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https://mathoverflow.net/questions/309132 | 2 | Let $S$ be the class of all $2$ by $2$ matrices of the form
$$\begin{bmatrix}
1 & a \\
a & 1
\end{bmatrix},\, |a|\leq 1.$$
Is there a single matrix $M\in S$ such that for any $N\in S$ and all $x>0$ we have
$$\mathbb{P}(||X||\_2 \geq x)\geq \mathbb{P}(||Y||\_2 \geq x),$$
where $X$ and $Y$ have the Gaussian distributi... | https://mathoverflow.net/users/24494 | Stochastic domination of Gaussian random vectors | $\renewcommand{\P}{\operatorname{\mathsf P}}\newcommand{\E}{\operatorname{\mathsf E}}$The answer is no. More specifically, let $Y\_a$ be a centered Gaussian random vector with covariance matrix $\begin{bmatrix}
1 & a \\
a & 1
\end{bmatrix}$.
Then there is some $u\_1\in(0,\infty)$ such that
\begin{equation}
\max\_... | 2 | https://mathoverflow.net/users/36721 | 309136 | 134,625 |
https://mathoverflow.net/questions/309141 | 3 | In the following paper (Example 2.1), it has been mentioned to K+M to provide an example of a pseudo valuation domain which is not a valuation ring, and its reference is Gilmer's book, but I have no access to Gilmer's book.
Can someone help me and explain what K+M is?
*Hedstrom, J. R.; Houston, E. G.*, Pseudo-valua... | https://mathoverflow.net/users/128190 | What is K+M structure? | A "valuation ring of the form $K+M$" is a valuation ring $V$ with maximal ideal $M$ such that $V$ contains a subring $K$ which is a field and one has $V=K+M:=\{k+m\,|\,k\in K,m\in M\}$.
In this case (see the paper), for every proper subfield $F$ of $K$, the set $F+M:=\{f+m\,|\,f\in F,m\in M\}$ --- call it $R$ --- is... | 3 | https://mathoverflow.net/users/86006 | 309143 | 134,627 |
https://mathoverflow.net/questions/309127 | 1 | Given an immersed submanifold $M$ of a Riemannian manifold $\overline{M}$, the *first normal space* of $M$ at a point $p \in M$ is defined as the linear subspace $N\_{p}^{1}M$ of $N\_{p}M$ spanned by the image of the second fundamental form $\alpha$ at $p$:
$$ N\_{p}^{1}M = \text{span of }\{\alpha(v,w) \in N\_{p}M \m... | https://mathoverflow.net/users/74033 | Definition of first normal space | OK I got confused really bad yesterday. The set $ \{\xi \in N\_{t}\gamma \mid A\_{\xi}=0 \}$ is indeed $(N-1)$-dimensional. Without loss of generality, assume $\gamma$ be unit-speed, and $\overline{D}\_{t}\dot{\gamma}$ never zero. Then $\overline{D}\_{t}\dot{\gamma}(t)$ is a non-zero vector in the normal space $N\_{t}\... | 0 | https://mathoverflow.net/users/74033 | 309152 | 134,630 |
https://mathoverflow.net/questions/309139 | 1 | I'm trying to prove that if every closed set in a topological space is regular $G\_\delta$, then the space is normal. By *regular $G\_\delta$*, I mean for any closed set $A$,
1. there exists a countable collection $\{U\_n:n\in\mathbb N\}$ such that for each $n$, $A \subset U\_n$.
2. $A = \bigcap\_{n\in\mathbb N}\over... | https://mathoverflow.net/users/106564 | Does regular $G_\delta$ imply normal? | A well-known characterisation of normality is useful here (proposition 1.5.15 in Engelking's *General Topology*), I wrote its proof [here](http://at.yorku.ca/p/a/c/a/07.pdf):
>
> $X$ is normal iff for each closed set $A$ of $X$ and each open set $O$ with $F \subseteq O$, there are open sets $W\_n$, $n \in \mathbb{N... | 4 | https://mathoverflow.net/users/2060 | 309153 | 134,631 |
https://mathoverflow.net/questions/309150 | 3 | Let $M$ be a smooth closed manifold, and let $g\_0$ be a Riemannian metric on $M$.
Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-g\_0|\_U\|\_{C^1} < \epsilon$ on $U$.
>
> Can we extend $g$ to a metric $\tilde g$ on $M$ such that $\| \tilde g-g\_... | https://mathoverflow.net/users/46290 | Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation? | Let $\chi$ be a smooth function that is identically $1$ on a neighborhood of $p$ and compactly supported on $U$. Let $\tilde{g} = (1-\chi)g\_0 + \chi g$.
| 3 | https://mathoverflow.net/users/613 | 309165 | 134,636 |
https://mathoverflow.net/questions/309166 | 3 | According to Bohr, the definition of the almost periodic function is:
A function $f:\mathbb{R}\rightarrow \mathbb{C}$ is called almost periodic if it is continuous and if for every positive $\epsilon$, there exists a positive number $l$ such that every closed interval of length $l$ contains an $\epsilon$-almost period.... | https://mathoverflow.net/users/113410 | Question on the definition of almost periodic function | It seems to me that if an interval is an $\epsilon$-almost period then it is trivially an $\epsilon'$-almost period for any $\epsilon' >\epsilon$. So as $\epsilon$ gets smaller the set of intervals which contain $\epsilon$-almost periods shrinks (i.e., fewer intervals have this property).
As to the second question, c... | 7 | https://mathoverflow.net/users/23141 | 309167 | 134,637 |
https://mathoverflow.net/questions/309162 | 7 | This is hard, so I am looking for partial results and how hard it is.
Let $n>4$. Is it true that the hyperelliptic curve $x^n=y(y+1)$
doesn't have rational point with $x \ne 0$?
If necessarily assume $n$ is prime.
Integral points on the curve are heavily studied.
It is one of the simplest exponential diophantin... | https://mathoverflow.net/users/12481 | Rational perfect power values of $y(y+1)$ | There are no such solutions. Let $x=a/b$ and $y=c/d$ be reduced fractions. Then
$a^n/b^n=(c(c+d))/d^2$ and since both sides are reduced fractions we get that
$a^n=c(c+d)$ and $b^n=d^2$. From the first equation we deduce that $c=e^n$ and $c+d=f^n$ since $c$ and $c+d$ are co-prime and their product is an $n$-th power.
... | 18 | https://mathoverflow.net/users/115052 | 309173 | 134,639 |
https://mathoverflow.net/questions/299759 | 7 | Let $\mathcal{A}$ be a $C^\*$-algebra and $p\in\mathcal{A}^{\*\*}$ be an open projection, that is, $p=p^\*=p^2$ and $p\in\overline{(p\mathcal{A}^{\*\*}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^\*}$, where $\hat{\mathcal{A}}$ is the canonical copy of $\mathcal{A}$ in $\mathcal{A}^{\*\*}$ and the closure is taken in the... | https://mathoverflow.net/users/25499 | Open projections and Murray-von Neumann equivalence |
>
> The answer is **no**.
>
>
>
*Proof (Thomas Schick)*. The idea of the proof is due to Thomas Schick. I thank him for allowing me to reproduce it here. Let $\mathcal{A}:=C([0,1])\otimes\mathbb{M}\_2$, where $\mathbb{M}\_2$ is the $W^{\star}$-algebra of $2\times2$ matrices with entries in $\mathbb{C}$. Since th... | 2 | https://mathoverflow.net/users/25499 | 309174 | 134,640 |
https://mathoverflow.net/questions/308737 | 3 | Let $(M,g\_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta\_{g\_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g\_0$-Laplacian. Let $H^k\_{g\_0}=\text{ker} \Delta\_{g\_0}$.
>
> Suppose $g\_{\epsilon}$ is close to $g\_0$ in the $C^1$ sense. Is it true that $H^k\_{g\_0}$ is "close... | https://mathoverflow.net/users/46290 | Does the space of harmonic forms change continuously with the metric? | I think the answer is positive.
Let $D$ be the subspace of smooth **closed** $k$-forms on $M$. Equip $D$ with the supremum- $C^1$ norm:
$$
\| \omega \|\_{C^1,sup}:=\max\{ \|\omega\|\_{sup}, \|T\omega\|\_{sup} \},
$$
All the norms are w.r.t $g\_0$.
Let $\delta\_g$ be the codifferential of $d$ w.r.t the metric $g$.... | 1 | https://mathoverflow.net/users/46290 | 309175 | 134,641 |
https://mathoverflow.net/questions/309142 | 11 | I am reading Infinite dimensional lie algebras by Kac. He starts with a $n \times n$ GCM (Generalized Cartan Matrix) $A$ of rank $l$, then he defines the realization associated with the matrix $A$ which is of dimension $d=2n-l$. I know that in the simple Lie algebra case this dimension is $n$ as $A$ is invertible
I d... | https://mathoverflow.net/users/33047 | Realisation of Kac-Moody Lie algebras | An equivalent definition of a *realisation* of a GCM $A=(a\_{ij})\_{1\leq i,j\leq n}$ of rank $\ell$ is as follows: it is a triple $(\mathfrak h, \Pi, \Pi^{\vee})$ where $\mathfrak h$ is a complex vector space, $\Pi=\{\alpha\_i \ | \ 1\leq i\leq n\}\subseteq\mathfrak h^\*$ and $\Pi^{\vee}=\{\alpha\_i^{\vee} \ | \ 1\leq... | 10 | https://mathoverflow.net/users/106751 | 309176 | 134,642 |
https://mathoverflow.net/questions/304905 | 5 | I am looking at the paper "p-adic Groups" by Bruhat (in the Boulder Proceedings, 1965). I have a question about one of the statements. Let $k$ be the quotient field of a complete discrete valuation ring $\mathcal{O}$. Let $V$ be a vector space over $k$. Let $L$ be a lattice in $V$. Choose a basis for $L$.
Bruhat stat... | https://mathoverflow.net/users/122801 | Integral structures via lattices | Here is a proof of (3). Hopefully there are no gaps.
I will write $K$ instead of $k$, which I usually reserve for the residue field.
Just to set the terminology:
* $G$ is an algebraic group over $K$.
* An $\mathcal{O}$-structure for $G$ is a finitely generated Hopf $\mathcal{O}$-subalgebra $A$ of $K[G]$ such that ... | 3 | https://mathoverflow.net/users/86006 | 309179 | 134,643 |
https://mathoverflow.net/questions/309130 | 2 | Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum
$-\infty<\lambda\_1\leq\lambda\_2\leq\cdots$ converging to $+\infty$. Do there exist reasonably convenient criteria on $V$ that can guarantee that the eigenvalues are all simple (i.e., $\lambda\_1<\l... | https://mathoverflow.net/users/50406 | Criteria for Schrödinger operator on real line to have simple spectrum | You don't need to assume anything. Only the absolutely continuous spectrum of a whole line operator (assumed to be in the limit point case at both endpoints, for convenience, though that, too, could be relaxed) can have multiplicity greater than $1$.
The result is usually attributed to I.S. Kac, On the multiplicity o... | 4 | https://mathoverflow.net/users/48839 | 309185 | 134,645 |
https://mathoverflow.net/questions/308691 | 50 | Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often see assertions of things like "the prime number theorem is equivalent to the fact that the Riemann zeta function has no ze... | https://mathoverflow.net/users/127902 | Motivated account of the prime number theorem and related topics | Let me record a pedestrian answer here. It all starts with Euler's formula
$$
\prod\_p\left(1-\frac{1}{p^s}\right)^{-1}=\sum\_{n\geq 1}\frac{1}{n^s}=:\zeta(s),\quad s>1,
$$
and the observation that since the RHS diverges for $s=1$, so does the LHS, and thus $\sum\_p p^{-1}=+\infty$. This is already interesting, since ... | 14 | https://mathoverflow.net/users/56624 | 309186 | 134,646 |
https://mathoverflow.net/questions/309184 | 7 | How many positive integer solutions of $$\sum\_{i=1}^{k}x\_i = N$$ for some positive integer $N$ given the constraints $n\_i\leq x\_i\leq m\_i$ for $i=1,\ldots,k$, where $n\_i$ and $m\_i$ are positive integers.
I know that it can be calculated by finding the coefficient of $y^N$ in the polynomial $\prod\_{i=1}^{K}(y^... | https://mathoverflow.net/users/120111 | Number of integer solutions of a linear equation under constraints |
>
> **Proposition 1:** The number of integer solutions of the equation
> $$
> \sum\_{i=1}^{k}x\_i = N
> $$
> where $x\_i\geq n\_i$ for $i=1,\ldots,k$, is given by
> $$
> {\small
> \binom{N+k-1-n\_1-n\_2-...-n\_k}{k-1}
> }
> $$
> if the upper index is non-negative and zero otherwise.
>
>
>
In the formula a... | 7 | https://mathoverflow.net/users/85967 | 309199 | 134,651 |
https://mathoverflow.net/questions/309160 | 19 | In [Jeffrey Lang, *A Jacobian identity in positive characteristic*, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409](https://projecteuclid.org/euclid.jca/1450102162), the following result is proven:
>
> **Theorem 1.** Let $p$ be a prime. Let $\mathbf{k}$ be a commutative $\mathbb{F}\_p$-algebra. Let $n$... | https://mathoverflow.net/users/2530 | Lang's Jacobian identity: slicker, elementary proof? | Awesome question! I haven't looked at Lang's paper yet, so I can't comment on whether this will be a different approach, but it is elementary. I will make use of Glynn's determinant formula at some point later on, so in order to keep this self contained I will start by giving a combinatorial proof of it.
**Lemma 1:**... | 14 | https://mathoverflow.net/users/2384 | 309203 | 134,652 |
https://mathoverflow.net/questions/309204 | 10 | In algebraic geometry, one has the notion of the spectrum of a commutative ring. These spectra serve as local charts for schemes.
In algebraic topology, a spectrum is a sequence of pointed spaces $X\_n$ $(n \in \mathbb{N})$ together with the structure maps $S^1 \wedge X\_n\rightarrow X\_{n+1}$. One can associate a g... | https://mathoverflow.net/users/nan | Etymology of 'spectrum' in algebraic geometry and algebraic topology | No, they are not etymologically related. The early development of stable homotopy theory happened simultaneously with the early developments of scheme theory, so certainly neither terminology was influenced by the other.
Grothendieck's choice of terminology of the "spectrum" of a ring comes from functional analysis. ... | 13 | https://mathoverflow.net/users/1310 | 309210 | 134,653 |
https://mathoverflow.net/questions/309170 | 4 | **In short:** The question is how to go from the first equation on page 8, [of this paper](https://pdfs.semanticscholar.org/748d/8a4af0ee6001a25395bac5b7077fc6ac5041.pdf) to the second equation.
**Some background**
I'm working in optimization and I am currently reading a paper
[see page 8,Proof of Proposition 2.3.]... | https://mathoverflow.net/users/nan | Exponential map/ Lie derivative in variation for constant formula for ODE | I think the key is that the variation of constants formula holds for any vector field $G$, i.e. the vector $G\_{\varphi\_H^t(v)}$ can be written as
$$G\_{\varphi\_H^t(v)}=\big(e^{tD\_H}G\big)\_v=\left(e^{tD\_T}G+\int\_0^te^{(t-s)D\_H}D\_Ve^{sD\_T}G\,ds \right)\_v. \tag1$$
Indeed, use the fact from page 7 that
$$\frac{d... | 1 | https://mathoverflow.net/users/69603 | 309213 | 134,654 |
https://mathoverflow.net/questions/309226 | 3 | Let $I\subset\mathbb{R}$ denote an open and bounded interval of the real line, $H\_0^1(I)$ all quadratic integrable Sobolev functions and $C(\bar{I})$ all continuous functions on said interval.
Since the embedding $H\_0^1(I)\hookrightarrow C(\bar{I})$ holds, we know that the delta distribution (point evaluation) is ... | https://mathoverflow.net/users/64457 | Measurability of specific function | It certainly is measurable. In fact, you may find an explicit formula for it.
If we take $I = (0,1)$, then $g\_s$ is simply given by $g\_s(t) = \operatorname{min}(s,t) - st$.
How did I find this? Well, at first you might think of trying to choose $g\_s$ so that $g\_s'(t) = 1\_{[0,s]}(t)$; then you'd have $\int\_0^1... | 2 | https://mathoverflow.net/users/4832 | 309237 | 134,660 |
https://mathoverflow.net/questions/309241 | 7 | In this [Berkovich](http://www.wisdom.weizmann.ac.il/~vova/Inven_1996_125_formalII.pdf)'s paper, the following kind of algebra is studied:
$$
A=A\_{m,n}=k^\circ \langle T\_1,\dots,T\_m \rangle [[S\_1,\dots,S\_n]]
$$
where $k$ is some non-archimedean field with non-trivial valuation and $k^\circ$ is the associated ring.... | https://mathoverflow.net/users/69190 | Formal power series in Berkovich geometry | In Berkovich's paper that you cite, he constructs a $k$-analytic space $\mathfrak{X}\_{\eta}$ associated to what he calls a ``special formal $k^{\circ}$-scheme'' $\mathfrak{X}$. Such a formal $k^{\circ}$-scheme is defined to be locally of the form
\begin{equation}
k^{\circ}\langle T\_1,\ldots,T\_n\rangle[[S\_1,\ldots,... | 1 | https://mathoverflow.net/users/47692 | 309262 | 134,665 |
https://mathoverflow.net/questions/309265 | 16 | I've studied some fundation of algebraic geometry, such as Hartshorne's "Algebraic Geometry", Liu's "Algebraic Geometry and Arithmetic Curves", Silverman's "The Arithmetic of Elliptic Curves", and some chapters of Mumford's "Abelian Varieties".
I would like to learn more advanced arithmetic, and I began reading Falting... | https://mathoverflow.net/users/128235 | Good introductory references on moduli (stacks), for arithmetic objects | If you want to learn about stacks, I can recommend 'Fundamental Algebraic Geometry: Grothendieck's FGA Explained'. Vistoli's exposition of the basic theory of stacks is hard to beat, I think. Moreover, the chapter about Picard schemes is also good if you want to learn when a functor is representable and what you might ... | 8 | https://mathoverflow.net/users/2039 | 309280 | 134,673 |
https://mathoverflow.net/questions/309002 | 19 | The journal *Research in the Mathematical Sciences* was founded in 2014 and originally published by SpringerOpen, a division of Springer supporting Open Access journals. In the [first article of the introductory issue](https://link.springer.com/article/10.1186/2197-9847-1-1) Ken Ono emphasizes the journal's commitment ... | https://mathoverflow.net/users/25028 | Why did _Research in the Mathematical Sciences_ change from open access to subscription-based? | this thread was forwarded to me. Together with Springer, the RMS and RNT Editors decided to abandon open access as very few authors have federal funding that pays publication charges. The decision was not based on a low number of submissions (note. the oa goal was to publish just 25-30 papers annually). The decision wa... | 26 | https://mathoverflow.net/users/128273 | 309297 | 134,677 |
https://mathoverflow.net/questions/309286 | 4 | Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}\_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have isomorphisms $\varphi\_0:\Bbbk\to\omega(\mathbb{I})$ and $\varphi=\left(\varphi\_{X,X}:\omega(X)\otimes \omega(Y)\to\o... | https://mathoverflow.net/users/105816 | Tannaka-Krein reconstruction and rigidity | It seems that asking helps in enlightening. Let $\mathcal{C}$ be an abelian monoidal category with exact tensor product and let $f:X\rightarrow Y$ be a morphism between (right) rigid objects in $\mathcal{C}$. We want to prove that $\ker \left( f\right) ^{{\star }}=\mathrm{coker}\left( f^{{\star }}\right) $ and that $\m... | 0 | https://mathoverflow.net/users/105816 | 309300 | 134,678 |
https://mathoverflow.net/questions/309260 | 25 | In the monograph *Equivariant Stable Homotopy Theory*, Lewis, May, and Steinberger cite a monograph "The homotopical foundations of algebraic topology" by Peter May, as "in preparation." It's their [107].
In his paper "When is the Natural Map $X \rightarrow \Omega\Sigma X$ a Cofibration?" Lewis also cited this monogr... | https://mathoverflow.net/users/11540 | Did Peter May's "The homotopical foundations of algebraic topology" ever appear? | An anonymous source told me this question is here. Dylan gave the quick answer and Tyler referred to it.
I'll use the question as an excuse to give a pontificating longer answer. When I first planned on writing that, maybe 45 or 50 years ago, I had not yet been converted to model category theory, let alone anything m... | 32 | https://mathoverflow.net/users/14447 | 309310 | 134,683 |
https://mathoverflow.net/questions/308694 | 2 | Fix integers $l\ge 1$ and $n \ge 3$, and let $P\_n$ denote the boundary of the regular $n$-sided polygon in the plane. We define a $(2l+1)$-*pointed equilateral star* to be a cyclically ordered list of points $\{v\_0,v\_1,\dots v\_{2l}, v\_{2l+1}=v\_0\} \subseteq P\_n$ such that the adjacent distances $\|v\_i-v\_{i+1}\... | https://mathoverflow.net/users/127904 | Monotonicity for the side lengths of stars inscribed in regular polygons | Instead of considering the *closed* stars, it is convenient to consider inscribed broken lines with constant leg lengths.
**Lemma.** Let $AOB$ be an angle, let the points $X$ and $Y$ move along $AO$ and $OB$ monotonically, so that $X$ moves with the constant speed, and $XY$ is constant. Then the coordinate of $Y$ cha... | 1 | https://mathoverflow.net/users/17581 | 309314 | 134,685 |
https://mathoverflow.net/questions/309317 | 9 | Let $(M,g)$ be a Riemannian manifold. The $LC$ connection associated to the metric gives an $n$ dimensional distribution $D$ for $TM$. Let $\omega$ be the symplectic structure of $TM$ which is obtained by pulling back of the standard structure of the cotangent bundle via the isomorphism between the tangent and cotangen... | https://mathoverflow.net/users/36688 | Does every manifold admit a Lagrangian Riemannian metric? | The distribution of horizontal subspaces is always Lagrangian not only for Riemannian metrics, but also for the Ehresmann connection associated to Finsler metrics and for the Levi-Civita connection of pseudo-Riemannian metrics. It is not always Lagrangian for general sprays though.
In the Riemannian case this is cla... | 12 | https://mathoverflow.net/users/21123 | 309324 | 134,690 |
https://mathoverflow.net/questions/308979 | 2 | I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite valued. There are a number of such results, perhaps most famously the Granas–Górniewicz–Kryszewski (G-G-K) theorem, when ... | https://mathoverflow.net/users/65956 | Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps? | No. Consider the very simple upper hemi-continuous correspondence $\phi$ from $[0,1]$ to $[0,1]$ such that
$$
\phi(x)=
\begin{cases}
\{0\} \text{ if }x<1/2\\
\{0,1\} \text{ if }x=1/2\\
\{1\} \text{ if }x>1/2.
\end{cases}
$$
Clearly, $[0,1]$ is an ANR. The correspondence $\phi$ looks almost like a discontinuous funct... | 1 | https://mathoverflow.net/users/35357 | 309327 | 134,691 |
https://mathoverflow.net/questions/309284 | 1 | $\newcommand{\M}{M}$
This is a [cross-post](https://math.stackexchange.com/questions/2885583/elliptic-regularity-of-harmonic-forms-in-l1). I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
Let $\sigma$ be a di... | https://mathoverflow.net/users/46290 | Elliptic regularity of harmonic forms in $L^1$ | Let me turn my comment into a general discussion for which a special case is an answer to your question.
First, on an open domain $D \subset \mathbb{R}^n$, there is a standard elliptic regularity result that says if $u$ is a distribution on $D$ satisfying weakly $$ a^{ij}\partial^2\_{ij}u + b^k\partial\_ku + cu= f, $... | 3 | https://mathoverflow.net/users/613 | 309335 | 134,694 |
https://mathoverflow.net/questions/309328 | 6 | Good morning,
I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected subgroup of $GL(\mathbb{R},d)$ generated by $\mathfrak{a}$.
Assume that for every $x\in\mathbb{R}^d\setminus \{0\}$, the or... | https://mathoverflow.net/users/nan | Existence of a real eigenvalue is a necessary condition for the density of all the orbits of a Lie subgroup of $GL(\mathbb{R},d)$ | It's true.
Since $A$ acts irreducibly, so does $\mathfrak{a}$, so the latter is reductive. Write $\mathfrak{a}=\mathfrak{z}\oplus\mathfrak{s}$, with $\mathfrak{z}$ its center and $\mathfrak{s}=[\mathfrak{a},\mathfrak{a}]$ being semisimple.
Assume that $A$ has no element with a nonzero real eigenvalue. This implies ... | 1 | https://mathoverflow.net/users/14094 | 309336 | 134,695 |
https://mathoverflow.net/questions/309340 | 10 | This is probably easy but it might be interesting. Here goes $\dots$
Let $P\in\mathbb{R}[x]$ be a polynomial of degree $n>2$ and $P'=\frac{dP}{dx}$. If $x\_1, x\_2, \dots, x\_n$ are the roots of $P(x)$, including multiplicities, consider the multi-variable expression
$$V\_n(P)=\sum\_{1\leq i<j\leq n}(x\_i-x\_j)^2.... | https://mathoverflow.net/users/66131 | Roots and relation between polynomials and their derivatives | Suppose that we have
$$P(x)=x^n-ax^{n-1}+bx^{n-2}+\cdots$$
where we can take $P$ to be monic since it doesn't affect $V\_n(P)$. From Vieta's formula we have
$$a=\sum\_{i=1}^n x\_i \quad , \quad b=\sum\_{1\le i<j\le n} x\_ix\_j$$
so we can find that $V\_{n}(P)=(n-1)a^2-2nb$. Similarly we have
$$V\_{n-1}(P')=(n-2)\frac... | 22 | https://mathoverflow.net/users/2384 | 309341 | 134,696 |
https://mathoverflow.net/questions/307490 | 1 | It is well-known that a continuous map $f:M\to\mathbb{R}^n$ from a Hilbert manifold can be closely approximated by a smooth map $g:M\to\mathbb{R}^n$ which has no critical points.
>
> But, can such a continuous map $f$ also be closely approximated by a map $h:M\to\mathbb{R}^n$ which has infinitely many critical poi... | https://mathoverflow.net/users/98896 | Can a continuous map on a Hilbert manifold be approximated by a map which has infinitely many critical points? | As suggested by Pietro, a continuous map $f:X\to\mathbb{R}^n$ on a Hilbert manifold $X$ may be approximated to have infinitely many points. That is, a a $C^0$-perturbation creates infinitely many local minima and maxima. However, this is, in general, not true for a $C^1$-perturbation. In particular, in local charts, su... | 0 | https://mathoverflow.net/users/98896 | 309343 | 134,697 |
https://mathoverflow.net/questions/309307 | 0 | Let $\mathfrak g$ be a Lie superalgebra.
If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same?
For me as sets they are linear span of $[a,x]$ and $[x,a]$ and hence they are same. But in book it is given they are differen... | https://mathoverflow.net/users/33047 | left ideals in Lie super algebras | The linear spans of $[a,x]$ and $[x,a]$, in a Lie superalgebra (i.e. a $\mathbb{Z}\_2$-graded Lie algebra) are generally not the same (unlike the Lie algebras case):
Since $\mathfrak a$ is not a graded subspace of $\mathfrak g$, then in general its elements are not homogeneous. So for $a\in \mathfrak a$ we generally ... | 3 | https://mathoverflow.net/users/85967 | 309344 | 134,698 |
https://mathoverflow.net/questions/242569 | 8 | There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper [here](http://arxiv.org/pdf/0907.0061v1.pdf)) and one for quasicategories (see the [Unstraightening and Straightening correspondence](https://ncatlab.org/nlab/show/(infinity,1)-Grothendieck+con... | https://mathoverflow.net/users/11546 | Interaction of Grothendieck Construction with Coherent Nerve | Answering this question took me some time. First of all, Liang Ze Wong and I had to write down a version of the enriched Grothendieck construction that worked for these purposes, as Tamaki's construction, which I linked to above, didn't quite work. That paper can be found [here](https://arxiv.org/abs/1804.03829).
Nex... | 4 | https://mathoverflow.net/users/11546 | 309351 | 134,700 |
https://mathoverflow.net/questions/265575 | 8 | $\newcommand{\M}{\mathcal{M}}$
Suppose I have a monoidal simplicial model category in which every object is cofibrant $(\M,\otimes,\mathbb{1})$ and I want to look at its underlying monoidal quasicategory, which I'll write as $N(\M)$, the simplicial nerve of $\M$. One way to do this is the following construction (foll... | https://mathoverflow.net/users/11546 | Lifting Strict Comonoids and Comodules to Quasicategories | The answer to this question is yes, and it's the main result of [this paper.](https://arxiv.org/pdf/1808.08020.pdf)
One thing to point out is that, even in the case that the tensor product of $\mathcal{M}$ preserves fibrant objects (so that the homotopy types of the mapping objects in the multicategory associated to ... | 4 | https://mathoverflow.net/users/11546 | 309352 | 134,701 |
https://mathoverflow.net/questions/309354 | 5 | An associative algebra $A$ is said to be Morita equivalent to another one $B$ if there is an equivalence $$\mathsf{Mod}\_A\simeq \mathsf{Mod}\_B$$ between its corresponding abelian categories of modules. Moreover, whenever $A$ and $B$ are commutative, they are Morita equivalent iff they are isomorphic. On the other han... | https://mathoverflow.net/users/nan | Derived Morita equivalence of associative algebras | This seems to be true by <https://arxiv.org/pdf/math/9810134.pdf> , theorem 2.7.
| 5 | https://mathoverflow.net/users/61949 | 309355 | 134,702 |
https://mathoverflow.net/questions/309320 | 9 | Let $E$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $R = \operatorname{End}(E)$ be its ring of endomorphisms. Then, it is known $R \otimes\_{\mathbb Z} \mathbb Q$ is an order in a quaternion algebra $D$. In particular, $D$ has a multiplicative norm function $N\col... | https://mathoverflow.net/users/36401 | Existence of certain endomorphism of supersingular elliptic curve | $\newcommand{\Z}{\mathbb{Z}}$
First, if you want to learn about quaternion algebras, their orders and their relation to supersingular elliptic curves, I suggest John Voight's [book](https://math.dartmouth.edu/~jvoight/quat.html).
The answer to your question is yes. To see this, you can use the fact that (because $R$ ... | 3 | https://mathoverflow.net/users/40821 | 309357 | 134,703 |
https://mathoverflow.net/questions/309338 | 6 | Suppose $(X,\omega,J)$ is a compact Kähler manifold, and $\beta\in H\_2(X,\mathbb Z)$ is given. Then, we can form the space $\overline{\mathcal M}:=\overline{\mathcal M}\_{0,0}(X,\beta)$ of stable maps $u:C\to X$ with $C$ a nodal curve of genus 0 and $u\_\*[C]=\beta$. Let $\mathcal M\subseteq\overline{\mathcal M}$ be t... | https://mathoverflow.net/users/110236 | Complex Analytic Structure on Moduli Space of Stable Maps | There's also a really beautiful approach by Salamon-Robbin-Ruan for integrable Js in their paper "The moduli space of regular stable maps"
<https://people.math.ethz.ch/~salamon/PREPRINTS/smUW.pdf>
building on the earlier paper by Salamon-Robbin "Construction of the Deligne-Mumford orbifold":
<https://people.math.... | 3 | https://mathoverflow.net/users/10839 | 309373 | 134,709 |
https://mathoverflow.net/questions/309358 | 6 | (I am a complete amateur in topology, so this is a question out of curiosity.)
The question was inspired by this post [Fake versus Exotic](https://mathoverflow.net/questions/108631/fake-versus-exotic) . What methods can, realistically, be used to construct a homeomorphism between, for example, $\mathbb{C}\mathbb{P}^... | https://mathoverflow.net/users/9833 | Tools for constructing homeomorphisms between 4-manifolds | Sadly, it seems that you need pretty much the full strength of Freedman's disc embedding theorem to construct such homeomorphisms. Wall's theorem builds an h-cobordism, essentially starting with the stabilization you mention, and then regluing by a diffeomorphism to get handles to algebraically cancel. But to get them ... | 5 | https://mathoverflow.net/users/3460 | 309378 | 134,711 |
https://mathoverflow.net/questions/308834 | 10 | This question was also asked [here](https://math.stackexchange.com/questions/2887659/i-really-dont-know-what-else-i-can-do-to-solve-this-integral) and [here](https://mathematica.stackexchange.com/questions/180192/why-cant-mathematica-compute-this-integral-int-0-pi-2-frac1-a-cos2x).
I have faced some difficulties to d... | https://mathoverflow.net/users/108503 | Difficult trigonometric integral | Here is an outline of the approach I have taken to solve this integral.
First rewrite the integral $(1)$ in Cartesian variables:
$$I=\int\_{-\infty}^{\infty} \mathrm{d}^3v~ \frac{3x^2y^2v\_z^2}{y^2v\_x^2+x^2v\_y^2+x^2y^2v\_z^2}\cos(uv\_x)\exp\left(-\frac{v\_x^2}{2}-\frac{v\_y^2}{2}-\frac{v\_z^2}{2}\right). $$
No... | 7 | https://mathoverflow.net/users/108503 | 309406 | 134,719 |
https://mathoverflow.net/questions/309403 | 4 | Let $G$ be a locally compact totally disconnected group and let $\phi$ be a surjective homomorphism from $G\to H$ (added later: where $H$ has the topology coinduced by $\phi$). Is H also locally compact and totally disconnected? If not is there a homomorphism from such a group to a Lie group?
More generally what hap... | https://mathoverflow.net/users/123459 | Is the image of a locally compact totally disconnected group also locally compact and totally disconnected? | The quotient group $H$ will be totally disconnected and locally compact if it is Hausdorff (or at least [$T\_1$](https://en.wikipedia.org/wiki/T1_space)) and $H$ inherits the quotient topology.
Let $\phi: G\to H$ be a surjective homomorphism of Hausdorff topological groups, $G$ locally compact and totally disconnect... | 8 | https://mathoverflow.net/users/1345 | 309412 | 134,721 |
https://mathoverflow.net/questions/309411 | 10 | Consider the following *somos-like* sequence
$$x\_n=\frac{x\_{n-1}^2+x\_{n-2}^2}{x\_{n-3}}.$$
It's known that $x\_n$ is a Laurent polynomial in $x\_0, x\_1$ and $x\_2$. I got interested in the denominators of the sequence $x\_n$. Some initial observations indicate particular structures regarding the exponents of the de... | https://mathoverflow.net/users/66131 | Denominators of certain Laurent polynomials | Yes, this is true. These are the *denominator vectors* or *$d$-vectors* of the cluster algebra associated to the Markov quiver. The Markov quiver has vertices $\{1,2,3\}$ to two arrows $i \to i+1$ for each $i$ taken modulo $3$ (i.e. a directed $3$-cycle with all double arrows). This quiver has the property that wheneve... | 11 | https://mathoverflow.net/users/51668 | 309414 | 134,722 |
https://mathoverflow.net/questions/309371 | 7 | Let $n,k,\ell$ be integers for which $0\leq k<\ell \leq n-6$. For a fixed $n$, think of $k,\ell$ as being allowed to vary. I believe the values
$$(n-k-5)(k+1)(k+2)\binom n{k+3}~~~\text{and}~~~(n-\ell-5)(\ell+1)(\ell+2)\binom n{\ell+3}$$
are not equal. A proof they are not equal is the goal, but insight as to why th... | https://mathoverflow.net/users/128140 | Conjectured combinatorial non-equality | It looks like we may simply say which of them is greater.
Denoting $k+3=t$ and $f(t)=(n-k-5)(k+1)(k+2)\binom n{k+3}=(n-t-2)(t-1)(t-2)\binom n{t}$ we get $$\frac{f(t+1)}{f(t)}=
\frac{t (n - t - 3) (n - t)}{(t - 2) (t + 1) (n - t - 2)}=\frac{n-t-1-\frac{2}{n-t-2}}{t-1-\frac2{t}}.
$$
If $t<n/2$, this is greater than 1, ... | 9 | https://mathoverflow.net/users/4312 | 309420 | 134,723 |
https://mathoverflow.net/questions/247399 | 30 | Let me first explain the statement of the question and then give some indication why the answer might be 'yes'.
By a space I mean, say, a simplicial set and by rational I mean rational in the sense of Bousfield, i.e. local with respect to the homology theory $H\mathbb{Q}$. If we denote the category of spaces by $\mathc... | https://mathoverflow.net/users/97202 | Is a filtered colimit of rational spaces again rational? | The answer to this, as a result of some discussions with Thomas, turns out to be no.
Consider the abelian groups $A\_n = \Bbb Q[x]/(x^n)$. Multiplication by $x$ embeds $A\_n$ into $A\_{n+1}$, and the direct limit of the sequence
$$
A\_0 \to A\_1 \to \dots
$$
is the group $A\_{\infty} \cong \Bbb Q[x^{\pm 1}] / \Bbb Q[... | 10 | https://mathoverflow.net/users/360 | 309423 | 134,724 |
https://mathoverflow.net/questions/309421 | 6 | Let $C$ be a (smooth, projective) curve over a finite field $\mathbb{F}\_q$, and let $J\_C(\mathbb{F}\_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.
Question 1: Are there curves $C$ for which $J\_C(\mathbb{F}\_q)$ is isomorphic, as a group, to $(\mathbb{Z}/2\mathbb{Z})^k$ for some $k$? Can on... | https://mathoverflow.net/users/31469 | 2-Torsion in Jacobians of Curves Over Finite Fields | I think $y^2=x^9-x$ over $\mathbb{F}\_3$ has $J\_C(\mathbb{F}\_3)$ isomorphic to $(\mathbb{Z}/2)^6$ but please check.
The $2$-torsion in $J\_C$ over the algebraic closure is $(\mathbb{Z}/2)^{2g}$ (or smaller in characteristic two). On the other hand, $\#J\_C(\mathbb{F}\_q) \ge (\sqrt{q} -1)^{2g}$, so for $q > 9$, the... | 11 | https://mathoverflow.net/users/2290 | 309426 | 134,726 |
https://mathoverflow.net/questions/309332 | 4 | I have come across the following easy-looking problem that is driving me mad.
I have a family of measures (on the real line $\mathbb R$) $\{\mu\_t\}\_{t>0}$ which is uniformly bounded (the measures being possibly signed). I know that $\mu\_0 = 0$ and that for every $\varphi \in C\_c^\infty([0,+\infty) \times \mathbb... | https://mathoverflow.net/users/111164 | Method of characteristics beyond the Lipschitz setting | Let us show that the condition ${\mu\_0}|\_{x>0} = 0$ implies that ${\mu\_T}|\_{x>0} = 0$ (for a.e. $T>0$). Let $\omega \in C\_0^\infty(\mathbb R)$ and $\delta\in (0,1)$ be such that $\omega(\xi) = 0$ for all $\xi\le \delta$. Define
$$
\varphi(t,x):=
\begin{cases}
0, & \frac{3}{2} x^{\frac{2}{3}} + t - T \le \delta\\
... | 6 | https://mathoverflow.net/users/44463 | 309427 | 134,727 |
https://mathoverflow.net/questions/308836 | 0 | Now we are writing a paper on minimal covers and minimal vertex-covers in hypergraphs and would like to know if there are any standard names for the following two (dual) properties of a hypergraph $(V,E)$, depending on an integer parameter $n\in\mathbb N$:
>
> **Property $1\_n$**: For any $n$-element family of edge... | https://mathoverflow.net/users/61536 | Standard names of two finitary properties of hypergraphs? | The problems you are discussing in your paper could just as well (or better) be stated in terms of a bipartite graph, *i.e.,* the vertex-edge incidence graph of your hypergraph. In terms of the bipartite graph, the properties might be called $K\_{\aleph\_0,n}$-free and $K\_{n,\aleph\_0}$-free, or $K\_{\omega,n}$-free a... | 2 | https://mathoverflow.net/users/43266 | 309429 | 134,728 |
https://mathoverflow.net/questions/309436 | 2 | I saw that "Over an algebraically closed field of characteristic 0, semisimple representations are isomorphic if and only if they have the same character" in the [Wikipedia page](https://en.wikipedia.org/wiki/Character_theory#Properties) , which does not mention the condition that the group is finite.
However, I can on... | https://mathoverflow.net/users/122681 | Character theory of representations of infinite groups | Yes, it is true, and one doesn't even need to assume the field $k$ is algebraically closed. Section 7 of Lam's "A First Course in Noncommutative Rings" is a good reference for character theory for $k$-algebras. In fact, Theorem 7.19 says exactly what you want: If $M$ and $M'$ are finite-dimensional semisimple represent... | 6 | https://mathoverflow.net/users/11791 | 309456 | 134,733 |
https://mathoverflow.net/questions/309446 | 4 | It was mentioned after Theorem 30.27 in Kanamori's Higher Infinite that Woodin constructed a model of $DC$ + there exists unboundedly many many $\kappa<\Theta$ such that $\kappa \to (\kappa)^\kappa\_{\alpha} \ \forall \alpha<\kappa$ and there exists a non-principal ultrafilter on $\omega$. In particular, this model is ... | https://mathoverflow.net/users/23835 | Strong partition property + DC + existence of non-principal ultrafilter on $\omega$ | The key reference for this is
>
> [MR0799042 (87d:03141)](https://mathscinet.ams.org/mathscinet-getitem?mr=799042). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. *[A barren extension](https://link.springer.com/chapter/10.1007%2FBFb0075312)*. In **Methods in mathematical logic (Caracas, 1983)**, C. A. Di Prisco,... | 7 | https://mathoverflow.net/users/6085 | 309457 | 134,734 |
https://mathoverflow.net/questions/309417 | 2 | I am not a hyperbolic geometer, so I apologize if I get anything wrong here, and please correct me.
The conformal compactification $\overline {\mathbb{H}^n}$ of hyperbolic $n$-space $\mathbb{H}^n$ can be obtained by viewing hyperbolic space as a subspace of projective space, since the boundary, which is the projecti... | https://mathoverflow.net/users/56938 | Is the conformal compactification of a convex-cocompact hyperbolic manifold $M$ conformally diffeomorphic to the convex core of $M$? | No, this follows from [Liouville's theorem](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)). A conformal diffeomorphism $\phi: \overline{M}\to K$ would have to be a restriction of an $n$-dimensional Möbius transformation in any chart. Lifting to the universal cover, we would get a conformal ma... | 4 | https://mathoverflow.net/users/1345 | 309461 | 134,736 |
https://mathoverflow.net/questions/309450 | 6 | We know iteration ${\mathbf X}\_k=\mathbf{A}{\mathbf X}\_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see [here](https://math.stackexchange.com/questions/126460/iteration-convergence)). Is there any known rule for iteration ${\mathbf X}\_k={\mathbf A}{\mathbf X}\_{k-1}{\mathbf B}$ to conver... | https://mathoverflow.net/users/70424 | Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$? | For every square matrix $C$, let $r(C)$ denote its spectral value. We say that a complex number $\lambda$ is
* a dominant eigenvalue of $C$ if $\lambda$ is the only eigenvalue of $C$ with modulus $r(C)$.
* a semisimple eigenvalue of $C$ if it is an eigenvalue of $C$ and its algebraic multiplicity coincides with its g... | 4 | https://mathoverflow.net/users/102946 | 309462 | 134,737 |
https://mathoverflow.net/questions/304130 | 1 | Review the main result of [mathoverflow.net/questions/297900](https://mathoverflow.net/questions/297900/coefficients-in-the-sum-sum-k-0n-1-sum-j-0ma-j-mn-kjkj-n2m1), that is the identity
\begin{equation}\label{f1}
n^{2m+1}=\sum\limits\_{1\leq k \leq n}\sum\limits\_{j\geq0}A\_{m,j}k^j(n-k)^j,
\end{equation}
where $A\_{m... | https://mathoverflow.net/users/113033 | Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\sum\limits_{0\leq k \leq m}(-1)^{m-k}U_m(n,k)\cdot n^k$ | First off, as I explained in the comments, the identity (1.3) should contain $U\_m(T,k)$ rather than $U\_m(n,k)$ (the latter does not make any sense), and so the correct identity (1.3) (for polynomials in $n$) states:
$$(1.3)\quad\sum\_{k=1}^T\sum\_{j=0}^m A\_{m,j}k^j(n-k)^j\equiv \sum\limits\_{0\leq k \leq m}(-1)^{m-k... | 3 | https://mathoverflow.net/users/7076 | 309470 | 134,738 |
https://mathoverflow.net/questions/309458 | 5 | **Definition.** A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be *soft* if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a homeomorphism $h$ of $c\mathbb N$ such that $h(A)\cap B$ is infinite and $h(x)=x$ for al... | https://mathoverflow.net/users/61536 | Is each compactification of $\mathbb N$ soft? | Let $A=\{0,2,4,\dots\}$ be the even numbers and let $B=\{1,3,5,\dots\}$ be the odd numbers. Topologize $A\cup \beta B$ so that $A$ is a sequence limiting to a unique point in $\beta B \setminus B $. This is a compactification of $\mathbb{N}$ that fails to be soft, since any homomorphism of the required form would give ... | 10 | https://mathoverflow.net/users/83901 | 309473 | 134,740 |
https://mathoverflow.net/questions/309467 | 9 | For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\setminus\mathbb N$. The group $\mathcal H(c\mathbb N,\mathbb N)$ determines the subgroup $$S\_{\mathbb N,c\mathbb N}:=\{h{\res... | https://mathoverflow.net/users/61536 | Is $\beta\mathbb N$ a unique compactification with the smallest possible permutation group? | Analyzing [the answer](https://mathoverflow.net/questions/309458/is-each-compactification-of-mathbb-n-soft/309473#309473) of @James Hanson to [my preceding question](https://mathoverflow.net/questions/309458/is-each-compactification-of-mathbb-n-soft), I realized that this question also has a simple negative answer: the... | 4 | https://mathoverflow.net/users/61536 | 309475 | 134,742 |
https://mathoverflow.net/questions/309454 | 14 | Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional.
**Question.** Does it follow that $A$ is finite dimensional?
**Remark.** Clearly, every element of $A$ is algebraic (i.e. annihilated by a polynomial) and thus has finite spectr... | https://mathoverflow.net/users/102946 | Criterion for a Banach algebra to be finite dimensional | I think it's true by Dixon's theorem ([JLMS 1974](https://doi.org/10.1112/jlms/s2-8.2.325)) which says (on the third page) that any Banach algebra consisting only of algebraic elements is nilpotent-by-finite. Thanks to this, we may assume $A$ is nilpotent. We moreover assume $A$ is infinite-dimensional and will constru... | 13 | https://mathoverflow.net/users/7591 | 309486 | 134,744 |
https://mathoverflow.net/questions/309433 | 9 | Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (in the case of a cubic surface the corresponding root system is $E\_6$). In particular this relation manifests itself in ... | https://mathoverflow.net/users/21620 | Del Pezzo surfaces and Picard-Lefschetz theory | Indeed you can see it this way. This is my symplectic geometer's
perspective on it (I blame Paul Seidel's [Lecture notes on four-dimensional Dehn twists](https://arxiv.org/abs/math/0309012 "Seidel Lecture notes on four-dimensional Dehn twists")).
Consider the $n$-point blow-up of $\mathbf{CP}^2$ at $n$ general
points... | 9 | https://mathoverflow.net/users/10839 | 309488 | 134,745 |
https://mathoverflow.net/questions/309445 | 3 | I am reading the seminal paper
>
> Stuart Geman and Donald Geman, *Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images*, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PAMI-6, no. 6, pp. 721-741, Nov. 1984. doi: [10.1109/TPAMI.1984.4767596](https://doi.org/10.11... | https://mathoverflow.net/users/78788 | Updating Geman and Geman (1984) on image restoration | Given that the paper has accumulated a stunning **21.850** citations at Google scholar to this date and 487 are from 2018 the work is clearly influential (zbMATH lists 913 citations and 17 from 2018, MATHSCINET does not have it, though). It is not straightforward to answer your question "Is there a more recent account ... | 3 | https://mathoverflow.net/users/9652 | 309493 | 134,748 |
https://mathoverflow.net/questions/309505 | 5 | Let $M$ be a very nice model category (cofibrantly generated, combinatorial or cellular and left proper simplicial model category). Let $f: X\rightarrow Y$ and $g: X\rightarrow Z $ be two morphisms in $M$ such that $g$ is weak equivalence. Suppose that the map $r: Z\rightarrow Y\cup\_{X} Z $ is a weak equivalence in th... | https://mathoverflow.net/users/128371 | Localization of a model category | No. Let $M$ be the category of simplicial sets with the Kan model structure. Let $S^1$ be $\Delta^1$ with its endpoints identified and let $f : \Delta^1 \to S^1$ be the obvious map. Let $g : \Delta^1 \to \Delta^0$ be the unique map. Then $r : \Delta^0 \to \Delta^0$ is the identity map, so it is a weak equivalence even ... | 6 | https://mathoverflow.net/users/62782 | 309506 | 134,751 |
https://mathoverflow.net/questions/309451 | 8 | I wonder whether such a result is known, and if so, whether the proof is trivial.
By polytope I mean the convex hull of finitely many points in $\Bbb R^n$. Assume the simplex to be symmetric and centered at the origin, so that the subspace goes through its center. The subspace can have any dimension $k\in\{0,...,n\}... | https://mathoverflow.net/users/108884 | Is every polytope combinatorially equivalent to the intersection of a simplex and a linear subspace? | The answer is yes.
The fact that any polytope is affinely equivalent to a section of a simplex is well-known (see the answer by Tobias Fritz).
Now in any simplex with vertices $(v\_i)$ we may consider projective transformations via reweighting barycentric coordinates: given positive numbers $(a\_i)$, such a transfo... | 11 | https://mathoverflow.net/users/908 | 309516 | 134,754 |
https://mathoverflow.net/questions/309517 | 1 | Sorry if this question is a bit broad. I would like to have examples of papers which have studied the surface singularity
$$x^4=yz,\quad(x,y,z\in\mathbb{C}).$$
I am trying to get a feel about what is known about it in the literature.
| https://mathoverflow.net/users/128380 | The surface singularity $x^4=yz$ | This singularity, and more generally the ones given by $x^n + yz=0$ are (well-)known as an ADE singularity. The ring $C[x,y,z] / x^n + yz$ is the ring of coordinates of the quotient of the natural $\mathbf Z/n \subset SL\_2(\mathbf C)$-action on $\mathbf C^2$. Check out some literature on the McKay-correspondence for f... | 4 | https://mathoverflow.net/users/18116 | 309518 | 134,755 |
https://mathoverflow.net/questions/309288 | 3 | Simplified question\*:
----------------------
Given $f(t)$ that satisfies $f'(t)>0$, $f'(t)=\omega\left(t^{-1}\right)$, $\log\left(f'(t)\right)=o\left(f(t)\right)$ we denote $F=\exp\left(f\left(t\right)\right)$. Let $H(t)$ be a solution of
$$ \dot{H}=F $$
Can we approximate H by F?
Specifically, I want to show that... | https://mathoverflow.net/users/128264 | Asymptotic solution for a first order ODE | The simplest cases are the linear cases where $f(t) = \alpha t$, and $H = \alpha^{-1}F$. Modelling on those cases, I will show that if asymptotically $f'(t)$ is bounded below (so if $f$ is asymptotically superlinear) than the desired conclusion hold.
---
The first step is to prove that the conclusion holds if $H... | 2 | https://mathoverflow.net/users/3948 | 309520 | 134,757 |
https://mathoverflow.net/questions/309487 | 0 | The notion of a *limiting recursive set* (Gold 1965, *J. Symb. Log.* **30**: 28–48) or *trial and error predicate* (Putnam 1965, *J. Symb. Log.* **30**: 49–57) is defined as follows. A *guessing function* is a total recursive $g: \mathbb{N} \times \mathbb{N} \rightarrow \{ 0,1 \}$.
A set $S \subseteq \mathbb{N}$ is ... | https://mathoverflow.net/users/91635 | Probabilistic generalization of trial-and-error predicates | I am not sure one should really call your family of sets "probabilistically limiting recursive" since there is no real randomness involved here, but in any case the answer to your question is yes. Suppose you have a function $g$ witnessing that $S$ is probabilistically limiting recursive. Consider the function $h$ defi... | 2 | https://mathoverflow.net/users/12126 | 309521 | 134,758 |
https://mathoverflow.net/questions/309515 | 65 | Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one passage that I remember vividly is this, for which I have little in the way of rigorous justification:
>
> The greatest ro... | https://mathoverflow.net/users/70848 | Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets? | First, there is indeed nothing mathematically very deep in this observation, and I agree that the word "breakthrough" might be exaggerated. But on the other hand lots of very deep ideas look trivial once spelled out explicitly. Moreover being younger than Voevosky I have never been really exposed to the idea that categ... | 68 | https://mathoverflow.net/users/22131 | 309524 | 134,759 |
https://mathoverflow.net/questions/309499 | 10 | Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., Haagerup U., *et al.*...
The main notion of this subject is a quantum space: $(E, \{\|\cdot\|\_n\})$ -- some normed sp... | https://mathoverflow.net/users/94631 | Quantum functional analysis | Okay, I'll take this one. First let me say that the English term is "completely bounded" (or "complete isometry", etc.).
About the term "quantum". The general principle is that analyzing some aspect of a physical system typically involves very different kinds of mathematical structures, depending on whether the syste... | 16 | https://mathoverflow.net/users/23141 | 309532 | 134,763 |
https://mathoverflow.net/questions/309496 | 3 | Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for instance projective spaces and intersections of quadrics, which admit full exceptional collections, are rational).
| https://mathoverflow.net/users/nan | A non-rational variety with a full exceptional collection? | Rationality of a variety with a full exceptional collection is a well-know folklore conjecture. In some form a similar open question is mentioned in the paper of Brown and Shipman "The McKay Correspondence, Tilting, and Rationality".
| 7 | https://mathoverflow.net/users/4428 | 309534 | 134,764 |
https://mathoverflow.net/questions/309540 | 5 | The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\infty)$ and discrete spectrum below zero.
It is known that the essential spectrum is preserved under relative compact ... | https://mathoverflow.net/users/128387 | Schrödinger operator with Coulomb potential | This has to be shown separately. There are potentials with this decay $V(x)=O(|x|^{-1})$ that have embedded (in the ac spectrum) eigenvalues. The most famous of these is the *von Neumann-Wigner potential* (search for it for more information).
This potential will be oscillating. The fact that for the Coulomb potential... | 5 | https://mathoverflow.net/users/48839 | 309541 | 134,767 |
https://mathoverflow.net/questions/309538 | 0 | Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map\_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. Suppose that for any fibrant object $R$, the induced map
$map\_{M}(B,R)\rightarrow map\_{M}(A,R)$ is a weak homotopy equiva... | https://mathoverflow.net/users/128371 | detecting weak equivalences in a simplicial model category | Yes. This is Proposition 9.7.1 in Hirschhorn's book. You don't even need $A$ to be cofibrant.
| 3 | https://mathoverflow.net/users/11540 | 309546 | 134,769 |
https://mathoverflow.net/questions/309549 | 2 | In the case that I'm working with a separable Hilbert space, $H$, on which I have a trace class operator, $K$, that's coming from a Gaussian (i.e., $K$ is self-adjoint, and for simplicity, has trivial kernel), how can I see the following two properties:
1. The Cameron-Martin space, defined as $K^{1/2}(H)$ in this cas... | https://mathoverflow.net/users/51335 | Properties of Cameron Martin Space | 1) To see that $K^{1/2}(H)$ is dense in $H$: if not, there is some nonzero $v$ orthogonal to it. But since $K^{1/2}$ is self-adjoint, that says $0 = (K^{1/2})^\* v = K^{1/2} v$, and then $K v = K^{1/2} K^{1/2} v = 0$, violating your assumption that the kernel is trivial.
2) Since the embedding $K^{1/2}$ is compact, i... | 1 | https://mathoverflow.net/users/13650 | 309551 | 134,771 |
https://mathoverflow.net/questions/309469 | 7 | Let $\kappa$ be the smallest cardinality of a family $\mathcal F$ of subsets of $\omega$ such that for any bijective function $f:A\to B$ between disjoint infinite subsets of $\omega$ there exists a set $F\in\mathcal F$ such that the set $\{x\in A\cap F:f(x)\notin F\}$ is infinite.
It can be shown that $\mathfrak s\le... | https://mathoverflow.net/users/61536 | A new cardinal characteristic of the continuum? | Recall that $\mathbf{non}(\mathcal{B})$ is the least cardinality of a non-meager subset of $\mathbb{R}$; the choice of presentation of $\mathbb{R}$ does not matter for this, so we take $\mathbb{R} = \mathcal{P}(\omega)$. It is well-known that consistently, $\mathbf{non}(\mathcal{B}) < \mathfrak{c}$; indeed, start from ... | 8 | https://mathoverflow.net/users/26705 | 309552 | 134,772 |
https://mathoverflow.net/questions/309555 | 4 | Counting edges easily shows that if $n$ is congruent to 2 or 3 modulo 4, there is no self-complementary graph on $n$ vertices. Is the converse true?
What I know: Paley graphs are self-complementary, so if $n$ is congruent to 1 mod 4 and is a prime power, then there is a self-complementary graph on $n$ vertices. Also,... | https://mathoverflow.net/users/39174 | How many vertices can a self-complementary graph have? | Note that if $G$ be a self-complementary graph with $n$ vertices, then the following gives a self-complementary graph $H$ on $n+4$ vertices:
Let $H$ be the graph obtained by adding 4 new vertices $\{a,b,c,d\}$ to $G$, with edges $(a,b),(b,c),(c,d), (a,x), (d, x)$ for any vertex $x$ of $G$.
Why $H$ is self-complementa... | 7 | https://mathoverflow.net/users/49822 | 309559 | 134,775 |
https://mathoverflow.net/questions/309567 | 1 | There is [a simple algorithm to pick a random point ON an $n$-dimensional hypersphere](https://mathoverflow.net/questions/136314/what-is-a-good-method-to-find-random-points-on-the-n-sphere-when-n-is-large).
Is there one to sample a point from inside it? (Sampling points from a hypercube and rejecting them if they are... | https://mathoverflow.net/users/30352 | Sampling a uniformly distributed point INSIDE a hypersphere? | Choose a uniform point $X$ on the unit hypersphere, then multiply it by $U^{1/n}$ where $U \sim U(0,1)$ is independent of $X$.
| 6 | https://mathoverflow.net/users/4832 | 309568 | 134,777 |
https://mathoverflow.net/questions/309562 | 0 |
>
> Question: Can we have a model of $ZF-\text {Regularity}$ where there exist an ordinal $\kappa$ such that $H\_{\kappa}$ exists and $H\_{\kappa}$ is not equinumerous to any well founded set?
>
>
>
The motivation for this question comes in connection with defining Cardinality under some situations beyond Regula... | https://mathoverflow.net/users/95347 | Can cardinality be defined with essentially no practical restriction on non-well-ordered combinatorics or ill-foundedness of sets? | It depends on what "well-founded set" means.
The most natural interpretation, in my opinion, is: $x$ is well-founded iff the transitive closure of $x$ is well-founded with respect to $\in$. Note that this means exactly that $x$ is a pure set! If this is what we mean, the answer to your question is **yes**: we can hav... | 4 | https://mathoverflow.net/users/8133 | 309571 | 134,778 |
https://mathoverflow.net/questions/309558 | 6 | Let $G\_1, G\_2$ be two lie groups, $V$ be a finite dimensional (continuous) irreducible complex representation of $G\_1 \times G\_2$, must $V \cong V\_1 \otimes V\_2$ for some irreducible representation $V\_i$ of $G\_i$?
If $G\_i$ are compact, this is true by Peter-Weyl theorem.
| https://mathoverflow.net/users/102104 | Irreducible representation of the product of two groups and tensor product | If the field is $\mathbb C$, There are many ways of seeing this. For $i=1,2$ we may replace $G\_i$ by its Zariski closure in $GL(V\_i)$ without changing the hypotheses or the conclusion. But if an algebraic subgroup $G\subset GL(V)$ is irreducible, then it is reductive (the unipotent radical will have a fixed space whi... | 9 | https://mathoverflow.net/users/23291 | 309579 | 134,782 |
https://mathoverflow.net/questions/309523 | 8 | Urysohn proved that any regular, Hausdorff, second-countable space $X$ is metrizable, i.e. there exists a metric space whose underlying topological space is $X$. But what if we ask the same question for *Lawvere metric spaces*?
**Definition:** Let $(X,d)$ be a Lawvere metric space. For any $\epsilon>0$ and point $x\i... | https://mathoverflow.net/users/2811 | Analogue of Urysohn metrization for Lawvere metric spaces? | According to [this SE-post](https://math.stackexchange.com/questions/1861611/on-the-separation-axiom-in-a-lawvere-or-generalized-metric-space), a *Lawvere metric* on a set $X$ is a function $d:X\times X\to[0,+\infty)$ satisfying two axioms:
1) $d(x,x)=0$ and
2) $d(x,z)\le d(x,y)+d(y,z)$
for all $x,y,z\in X$.
Th... | 8 | https://mathoverflow.net/users/61536 | 309587 | 134,785 |
https://mathoverflow.net/questions/309588 | 2 | let $\Delta$ be the triangle whose corners $A$, $B$, $C$ points in general position in Euclidean plane and, let $D$ be a fourth point inside $\Delta$.
>
> **Question:**
>
>
> what is known about the construction of $D$ with
> $$\|D-A\|+\|C-B\|\ =\ \|D-B\|+\|A-C\|\ =\ \|D-C\|+\|B-A\|$$ i.e. for which all matchin... | https://mathoverflow.net/users/31310 | Triangle Center from Weighted Perfect Matchings | This is known as the [point(s) of equal detour](http://mathworld.wolfram.com/EqualDetourPoint.html). This is usually defined as the point(s) $D$ such that
$$|DA|+|DB|-|AB|=|DA|+|DC|-|AC|=|DB|+|DC|-|BC|$$
but this is easily seen to be equivalent to your definition. The reason I wrote "point(s)" is that sometimes a trian... | 4 | https://mathoverflow.net/users/2384 | 309591 | 134,786 |
https://mathoverflow.net/questions/309531 | 15 | Let $Q\_8$ be the group of quaternions of order $8$. It is a non-abelian $2$-group such that $H^3(Q\_8,\mathbb{Z})=0$, where $\mathbb{Z}$ has the trivial action. For a proof, see the book "Homological Algebra" of Cartan and Eilenberg, Chapter XII, Section 7 (Examples), where the case of cyclic groups and generalized qu... | https://mathoverflow.net/users/128384 | $p$-groups with trivial $H^3$ | For $G$ a finite group, $H^3(G,\mathbb{Z})$ is isomorphic to the Schur multiplier, and you’ll find lots of examples using that as a search term (also, “Schur-trivial” is sometimes used to mean “having trivial Schur multiplier”).
For an example of order $p^3$, see *[The integral cohomology rings of groups of order $p^... | 10 | https://mathoverflow.net/users/22989 | 309601 | 134,788 |
https://mathoverflow.net/questions/309550 | 10 | Let $A$ be a Noetherian local ring, $f:A \rightarrow A$ be a local ring morphism. Assume some power of $f$ is a flat morphism, must $f$ be flat as well?
Motivation: Kunz's theorem shows the result is true for a positive characterestic ring $A$ and its Frobenius morphism.
| https://mathoverflow.net/users/102104 | Iteration of a morphism and flatness | Yes. Assume $f^n$ is flat for some $n>1$. Then since $f^n$ is local, it is faithfully flat. For any $A$-module $M$, put $M\_1:=A\otimes\_{f,A} M$ and recursively $M\_i:=(M\_{i-1})\_1$. Let $u:E\to F$ be an injective $A$-module homomorphism. We need to show that $u\_1:E\_1\to F\_1$ is injective. Let $K$ be its kernel. W... | 12 | https://mathoverflow.net/users/7666 | 309602 | 134,789 |
https://mathoverflow.net/questions/309593 | 0 | Reading a [paper](https://www.sciencedirect.com/science/article/pii/0001870874900218) about eta invariants I came across a zeta-like function.
I'm looking for the analytic continuation of $$\sum\_{k=1}^\infty k(k+a)^{-s}$$ at $s=0$, where $a$ is positive.
In the paper he just says "The [...] term causes no problem ... | https://mathoverflow.net/users/128226 | Analytic Continuation of Zeta-like function | Let $a>0$. We can write
$$f(s,a):=\sum\_{k=1}^\infty k(k+a)^{-s}=\sum\_{k=0}^\infty (k+a)^{-s+1}-a\sum\_{k=0}^\infty (k+a)^{-s}=\zeta(s-1,a)-a\zeta(s,a).$$
Hence, $f$ has a meromorphic continuation to $\mathbb{C}$ with simple poles at $s=1$ and $s=2$.
Now, if $s=-n$ is a non-positive integer, it is known that
$$\zet... | 1 | https://mathoverflow.net/users/109085 | 309607 | 134,790 |
https://mathoverflow.net/questions/309128 | 0 | I'm trying to understand the proof of theorem 1.6 from the [paper "A Matrix Expander Chernoff Bound"](https://arxiv.org/pdf/1704.03864.pdf).
In the proof they say: "Iterating this construction on the remainder a total of $T ≤ k$ times" and later they choose a value for $T$.
$k$ is the length of a given walk on the ... | https://mathoverflow.net/users/128183 | Proof of reduction from random walks to martingales - why $T\le k$? | One of the authors of the paper has answered me and he said that they implicitly assume $$k \ge \frac{2 log(\frac{F}{ε})}{1 − λ}$$.
He also said that actually the Theorem would still be true if $k \lt 2 log(F/ε)/(1 − λ)$. In this case, with the same $Z\_i$’s, one will actually get $W = 0$ and $|Z\_i|\_\* \le k max\_v... | 0 | https://mathoverflow.net/users/128183 | 309611 | 134,792 |
https://mathoverflow.net/questions/309577 | 30 | I have a preprint X that is sitting in the ArXiv for which I am not sure if it is still worth publishing. It turns out the paper I wrote has considerable overlap with another preprint Y after one of its authors informed me about it through email. Consider the following:
1. Paper Y was posted in the ArXiv just a month... | https://mathoverflow.net/users/73942 | Should I publish a paper if its results overlap significantly with an earlier paper? | I once wrote a paper with an undergraduate that I thought was very nice. After it was **accepted** for publication, we found a paper not only proving our results, but going a step further. We hadn't found it previously because, similar to your situation, they used different terminology. In our case, our proofs didn't a... | 25 | https://mathoverflow.net/users/3199 | 309612 | 134,793 |
https://mathoverflow.net/questions/309542 | 14 | In Milnor's book Morse Theory, it is proved that the loop space $\Omega S^n$ of the n sphere has the homotopy type of a CW complex with one cell each in the dimensions 0, n-1, 2n-2, 3n-3, ... Or more generally, given non conjugate points p, q on a complete Riemannian Manifold M, the path space $\Omega(M,p,q)$ (of all c... | https://mathoverflow.net/users/74664 | CW complex of iterated loop spaces | By a result of [Milnor](https://www.ams.org/journals/tran/1959-090-02/S0002-9947-1959-0100267-4/S0002-9947-1959-0100267-4.pdf), the space of maps from a finite CW complex to any CW complex is homotopy equivalent to a CW complex. This gives a general reason why spaces like $\Omega^k M$ have a CW structure.
There is a ... | 19 | https://mathoverflow.net/users/6668 | 309616 | 134,795 |
https://mathoverflow.net/questions/309605 | 11 | I am looking for interesting examples of categories admitting multiple *monoidally inequivalent* [closed](https://ncatlab.org/nlab/show/closed+monoidal+category) (or [compact closed](https://en.wikipedia.org/wiki/Compact_closed_category)) symmetric monoidal structures.
We know how to construct disconnected toy models ... | https://mathoverflow.net/users/128347 | Inequivalent compact closed symmetric monoidal structures on the same category | A pretty interesting class of examples comes about by classifying compact monoidal *groupoids*. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G \times G \times G \to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) \i... | 15 | https://mathoverflow.net/users/2926 | 309620 | 134,798 |
https://mathoverflow.net/questions/309592 | 0 | Let $f : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}$ be a smooth and convex function. Let us assume that $\Gamma\_f = \mathrm{graph}(f) $ is a complete hypersurface of $\mathbb{R}^{n+1}$. Then I know that $\Gamma\_f$ must be **properly** embedded.
I can prove this simple fact with an argument by contradiction, but... | https://mathoverflow.net/users/86341 | On the properness of the graph of a convex function | A fairly elementary proof is as follows (I assume you are using the convention that $f$ convex means $\Omega$ is convex and $f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$). For $x,y\in \Omega$ let $[x,y]\subset \Omega$ be the segment connecting them.
Idea, if $p=(x,f(x)),q=(y,f(y))\in \Gamma\_f$, then one has
$$
d(p,q)\leq \int\... | 1 | https://mathoverflow.net/users/127803 | 309621 | 134,799 |
https://mathoverflow.net/questions/309619 | 0 | Consider the sum of $k^{th}$-power of divisors of $n$, denoted
$$\sigma\_k(n)=\sum\_{d\vert n}d^k.$$
Let $\nu\_p(x)$ stand for the $p$-adic valuation of the integer $x$.
The following appears to be true but is it?
>
> **Question:** Fix $k, \ell\in\mathbb{N}$. If $k$ and $\ell$ have the same parity then
> $$\nu... | https://mathoverflow.net/users/66131 | $2$-adic valuations and sum of divisor function | As noted in the comment by user44191, one needs only check this for prime powers. Note that this is trivial for $q=2$ and so we may assume that we have an odd prime $q$. Then the claim is that when $k \equiv \ell$ one has that $$v\_2(\sigma\_k(q^m)) = v\_2\sigma\_\ell (q^m).$$
This is the same as asserting that $$v\_... | 3 | https://mathoverflow.net/users/127690 | 309622 | 134,800 |
https://mathoverflow.net/questions/309001 | 21 | It is extensively used and cited the following statement due to Giroux:
>
> Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and open book decompositions of $M$ up to positive stabilization.
>
>
>
Given such a contact structure, the exist... | https://mathoverflow.net/users/43097 | Proof of Giroux's correspondence | As far as I know, there is no publicly available written proof of uniqueness. Goodman's thesis pointed out by Chris proves neither uniqueness nor existence. What he did was to provide some of the first steps towards understanding the link between open books and tightness. Before that, he does sketch a proof of the open... | 10 | https://mathoverflow.net/users/58618 | 309623 | 134,801 |
https://mathoverflow.net/questions/309553 | 4 | Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be a character of $T^F$. Then there is a Deligne-Lusztig character $R\_{T,\theta}$ of $G^F$. It is known that if $T^F$ is ... | https://mathoverflow.net/users/13466 | a question on Deligne-Lusztig characters | No, your parenthetic comment at the end indicates some confusion about the nature of Deligne-Lusztig virtual (= generallized) characters: these are defined to be $\mathbb{Z}$-linear combinations of actual characters, not necessarily "alternating sums" (meaning coefficients are $\pm 1$). Already in 1974 Chang-Ree (befor... | 6 | https://mathoverflow.net/users/4231 | 309626 | 134,804 |
https://mathoverflow.net/questions/286870 | 5 | Consider the action of $G = SL(n+1)$ on $\mathbb{P}^N$, and embed $\mathbb{P}^n$ in $\mathbb{P}^N$ via the degree two Veronese embedding. Let $V\subset\mathbb{P}^N$ be the corresponding Veronese variety. Then the ideal $I(Sec\_k(V))$ of the $k$-secant variety of $V$ is a $G$-module.
Now, let $f\_1,\dots, f\_r$ be gen... | https://mathoverflow.net/users/nan | G-modules and ideals of secant varieties | In the symmetric case the representation is not irreducible.
For instance, consider a $4\times 4$ symmetric matrix $Z^{+}$ with entries $z\_{i,j}$. Then $\wedge^{2}Z^{+}$ is given by
$$
\left(\begin{array}{cccccc}
z\_{0,0}z\_{1,1}-z\_{0,1}^2 & z\_{0,0}z\_{1,2}-z\_{0,1}z\_{0,2} & z\_{0,0}z\_{1,3}-z\_{0,1}z\_{0,3} & z... | 0 | https://mathoverflow.net/users/14514 | 309628 | 134,805 |
https://mathoverflow.net/questions/309634 | 10 | Let $p$ be an odd prime. Does the equation
$$2^x-3p^y=5$$
only have finitely many solutions in positive integers $x$ and $y$?
| https://mathoverflow.net/users/128426 | Does $2^x-3p^y=5$ (with $p$ an odd prime) have only finitely many positive integer solutions? | The solutions of your equation can be injected into the solutions of the $S$-unit equation over $\mathbb{Q}$, where $S=\{\infty,2,3,5,p\}$. As the latter is known to have finitely many solutions by the results of Siegel, Mahler, Lang (see Chapter 5 in Bombieri-Gubler: Heights in Diophantine geometry), your equation als... | 25 | https://mathoverflow.net/users/11919 | 309637 | 134,807 |
https://mathoverflow.net/questions/309629 | 3 | I am working in data science and I have to deal with the following problem for which I would like to find a simplification:
We call a function almost positive if $f(x\_1,y\_1)f(x\_2,y\_2)-f(x\_1,y\_2)f(x\_2,y\_1) \ge 0$ for all $0< x\_1\le x\_2 < \infty$ and $0 < y\_1\le y\_2 < \infty.$
**I would like to know:** Ar... | https://mathoverflow.net/users/128387 | Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$ | You say your function is smooth, so letting $x\_1 = x, y\_1 = y, x\_2 = x + \Delta x, y\_2 = y+\Delta y,$ we get in the limit as the deltas go to zero, if we ignore the second order terms, then
$$ \dfrac{\partial f}{\partial x} \dfrac{\partial f} {\partial x} d x d y < 0.$$ This indicates that we cannot ignore the s... | 1 | https://mathoverflow.net/users/11142 | 309642 | 134,810 |
https://mathoverflow.net/questions/309645 | 7 | Consider the function given by
$$f(x)=1-a\_1x-a\_2x^2-a\_3x^3-\cdots$$
where each $a\_k\geq0$ and some $a\_j>0$. If $f(x)$ is a polynomial then [Descartes' Rule of signs](https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs) tells us there is exactly one positive zero, i.e. root of $f(x)=0$.
Assume $f(x)$ is a (... | https://mathoverflow.net/users/66131 | Descartes' rule of signs for infinite series | $f$ is strictly decreasing on $[0,R)$, so if there is any positive zero there is only one. There is a positive zero in $[0,R)$ iff $\lim\_{x \to R-} f(x) < 0$, which may or may not be true. For an example where it is not, consider
$$ 1 - \sum\_{n=2}^\infty \frac{x^n}{n^2}$$
| 11 | https://mathoverflow.net/users/13650 | 309646 | 134,811 |
https://mathoverflow.net/questions/309657 | 0 | Let $(M,g)$ be a Riemannian manifold which admit a non vanishing vector field.(That is $\chi(M)=0$ when $M$ is a compact manifold). We pull back The symplectic structure of the cotangent bundle to the $2$-form $\omega$ on $TM$.
>
> Is there necessarily a non vanishing vector field $X$ on $M$ for which the following... | https://mathoverflow.net/users/36688 | Symplectic submanifolds of the tangent bundle $TM$ which have the form of a vector or fiber bundle | I probably don't understand your question correctly, because the answer to the boxed question seems to be: obviously $X$ never exists if $M$ is compact. More generally, there is no closed manifold $V$ and map $f : V \to T^\*M$ such that $f^\*\omega$ is symplectic. Otherwise you would get an exact symplectic form on a c... | 3 | https://mathoverflow.net/users/58618 | 309674 | 134,820 |
https://mathoverflow.net/questions/309651 | 4 | Let $f:S \to T$ be a surjective, unramified, holomorphic map between connected Riemann surfaces. If $S$ is not compact is it always true that $f$ is a covering?
This is of course true if $S$ is compact or, more generally, if $f$ is proper. However, I can not see why this should be true in general.
| https://mathoverflow.net/users/11392 | Unramified map of Riemann surfaces | The simplest "non-trivial" example is $$z\mapsto \int\_0^ze^{-\zeta^2}d\zeta:\quad C\to C.$$
It is surjective, and not ramified. But it is certainly not a covering because every covering over a simply connected surface is a homeomorphism.
You can make the target surface compact if you wish. Consider the map from $C$ ... | 2 | https://mathoverflow.net/users/25510 | 309677 | 134,822 |
https://mathoverflow.net/questions/289356 | 6 | Let $X$ be a compact Alexandrov space with $curv\geq 1$ (and without boundary). Does $X$ always have a nontrivial compact convex subset without boundary?
Definition of a convex subset: $A\subseteq X$ is called convex if for every two points $p ,q\in A$, there exists a minimizing geodesic between them which is complet... | https://mathoverflow.net/users/38302 | Convex sets in Alexandrov spaces | This is extremally rare, even if $X$ is a Riemannian manifold.
If the convex set $A$ has interior points, then any boundary point of the subset $A$ in $X$ lies on the boundary of Alexandrov space $A$.
So if $X\ne A$ then $\dim A<\dim X$.
Note that $A$ has to be totally geodesic, otherwise an end of geodesic would b... | 3 | https://mathoverflow.net/users/1441 | 309690 | 134,828 |
https://mathoverflow.net/questions/309689 | 10 | What is the topological dimension of a (locally analytic) $p$-adic manifold over a non Archimedean field $K$?
Is the topological dimension of $K^n$, $n$?
| https://mathoverflow.net/users/nan | Topological dimension of $p$-adic manifolds | $p$-adic numbers are locally compact, Hausdorff and totally disconnected (see this [nLab page](https://ncatlab.org/nlab/show/p-adic+number#Disconnectedness)), hence they are [zero-dimensional](https://en.wikipedia.org/wiki/Zero-dimensional_space). This means that---at least naively---topological dimension of $p$-adic m... | 10 | https://mathoverflow.net/users/128347 | 309691 | 134,829 |
https://mathoverflow.net/questions/157067 | 3 | I have this question also in MSE (see: <https://math.stackexchange.com/questions/666053/centralizers-and-containment-of-c-0>), but I have not got an answer there. So I thought I try my luck here.
---
Let $X$ be a Banach space over $\mathbb{R}$ or $\mathbb{C}$.
By a *multiplier* on $X$ we mean a bounded linear ... | https://mathoverflow.net/users/46114 | Centralizers and containment of $c_0$ | Cameron's answer indicates why $Z(X)$ rather than $X$ contains an isomorphic copy of $c\_0$. Using E. Behrends's function module representation theory, one actually obtains an isometric copy of $c\_0$ in $X$. This can be found explicitly in E. Behrends, $M$-Structure and the Banach-Stone Theorem; LNM 736 (1979), Prop. ... | 3 | https://mathoverflow.net/users/127871 | 309700 | 134,834 |
https://mathoverflow.net/questions/309699 | 7 | Fix $k\in\mathbb{N}$ and assume $f(x)$ is a real polynomial of degree $n$ such that we have the normalization
$$\int\_{-1}^1f(x)^2\,(1-x)^kdx=1.$$
I am interested in the optimal size of the sum of the coefficients of $f(x)$. To this end, I ask:
>
> **Question 1:** It appears to me that the maximum value
> $$\max\... | https://mathoverflow.net/users/66131 | Bound on sum of coefficients of polynomials w.r.t a weighted integral | Consider the $(k,0)$ Jacobi polynomials $P\_n$, which are orthogonal with respect to the weight $(1-x)^k$ on $[-1,1]$. They have squared norm $c\_m=\langle P\_m,P\_m\rangle=\frac{2^{k+1}}{2m+k+1}$ and $P\_m(1)={m+k\choose k}$.
Expand $f$ as the sum $\sum\_{m=0}^n a\_m P\_m$. The constraint is that $\sum\_{m=0}^n a\_m... | 6 | https://mathoverflow.net/users/112641 | 309704 | 134,837 |
https://mathoverflow.net/questions/309633 | 2 | In unpublished notes by Yi Hu (which appear to be no longer online), I found the following:
>
> Corollary 2.4.5. Let the characteristic of $k$ is zero. Assume that a reductive group $G$ acts rationally on a finitely generated $k$-algebra $R$. Let $J$ be an ideal in $R$, invariant under $G$. Then $(R/J)^G = R^G /(J ... | https://mathoverflow.net/users/128424 | Reference on reductive group acting on quotient algebra | With some help, I now see that this result is not so hard. As Jason points out, we get a $k[G]$-module complement to $J$ in $R$, call it $C$. Obviously the projection $R \rightarrow C$ restricts to give a surjective map $R^G \rightarrow C^G$. But this is the same as the natural map $R^G \rightarrow (R/J)^G$, so that ma... | 0 | https://mathoverflow.net/users/128424 | 309708 | 134,838 |
https://mathoverflow.net/questions/309683 | 4 | Is there a finite, connected, simple, undirected graph $G=(V,E)$ such that
1. $G$ is not complete, and
2. whenever two vertices of distance $2$ are identified ("folded"), then the chromatic number increases?
| https://mathoverflow.net/users/8628 | Increasing the chromatic number by "folding" two vertices of distance 2 | The answer is no.
We may assume $G$ is not complete. If $G$ is a cycle, then identifying any two vertices at distance 2 does not change the chromatic number. Now assume that $G$ is not a cycle.
Consider a colouring of $G$ with $k:=\chi(G)$ colours.
Let $v$ be a vertex of maximum degree $d$.
By Brooks' Theorem, $k... | 9 | https://mathoverflow.net/users/25980 | 309725 | 134,846 |
https://mathoverflow.net/questions/297577 | 2 | Let $A=\{a\_1,\ldots,a\_m\}.$ Let a choice function $f:\mathcal{P}(A) \mapsto A,$ be such that, for all $B \in \mathcal{P}(A),$ $f(B)=x$ for some $x\in B.$ For instance, with $A=\{a\_1,a\_2,a\_3\}$ one such function is $f(\{a\_1,a\_2,a\_3\})=a\_1, f(\{a\_1,a\_2\})=a\_1, f(\{a\_1,a\_3\})=a\_1, f(\{a\_2,a\_3\})=a\_2.$ [$... | https://mathoverflow.net/users/89007 | Configuration of vectors satisfying some constraints | This shows up in the literature under the name *score sequence*. For the $r=2$ case you can find more information on [MathWorld](http://mathworld.wolfram.com/ScoreSequence.html) and on the page of [A000571](http://oeis.org/A000571) in the OEIS.
For general values of $r$ a characterization is given in [On Score Seque... | 3 | https://mathoverflow.net/users/51668 | 309735 | 134,848 |
https://mathoverflow.net/questions/309741 | 1 | I have fundamental questions on Dirichlet Laplacians.
Let $D \subset \mathbb{R}^d$ be an open subset and $\mathcal{L}$ be the (non positive) Dirichlet laplacian on $D$.
We denote by $T\_t=e^{t\mathcal {L}}$ the semigroup on $L^{2}(D,m)$.
Here $m$ is the Lebesgue measure on $D$.
I am concerned with when $T\_t$ bec... | https://mathoverflow.net/users/68463 | Disreteness of spectra of Dirichlet laplacians | **Yes**. Take $D$ to be the union of disjoint intervals $(n, n + a\_n)$, where $a\_n$ takes values in $(0, 1)$ slowly converges to zero. (Or, in higher dimensions, take $D$ to be the union of disjoint balls with slowly decreasing radii. One can even make $D$ connected by joining the balls using sufficiently narrow rods... | 2 | https://mathoverflow.net/users/108637 | 309746 | 134,851 |
https://mathoverflow.net/questions/309379 | 2 | When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C\_k(X)$ is $$\partial\_k([v\_0,...,v\_k])=\sum\_{i=0}^k (-1)^i[v\_0,...,v\_{i-1},v\_{i+1},...,v\_k],$$
and the coboundary map $\delta\_k$ based on the space of cochains $C^k(X)$ is the... | https://mathoverflow.net/users/128309 | How does the high-dimensional combinatorial Laplacian work? | A lot is lost in the abstract definitions of coboundary maps and cohomology (at least in the finite dimensional case, which I'm restricting to in my answer). But of course, any finite dimensional vector space with basis $e\_1, \dots e\_n$ is isomorphic to its dual via the map sending $e\_i$ to the map that is $1$ on $e... | 5 | https://mathoverflow.net/users/19729 | 309761 | 134,856 |
https://mathoverflow.net/questions/309755 | 2 | I am reading the book "[Lie superalgebras and enveloping algebras](http://www.ams.org/books/gsm/131)" by Ian M. Musson.
The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $n = 1$ the definition makes perfect sense.
My question is, $n = 1$ case is not ... | https://mathoverflow.net/users/33047 | $P(1)$ strange type classical Lie superalgebras | $P(1)$ is not simple:
To see why, consider the strange, type I, classical, simple, complex, LS $P(n)$, $n\geq 2$ realized as the set of complex, $(2n+2)\times(2n+2)$ matrices $\mathbf{M}$, with grading partitioning:
$$
\mathbf{M}=\begin{bmatrix}
\mathbf{A} & \mathbf{B} \\
\mathbf{C} & -\mathbf{A}
\end{bmatrix}
$$
wh... | 2 | https://mathoverflow.net/users/85967 | 309769 | 134,861 |
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