C_0$-semigroup | Yes the theorem is true in the general case. The semigroup is assumed to be contractive for the sake of simplicity. Of course the assumption is not restrictive since there is always an equivalent norm which makes the semigroup contractive (this is the same idea in the proof of **Hille-Yosida theorem** to pass from the ... | 3 | https://mathoverflow.net/users/124904 | 356011 | 150,225 |
https://mathoverflow.net/questions/355987 | 1 | Let ($X$,$\Delta$) be projective klt pair and $f \colon X \rightarrow Z$ be contraction of ($K\_X + \Delta$) - negative extremal ray $R$.
If $X$ is $\mathbb{Q}$ -factorial and $\mathrm{dim}Z < \mathrm{dim} X $, then Z is $\mathbb{Q}$- factorial?
I have read proof of above claim in Kollar and Mori "Birational geometry... | https://mathoverflow.net/users/146728 | Is $\mathbb{Q}$-factoriality preserved under contraction? | Let $D''$ be a Cartier divisor on $Z$ such that $f^\*\mathscr O\_Z(D'')\simeq \mathscr O\_X(mD')$. Then by the projection formula $\mathscr O\_Z(D'')\simeq f\_\*\mathscr O\_X(mD')$ (since $f\_\*\mathscr O\_X\simeq \mathscr O\_Z$).
On the other hand, from the construction we see that $\mathscr O\_Z(D'')|\_{Z\_0}\sime... | 1 | https://mathoverflow.net/users/10076 | 356017 | 150,226 |
https://mathoverflow.net/questions/355981 | 6 | Let $n\in N$, where $n = p\_{1}^{k\_{1}}p\_{2}^{k\_{2}}...p\_{m}^{k\_{m}}$ for $p\_{i}$ prime.
Define the 'density' of $n$ as:
$d(n) = \frac{(p\_{1}+1)^{k\_{1}}(p\_{2}+1)^{k\_{2}}...(p\_{m}+1)^{k\_{m}}}{n}$
Notice that $d(n)$ gives us a measure of the 'compositeness' of a number - relative to other numbers of a ... | https://mathoverflow.net/users/155247 | Is there a connection between the average 'compositeness' of a rational number and $\phi$ (golden ratio)? | Probably not. I can tell you what the limiting value is when taking averages over $m\times m$ grids themselves, rather than diagonal-counting-sequences; but I suspect the averages are the same.
The average of $d(r)$ over the $m\times m$ grid is simply
$$
\frac1{m^2} \sum\_{a=1}^m \sum\_{b=1}^m d\big( \tfrac ab\big) =... | 7 | https://mathoverflow.net/users/5091 | 356023 | 150,227 |
https://mathoverflow.net/questions/355881 | 4 | Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0.
We consider the adjoint representation
$$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$
where ${\mathfrak g}={\rm Lie}\ G$.
I am looking for a *reference* to a proof of the following assertion:
>
> **P... | https://mathoverflow.net/users/4149 | Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request | A reference: [Steinberg, Torsion in reductive groups,
Advances in Math. 15 (1975), 63–92](https://www.sciencedirect.com/science/article/pii/0001870875901255?via%3Dihub), Corollary 3.11.
In positive characteristic $p$: see *loc. cit.*, Theorem 3.14. It says that if (and only if) $p$ is not a torsion prime for $G$, t... | 6 | https://mathoverflow.net/users/4149 | 356025 | 150,228 |
https://mathoverflow.net/questions/355983 | 3 | What are some "natural" motivating examples of the following:
i) A strict monoidal category,
ii) A monoidal with non-trivial associatots?
For i) the only examples I know are categories which have been strictified, are there any examples occuring "in nature" which are strict, or is strictness in some sense an "un... | https://mathoverflow.net/users/153228 | Examples of strict monoidal categories and monoidal categories with nontrivial associators | i) [A monoid](https://en.wikipedia.org/wiki/Monoid)
ii) Representations of [a quasi-Hopf algebra](https://en.wikipedia.org/wiki/Quasi-Hopf_algebra)
| 3 | https://mathoverflow.net/users/5301 | 356028 | 150,229 |
https://mathoverflow.net/questions/355944 | 3 | Let $A$ be a finite dimensional quiver algebra such that any two indecomposable modules with the same dimension vector are isomorphic.
>
> Question: In case $A$ has $n$ simple modules and finite global dimension, does $A$ have Loewy length at most $n$?
>
>
>
| https://mathoverflow.net/users/61949 | On algebras where indecomposable modules are determined by their dimension vectors | Yes.
Let $A=kQ/I$ be a quiver algebra with $n$ simple modules and finite global dimension such that $\text{rad}^nA\neq0$. For a vertex $i$ of the quiver $Q$, I'll use $e\_i$ to denote the corresponding primitive idempotent of $A$, and $S\_i$ to denote the corresponding simple (right) $A$-module.
Since $\text{rad}^n... | 3 | https://mathoverflow.net/users/22989 | 356031 | 150,231 |
https://mathoverflow.net/questions/356038 | 5 | I have calculated some real quadratic field 's Hilbert class field with class number $2$,and I found they were satisfied $Gal(H\_{K}/Q)\cong Z/2Z\oplus Z/2Z$,here $H\_{K}$ is the Hilbert class field of a real quadratic field $K$ whose class number is $2$.Is it true for all quadratic field with class number $2$? How to ... | https://mathoverflow.net/users/147080 | A problem about real quadratic field | This is a consequence of "genus theory". If $K / \mathbf{Q}$ is an abelian extension, then the "genus field" of $K$ is the maximal extension $L / K$ such that $L/K$ is unramified and $L/\mathbf{Q}$ is abelian. You can read more about genus theory here: <https://en.wikipedia.org/wiki/Genus_field>.
In your case, the fa... | 11 | https://mathoverflow.net/users/2481 | 356039 | 150,232 |
https://mathoverflow.net/questions/355993 | 1 | My question concerns Section 2 of the article "The Simplicial Model of Univalent Foundations (after Voevodsky)" (<https://arxiv.org/pdf/1211.2851.pdf>).
Let $\alpha$ be a strongly inaccessible cardinal.
Let $f : X \to Y$ be a map of simplicial sets.
1. We say that $f$ is *well-ordered* if it is equipped with a we... | https://mathoverflow.net/users/115055 | Showing that a certain simplicial set has levelwise small cardinality | $\mathcal{U}\_\beta(\Delta[n])$ is the set of isomorphism classes of $\beta$-small well-ordered fibrations over $\Delta[n]$. Such a thing is uniquely determined by an isomorphism class of $\beta$-small well-orderings over each element of $\Delta[n]$, together with the face and degeneracy maps between them. There are $\... | 1 | https://mathoverflow.net/users/49 | 356048 | 150,234 |
https://mathoverflow.net/questions/355932 | 6 | Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}\_n(\mathbb{Z})$ on $\operatorname{SL}\_n(\mathbb{R})/\operatorname{SO}\_n(\mathbb{R})$ for each $n \leq 6$. In the case where $n = 2$, one obtains the familiar fundamental domain for the action of $\operatorname{SL}\_2(\mathbb{Z})$ on ... | https://mathoverflow.net/users/76440 | Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$ | For information about these groups up through dimension 17, see:
Vinberg, È. B.
The groups of units of certain quadratic forms. (Russian)
Mat. Sb. (N.S.) 87(129) (1972), 18–36.
English translation [Math. USSR-Sb. 87 (1972), 17–35].
Vinberg shows up through dimension 17 that $O(n,1; \mathbb{Z})$ has a finite index s... | 9 | https://mathoverflow.net/users/142269 | 356050 | 150,235 |
https://mathoverflow.net/questions/356029 | 3 | Let $\mathcal F$ be a field of subsets of a set $\Omega$. Equip the space $[0,1]^\mathcal F$ of functions from $\mathcal F$ into $[0,1]$ with the product topology. Then, the set $\Delta$ of finitely additive probability measures on $\mathcal F$ is a convex and compact subset of $[0,1]^\mathcal F$.
If $\mu \in \Delta$... | https://mathoverflow.net/users/96899 | Is the inner/outer measure mapping continuous? | Let $\Omega = \mathbb{N}$ and let $\mathcal{F}$ be the field consisting of all finite and cofinite subsets of $\mathbb{N}$. Let $\mu\_n = \delta\_{2n}$ be a point mass at the integer $2n$, and let $\mu$ be the finitely additive measure that assigns measure $0$ to every finite set and $1$ to every cofinite set. Then $\m... | 2 | https://mathoverflow.net/users/4832 | 356052 | 150,236 |
https://mathoverflow.net/questions/356045 | 0 | Does there exist a power series $\sum\_i a\_i x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum\_i |a\_i|$ is polynomial in $n$?
I feel the answer might be no but I'm not sure how to prove it.
| https://mathoverflow.net/users/112954 | Can this function be interpolated with a small power series | Take an entire function $f$ such that $f(0)=1$ and $f(j) = 0$ for all nonzero integers: an example is $f(z) = \sin(\pi z)/(\pi z) $ for $z \ne 0$, $1$ for $z=0$.
The Maclaurin series of $f$ satisfies $\sum\_{i} |a\_i| < \infty$.
| 3 | https://mathoverflow.net/users/13650 | 356056 | 150,238 |
https://mathoverflow.net/questions/356063 | 0 | Does there exist a simple non-abelian 2-generated group $G$ and two elements $a, b \in G$, such that $\langle \{a, b\} \rangle = G$, $a^2 =1$ and $\forall c, d \in G$ $\langle \{c^{-1}bc, d^{-1}bd \} \rangle \neq G$?
| https://mathoverflow.net/users/74657 | about simple non-abelian 2-generated group | No. Note that $\langle b ,a^{-1}ba \rangle$ is normalized by $b$, and by $a$. Hence
$\langle b, a^{-1}ba \rangle$ is normalized by $\langle a,b \rangle = G$. Since $G$ is simple non-Abelian, $G = \langle b, a^{-1}ba \rangle .$
| 10 | https://mathoverflow.net/users/14450 | 356065 | 150,241 |
https://mathoverflow.net/questions/356076 | 3 | $G$ is a finitely presented group (but not a finite group), and $\mathbb{Z}G$ is the corresponding group ring.
$I$ is the kernel of the augmentation morphism $\mathbb{Z}G\rightarrow \mathbb{Z}$.
Is $I$ (always) a finitely generated $\mathbb{Z}G$-module (let say right module).
| https://mathoverflow.net/users/129583 | augmentation ideal is always finitely generated? | The augmentation ideal is finitely generated as a left (or right) ideal if and only if the group is finitely generated. It is obvious the augmentation ideal is generated as an abelian group by all elements of the form $g-1$ with $g\in G$. Then from the computation $ab-1=a(b-1)+a-1$ one easily deduces by induction on wo... | 5 | https://mathoverflow.net/users/15934 | 356081 | 150,244 |
https://mathoverflow.net/questions/350988 | 3 | I cannot really state my question in an incredibly precise way as I'm very new to this area, but I hope what I'm asking will be clear. Let $\mathcal{C}$ be the infinity category of sheaves of quasi-coherent spectra on $\mathbb{P}^1$ as a scheme. Then, I think this should be a stable infinity category. Higher Algebra 7.... | https://mathoverflow.net/users/136287 | Is the category of spectra on $\mathbb{P}^1$ a module category? | $\newcommand{\PP}{\mathbf{P}} \newcommand{\QCoh}{\mathrm{QCoh}} \newcommand{\cf}{\mathcal{F}} \newcommand{\cg}{\mathcal{G}} \newcommand{\Map}{\mathrm{Map}} \newcommand{\co}{\mathcal{O}}$ The ordinary abelian category of quasicoherent sheaves on $\PP^1$ is not the category of modules over any ring. We shall prove the fo... | 3 | https://mathoverflow.net/users/102390 | 356089 | 150,247 |
https://mathoverflow.net/questions/356086 | -1 | When I tried to give bounds for $\zeta(0.5+it)$ using some transformations over Gamma function using the function $f(x)=\exp(-n x)$ over the range $(0,+\infty)$ , For $ Re(s)=\frac12 $ and $t >0$ I come up to the final Bounds for $\zeta(0.5+it) $ which is represented by the following formual :For $t\geq 1.22$: $$|\zeta... | https://mathoverflow.net/users/51189 | Any simplification of this inequality if it is true? :For $t\geq 1.22$: $|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}$ | The RHS of (1) is $\sim |t|$ by Stirling's formula. The Weyl bound states that there exists a constant $c>0$ (which one can compute, but I won't) such that if $t\in\mathbb{R}$, then $|\zeta(\frac{1}{2}+it)|\leq c(|t|+1)^{1/6}$. One can do better under assuming the Riemann hypothesis, see [Chandee and Soundararajan](htt... | 1 | https://mathoverflow.net/users/111215 | 356091 | 150,248 |
https://mathoverflow.net/questions/356049 | 25 | This is a follow-up on the following [question](https://mathoverflow.net/questions/356042/spaces-with-unique-endomorphism-monoids). Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition).
What is an example of a topolo... | https://mathoverflow.net/users/8628 | What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$? | No such space exists. We actually get the stronger statement that every isomorphism $\operatorname{End}(X) \stackrel\sim\to \operatorname{End}(\mathbf R)$ is induced by an isomorphism $X \stackrel\sim\to \mathbf R$ (unique by Observation 1 below). In contrast, in Emil Jeřábek's beautiful construction in [this parallel ... | 27 | https://mathoverflow.net/users/82179 | 356095 | 150,249 |
https://mathoverflow.net/questions/355978 | 3 | Let $X$ be a smooth separated integral variety over an algebraically closed field.
>
> **Question:**
>
>
> Is it true that a basis for the Zariski topology is given by the family $$\mathcal{U}=\{X\setminus \mathrm{Supp}(\mathrm{div}(f))\mid f\in K(X)\}$$
> where $\mathrm{div}(f)$ is the principal divisor defined... | https://mathoverflow.net/users/114772 | The Zariski topology of a variety is determined by its principal Cartier divisors | Let $U$ be a an affine open set of $X$ containing $x$. If $U$ contains $D$, we choose $f$ as in your first bullet point vanishing on $D$ but not on $x$. If $U$ does not contain $D$, there must exist some $f$ on $U$ with a pole at $D$. If $f$ vanishes at $x$, add $1$ to $f$.
Why must there exist some $f$ on $U$ with a... | 2 | https://mathoverflow.net/users/18060 | 356107 | 150,251 |
https://mathoverflow.net/questions/356099 | 1 | Let $ K $ be a $ p $-adic field. Suppose we have an isogeny of elliptic curves $ \phi : E \to E' $ defined over $ K $, where $ E $ and $ E' $ both have multiplicative reduction.
1) Is there anything we can say about the structure of the induced map on the Tate modules $ V\_l(E) \to V\_l(E') $? Mostly I'm interested ... | https://mathoverflow.net/users/154188 | isogenies between elliptic curves with multiplicative reduction | I believe this is the answer in the split case: Let $E$ be the Tate curve with parameter $q$. Let $n>1$. We look for isogenies with cyclic kernel of order $n$. We may suppose that $n$ is prime.
First, there is the isogeny to the Tate curve $E'$ with parameter $q' = q^n$ and the map is induced from $K\to K$ sending $... | 1 | https://mathoverflow.net/users/5015 | 356111 | 150,252 |
https://mathoverflow.net/questions/356117 | 3 | Lat $A$ be a set and $\underline{A}$ the associated constant Zariski sheaf on the category $Sm/S$ of schemes which are smooth over $S$ for a fixed base scheme $S$. Is $\underline{A}$ already a (constant) sheaf for the Nisnevich topology on $Sm/S$?
I ask this because constant Zariski sheaves are easier to describe, wh... | https://mathoverflow.net/users/42571 | Is any constant Zariski sheaf already a Nisnevich sheaf? | Yes. This is fine for every topology in which covers are collections of morphism that are open for the Zariski topology and surjective on points.
To see this, because $\underline{A}$ satisfies the sheaf condition for disjoint unions, it suffices to show for $f: Y \to X$ open and surjective on points, $\underline{A}(... | 4 | https://mathoverflow.net/users/18060 | 356120 | 150,254 |
https://mathoverflow.net/questions/356114 | 1 | Let $\pi:X \rightarrow Y$ be a surmersion (surjective submersion) between closed manifolds.
1) Is there any obstruction to the existence of a "multi-valued" section $s$ of $\pi$ such that $\pi \circ s$ is a smooth covering of $Y$ ?
By a multi-valued section I was thinking about gluing local section of $\pi$, where ... | https://mathoverflow.net/users/150199 | About submersion and sections | First, since you are dealing with closed manifolds, a submersion $\pi:X\rightarrow Y$ (assuming that the base is connected) is surjective because it is simultaneously open and closed. Moreover, due to Ehresmann's Theorem, the submersion $\pi:X\rightarrow Y$ is a fiber bundle as it is proper.
Here is an approach for ... | 2 | https://mathoverflow.net/users/128556 | 356124 | 150,256 |
https://mathoverflow.net/questions/356082 | 1 | Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that is
$$\lim\_{r\to0^+}\frac{\lambda^{n}(E\cap B\_{r}(x))}{\lambda^{n}(B\_r(x))}=0,$$
where $\lambda ^{n}$ is the Lebesg... | https://mathoverflow.net/users/155314 | Integral average near a point of dispersion | $\newcommand{\tb}{\tilde B}$
Let $d:=n$.
The dispersion condition
\begin{equation\*}
\lim\_{r\downarrow0}\frac{|E\cap B\_r(x)|}{|B\_r(x)|}=0
\end{equation\*}
is of no help, where $|\cdot|$ denotes the Lebesgue measure on $\mathbb R^d$.
More specifically, the following is true:
>
> **Theorem** Suppose that $f$ ... | 1 | https://mathoverflow.net/users/36721 | 356126 | 150,257 |
https://mathoverflow.net/questions/355985 | 2 | Consider the integers $[1,n]=\{1,\dots,n\}$ and call subsets of the type $[a,b]=\{a,\dots,b\}$ with $1\le a < b\le n$ *intervals*. We say that two intervals $[a,b],[c,d]$ are *crossing* if either $a<c<b<d$ or $c<a<d<b$. Otherwise we say that the intervals are non-crossing. So $[1,3],[2,4]$ are crossing, but $[1,3],[2,3... | https://mathoverflow.net/users/89934 | Number of non-crossing sets of intervals | As discussed in the comments, the number in question is clearly the same as the number of graphs on $n$ vertices drawn on a circle without crossing edges, which is in the OEIS at <https://oeis.org/A054726>. As discussed in that OEIS entry, the number is $2^n$ times a "[little Schroeder number](https://en.wikipedia.org/... | 4 | https://mathoverflow.net/users/25028 | 356133 | 150,260 |
https://mathoverflow.net/questions/356102 | 1 | I am trying to understand the following argument: Let $\mathcal{L}:L^2(\mathbb{R})\to L^2(\mathbb{R})$ be an essentially self-adjoint unbounded linear operator with domain $D(\mathcal{L})=H^s(\mathbb{R})$ for some $s>0$. Let us assume that $\mathcal{L}$ has only one negative eigenvalue (which is simple) with associated... | https://mathoverflow.net/users/129131 | Coercivity of linear operators | Yes, the inequality is true. Let $\lambda$ be large enough so $(L+\lambda I)$ is positive definite. We have $$((L+\lambda I)y,y)=((L+\lambda I)^{1/2}y,(L+\lambda I)^{1/2}y)\ge C\|y\|^2\_{H^{s/2}},$$
since the domain of $(L+\lambda I)^{1/2}$ is $H^{s/2}$ by the general theory of interpolation spaces. It follows that
$$(... | 2 | https://mathoverflow.net/users/12120 | 356143 | 150,263 |
https://mathoverflow.net/questions/356123 | 2 | Suppose that $\mathcal{C}$ is a locally cartesian closed right proper Quillen model category for which all objects are fibrant. Let $x$ be an object of $\mathcal{C}$. Let $\mathbb{F}$ denote the class of fibrations and let $\mathbb{F}\_x$ denote the category of all fibrations with codomain $x$. Consider the inclusion f... | https://mathoverflow.net/users/115055 | Conditions for certain inclusion functor to preserve internal homs | You also need to assume that cofibrations are monos since this implies that (trivial) cofibrations are stable under pullbacks along fibrations. In turn, this implies that the exponent of two fibrations over $x$ is also a fibration. It is easy to see that the exponent also has its universal property in $\mathbb{F}\_x$. ... | 1 | https://mathoverflow.net/users/62782 | 356176 | 150,276 |
https://mathoverflow.net/questions/328747 | 2 | **Background**
I am currently reading "Modularity and community structure in networks" (2006) by Newman `[1]`.
In it, he derives a score for the modularity of a graph that, intuitively, is based on finding a division for the vertices, so that there are as many "unexpected" `[2]` edges as possible between vertices i... | https://mathoverflow.net/users/138750 | Modularity in a graph -- derivation of modularity score | Of course the comments and other answers are right, but for the sake of completeness and for those who are new to Linear Algebra (like me) I'll try to explain how this entire thing comes together.
>
> $\mathbb{R}^n$ has a basis of $n$ real eigenvectors $(u\_1, ...., u\_n)$ of $\mathbf{B}$.
>
>
>
*Proof sketch*... | 0 | https://mathoverflow.net/users/138750 | 356178 | 150,277 |
https://mathoverflow.net/questions/356171 | 2 | I have a result that holds for cocomplete and finitely complete categories such that pullbacks preserve directed colimits, by which I mean $A \times\_B (\operatorname{colim}\_{i \in I} C\_i) = colim\_{i \in I} (A \times\_B C\_i)$, where $I$ is a directed category.
Thanks to a result by Adamek and Rosicky [locally pr... | https://mathoverflow.net/users/68468 | Cocomplete and finitely complete category with nice pullbacks that is not locally presentable | If you look at the article on [quasitoposes](https://ncatlab.org/nlab/show/quasitopos#examples), which are locally cartesian closed and therefore satisfy your exactness condition, you'll find a number of examples that are not locally presentable. For example, the category of pseudotopological spaces, the category of bo... | 11 | https://mathoverflow.net/users/2926 | 356179 | 150,278 |
https://mathoverflow.net/questions/356184 | 1 | I consider a map germ $f: (\mathbb{R}^n,0) \to \mathbb{R} $ which is $k$-determined for some $k \in \mathbb N$, i.e. for all map germs $g: (\mathbb{R}^n,0) \to \mathbb{R} $ having the same $k$-jet as $f$ , there exists a (local) diffeomorphism $\varphi: (\mathbb{R}^n,0) \to (\mathbb{R}^n,0)$ such that $g = f \circ \var... | https://mathoverflow.net/users/101434 | Tangent space to subspace of orbit in jet spaces | Those are not monomials. The space $m^{k+1}/m^{k+2}$ is the space of functions vanishing up to order $k$, modulo those vanishing up to order $k+1$, so Taylor series of homogeneous polynomials of degree exactly $k+1$. The space $L$ is, as you defined it, the translate of $m^{k+1}/m^{k+2}$ by $j^{k+1}f(0)$ inside $J^{k+1... | 3 | https://mathoverflow.net/users/13268 | 356185 | 150,281 |
https://mathoverflow.net/questions/356190 | 12 | I am teaching a graduate course in Complex Analysis and I am covering Newman's proof of the prime number theorem. I have been using the simplified version in the papers of
[Zagier](https://www.jstor.org/stable/2975232?seq=1) and [Korevaar](https://link.springer.com/article/10.1007/BF03024240). However, I ran into a pro... | https://mathoverflow.net/users/101392 | Newman's proof of the prime number theorem | It doesn't seem to me that either article follows the line of reasoning as you have presented it. Indeed, we do not take the integral over the union of integrals $(x\_n,(1+\varepsilon)x\_n)$. We do get that the integral over those intervals is infinite, but you correctly note this does not give a contradiction. Instead... | 13 | https://mathoverflow.net/users/30186 | 356192 | 150,282 |
https://mathoverflow.net/questions/356188 | 48 | Given a metric space $(X,d)$ and three points $x,y,z$ in $X$, say that $y$ is *between* $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$, and write $[x,z]$ for the set of points between $x$ and $z$.
Obviously, we have
1. $x,z\in[x,z]$;
2. $[x,z]=[z,x]$;
3. $y \in [x,z]$ implies $[x,y] \subseteq [x,z]$;
4. $w,y \in [x,z]$ ... | https://mathoverflow.net/users/148575 | What happens if you strip everything but the “between” relation in metric spaces | There is a wide body of work on this in connection with the classic [De Bruijn–Erdős theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(incidence_geometry)).
>
> **De Bruijn–Erdős Theorem.** Every set of $n$ points in the
> plane (not all lying on the same line) determine at least $n$ li... | 41 | https://mathoverflow.net/users/2233 | 356193 | 150,283 |
https://mathoverflow.net/questions/356204 | 3 | Let $(X, \mathcal{B}, \mu, S)$ and $(Y, \mathcal{C}, \nu, T)$ be invertible probability-measure-preserving systems, with a measurable factor map $\pi: X \to Y$, i.e. $\pi \circ S = T \circ \pi$. Suppose that there is some $N \in \mathbb{N}$ such that $\nu$-almost every $y \in Y$ has at most $N$ preimages under $\pi$, i... | https://mathoverflow.net/users/106151 | Applying the Abramov-Rokhlin skew product entropy formula to a bounded-to-one factor | One should always keep in mind that the natural substrate of ergodic theory is **Lebesgue-Rokhlin** (aka Lebesgue or standard) measure spaces which enjoy a lot of properties not necessarily present in general measure spaces. One of these properties is an explicit description of homomorphisms of such spaces obtained by ... | 2 | https://mathoverflow.net/users/8588 | 356211 | 150,287 |
https://mathoverflow.net/questions/356042 | 6 | If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the *endomorphism monoid* $(\text{End}(X), \circ)$.
We say that the space $X$ has a *unique endomorphism monoid* if $\text{End}(X) \cong \text{End}(Y)$ as monoids, for som... | https://mathoverflow.net/users/8628 | Spaces with unique endomorphism monoids | $\DeclareMathOperator\End{End}$As shown in Todd Trimble’s comment, the set of constant maps $X\to X$ is definable in $\End(X,\tau)$, as it consists of exactly the left-absorbing endomorphisms (i.e., $\phi\in\End(X,\tau)$ such that $\phi\circ\psi=\phi$ for all $\psi\in\End(X,\tau)$). Thus, an isomorphism $F\colon\End(X,... | 13 | https://mathoverflow.net/users/12705 | 356219 | 150,290 |
https://mathoverflow.net/questions/355777 | 3 | This is an elementary question in differential geometry. We know that for a smooth real-valued function $f$ defined on an open geodesically convex set of a Riemannian manifold $ \mathcal{X} \subset \mathcal{M}$, $f$ is geodesically convex if and only if its Riemannian Hessian is positive semidefinite on $\mathcal{X}$. ... | https://mathoverflow.net/users/135962 | Geodesic convexity and the Geometric Hessian | Thanks to Prof. Absil for verifying the following proof.
Let $\mathfrak{X}(\mathcal{M})$ and $\mathfrak{F}(\mathcal{M})$ be the set of smooth vector fields and scalar functions on $\mathcal{M}$, respectively. By Theorem 6.2 in [1], $f$ is geodesically convex in $\mathcal{X}$ if and only if its second covariant deriva... | 1 | https://mathoverflow.net/users/135962 | 356224 | 150,292 |
https://mathoverflow.net/questions/356144 | 3 | For each $a\in \mathbb C$ define $f\_a:\mathbb C\to \mathbb C$ by $f\_a(z)=\exp(z)+a$. I am primarily interested in real values $a\in (-\infty,-1)$.
For each $r\in [0,\infty)$ define $M\_a(r)=\max\{|f\_a(z)|:|z|=r\}$. So $M\_a(r)=|z'|$ for some point $z'$ in the image of the circle of radius r. Is there a more explic... | https://mathoverflow.net/users/95718 | Computing the maximum modulus | A partial solution: As mentioned by @MargaretFriedland, the desired $M\_a(r)$ is the absolute maximum of $g(t):=\sqrt{{\rm{e}}^{2r\cos t}+2a{\rm{e}}^{r\cos t}\cos(r\sin t)+a^2}$. Notice that this is an even function, so it suffices to maximize over $[0,\pi]$. If $t\in\left[\frac{\pi}{2},\pi\right]$, then $\cos t<0$. So... | 4 | https://mathoverflow.net/users/128556 | 356245 | 150,301 |
https://mathoverflow.net/questions/355764 | 2 | **O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval.**
Using $\{x\}$ to denote the fraction part of $x$ we can define for any $I\subset [0,1]$,
$$E(n,\theta, I) ={ \left|\{\,\{\theta\},\{2\theta\},\dots,\{n\theta\} \,\} \cap I \right|}-n|I|$$
$ $
$$\Delta\_{sup}(n,\theta)=\sup... | https://mathoverflow.net/users/7113 | O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval | I think you are asking for so-called "bounded remainder sets". Given $\theta$, there exist intervals $I$ having discrepancy $O(1)$, namely those whose length is in $\mathbb{Z} + \theta \mathbb{Z}$. The classical reference is a paper of Kesten:
H. Kesten,On a conjecture of Erdös and Szüsz related to uniform distribut... | 1 | https://mathoverflow.net/users/46852 | 356255 | 150,303 |
https://mathoverflow.net/questions/352448 | 25 | The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's *Lectures in Condensed Mathematics*), have enough injectives ?
Does it, in fact, have any nontrivial injective ?
Recall that $Cond(\mathbf{Ab})$ is defined to be the colimit over strong limit card... | https://mathoverflow.net/users/102343 | Are there (enough) injectives in condensed abelian groups? | Indeed, there are no nonzero injective condensed abelian groups.
Let $I$ be an injective condensed abelian group. We can find some surjection
$$ \bigoplus\_{j\in J} \mathbb Z[S\_j]\to I$$
for some index set $J$ and some profinite sets $S\_j$, where $\mathbb Z[S\_j]$ is the free condensed abelian group on $S\_j$ -- th... | 20 | https://mathoverflow.net/users/6074 | 356261 | 150,304 |
https://mathoverflow.net/questions/356258 | 2 | Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a function $\psi\in L^2(\mathbb{R}^3,\langle x\rangle^{-1-\varepsilon}dx)\setminus L^2(\mathbb{R}^3)$ which satisfies $(-... | https://mathoverflow.net/users/54552 | Resonances for Schrodinger operators with radial potentials | Expand your resonance in spherical harmonics: $\psi = \sum\_{\ell=0}^\infty \sum\_m \psi\_{\ell m}(r) Y\_{\ell m}(\theta,\phi)$. Then each coefficient satisfies the radial Schrödinger equation
$$ -\frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial \psi\_{\ell m}}{\partial r} + \frac{\ell(\ell+1)}{r^2}\psi\_{... | 5 | https://mathoverflow.net/users/2622 | 356269 | 150,305 |
https://mathoverflow.net/questions/356197 | 3 | Suppose $A$ is an abelian variety over an algebraically closed field $k$. I'm confused about the notion of translates of abelian subvarieties. I looked over a few related papers, but I couldn't find a precise algebro-geometric definition. People simply used $x+B$, and I assume they are interested in functor of points. ... | https://mathoverflow.net/users/70360 | Translates of abelian subvarieties | Usually, when one speaks of "translate of an abelian subvariety" it really is in the sense "translate of an abelian subvariety by a closed point".
To answer your question: Let $X\subset A$ be a closed subvariety of an abelian variety $A$ over $k$. Assume that $k$ is algebraically closed of characteristic zero. Let $... | 2 | https://mathoverflow.net/users/4333 | 356281 | 150,310 |
https://mathoverflow.net/questions/356212 | 2 | Recently I met an ODE problem but after thinking for quite a while I still could not find an answer. Here is the question, which looks very simple:
---
Let $y=y(t)$ be a smooth function defined on $[0,\infty)$. If $y(t)$ satisfies the differential inequality
$$(ty')' \le \frac{4t}{(1+t^2)^2}(1-e^{2y})$$with the i... | https://mathoverflow.net/users/51546 | An ODE comparison problem | $\newcommand{\de}{\delta}
\newcommand{\vp}{\varepsilon}$
The conjecture is false. The previous answer, using numerics, was convincing enough for me, but perhaps not for others. So, here is a rigorous answer, based on the same general idea.
Let
\begin{equation\*}
L(y)(t):=(ty'(t))'-\frac{4t}{(1+t^2)^2}(1-e^{2y(t)})... | 2 | https://mathoverflow.net/users/36721 | 356292 | 150,312 |
https://mathoverflow.net/questions/356270 | 9 |
> As the title asks: does there exist $N$ such that, for any prime $p$ larger than $N$, the expression $x^4 +y^4$ takes on all values in $\mathbb{Z}/p\mathbb{Z}$?
>
I have been thinking about this problem for days, but I failed to solve it. Does anyone know whether this is true, or does anyone know any partial resu... | https://mathoverflow.net/users/153530 | Does the expression $x^4 +y^4$ take on all values in $\mathbb{Z}/p\mathbb{Z}$? | emtom has found the right reference, but there is a more explicit result in that book (Ireland and Rosen, *A Classical Introduction to Modern Number Theory*). In fact, Theorem 5 of Chapter 8 (on page 103) directly implies that the number $N = N\_{p,\alpha}$ of solutions to $x^4+y^4=\alpha$ in $\mathbb{F}\_p$ satisfies ... | 6 | https://mathoverflow.net/users/17907 | 356293 | 150,313 |
https://mathoverflow.net/questions/356299 | 1 | Given a radius $r > 0$, the internal covering number of a subset $T$ of a metric space $(X, d)$ is denoted $N\_r(T)$ and is defined to be the smallest number of balls of radius $r$ (under $d$) with centres in $T$ such that $T$ is contained in the union of the balls.
Given another subset of $X$, $U$, which is a supers... | https://mathoverflow.net/users/102255 | When is the internal covering number of a metric space monotonic? | The most obvious example:
Suppose $U$ is a closed ball of radius $1$ in $\mathbb R^d$, and $T$ is the corresponding sphere. Then if $r = 1$, $N\_r(U) = 1$ but $N\_r(T) > 1$.
EDIT:
Let $X$ be any metric space such that there are three points $a,b, c$ with $d(a,b) \le d(a,c) < d(b,c)$. Then take $d(a,c) \le r < d(b,c)... | 2 | https://mathoverflow.net/users/13650 | 356302 | 150,316 |
https://mathoverflow.net/questions/355153 | 2 | Let M be a 2-dimensional (complex dimension) Kähler manifold and $\phi$ be a real $(1,1)$-form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$?
| https://mathoverflow.net/users/147073 | Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on a Kähler manifold? | If $M$ is compact this is obviously false: if we can write $\phi=u\sqrt{-1}\partial\overline{\partial}u$ then $\int\_M \omega\wedge\phi\leq 0$ integrating by parts. So for every $\phi$ which does not satisfy this (for example $\phi=\omega$) it will not be possible to find such $u$.
| 2 | https://mathoverflow.net/users/13168 | 356309 | 150,319 |
https://mathoverflow.net/questions/356296 | 2 | Let $X$ be a variety and $Y \subset X$ a closed subvariety.
**Edit:** Assume they are both smooth.
Denote $N\_{Y / X}$ the normal bundle of $Y$ in $X$. The formal neighbourhood of $Y$ in $X$ is the formal scheme $X\_Y = \text{Spec}\left( \lim\_{n} \mathcal{O}\_X \left/ \mathcal{I}\_Y^n \right. \right)$.
The ques... | https://mathoverflow.net/users/91572 | Formal neighbourhood of a closed subscheme | If you look at a family of elliptic curves like $y^2 = x^3 - x-t^n$ as a surface $X$ mapping to a curve $C$ with parameter $t$, and $Y$ the fiber over $0$, then the neighborhood modulo $\mathcal I\_Y^n = (t)^n= (t^n)$ is trivial, hence equal to the $n$'th formal neighborhood of $Y$ in the total space of $N\_{Y/X}$, but... | 2 | https://mathoverflow.net/users/18060 | 356318 | 150,321 |
https://mathoverflow.net/questions/356312 | -1 | I am really interested in sufficient conditions on $a\_i, b\_i$ guaranteeing that the linear forms $a\_1\phi(n)+b\_1,\dots, a\_k\phi(n)+b\_k$ become simultaneously prime for infinitely many positive integers $n$. Here, $\phi(n)$ is the Euler's totient function and $a\_i, b\_i$ are arbitrary positive integers.
**Back... | https://mathoverflow.net/users/51189 | Sufficient conditions on $ a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often? | Most probably $\gcd(a\_i,b\_i)=1$ and $a\_i \geq 1$ and $b\_i$ to be odd for $i=1,...,k$ is enough.
Since $\{\varphi(n): n \in \mathbb N\}$ can be partitioned into countably many subsets $C\_l=\{\varphi(w\_{rl}): r \in \mathbb N\}$ such that $\varphi(w\_{ml})<\varphi(w\_{nl})$ if $m<n$ the $k$ forms $a\_j \cdot \varp... | 5 | https://mathoverflow.net/users/nan | 356322 | 150,322 |
https://mathoverflow.net/questions/348635 | 0 | <http://cantorsattic.info/index.php?title=J%C3%A4ger%27s_collapsing_functions_and_%CF%81-inaccessible_ordinals&action=edit>
Sadly, Cantor's attic is making an error. This is all I know about Jager's ordinal collapsing functions.
I know it is in [the paper](https://link.springer.com/article/10.1007/BF02007140), but ... | https://mathoverflow.net/users/150063 | Can I find the definition of Jager's ordinal collapsing functions? | I thank Carlo Beenakker for the comment.
I can see Jäger's function in [Googology Wiki](https://googology.wikia.org/wiki/J%C3%A4ger%27s_function?oldid=278807), now. This page is the revival of Cantor's Attic.
| 0 | https://mathoverflow.net/users/150063 | 356328 | 150,325 |
https://mathoverflow.net/questions/356310 | 2 | What is the covering number $N(\epsilon, B\_2, ||\cdot||\_2)$ of a ball $B\_2$ in $\mathbb{R}^d$ of radius $r$ under the $l\_2$ norm?
| https://mathoverflow.net/users/134168 | Covering number of $l_2$ Ball in $\mathbb{R}^d$ | The $\epsilon$-covering number of the euclidean unit ball in $\mathbb R^d$ scales like $(1/\epsilon)^d$. More formally,
>
> **Lemma.** If $\epsilon < 1$, then $(1/\epsilon)^d \le N(\epsilon, B\_2) \le (3/\epsilon)^d$. Else $N(\epsilon,B\_2) = 1$.
>
>
>
*Proof.* See **Theorem 4.2** and **Example 14.1** of this ... | 3 | https://mathoverflow.net/users/78539 | 356341 | 150,329 |
https://mathoverflow.net/questions/355776 | 5 | If $G$ is a (say) compact group and $V=\bigoplus\_{i\in I}V\_i$ the isotypic (a.k.a. primary) decomposition of a $G$-module, then
>
> any $G$-invariant subspace $W\subset V$ writes $W=\bigoplus\_{i\in I}(W\cap V\_i)$.
>
>
>
While this isn’t hard to prove (similar to Hoffman-Kunze 1971, [§7.5](//archive.org/de... | https://mathoverflow.net/users/19276 | Invariant subspaces and isotypic decomposition (reference request) | As suggested by the OP, I am turning my comment into an answer: a better Bourbaki reference is Algèbre VIII (new edition), §4, Proposition 4 d) (unfortunately not yet translated, as far as I know). It works for semi-simple modules over an arbitrary ring.
| 2 | https://mathoverflow.net/users/40297 | 356359 | 150,335 |
https://mathoverflow.net/questions/356354 | 11 | **Disclaimer:** When I came up with this question yesterday, I suspected it to be trivial (trivially true or trivially false). Then it kept me awake several hours tonight... (I still hope, though, this is just due to my ignorance.)
**Question.** Let $E,F$ be Banach space and suppose that $E$ embeds densely and contin... | https://mathoverflow.net/users/102946 | On dense embedding of Banach spaces | Great question! What you need is Sandy Grabiner's approximation lemma:
**Lemma**: *Let $E$ and $F$ be Banach spaces, let $T \in B(E,F)$, and let $M > 0$ and $0 < r < 1$. Suppose that for each $y \in [F]\_1$ there exists $x\_0 \in [E]\_M$ with $\|y - Tx\_0\| \leq r$. Then for each $y \in F$ there exists $x \in E$ with... | 9 | https://mathoverflow.net/users/23141 | 356364 | 150,336 |
https://mathoverflow.net/questions/356362 | 2 | Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M\_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$ restricted to $X\setminus Y$ is a line bundle. $M\_X$ becomes a monoid via the tensor product. Now let $G\_X$ be the ... | https://mathoverflow.net/users/36563 | Picard group modulo codimension 2 | The group $G\_X$ can be identified with the group of rank 1 reflexive sheaves on $X$ ($F$ is reflexive if the canonical morphism $F \to F^{\vee\vee}$ is an isomorphism) by taking a sheaf $F$ to the reflexive sheaf $F^{\vee\vee}$. The monoidal structure on the set of all reflexive sheaves is given by
$$
(F,G) \mapsto (... | 6 | https://mathoverflow.net/users/4428 | 356370 | 150,338 |
https://mathoverflow.net/questions/355704 | 4 | Let $(M,g)$ be a Riemannian manifold and let $N$ be a compact hypersurface isometrically embedded into $M$ and let $\eta$ denote a choice of unit normal vector field on $N$. It is then true that $N$ admits an $\varepsilon$-tubular neighborhood. That is, there exists some $\varepsilon>0$ such that the normal exponential... | https://mathoverflow.net/users/135839 | Injectivity radius of parallel hypersurfaces | Yes, the two injectivity radii are related.
Let $r>0$ be the injectivity radius from $N$. Then there is a neighborhood $U \subset M$ of $N$ and a Riemannian isometry $\phi : U \to (-r,r) \times N$. Each hypersurface $N\_t$ as you describe, for $|t|<r$ will correspond to the slice $\{t\} \times N$ with the identificat... | 1 | https://mathoverflow.net/users/13915 | 356381 | 150,341 |
https://mathoverflow.net/questions/355460 | 7 | In the paper "Hilbert's inequality", Montgomery and Vaughan proved that a generalization of the discrete Hilbert transform is bounded in $\ell^2$. The inequality reads as follows
$$ \Big| \sum\_{k\neq n}\frac{a\_k \overline{b\_n}}{\lambda\_k-\lambda\_n} \Big| \leq \frac{\pi}{\delta} \Big(\sum\_{k=1}^{\infty} |a\_k |^2 ... | https://mathoverflow.net/users/153260 | A generalization of discrete Hilbert's transform (Montgomery's inequality) | One can transfer the continuous $L^p$ theory to this discrete setting without difficulty.
Let's normalise $\sum\_k |a\_k|^p = \sum\_n |b\_n|^q = 1$. Consider the two quantities
$$ X\_1 := \sum\_{k \neq n} \frac{a\_k \overline{b\_n}}{\lambda\_k - \lambda\_n}$$
$$ X\_2 := \sum\_{k, n} p.v. \int\_{{\bf R}^2} \varphi... | 8 | https://mathoverflow.net/users/766 | 356384 | 150,343 |
https://mathoverflow.net/questions/356383 | 5 | By a *dynamical system* I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$.
We say that a dynamical system $(K,G)$
$\bullet$ is *topologically transitive* if for every non-empty open set $U\subseteq K$ its orbit $GU=\{g(x):g\in G,\;x\in U\}$ is de... | https://mathoverflow.net/users/61536 | A topologically transitive dynamical system without dense orbits | The answer is yes:
there is a topologically transitive dynamical system without dense orbits.
***Indeed***,
let X be a topological space that is not separable. Let $\ K=X^{\Bbb Z},\ $ and let $\ G\ $ be the group of homeomorphism of $\ K,\ $ induced by shifts $\ s\_n\ (n\in\Bbb Z)\ $ of $\ \Bbb Z:\ $
$$ \forall... | 6 | https://mathoverflow.net/users/110389 | 356389 | 150,345 |
https://mathoverflow.net/questions/356353 | 1 | The quadratic subfield of $\mathbb{Q}(\zeta\_p)$ is given by $\mathbb{Q}(\sqrt{p^\*})$, where $p^\*$ is the choice of $\pm p$ which is $1$ mod $4$. By some elementary Galois theory, the cyclotomic polynomial $\Phi\_p = \frac{x^p-1}{x-1}$ factors into two irreducible polynomials of degree $\frac{p-1}{2}$ over this quadr... | https://mathoverflow.net/users/39747 | Factoring cyclotomic polynomials over quadratic subfield | Some trivial observations. We have
$$P\_{QR}(1/x) x^{(p-1)/2} = \prod\_{QR} (1 - x \zeta^k),$$
$$P\_{QNR}(1/x) x^{(p-1)/2} = \prod\_{QNR} (1 - x \zeta^k),$$
which are easier to work with. On the other hand,
if $p \ge 5$, the product of $\zeta^k\_p$ over quadratic residues is one,
and the product over non-residues is a... | 3 | https://mathoverflow.net/users/155528 | 356390 | 150,346 |
https://mathoverflow.net/questions/356253 | 5 | Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the topology of [uniform convergence on compacta][2].
Let $N'\subseteq N$ be a dense subset which is homeomorphic to $\mathbb{R... | https://mathoverflow.net/users/36886 | Non-density of continuous functions to interior in set of all continuous functions | 1) About Erz' answer, and Annie's second question. It is a general property that for two differentiable manifolds $M$, $N$, such that $M$ is compact, and given a continuous map $f:M\to N$, any continuous map $g:M\to N$
close enough to $f$ is homotopic to $f$. This can be proved by many different ways, depending on tast... | 3 | https://mathoverflow.net/users/105095 | 356395 | 150,348 |
https://mathoverflow.net/questions/347760 | 5 | I am trying to understand the correspondence between Donaldson invariants and different correlation functions in certain topological quantum field theories. To be exact, among others I am reading Witten's "Supersymmetric Yang-Mills Theory on a Four-Manifold". However, topic seems to be a difficult one for me, and so I ... | https://mathoverflow.net/users/108687 | Donaldson Invariants in 2 dimensions | It depends what you mean by the Donaldson invariants. The smooth structure on a two manifold is unique, so there aren't any analogues of the Donaldson invariant in that sense. But there are 2d physics analogues of the Donaldson invariants, in that there are two-dimensional topologically twisted quantum field theories s... | 2 | https://mathoverflow.net/users/35508 | 356398 | 150,350 |
https://mathoverflow.net/questions/356396 | 2 | I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:
The Thom isomorphism in Bott & Tu is obtained as $H\_{cv}^{\*+n}(E)\rightarrow H^\*(M)$, where $\pi\colon E\to M$ is $n$ plane bundle over the manifold of dimension $m$ manifold $M$ and the isomorphism is given by the ... | https://mathoverflow.net/users/151571 | Relation between compact vertical cohomology and local cohomology groups | Compactly supported homology can be seen to be equal to the reduced homology of the one point compactification, essentially by definition of the topology on the one point compactification.
A similar argument shows that vertical compactly supported homology is the reduced homology of the space obtained by compactifyin... | 6 | https://mathoverflow.net/users/134512 | 356402 | 150,352 |
https://mathoverflow.net/questions/356134 | 7 | My questions is about Schauder bases and more specifically about coefficient functionals.
Let $(x\_n)$ be a Schauder basis of a Banach space $X$. Thus for all $x$ in $X$, $x = \sum f\_n(x) x\_n$. The $f\_n$ are called coefficient functionals.
They are continuous. If $X$ is reflexive, they form a basis of $X^\*$ (wit... | https://mathoverflow.net/users/155355 | Completeness of coefficient functionnals | M. Zippin [showed](http://link.springer.com/article/10.1007/BF02771607) that for a Banach space $X$ with a basis, if every basis of $X$ is boundedly complete or if every basis of $X$ is shrinking, then $X$ is reflexive.
The result of Zippin answers you question in the negative (or, rather, the answer is, *"not necess... | 6 | https://mathoverflow.net/users/848 | 356414 | 150,353 |
https://mathoverflow.net/questions/355526 | 22 | The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.
* Drinfeld modules are like the function field analogue of CM elliptic curves. To see this, complexify an elliptic curve $E$ to get a... | https://mathoverflow.net/users/151698 | How can I see the relation between shtukas and the Langlands conjecture? | I think shtukas are best understood ahistorically. I would start with the modular curves, but specifically with the (geometric) Eichler-Shimura relation. This says that the Hecke operator at $p$, viewed as a correspondence on $X\_0(N) \times X\_0(N)$, when reduced at characteristic $p$, is just the graph of Frobenius p... | 13 | https://mathoverflow.net/users/18060 | 356422 | 150,355 |
https://mathoverflow.net/questions/356394 | 3 | I am working on an enumerative problem related to knot theory, and I have found the following generating function
$$F(z)=\prod\_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$
I am interested on getting asymptotic estimates for this GF. I am not an expert on partition function theory, but it seems that trying to apply Meinardu... | https://mathoverflow.net/users/46573 | Meinardus theorem at use: problems with conditions | This is a postscript to Carlo Beenakker's answer, a combinatorial explanation for the relationship between $a\_n$ and $c\_n$.
Your $F(z)=\sum\_n c\_nz^n$ is the generating function for the number of partitions of $n$ into odd parts 3 or greater where there are two kinds of each part. Carlo's $G(z) = \sum\_n a\_nz^n$ ... | 2 | https://mathoverflow.net/users/14807 | 356432 | 150,357 |
https://mathoverflow.net/questions/356401 | 1 | Let $G$ be a subgroup of the permutation group $S\_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech compactification $\beta\omega$ of $\omega$. The homeomorphism $\bar g$ induces a homeomorphism of the remainder $\omega^\*=\b... | https://mathoverflow.net/users/61536 | A permutation group inducing a topologically transitive action without dense orbits on $\omega^*$ | It turns out that this problem is independent of ZFC because of the following simple
>
> **Theorem.** Under $\mathfrak t=\mathfrak c$, every topologically transitive continuous action of a group $G$ on $\omega^\*$ has a dense orbit.
>
>
>
*Proof.* Let $(A\_\alpha)\_{\alpha\in\mathfrak c}$ be an enumeration of ... | 4 | https://mathoverflow.net/users/61536 | 356452 | 150,365 |
https://mathoverflow.net/questions/356451 | 7 | I am researching closed random walks on graphs and have the following problem that I haven't been able to find a reference for.
Consider a random walk on $\mathbb Z$ starting at 0 and at each step it moves $-1$ or $+1$ each with probability $1/2$. If the walk has length $2n$ it is well-known that the support (or how... | https://mathoverflow.net/users/152267 | Support of closed random walk on $\mathbb Z$ | One way to do this is as follows. We have to show that
$$P(M\_n\ge x|S\_n=0)\to1$$
(as $n\to\infty$) if $x=o(\sqrt n)$, where $S\_n$ is the position of the walk at time $n$ and $M\_n:=\max\_{0\le k\le n}S\_k$. By the reflection principle (see e.g. [Theorem 0.8](http://cgm.cs.mcgill.ca/~breed/MATH671/lecture2corrected.... | 4 | https://mathoverflow.net/users/36721 | 356457 | 150,367 |
https://mathoverflow.net/questions/356415 | 21 | I noticed the following strange (to me) fact. If $M$ is a real manifold (smooth or not) and $R = C(X, \mathbb{R})$ is the ring of real functions (smooth functions in the smooth case) then the affine scheme $X = \mathrm{Spec}(R)$ has a natural map $M \to X$ which is a homeomorphism on real points i.e. $M \to X(\mathbb{R... | https://mathoverflow.net/users/154157 | Real manifolds and affine schemes | (1) This is a highly productive way of looking at smooth manifolds.
It is responsible for [synthetic differential geometry](https://ncatlab.org/nlab/show/synthetic%20differential%20geometry) and [derived smooth manifolds](https://ncatlab.org/nlab/show/derived%20differential%20geometry).
Both of these subjects heavily r... | 20 | https://mathoverflow.net/users/402 | 356474 | 150,370 |
https://mathoverflow.net/questions/356476 | 4 | Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and that $B$ is positive and rank-1. I'm interested in conditions when $A - \epsilon B \ge 0$ for some strictly positive $\eps... | https://mathoverflow.net/users/76565 | When is rank-1 perturbation to a positive operator still positive? | In "On Majorization, Factorization, and Range Inclusion of Operators on Hilbert Space (1966)", R. G. Douglas proved the following result (Theorem 1 in the paper):
**Theorem.** Let $C$ and $D$ be bounded linear operators on a real or complex Hilbert space $\mathcal{H}$; then the following are equivalent:
(i) $C\math... | 3 | https://mathoverflow.net/users/102946 | 356478 | 150,372 |
https://mathoverflow.net/questions/355968 | 1 | I am trying to understand a (straightforward I guess) statement of an old (but outstanding) paper on stability of solitary waves. Let us consider the following functionals:
$$
V(u)=\dfrac{1}{2}\int\_\mathbb{R} u^2dx \quad \hbox{and}\quad E(u)=\int\_\mathbb{R}(\tfrac{1}{2}uMu-\tfrac{1}{2}u^2-F(u))dx.
$$
As you can gues... | https://mathoverflow.net/users/129131 | Understanding traveling waves as critical points of the constrained energy | The problem is precisely that you should not fix a priori the parameter $c$, because it will be exactly the Lagrange multiplier (or rather, $\lambda=-c$ will).
Think of it like this: Let me introduce a new parameter $R>0$, which I think of as a prescribed $L^2$ energy level.
Then the minimization problem
$$
\min\limits... | 2 | https://mathoverflow.net/users/33741 | 356480 | 150,373 |
https://mathoverflow.net/questions/356406 | 2 | Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (Proposition 3.3 [here](https://iopscience.iop.org/article/10.1070/IM1990v035n03ABEH000716)). A different proof was given ... | https://mathoverflow.net/users/1181 | Why are Serre functors always exact? | (Notations as in Huybrechts' book.) Consider the maps $\mathrm{Hom}(C,S(B))\rightarrow\mathrm{Hom}(C,C\_0)$ and $\mathrm{Hom}(A[1],C\_0)\rightarrow\mathrm{Hom}(C,C\_0)$, and let $I$ and $J$ be their images. Huybrechts tells you how to define linear forms $\Xi\_I:I\rightarrow k$ ("condition $\mathrm{i}')$") and $\Xi\_J:... | 2 | https://mathoverflow.net/users/104669 | 356481 | 150,374 |
https://mathoverflow.net/questions/356461 | 7 | I am trying to understand and further formalize Witten's proof of the positive mass theorem. Dan Lee, in his book "[Geometric relativity](https://bookstore.ams.org/gsm-201)" did a wonderful job with formalizing and carrying out the details of [Parker and Taubes' work](https://projecteuclid.org/euclid.cmp/1103921154), w... | https://mathoverflow.net/users/141944 | Completeness hypothesis in the positive mass theorem | Completeness is necessary. Otherwise you can just take a maximal spatial slice of the negative-mass Schwarzschild solution (i.e. a constant $t$ slice in Boyer-Lindquist coordinates) and it has vanishing scalar curvature with $m < 0$. There are however no *complete* maximal spatial sections of the negative-mass Schwarzs... | 7 | https://mathoverflow.net/users/3948 | 356491 | 150,376 |
https://mathoverflow.net/questions/356475 | 9 | The question is motivated by my failed [comment](https://mathoverflow.net/questions/356253/non-density-of-continuous-functions-to-interior-in-set-of-all-continuous-functio#comment895364_356253) to [this one](https://mathoverflow.net/questions/356253/non-density-of-continuous-functions-to-interior-in-set-of-all-continuo... | https://mathoverflow.net/users/53155 | Is limit of null-homotopic maps null-homotopic? | I'll provide a general theorem (then one has to apply it to specific circumstances). There is a micro-dictionary/Notation at the bottom of this note.
**B-assumption**: Space $\ N\times N\times[0;1]\ $ is normal.
Every metric space $\ N\ $ satisfies **B-assumption**.
**Notation** Let $\ \mathcal W\_N\ $ be the... | 5 | https://mathoverflow.net/users/110389 | 356504 | 150,382 |
https://mathoverflow.net/questions/356506 | 1 | If $A(z) :=[A\_{ij}(z)] $ and $B(z) :=[B\_{ij}(z)] $ are two invertible $n\times n$ matrices of entire complex valued functions entries $A\_{ij}(z)$, and $B\_{ij}(z) $ with
(1). $AA^{\#}=A^{\#}A$ where $A^{\#}(z)=\left(\overline{A(\bar{z})} \right)^{T}$.
(2). $BB^{\#}=B^{\#}B$
(3).$ AB=BA$
Is it true that
$$... | https://mathoverflow.net/users/149682 | Commuting matrices of complex functions | Let $z\_0\in\Bbb{R}$ be arbitrary. The matrices $A\_0:=A(z\_0)$ and $B\_0:=B(z\_0)$ are *normal*; in view of $A\_0^\*=A^\#(\bar{z\_0})=A^\#(z\_0)$ and $B\_0^\*=B^\#(\bar{z\_0})=B^\#(z\_0)$ they commute with their Hermitian adjoints. They also commute with each other. So $A\_0$ and $B\_0$ could be simultaneously diagona... | 4 | https://mathoverflow.net/users/128556 | 356509 | 150,384 |
https://mathoverflow.net/questions/356510 | 3 | I am learning a little bit about Dehn functions of group presentations and I came across a question that is probably pretty basic but that I was giving me trouble. I'll set some notation but essentially I want to understand why being able to compute an upper bound for the Dehn function means that the word problem is so... | https://mathoverflow.net/users/99414 | Bounding the size of the conjugating elements given the Dehn function | It is, in particular, in my book "Combinatorial algebra: syntax and semantics" which can be found on my Web site. The length of each "petal" is at most the length of the word plus the Dehn function multiplied by a constant. The book contains a proof (in fact a stronger result is proved). It is essentially a copy of our... | 2 | https://mathoverflow.net/users/nan | 356513 | 150,385 |
https://mathoverflow.net/questions/356512 | 18 | I am totally new to academia so I am really not sure how mathematicians works together, can more experienced mathematicians here shed some light on how you find coauthors? I guess one way to do this is to attend conference. But if one doesn't have the chance to do so, are there other options?
| https://mathoverflow.net/users/155590 | How do mathematicians find coauthors? | Alright, I'll try to outline several ways of how mathematicians find co-authors (since I'm still quite young, I lack the experience to judge how things might change later during a career, but given that you're new to academia, I hope that what follows fits your situation):
I'll try to write my answer as a kind of cla... | 31 | https://mathoverflow.net/users/102946 | 356521 | 150,388 |
https://mathoverflow.net/questions/356410 | 1 | Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed quotient of a zero-dimensional space may not be zero-dimensional, even if the equivalence classes are compact. Note that in ... | https://mathoverflow.net/users/95718 | Quotients of the irrationals | The answer to this question is negative and follows from the characterization:
>
> **Theorem.** A topological space $X$ is an image of the space of irrationals $\mathbb P$ under a perfect map $f:\mathbb P\to X$ if and only if $X$ is Polish and nowhere locally compact.
>
>
>
*Proof.* To prove the "only if" part... | 3 | https://mathoverflow.net/users/61536 | 356523 | 150,389 |
https://mathoverflow.net/questions/356493 | 5 | Let $C$ be a smooth projective curve over a finite field $\mathbb F\_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1\_{et}(C,\mathbb Q\_{\ell})$ (that is, on $\ell$-adic etale cohomology, where $\ell$ is a prime distinct from $p$) then $\alpha... | https://mathoverflow.net/users/2191 | Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible? | Every value $c \in \big[0,\tfrac{1}{2}\big] \cap \mathbf Q$ can occur as the smallest slope of an abelian variety over $\mathbf F\_q$; see the corollary below.
What Honda actually proves [Hon68] (see [Mil94, Prop. 2.6] for a motivic reinterpretation) is:
**Theorem** (Honda). *Let $q$ be a power of a prime $p$. Then... | 5 | https://mathoverflow.net/users/82179 | 356524 | 150,390 |
https://mathoverflow.net/questions/356463 | 3 | Let $d\in\mathbb N$ and $0<\alpha<d$. Define the Riesz kernel $K\_\alpha(x):=|x|^{\alpha-d}$, and the associated convolution operator
$$K\_\alpha f(x):=\int\frac{f(y)}{|x-y|^{d-\alpha}}~dy.$$
The classical [Hardy-Littlewood-Sobolev inequality](https://en.wikipedia.org/wiki/Sobolev_inequality#Hardy%E2%80%93Littlewood%E2... | https://mathoverflow.net/users/50406 | Hardy-Littlewood-Sobolev for "componentwise product" of Riesz kernels | I realise I was too optimistic in my comment: no such ineqaulity holds in general. Indeed, consider $f(x) = \prod\_j f\_j(x\_j)$. Then
$$ K\_{(\alpha\_1,\ldots,\alpha\_d)} f(x) = \prod\_j K\_{\alpha\_j} f\_j(x\_j) $$
and so
$$ \|f\|\_p = \prod\_j \|f\_j\|\_p , \qquad \|K\_{(\alpha\_1,\ldots,\alpha\_d)} f\|\_q = \prod\_... | 2 | https://mathoverflow.net/users/108637 | 356533 | 150,392 |
https://mathoverflow.net/questions/356534 | 9 | Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer, but he just wants to hear my views. However, I'm not a number theorist, hence I couldn't answer him. So below is the q... | https://mathoverflow.net/users/155609 | A question on the Riemann zeta function | The identity
$$\sum\_{n=1}^\infty\frac{\mu(n)\log n}{n}=-1$$
was conjectured by Möbius (1832) and proved by [Landau (1899)](https://archive.org/details/ComptesRendusAcademieDesSciences0129/page/n812/mode/1up). It is a consequence of the prime number theorem. Not surprisingly, the rate of convergence is determined by th... | 7 | https://mathoverflow.net/users/11919 | 356540 | 150,394 |
https://mathoverflow.net/questions/356544 | 4 | I was reading the paper "[Short Star-Products for Filtered Quantizations](https://www.emis.de/journals/SIGMA/2020/014)" by Pavel Etingof and Douglas Stryker ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=4074404)), where in the introduction they claim that the algebra of regular functions on the quadratic cone... | https://mathoverflow.net/users/155568 | Algebra of regular functions on the quadratic cone and SU(2) representations | The ring $\Bbb C[x,y,z]/(xy-z^2)$ is $\Bbb N$-graded because $xy-z^2$ is homogeneous. Its $m$-th component has dimension $2m+1$, because a basis is given by $\{ x^ay^b \mid a+b=m \} \cup \{ x^ay^bz \mid a+b=m-1 \}$.
It's easy to see that the representation is irreducible (I assume the action is the adjoint action of... | 7 | https://mathoverflow.net/users/104742 | 356546 | 150,395 |
https://mathoverflow.net/questions/121483 | 9 | I'm interested in tilings of the plane by squares, with labels on the edges. It's well known that (1) the question "can one tile the plane with the following finite set of tiles?" is undecidable, and (2) there are finite sets of tiles that tile the plane, but only aperiodically. Also, (1) implies (2).
Is the question... | https://mathoverflow.net/users/10481 | Decidability of periodic tilings of the plane | Deciding whether a set of tiles admits a periodic tiling or no tiling at all is undecidable as well.
This has been shown in [Y.S. Gurevich, I.I. Koryakov, Remarks on Berger's paper on the domino problem, J Sib Math J 13, 319–321 (1972)](https://scholar.google.com/scholar?cluster=2757770576292515424). The results can ... | 4 | https://mathoverflow.net/users/16710 | 356547 | 150,396 |
https://mathoverflow.net/questions/356532 | 2 | I need a proper reference to the following obvious fact:
>
> An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $U$ is dense) if and only if it the orbit $Gx$ of some point $x\in K$ is dense in $K$.
>
>
>
For a cyclic group... | https://mathoverflow.net/users/61536 | A reference to the fact that a topologically transitive action of a group on a compact metrizable space has a dense orbit | See Theorem 9.20 of "Topological Dynamics" by Gottschalk and Hedlund. It states that, for systems $(X,G)$ whose phase space is non-empty complete separable metric, point transitivity (a point having a dense orbit) and topological transitivity (every non-empty open set having dense orbit) are equivalent.
| 4 | https://mathoverflow.net/users/33039 | 356552 | 150,399 |
https://mathoverflow.net/questions/356537 | 23 | In a typically lucid and helpful page of notes for students, [A beginner’s guide to countable ordinals](https://www.dpmms.cam.ac.uk/~wtg10/ordinals.html), Tim Gowers explains how the countable ordinals can be “constructed rigorously in a way that requires more or less no knowledge of set theory”, and this is enough for... | https://mathoverflow.net/users/14111 | What's the use of countable ordinals? (prompted by a remark of Tim Gowers) | So, we know from reverse mathematics that nearly all "bread-and-butter" theorems are, suitably encoded, provable in proof-theoretically weak subsystems of second order arithmetic. If a theorem is provable in a system whose proof theoretic ordinal is $\alpha$, then in some sense ordinals larger than $\alpha$ need not co... | 13 | https://mathoverflow.net/users/23141 | 356555 | 150,401 |
https://mathoverflow.net/questions/356545 | 3 | I am looking for approximations, or a closed form, if available, for the sum
$$S(n,a,b)=\sum\_{1\leq x,y,\leq n} \frac{x^a}{\mathrm{lcm}(x,y)^{b}},$$
where $\mathrm{lcm}(x,y)$ is the least common multiple of integers $x,y$ and $a,b$ are positive quantities. I am in particular interested in $a=b=1.$ For this case nu... | https://mathoverflow.net/users/17773 | Estimating the sum $\sum_{1\leq x,y\leq n} \frac{x}{ \mathrm{lcm}(x,y)}$ | The original sum can be written as
$$T(\alpha,\beta,\gamma,n)=\sum\_{x,y\le n}x^\alpha y^\beta(x,y)^\gamma,$$
where $(x,y)=\mathrm{gcd}(x,y)$. One can find asymptotic formula for this sum using standart approach. Let $d=(x,y)$. Then
$$T(\alpha,\beta,\gamma,n)=\sum\_{d\le n}d^\gamma\sum\_{{x,y\le n\atop (x,y)=d}}x^\alph... | 6 | https://mathoverflow.net/users/5712 | 356568 | 150,405 |
https://mathoverflow.net/questions/356577 | -1 | This is linked to [my question](https://math.stackexchange.com/questions/3605201/examples-and-references-of-connections-on-elliptic-curves) on math.Stackexchange for which I had no answer.
I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me fully gra... | https://mathoverflow.net/users/101411 | Connections on vector bundles over elliptic curves - concrete computations | Perhaps the following very elementary discussion will help you to understand the relationship between systems of first order ODE's, connections, and vector bundles. If what I write below isn't helpful, I apologize for wasting your time.
Let $u: \mathbb{C}\rightarrow \mathbb{C}^{n}$ be a vector valued holomorphic func... | 1 | https://mathoverflow.net/users/49247 | 356584 | 150,410 |
https://mathoverflow.net/questions/356502 | 0 | Does a left translation of an automorphic form satisfy left $K$-finiteness?
Let $F$ be a number field and $G$ is an algebraic group. Let $\phi$ is an automorphic form on $G$. Let $K$ be a maximal compact subgroup of $G(A)$.
Then $\phi$ has $K$-finiteness.
For arbitrary $x$, let $\phi\_x(g)=\phi(xg)$. Then I am wo... | https://mathoverflow.net/users/29422 | Left translation of automorphic form satisfies $K$-finiteness? | Presumable (in an automorphic forms context with contemporary left-right conventions) you mean that $\varphi$ is *right* $K$-finite. This could apply to any (complex-valued) function $\varphi$ on a topological group $G$, with compact subgroup $K$. It means that the space of functions obtained by right translation $R\_k... | 2 | https://mathoverflow.net/users/15629 | 356588 | 150,411 |
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