parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/407086 | 1 | Let $[\omega]^{<\omega}$ denote the collection of finite subsets of the integers, and let us call $E\subseteq [\omega]^{<\omega}$ *non-nested* if $a\not\subseteq b$ whenever $a\neq b\in E$.
Is there a non-nested set $E \subseteq [\omega]^{<\omega}$ with the following properties?
1. Every $n\in \omega$ is contained ... | https://mathoverflow.net/users/8628 | Non-summable subsets of $[\omega]^{<\omega}$ | Theorem: There is no such $E$.
Claim: for each $a\in E$ there exists $b\in E$ with $|b\setminus a|\ge 2$.
Proof of Claim: Let $a$ be a counterexample. Then all $b\ne a$ contain exactly one element each that's not in $a$. Let $n\not\in a$. Since $n$ is in infinitely many $b$'s, some $b$'s must be equal, contradiction.... | 2 | https://mathoverflow.net/users/4600 | 407091 | 166,813 |
https://mathoverflow.net/questions/407110 | 4 | Let $G$ be a planar graph, which we may assume to be a triangulation, with vertex set $V$ and edge set $E$. Suppose the minimum vertex degree is at least 3, and suppose any two distinct edges share at most one vertex.
**Definition.** Given an edge $e \in E$ with vertices $v\_1$ and $v\_2$, define $D(e) = \mathrm{deg}... | https://mathoverflow.net/users/6040 | Product of vertex degrees of an edge in a planar graph | Regarding **Question 3**, here is a proof that $f(n)=30$ for all $n \geq 40$. Let $H$ be a $2$-connected planar graph with minimum degree $5$. Let $G$ be obtained from $H$ by adding a new vertex inside each face of $H$, and making it adjacent to all vertices of the face. Let $V(G)=X \cup Y$, where $X=V(H)$, and $Y$ are... | 4 | https://mathoverflow.net/users/2233 | 407113 | 166,819 |
https://mathoverflow.net/questions/407133 | 9 | Let $X$ be a complex variety. By Poincare's lemma, its singular cohomology can be computed as hypercohomology of the holomorphic de Rham complex (viewing $X$ as a complex manifold)
$$\text{H}^\cdot(X,\mathbf{C})\ \simeq\ \text{H}^\cdot(X,\Omega^\bullet).$$
By GAGA the right side is the same as the hypercohomology of th... | https://mathoverflow.net/users/119012 | Integrating hypercohomology classes | One way to see this is algebraically. Let $\omega\_X = \Omega^n\_X$. The map you seek to describe is
$$
\int\_X \colon \mathrm{H}^n(X,\omega\_X) \to \mathbb{C}
$$
Abstractly, it arises as the counit of the Serre duality adjunction
$$
\mathrm{H}^n(X,-) \dashv -\otimes \omega\_X
$$
By the compatibility between local and ... | 8 | https://mathoverflow.net/users/6348 | 407136 | 166,824 |
https://mathoverflow.net/questions/407139 | 0 | Consider an arbitrary simplex $\mathcal{S} \subseteq \mathbb{R}^n$ ($\mathcal{S}$ is a polytope in $\mathbb{R}^n$ with $n+1$ vertices and non-empty interior). Let ${\bf P} \in \mathbb{R}^{m \times n}, m<n$ be an orthogonal matrix (in the sense that ${\bf P} {\bf P}^T = {\bf I}$, where ${\bf I} \in \mathbb{R}^{m \times ... | https://mathoverflow.net/users/106178 | Maximum vertex amount of low-dimensional simplex projection | For every simplex $\Delta\subset\Bbb R^n$ with $n+1$ vertices and every number $m\le n$ there is an $m$-dimensional subspace $W\subseteq\Bbb R^n$ so that the orthogonal projection of $\Delta$ onto $W$ has $n+1$ vertices.
*Proof*.
Fix a polytope $P\subset\Bbb R^m$ with $n+1$ vertices.
First, we show that $P$ is the ... | 1 | https://mathoverflow.net/users/108884 | 407161 | 166,827 |
https://mathoverflow.net/questions/162593 | 13 | Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) which exposes the theory completely in the setting of smooth sections, that is using Fréchet space techniques.
In particu... | https://mathoverflow.net/users/17047 | Reference Request: Elliptic differential operators in the Fréchet setting | A few bits and pieces of Fredholm theory in the locally convex (and, in particular, Fréchet) setting are discussed in the literature. The most comprehensive treatment I could find were two old articles by Schaefer:
* [Über singuläre Integralgleichungen und eine Klasse von Homomorphismen in lokalkonvexen Räumen, Math.... | 1 | https://mathoverflow.net/users/17047 | 407162 | 166,828 |
https://mathoverflow.net/questions/407153 | 2 | Let $U\to X$ be an open immersion of schemes and denote by $D$ the (say reduced) complement. Then by applying the de Rham functor, we get morphisms
$$U\_{dR}\to X\_{dR}\leftarrow D\_{dR}$$
of the associated de Rham spaces. Then in [this](https://people.math.harvard.edu/%7Egaitsgde/GL/Crystalstext.pdf) (in 2.5.2) Gaitsg... | https://mathoverflow.net/users/152554 | Open/closed embeddings and the de Rham space | I believe the following is a counter example to the claim in the last paragraph. Let $X$ be a smooth affine curve for simplicity, and let $pt \to X$ be a closed point. Consider the canonical map $X \to X\_{dR}$. Then $D := X \times\_{X\_{dR}} pt$ is the formal completion of the point, and the map
$$D \longrightarrow ... | 4 | https://mathoverflow.net/users/101861 | 407163 | 166,829 |
https://mathoverflow.net/questions/407115 | 4 | I was reading the book "Differentiable Periodic Maps" by P.E. Conner (1979). I am stuck at the following problem given at the end of section 21:
Let $\xi\to V^n$ be a $k$-plane bundle over a closed n-manifold then $[\mathbb{R}P(\xi)]\_2=0$ if either $k=2$ or $n=1$.
I was able to solve the $k=2$ case just by using t... | https://mathoverflow.net/users/126899 | Cobordism class of projectivization of a bundle | First, we can assume wlog that $\xi$ has an inner product.
Next, given a $2$-dimensional real vector space $U$ with inner product, let $QU$ be the space of self-adjoint endomorphisms of trace zero. Given a unit vector $u$, define $\phi(u)\in QU$ by $\phi(u)(v)=\sqrt{2}(\langle v,u\rangle u - \langle u,u\rangle v/2)$.... | 4 | https://mathoverflow.net/users/10366 | 407169 | 166,832 |
https://mathoverflow.net/questions/407067 | 4 | I'm trying to understand some things about quotients of braid groups, and particularly I'd like to solve the word problem for some elements of these quotients. I'm using MAGMA to try to access this, but I can't seem to find whether MAGMA's `GrpBrd` has a quotient construction. The more general finitely-presented group ... | https://mathoverflow.net/users/151664 | Good algorithmic properties for quotients of braid groups | Patrick Dehornoy's handle reduction [1] technique that solves the word problem in the braid group can be modified so that it seems to solve the word problem in the groups of the form $B\_{n}(d)$. In practice, for all the braid words $w$ that I have tested that are in the normal subgroup generated by $\sigma\_{i}^{d}$, ... | 1 | https://mathoverflow.net/users/22277 | 407181 | 166,837 |
https://mathoverflow.net/questions/407159 | 2 | Consider the specific element of the corresponding Lie algebra $\mathbb{1}\_3 \times \sigma^3$, where $\mathbb{1}\_3$ is the unit matrix in 3 dimensions, $\sigma^3$ is the 3rd Pauli matrix and $\times$ stands for Kronecker product. I want to find the space
$$
\left\{ U\in U(6) \; \text{ such that} \; U\left( \mathbb{1}... | https://mathoverflow.net/users/159498 | Set of $U(6)$ elements which leave a Lie-algebra element invariant under conjugation | It is a theorem (of Hopf, I believe) that the centralizer of any member of the Lie algebra (not just $\mathbb{1}\_3 \otimes \sigma^3$) is connected. See Bourbaki, Lie groups, Chap. 9, §2, nº2, [Corollary 5](https://books.google.com/books?id=oCO0xOzNLhAC&pg=PA290).
In your case $\mathbb{1}\_3 \otimes \sigma^3$ is conj... | 4 | https://mathoverflow.net/users/19276 | 407199 | 166,842 |
https://mathoverflow.net/questions/407209 | 8 | In [Between $T\_1$ and $T\_2$](https://doi.org/10.2307/2316017), Albert Wilansky mentioned in **6.** that it was not known whether or not every locally compact US space is $T\_2$.
Is this matter still an open problem?
| https://mathoverflow.net/users/146942 | Is this question about US spaces still an open problem? | No, it is not. S. Franklin gave an example in his [review of Wilansky's paper](https://mathscinet.ams.org/mathscinet-getitem?mr=208557): take a compact space with a point $x$ that is not the limit of a non-trivial sequence, for example the ordinal $\omega\_1+1$ with $x=\omega\_1$, and double that point. The set of our ... | 19 | https://mathoverflow.net/users/5903 | 407210 | 166,845 |
https://mathoverflow.net/questions/402225 | 21 | For all primes up to $p=89$ there exists a product $Q=\prod\_{j=1}^d(x-a\_j)$ involving $d\geq (p-1)/4$ distinct linear factors $x-a\_j$ in $\mathbb F\_p[x]$ such that $Q'$ has all its roots in $\mathbb F\_p$. Since $Q$ has only simple roots,
the product $QQ'$ has also only simple roots.
(It is obvious that a factori... | https://mathoverflow.net/users/4556 | Existence of a polynomial $Q$ of degree $\geq (p-1)/4$ in $\mathbb F_p[x]$ such that $QQ'$ factorizes into distinct linear factors | Such polynomials always exist, examples are [Dickson polynomials](https://en.wikipedia.org/wiki/Dickson_polynomial) of the first kind (with parameter $1$). For a positive integer $n$ these degree $n$ polynomials $D\_n$ are most conveniently defined implicitly by $D\_n(z+1/z)=z^n+1/z^n$.
For $p=4m\pm1$ the polynomial ... | 15 | https://mathoverflow.net/users/18739 | 407212 | 166,846 |
https://mathoverflow.net/questions/398269 | 1 | let $X,Y:M\to TM$ be vector fields on $M$. $\nabla\_XY$ is the change in $Y$ along the flow curves of $X$. so for a point $p \in M$ let $\phi^X(t):\mathbb{R}\to M$ be a flow curve of $X$ passing through $p$ :
$$(\phi^X)'=X \; \; ; \; \; \phi^X(0)=p$$
since I can't measure the change in $Y$ at two different points direc... | https://mathoverflow.net/users/311976 | Approximating the parallel transport map on a curve with the covariant derivative | Defining
\begin{equation}
\nabla\_XY|\_p := \lim\_{t\to 0} \frac{\Pi\_{tX}^{-1}(Y\_{\phi^X(t)})-Y\_p}{t}
\end{equation}
the approximation
$\nabla\_XY|\_p \approx \frac{\Pi\_{tX}^{-1}(Y\_{\phi^X(t)})-Y\_p}{t}$, for $t$ small, holds and now both sides are tangent vectors in $T\_pM$. It also follows that $\Pi\_{tX}^{-1}(Y... | 1 | https://mathoverflow.net/users/332652 | 407224 | 166,847 |
https://mathoverflow.net/questions/396890 | 38 | Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (<https://arxiv.org/abs/1909.09796>, <https://arxiv.org/abs/1502.05585>).
They seem to be mainly motivated by the idea that this helps with the Riemann hypo... | https://mathoverflow.net/users/143968 | Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry | I know very little about the absolute/algebraic geometry side, but I think I understand the gist of the category theory going on here. I guess this answer might require one to know a bit of both the stable homotopy story and the $\mathbb{F}\_1$-geometry story.
tl;dr is that, no, algebra over $\mathbb{S}$ requires tha... | 18 | https://mathoverflow.net/users/11546 | 407227 | 166,849 |
https://mathoverflow.net/questions/407213 | 14 | For a model category $C$, I'm denoting $h\_\infty(C)$ the associated $\infty$-category (for example its Dwyer-Kan localization at weak equivalences, or if $C$ is simplicial the simplicial nerve of the category of bifibrant objects, or any other equivalent construction...).
If $C$ is a combinatorial model structure an... | https://mathoverflow.net/users/22131 | On diagrams in model categories and rectification | If we look at the proof that I know of in the case this is true - see below - we need in fact that, for any small $1$-category $I$, $h\_\infty(C^I)$ has small (co)limits and that they can be computed termwise, and we need to prove the particular case where $I$ is discrete as a first step, which usually comes from a var... | 11 | https://mathoverflow.net/users/1017 | 407229 | 166,850 |
https://mathoverflow.net/questions/407231 | 5 | For a simple complex Lie algebra $\frak{g}$, let $V$ be an irreducible $\frak{g}$-module. Is it true that the weights of the non-zero weight vectors in $V$ are less than the highest weight vector and greater than the lowest weight vector with respect to the partial order on weights? If not, what is a simple counterexam... | https://mathoverflow.net/users/371382 | Do weight vectors live between the highest and lowest weights? | One nice way to see this is using the [PBW theorem](https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem). Write $\mathfrak{g} = \mathfrak{n}\_- \oplus \mathfrak{h} \oplus \mathfrak{n}\_+$ in the usual way. Take an ordered basis of $\mathfrak{g}$ consisting of, first, a basis for $\mathfrak... | 6 | https://mathoverflow.net/users/297 | 407236 | 166,852 |
https://mathoverflow.net/questions/407207 | -1 | Can someone explain to me the proof on page 7/20 of the original paper about potential games ([https://www.cs.tau.ac.il/~mansour/sem-game-02-03/monderer-potential-96.pdf](https://www.cs.tau.ac.il/%7Emansour/sem-game-02-03/monderer-potential-96.pdf))?
It is about why two potentials of a game G can only differ by a con... | https://mathoverflow.net/users/434074 | Why do two potentials of a game only differ by a constant? | This is what it looks like to me: to get $H(y)$, we change strategy profile $z$ into $y$ one player at a time, summing the changes in utility. For any exact potential $P$ (which is actually equation 2.2 with weights $1$), each change in utility is equal to $P(a\_{i-1}) - P(a\_{i})$. This is a telescoping sum, so $H(y) ... | 1 | https://mathoverflow.net/users/29697 | 407241 | 166,854 |
https://mathoverflow.net/questions/407187 | 1 | Consider the symplectic group $\text{Sp}\_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C\_g$ and associated root subgroups $U\_\varphi$ for $\varphi\in C\_g$. These subgroups are defined by the [Bourbaki tables](https://books.google.nl/books?id=oCO0xOzNLhAC&pg=PA205&redir_esc=y#v=onepage&q&f=fal... | https://mathoverflow.net/users/nan | The principal congruence subgroup of the symplectic group over the integers | If we assume that $g \geq 2$, then it is known by a Theorem of Tits
( Tits, Jacques :
Systèmes générateurs de groupes de congruence. (French. English summary)
C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 9, Ai, A693–A695)
that the group $\Delta$ generated by $T$ and $2U\_{\phi}$ (your notation) has finite index i... | 3 | https://mathoverflow.net/users/23291 | 407244 | 166,855 |
https://mathoverflow.net/questions/405335 | 4 | While browsing through Davenport's lecture notes "Analytic methods for diophantine equations and Diophantine inequalities", near the end of chap 8 I came across the statement that for the equation
$c\_1 x\_1^k + ... + c\_s x\_s^k = 0$,
if the $c\_i$ are non-zero and, if $k$ is even, the sign of $c\_i$ are not all t... | https://mathoverflow.net/users/66397 | bounds for circle method | In general, what the circle method gives you is an asymptotic formula for the number of solutions to the equation, which will be of the form $C P^{s-k}$ for some non-negative $C$. We are currently (with the help of recent advances on Vinogradov's mean value theorem) able to establish such asymptotic formulae when $s>k^... | 4 | https://mathoverflow.net/users/435788 | 407257 | 166,859 |
https://mathoverflow.net/questions/407253 | 2 | A polynomial in the complex variable $z$, whose coefficients are themselves complex polynomials in another complex variable $a$, looks like
$$
f\in\Bbb C[A][Z],\;\;f(z,a)=c\_0(a)+c\_1(a)z+\cdots+c\_n(a)z^n
$$
with $c\_j\in\Bbb C[A]$ of degree less or equal than $1$. Assume not all $c\_j$ are constant and the leading co... | https://mathoverflow.net/users/70148 | Simple zeroes of complex polynomial $f(\cdot,a)$: condition on $P(a)=\operatorname{Res}_z(f,f')$ | No, it's not true, as is shown by the polynomial $f(z,a):=(z-a)(z-1)^2$.
| 3 | https://mathoverflow.net/users/24309 | 407271 | 166,862 |
https://mathoverflow.net/questions/407266 | 4 | Let $A$ be a differentially graded augmented algebra. Then $\mathbf{B}A$ can be equipped with the structure of a coalgebra. This is proved in, for example, Loday and Vallette's book on *Algebraic Operads.*
The $n$-lab page <https://ncatlab.org/nlab/show/bar+and+cobar+construction> points out that the coalgebra struct... | https://mathoverflow.net/users/112756 | Why is the bar construction of a DG algebra a coalgebra? | This is the type of question with multiple correct answers, because, as you say, it depends very much on what you think the bar construction "is" initially.
You say you want to think of $\mathbf{B}A$ as $B(1,A,1)$. Well, I don't know what you think $B(X,A,Y)$ "is", but one thing that it "is" is a specific nice resolu... | 6 | https://mathoverflow.net/users/78 | 407272 | 166,863 |
https://mathoverflow.net/questions/407261 | 3 | Let $M$ be a Riemannian manifold and $S \subset M$ a compact submanifold of strictly lower dimension. Does every smooth function on $S$ extend to a harmonic function on a neighborhood of $S$?
| https://mathoverflow.net/users/409915 | Dirichlet-type condition on Riemannian manifold | This is not possible in general, for example because of constraints related to the analyticity of the extension.
For a concrete example, let $M = \mathbf{R}^2$, and $S \subset \mathbf{R}^2$ be a smooth, simple closed curve in the plane so that in the unit disc
\begin{equation}
S \cap D\_1 = \{ x\_2 = 0 \} \cap D\_1 =... | 5 | https://mathoverflow.net/users/103792 | 407278 | 166,864 |
https://mathoverflow.net/questions/407218 | 6 | According to the prescription of [functorial quantum field theory](https://ncatlab.org/nlab/show/functorial+field+theory), a quantum field theory can be viewed as a monoidal functor from some monoidal category of $n$-cobordisms to some monoidal category of vector spaces (typically a category of Hilbert spaces).
Now, ... | https://mathoverflow.net/users/432464 | Functor category of quantum field theories? | The question of what "natural transformation of QFTs" should be is a somewhat subtle one. The issue is most apparent if you work with TQFTs, but it doesn't completely go away if you work with dynamical theories.
Suppose, for example, that you have two $n$D TQFTs $\cal{V},\cal{W}$ and an $(n-1)$D closed manifold $\Sig... | 8 | https://mathoverflow.net/users/78 | 407280 | 166,865 |
https://mathoverflow.net/questions/407219 | 4 | Consider the following wave-type equation,
$$u\_{tt}-\frac{2}{t}u\_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u\_{t})=(0,0)$ at $t=\epsilon.$ Then define the energy of the solution $u$ as follows,
$$E(t) = E(u(t))=t^2 \int (u\_t)^2+|\nabla u|... | https://mathoverflow.net/users/68232 | Energy estimates for nonlinear wave type equation | Define using $H(t) = \int (u\_t)^2 + |\nabla u|^2 ~dx $ the standard energy.
Taking the time derivative you find
$$ \frac{d}{dt}H(t) = 2 \int u\_t( g + \frac{2}{t} u\_t)$$
Writing $\|\cdot \|$ for the $L^2$ integral, you have then
$$ \frac{d}{dt} H(t) \geq - 2 \|u\_t\|\cdot \|g\| + \frac{4}{t} \|u\_t\|^2 \tag{A}$$
by C... | 2 | https://mathoverflow.net/users/3948 | 407284 | 166,866 |
https://mathoverflow.net/questions/281771 | 10 | I couldn't add to the well-written $n$Lab page about [relative adjoints](https://ncatlab.org/nlab/show/relative+adjoint+functor), so let me start taking that definition for granted.
I have a few questions about how the classical results on adjoints remain true, even in this fairly asymmetric setting.
I would like t... | https://mathoverflow.net/users/7952 | What is known about relative adjunctions? | 1. Using the answer to (7), if the natural transformation $J \Rightarrow J'$ is invertible, and $G$ and $F'$ are fully faithful, then the pasting square will commute up to natural isomorphism (assuming $J$ is dense and fully faithful).
2. This condition was called being a "symmetric lift" in Lewicki's [Categories with ... | 3 | https://mathoverflow.net/users/152679 | 407287 | 166,868 |
https://mathoverflow.net/questions/407250 | 3 | A ternary $C^{\ast}$-ring is a complex Banach space $X$, equipped with a ternary product $[\cdot,\cdot,\cdot]:X^3 \to X$ which is linear in outer variables and conjugate linear in middle variable. Also $X$ is associative i.e. $$[[a,b,c],d,e]=[a,[d,c,b],e]=[a,b, [c,d,e]].$$ Moreover, $\lVert[a,a,a]\rVert= \lVert a\rVert... | https://mathoverflow.net/users/129638 | Example of a ternary $C^{\ast}$-ring which is not an operator space | According to Zettl [1], a *ternary ring of operators* (TRO) is a
ternary $C^\*$-ring which is isomorphic to a closed
subspace $X\subseteq B(H)$, such that $XX^\*X\subseteq X$, equipped with the ternary multiplication
$$
[x,y,z] := xy^\*z.
$$
On the other hand, an *anti-TRO* is a ternary $C^\*$-ring defined as above, ... | 4 | https://mathoverflow.net/users/97532 | 407306 | 166,874 |
https://mathoverflow.net/questions/407302 | 1 | Here is the definition of Hajós construction.
>
> Let $G$ and $H$ be two undirected graphs, $vw$ be an edge of $G$, and $xy$ be an edge of $H$.
> Then the Hajós construction forms a new graph that combines the two graphs by identifying vertices $v$ and $x$ into a single vertex, removing the two edges $vw$ and... | https://mathoverflow.net/users/384338 | If $G$ and $H$ are $k$-critical, then applying Hajós construction to $G$ and $H$ makes $k$-critical graph | Let $G$ and $H$ be $k$-critical graphs and $G +\_h H$ be the Hajós construction applied to $G$ and $H$ with respect to $vw \in E(G)$ and $xy \in E(H)$. Since $G$ and $H$ are $k$-critical, $G-vw$ and $H-xy$ both have $(k-1)$-colourings. Permute the colours so that the colour of $v$ is the same as the colour of $x$. Then... | 1 | https://mathoverflow.net/users/2233 | 407307 | 166,875 |
https://mathoverflow.net/questions/407289 | 15 | Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - $\mathbb{R}$ and $\mathbb{C}$?
| https://mathoverflow.net/users/148161 | Are there only two smooth manifolds with field structure: real numbers and complex numbers? | Here is a series of standard arguments.
Let $(\mathbb{F},+,\star)$ be such a field. Then $(\mathbb{F},+)$ is a finite-dimensional (path-)connected abelian Lie group, hence $(\mathbb{F},+) \cong \mathbb{R}^n \times (\mathbb{S}^1)^m$ as Lie groups. Since $\mathbb{F}$ is path-connected, there is in particular a path $\g... | 22 | https://mathoverflow.net/users/1849 | 407309 | 166,876 |
https://mathoverflow.net/questions/407200 | 2 | In Neukirch’s book “Algebraic Number Theory”, Proposition II.5.7, the following is insisted: for a mixed characteristic local field $K$ with a residue field $\mathbb{F}\_q$, $q = p^f$, then one has an isomorphism $K^\times \cong \mathbb{Z} \times \mathbb{Z}/(q-1)\mathbb{Z} \times \mathbb{Z}/p^a\mathbb{Z} \times \mathbb... | https://mathoverflow.net/users/433725 | Topology of multiplication groups of local fields | Let $\mathfrak m$ be the maximal ideal of the ring of integers (valuation ring) of $K$, so your $U^{(1)}$ is $1 + \mathfrak m$. You want to prove $1+\mathfrak m$ looks like $\mathbf Z/p^a\mathbf Z \times \mathbf Z\_p^d$ topologically for some $a \geq 0$ and $d = [K:\mathbf Q\_p]$. You said you agree there is such an is... | 4 | https://mathoverflow.net/users/3272 | 407320 | 166,878 |
https://mathoverflow.net/questions/407188 | 3 | Let $\Delta$ be the Laplacian on a smooth domain $\Omega\subset \mathbb{R}^2$ with Dirichlet boundary conditions. I am interested in whether the implication
\begin{align}
\Omega \text{ is asymmetric } \Rightarrow \Delta \text{ only has single eigenvalues}
\end{align}
holds, where by asymmetric I mean that the only plan... | https://mathoverflow.net/users/430002 | Multiplicity of Dirichlet Laplacian eigenvalues of asymmetric domains | A counterexample is given by $\Omega = (0,2\pi)^2\setminus [0,\pi]^2$. Then $u\_{mn}(x,y)=\sin mx\sin ny$ is an eigenfunction with eigenvalue $m^2+n^2$, and so is $u\_{nm}$.
You can build many other examples in this way by looking at the zero set of eigenfunctions of the square (in particular, the reflection symmetry... | 1 | https://mathoverflow.net/users/48839 | 407325 | 166,881 |
https://mathoverflow.net/questions/407322 | 3 | Suppose that we are given an AF-algebra $A$ and a sequence of finite-dimensional subalgebras $\mathbb{C}=A\_0\subset A\_1\subset A\_2\subset\ldots$ such that $A=\overline{\bigcup\limits\_{n\geq 0}A\_n}$. Let me denote this dense subalgebra of $A$ by $A^{LS}$, i.e. $A^{LS}= \bigcup\limits\_{n\geq 0}A\_n$.
Next, we def... | https://mathoverflow.net/users/170524 | Monotone approximation of elements in AF-algebras | No, this is not possible in general. $A^{LS}$ might have trivial intersection with a non-zero hereditary $C^\ast$-subalgebra of $A$, and thus any non-zero positive element in such a hereditary $C^\ast$-subalgebra cannot be approximated from below by elements in $A^{LS}$.
For a concrete example, I will give a non-unit... | 8 | https://mathoverflow.net/users/126109 | 407329 | 166,883 |
https://mathoverflow.net/questions/407283 | 5 | Let $t\_1$ and $t\_2$ be lower semicontinuous semifinite densely defined traces on a $C^\*$-algebra $A$. Let us denote by $\mathcal{R}\_1$ and $\mathcal{R}\_2$ their ideals of definition, i.e. $\mathcal{R}\_i=\left\{x\in A\ \middle|\ t\_i(x^\*x)<+\infty \right\}$. Next, let $B$ be an involutive dense subalgebra of $A$ ... | https://mathoverflow.net/users/170524 | Two densely defined traces on a $C^*$-algebra coinciding on a dense subalgebra are equal | Yes, this is true.
Let $a\in A\_+$. By lower semicontinuity it suffices to show that $t\_1((a-\delta)\_+) = t\_2((a-\delta)\_+)$ for all $\delta>0$ (where $(a-\delta)\_+$ is the positive part of $a-\delta 1$ in the unitisation of $A$ (which is an element of $A$ and not actually in the unitisation)). Fix $\delta>0$.
... | 5 | https://mathoverflow.net/users/126109 | 407333 | 166,884 |
https://mathoverflow.net/questions/407332 | 0 | Let $E$ be a (right) Hilbert module over the $C^\*$-algebra $B$. Let $\phi$ be a state on the $C^\*$-algebra $B$. Then consider
$$N\_\phi:= \{x \in E: \phi(\langle x,x\rangle)=0\}.$$
I want to show that $N\_\phi$ is a submodule of $E$, but for this I need to show that
$$b \in B, x \in N\_\phi \implies \phi(b^\*\langl... | https://mathoverflow.net/users/216007 | Is $N_\phi = \{x \in E: \phi(\langle x,x\rangle)=0\}$ a Hilbert submodule of $E$? | It is not true. Take $B= M\_2(\mathbb C)$ (with standard matrix units $e\_{i,j}$), $E= B$ as a Hilbert $B$-module in the usual way, and let $\phi \in B^\ast$ be compression to the $(1,1)$-corner. Then $x = e\_{2,2} \in E$ and $b = e\_{2,1} \in B$ satisfy $x\in N\_\phi$ and $xb \notin N\_\phi$.
| 5 | https://mathoverflow.net/users/126109 | 407335 | 166,886 |
https://mathoverflow.net/questions/407347 | 1 | Let $f(x)\in \mathbb{Z}[X]$ be a polynomial of degree at least $2$. We denote the set of primes $p$ for which $f(x)$ is injective modulo $p$ as $\mathcal{T}$. Then, can we say something about the proportion of polynomials $f(x)$ for which cardinality of the set
$$\#\mathcal{T}(y)\ll \frac{y}{(\log{y})^2}.$$
| https://mathoverflow.net/users/160943 | Estimating the size of set of primes $p$ for which the polynomial is bijective in $\mathbb{F}_p[X]$ | I'll write $T\_f$ for the set of primes $p$ such that $f(x)$ is a bijection $\mathbb{F}\_p \to \mathbb{F}\_p$. I claim that $T\_f$ is always either finite or else $\# \{ p \in T\_f : p \leq y \} \sim c \tfrac{y}{\log y}$ for some $c>0$.
The following result is known as Schur's conjecture; a flawed proof was given by ... | 5 | https://mathoverflow.net/users/297 | 407353 | 166,891 |
https://mathoverflow.net/questions/407158 | 4 | $\DeclareMathOperator\gr{gr}$Let $A = \cup\_{i=0}^\infty F\_i A$ be a filtered commutative ring, $I \subseteq A$ an ideal. Then we have a canonical surjection
$$ \gr(A)/\gr(I) \to \gr(A/I).$$
Under what conditions is this surjection an isomorphism?
I most wish to know about the following special case: $A = \mathbb C[... | https://mathoverflow.net/users/125523 | Quotients and associated graded | I follow user @Z. M's comment.
If $M = \cup\_{i=0}^\infty F\_i M$ is a filtered module and
$$0 \to M' \to M \to M'' \to 0$$
is a short exact sequence such that $M', M''$ have the *induced filtrations*, then we get a short exact sequence of Rees modules
$$0 \to R\_hM' \to R\_hM \to R\_hM'' \to 0,$$
which yields the sh... | 0 | https://mathoverflow.net/users/125523 | 407362 | 166,896 |
https://mathoverflow.net/questions/407311 | 3 | I read Neukirch’s book “Algebraic Number Theory”, and its remark following to proposition VI.2.3, there is an assertion that natural map $\mathbb{A}\_K \otimes\_K L \to \mathbb{A}\_L$ is isomorphism. How can it be proved ?
| https://mathoverflow.net/users/433725 | base change of adele rings | This is Theorem 1 in Chapter IV-1 of Weil: Basic Number Theory. See also Corollaries 1-2 after the proof of this theorem.
| 4 | https://mathoverflow.net/users/11919 | 407364 | 166,897 |
https://mathoverflow.net/questions/407155 | 8 | Let $D$ be a central division (or maybe just simple) algebra over $\mathbb{Q}$. Let $\mathcal{O} \subset \mathcal{O}\_m$ be an order inside a fixed maximal order and denote by $\mathcal{O}^1$ its group of units with norm $1$. If $[\mathcal{O}\_m : \mathcal{O}] = N$, then can we say in general that $[\mathcal{O}^1\_m : ... | https://mathoverflow.net/users/168129 | Covolumes of unit groups of division algebras | First, you need to avoid definite quaternion algebras over $\mathbb{Q}$: in this case, the unit groups are finite, so the index cannot grow with $N$.
With that out of the way, your algebra $D$ satisfies strong approximation (chapter 28 of my book, I'm sorry I only discuss the quaternion case, but see Remark 28.6.11; ... | 11 | https://mathoverflow.net/users/4433 | 407367 | 166,899 |
https://mathoverflow.net/questions/407032 | 6 | Let $\mathcal{U}$ be an ultrafilter over $\omega$, and let $\mathcal{X} \subseteq [\omega]^\omega$. In two separate texts, there are two possible interpretations of a $\mathcal{U}$-Ramsey set, as described below (Definition 7.37 and Definition 3). My question is:
>
> Do these two definitions coincide? What if we re... | https://mathoverflow.net/users/146831 | Are these two definitions of $\mathcal{U}$-Ramsey set equivalent? | If I understand your question correctly, you ask if **Defintion 7.37.** and **Definition 3.** are equivalent for a Ramsey ultrafilter $\mathcal{U}(=\mathcal{H})$. The short answer is no, but let me elaborate.
Since for a $\mathcal{U}$-tree $T$, the set $[T]$ is technically a subset of $\omega^\omega$, we will only co... | 3 | https://mathoverflow.net/users/134910 | 407372 | 166,901 |
https://mathoverflow.net/questions/407350 | 2 | * [Li-Yau 1983\_Article](https://link.springer.com/content/pdf/10.1007/BF01213210.pdf)
* The second part of above paper used the discrete eigenvalues of $\frac{-\Delta}{q}$ where $q>0$ to proof the the number of non-positive eigenvalues of
Schrödinger operator $-\Delta+V$ can be bounded by the $L\_{\frac{n}{2}}$-norm o... | https://mathoverflow.net/users/166368 | On the Schrödinger equation and the eigenvalue problem | Assuming $q>0$ the Schroedinger operator $-\Delta/q$ is associated to the form $a(u,v)=\int\_{\mathbb R^n} \nabla u \cdot \nabla v$ in $L^2(\mathbb R^n, q\, dx)$. The form domain consists of all $u\in L^2(\mathbb R^n, q\, dx)$ such that $u \in \dot H^1:=\{u \in L^{2^\*}(\mathbb R^n), \nabla u \in L^2(\mathbb R^n)\} $ a... | 2 | https://mathoverflow.net/users/150653 | 407383 | 166,905 |
https://mathoverflow.net/questions/407259 | 3 | Let $F$ be a $p$-adic field with residue field $k$ and let $G$ be a connected reductive group over $F$. Let us assume that $G$ is simply connected as an algebraic group over an algebraic closure of $F$. If I am not mistaken, it will insure that maximal compact subgroups and maximal parahoric subgroups of $G(F)$ are the... | https://mathoverflow.net/users/125617 | Is it possible to detect when a maximal parahoric subgroup is (hyper)special from its finite reductive quotient? | (I write this answer in quite a haste, so there will probably be some inaccuracies. Sorry for that and, please, let me know, when you find them.)
First, I think, that if the group is not almost simple, then the question makes few sense. For example, each of the split groups $G\_1 = Sp\_4(F)$ and $G\_2 = {\rm SL}\_2(F... | 2 | https://mathoverflow.net/users/148992 | 407385 | 166,906 |
https://mathoverflow.net/questions/407388 | 6 | **For reference, my motivation:** It's of interest to classify free actions of groups on spheres of positive even dimension. Establishing such a classification up to homotopy is not too difficult: Every free group action on a sphere of even dimension is homotopic to either the trivial action by the trivial group or the... | https://mathoverflow.net/users/131309 | Do all spaces doubly covered by $S^{2n}$ have the homeomorphism type of $\mathbb{P}^{2n}_{\mathbb{R}}$? | One approach to a homeomorphism classification of closed manifolds simply homotopy equivalent to a closed manifold $X$ of dimension $>4$ is to compute the topological structure set $\mathcal S^s\_\text{TOP}(X)$ and the group of homotopy classes of simple homotopy equivalences $\text{Aut}\_s(X)$. Then $\text{Aut}\_s(X)$... | 16 | https://mathoverflow.net/users/1573 | 407390 | 166,907 |
https://mathoverflow.net/questions/407407 | 2 | Let $\frak{g}$ be a complex simple Lie algebra and let $\frak{k}$ be a non-zero semisimple Lie subalgebra of $\frak{g}$. Is it possible to realize every simple $\frak{k}$-module $W$ as a $\frak{k}$-submodule of a some $\frak{g}$-module $V$. (Of course we are thinking about $V$ as a $\frak{k}$-module by restriction). I ... | https://mathoverflow.net/users/371382 | Extending representations of Lie subalgebras to the whole Lie algebra | Suppose you have an inclusion of algebraic objects $A \subset B$. In this case, $A = \mathfrak{k}$ and $B = \mathfrak{g}$ are Lie algebras, but it doesn't make a big difference — you can read "algebraic object" as "group" or "Lie group" or "algebraic group" or "Hopf algebra" or many other things. What's important is th... | 4 | https://mathoverflow.net/users/78 | 407409 | 166,912 |
https://mathoverflow.net/questions/407369 | 0 | Let $\mathcal{X}\_n=\{ X\_{n,\lambda}, \lambda \in \Lambda\}$ be a collection of random variables (defined on the same probability space) indexed by a deterministic index $\lambda$ over an index space $\Lambda$. Assume that for any $\lambda \in \Lambda$ it is known that $X\_{n,\lambda}\to 0$ almost surely, i.e. for any... | https://mathoverflow.net/users/148849 | Almost sure convergence of the supremum over a class of random variables | The fundamental issue here is bounding the distribution of the supremum of a collection of random variables. The book "Upper and Lower Bounds for Stochastic Processes" by Michel Talagrand is largely devoted to this issue, <https://link.springer.com/book/10.1007/978-3-642-54075-2>
as is his earlier related book "the gen... | 1 | https://mathoverflow.net/users/7691 | 407411 | 166,913 |
https://mathoverflow.net/questions/407230 | 3 | The other day I was playing a game called Trans Europa (or Trans America) which is quite graph theoretic in flavour. The game takes place on a triangular lattice graph with certain distinguished nodes (called cities). Each player is given five cities at random. The players then take it in turns to 'claim' an edge by la... | https://mathoverflow.net/users/119114 | Game theory approach to Trans Europa | This is a non-answer that's growing a little large for a comment.
First, a mathematical statement of a slightly generalised version of the game. The board is a graph. Each player has a starting vertex and a set of goal vertices. At any stage there is a subgraph of marked edges. A legal move for a player is to mark an... | 1 | https://mathoverflow.net/users/25485 | 407414 | 166,916 |
https://mathoverflow.net/questions/407368 | 4 | For a compact manifold $M$ the space of smooth functions $C^{\infty}(M)$ is a Fréchet space where the seminorms are the suprema of the norms of all partial derivatives. Is there some way to characterise those Fréchet subalgebras coming from a differential structure? As a naive guess, how about the following:
Conjectu... | https://mathoverflow.net/users/438034 | A Fréchet space characterization of smooth structures on topological spaces? | The book *Smooth Manifolds and Observables* by the pseudonymous Jet Nestruev may be of interest, as it defines and studies smooth manifolds using only the algebra of smooth functions. However, their definition of smooth structure is somewhat disappointing, as it requires the algebra to be locally isomorphic to $C^\inft... | 4 | https://mathoverflow.net/users/226696 | 407418 | 166,917 |
https://mathoverflow.net/questions/407404 | 19 | $f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$.
Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ?
---
Source: [AoPS](https://artofproblemsolving.com/community/c7h1570936)
| https://mathoverflow.net/users/110301 | Strange result about convexity | The answer is yes, and it follows immediately from a simple description of the extreme rays of the convex cone of the functions with a convex second derivative.
Indeed, for $x\in(0,1]$, let $f'''(x)$ denote the left derivative of the convex function $f''$ at $x$, so that $f'''$ is non-decreasing on $(0,1]$.
Take an... | 15 | https://mathoverflow.net/users/36721 | 407431 | 166,922 |
https://mathoverflow.net/questions/407377 | 4 | If $f$ is any monic polynomial/$\mathbb{Z}$ with non-zero constant coefficient. I wish to study the quantities
$$t\_n=\sum\_{i}\theta\_i^n\in\mathbb{Z}$$
where $(\theta\_i)\_{i=1}^{d}$ are the roots of $f$ counted with multiplicity.
The main question I am interested is finding all the primes $p$ such that $p\mid ... | https://mathoverflow.net/users/159298 | A condition such that $p\mid\sum_{f(\theta)=0}\theta^n$ for all $n$? | You have $t\_k=0$ for all $k$ if and only if $f(x) \bmod p$ is a $p$-th power.
Let $g(x)$ be the image of $f(x)$ in $\mathbb{F}\_p[x]$; let $\alpha\_1$, $\alpha\_2$, ..., $\alpha\_n$ be the roots of $g$ (with multiplicity) in $\overline{\mathbb{F}\_p}$, let $e\_k$ be the $k$-th elementary symmetric function in the ro... | 7 | https://mathoverflow.net/users/297 | 407434 | 166,923 |
https://mathoverflow.net/questions/407391 | 2 | Let $f$ vanishes on an open set containing 0. So there exists $l>0$ such that $f$ vanishes on $B(0,2l).$ So we can choose $g\in C\_c^\infty (\mathbb{R}^n)$ (supported on $B(0,l)$) such that $f\*g$ vanishes on an open set ( vanishes on $B(0,l)$).
My question: Is it remains true if we replace open set by set of positiv... | https://mathoverflow.net/users/184109 | Problem regarding vanishing set of convolution | As noted in Pietro Majer's comment, you can trivially take $g=0$ to get the "yes" answer.
To avoid this and make the question less trivial, one may additionally require that the function $g$ be nonnegative and nonzero. Then the answer becomes "no".
Indeed, e.g. let
$$f:=1\_{D},$$
where $D:=\mathbb R\setminus C$ and... | 0 | https://mathoverflow.net/users/36721 | 407459 | 166,931 |
https://mathoverflow.net/questions/407455 | 1 | Let $A$ be a $C^\*$-algebra and $(a\_{ij}) \in M\_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a\_{ij}$) such that
$$\lVert(a\_{ij})\rVert \le C \Bigl\lVert\sum\_i a\_{ii}\Bigr\rVert?$$
**Attempt**: Choose a faithful representation $A \subseteq B(H)$. Then $M\_n(A) \subsete... | https://mathoverflow.net/users/216007 | $C\lVert\sum_i a_{ii}\rVert \ge \lVert(a_{ij})\rVert$ for matrices with entries in a $C^*$-algebra | This answer arose by a discussion with @JamieGabe in the comments.
One can prove that
the map
$$\Phi: M\_n(A) \to M\_n(A): A \mapsto n \operatorname{Diag}(A)-A$$
is completely positive [Paulsen, "Completely bounded maps and operator spaces", exercise 3.6].
In particular, it is positive. Hence, writing $A= (a\_{i,j}... | 3 | https://mathoverflow.net/users/216007 | 407461 | 166,932 |
https://mathoverflow.net/questions/407464 | 1 | Motivated by the answer to this [question](https://mathoverflow.net/questions/407407/extending-representations-of-lie-subalgebras-to-the-whole-lie-algebra), I will ask the following question: Let $\mathcal{A}$ and $\mathcal{B}$ be small semisimple abelian categories. Let $U:\mathcal{A} \to \mathcal{B}$ be a functor tha... | https://mathoverflow.net/users/371382 | Adjoints of exact functors between semisimple abelian categories | This was too big to fit as a comment. Here is a cute, completely trivial, but incredibly useful fact.
Suppose $(f^\*, f\_\*)$ is an adjoint pair of functors between additive categories (not necessary abelian or anything of that nature).
If $X$ is an object such that $f^\*X \neq 0$, then the unit map $\eta\colon X \... | 4 | https://mathoverflow.net/users/392998 | 407467 | 166,935 |
https://mathoverflow.net/questions/407465 | 5 | For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this:
>
> Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random
> variable $u \in H$ is said to be a Gaussian random variable if
> $\langle u,v \rangle$ is Gaussian (i.e., $\langle u,v \rangle$ foll... | https://mathoverflow.net/users/170508 | Definition of infinite-dimensional Gaussian random variable | Even in finite dimensions this definition is more convenient since it is independent of coordinates. If you are interested in geometric applications this is what you need.
This definition has the advantage that clarifies the nature of the the various invariants. Here are some more details.
A Gaussian measure on a a... | 9 | https://mathoverflow.net/users/20302 | 407468 | 166,936 |
https://mathoverflow.net/questions/407439 | 16 | Let $\mathcal P$ be the set of finite subsets of $\mathbb Z\_{\geq 0}$ , each of them contains $0$. We say that $A \in \mathcal P$ is *indecomposable* if it is not $B+C$ (the sum set of $B,C$) with $B,C\in \mathcal P$ and $B,C\neq \{0\}$.
It is easy to see which small cardinality sets are indecomposable. If $|A|=2$, ... | https://mathoverflow.net/users/2083 | Sets that are not sum of subsets | There are three questions in the OP, and I'll try to address each of them in as comprehensive a way as I can. I'll be glad to add in further details (and references) if requested.
◇ **Preliminaries on free monoids.**
We write $\mathscr F(X)$ for the [free monoid](https://en.wikipedia.org/wiki/Free_monoid) on an alp... | 17 | https://mathoverflow.net/users/16537 | 407471 | 166,937 |
https://mathoverflow.net/questions/407476 | 1 | Definition of 'replication of $v$' is
>
> Suppose $v \in V(G)$. Replication of $v$ is constructing $G'$ by adding a new vertex $v'$ such that $N\_{G'}(v')=N\_G(v) \cup \{v\}$.
>
>
>
And the following statement is well-known fact(you can find a proof by googling).
>
> Replication of a single vertex conserve... | https://mathoverflow.net/users/384338 | Pseudo-replication of a vertex in a perfect graph | If we pass to the complement, this is just a replication. But passing to a complement preserves the class of perfect graphs (a theorem of Lovasz, previously conjecture of Berge).
| 2 | https://mathoverflow.net/users/4312 | 407477 | 166,938 |
https://mathoverflow.net/questions/407480 | 11 | In "[Toward a Galoisian interpretation of homotopy theory](https://arxiv.org/abs/math/0007157)" (2000), B. Toën wrote:
>
> Pour expliquer notre point de vue sur la notion de champs rappelons une construction (non conventionnelle) du topos de l’espace $X$ (i.e. d’une catégorie qui est naturellement équivalente à la ... | https://mathoverflow.net/users/429204 | A non-conventional definition of topoi | This idea originates in homotopy theory and is due to Jardine, "Simplicial presheaves", JPAA **47** Issue 1 (1987) pp 35–87, <https://doi.org/10.1016/0022-4049(87)90100-9> ([pdf](https://core.ac.uk/download/pdf/82485559.pdf)).
One can not take for $X$ any site, this definition only works in a topos with "enough point... | 13 | https://mathoverflow.net/users/1310 | 407482 | 166,940 |
https://mathoverflow.net/questions/407427 | 3 | The definition of a line graph is as follows:
>
> Given a graph $G$, its line graph $L(G)$ is a graph such that
>
>
> 1. each vertex of $L(G)$ represents an edge of $G$.
> 2. two vertices of $L(G)$ are adjacent if and only if their corresponding edges share a common endpoint ("are incident") in $G$.
>
>
>
I ... | https://mathoverflow.net/users/384338 | What kind of graph has more edges than its line graph? | For the case when $G$ is connected, we can argue as follows:
Since $\lvert E(G)\rvert=\lvert V(L(G))\rvert$, the inequality $\lvert E(L(G))\rvert<\lvert E(G)\rvert$ can be read as "$L(G)$ has more vertices than edges". Since $L(G)$ is connected as well, it must therefore be a tree. In particular, $L(G)$ cannot contai... | 5 | https://mathoverflow.net/users/108884 | 407483 | 166,941 |
https://mathoverflow.net/questions/407365 | 6 | For a function $x:\omega\to\mathbb R$ let $|x|$ denote the function $|x|:\omega\to[0,\infty)$, $|x|:n\mapsto|x(n)|$.
It is well-know that a series $\sum\_{n\in\omega}r\_n$ of real numbers converges unconditionally if and only if $\sum\_{n\in\omega}|r\_n|<\infty$.
On the other hand, there exists an unconditionally c... | https://mathoverflow.net/users/61536 | Unconditionally convergent series in $\ell_2$ consisting of $\ell_1$-small vectors | If I understand correctly what you are asking, the answer is "certainly not".
Consider $k$ orthogonal vectors $v\_j$ with $k$ coordinates $\pm 1$ (the Hadamard matrix). Now multiply them by $t$ and take $n$ such identical bunches. Then the sum of $\ell^1$ norms squared is $nt^2k^3$, the squared $\ell^2$ norm of the s... | 6 | https://mathoverflow.net/users/1131 | 407485 | 166,942 |
https://mathoverflow.net/questions/407491 | 5 | Let $M^n$ be an $n$-manifold with nonempty boundary and let $\partial\_0 M$ be a specific connected component of $\partial M$. I am interested in the set of continuous maps $f : [0,1] \to M$ such that $f^{-1}(\partial M) = \{ 0,1\}$ and such that $f(0),f(1) \in \partial\_0 M$. Actually, I am interested in this set cons... | https://mathoverflow.net/users/99414 | Set of proper homotopy classes of arcs in a manifold | Yes, this map is a bijection.
The condition $f^{-1}(\partial M)=\{0,1\}$ is superfluous. Using a collar neighborhood on $\partial M$ you can show that the space of continuous maps that you defined is a deformation retract of the space of maps $f\colon [0, 1]\to M$ that satisfy just the condition $f(0), f(1)\in \parti... | 7 | https://mathoverflow.net/users/6668 | 407497 | 166,945 |
https://mathoverflow.net/questions/407475 | 8 | Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f\_1,f\_2,\ldots,f\_n$ be non-constant smooth functions on $\partial \Omega$ that are linearly independent from each other. Given any $j=1,2,\ldots,n$, we denote by $u\_j$, the unique har... | https://mathoverflow.net/users/50438 | Points where harmonic functions fail to give a coordinates system | In dimension two, the [Rado-Kneser-Choquet theorem](https://en.wikipedia.org/wiki/Rad%C3%B3%E2%80%93Kneser%E2%80%93Choquet_theorem) explains how to choose boundary data to obtain a non vanishing Jacobian in the interior. [Lewy's Theorem](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/v... | 4 | https://mathoverflow.net/users/40120 | 407503 | 166,948 |
https://mathoverflow.net/questions/407235 | 2 | $\DeclareMathOperator\SU{SU}$Consider the Lie group $\SU(6)$, its Lie algebra $\mathfrak{su}(6)$ and the $\mathfrak{su}(2)$ subalgebra spanned by $\mathbb{1}\_3 \otimes \sigma^i$, where $\sigma^i$ are the 3 Pauli matrices, $\mathbb{1}\_3$ is the unit matrix in 3 dimensions and $\otimes$ stands for Kronecker product. I ... | https://mathoverflow.net/users/159498 | Set of $\mathrm{SU}(6)$ matrices which conjugate $\mathbb{1}_3 \otimes \sigma^3$ subalgebra element into $\mathfrak{su}(2)$ | The set in question is isomorphic to the fiber product $F\times^SH$ where $F=SU(2)$, $H=S(U(3)\times U(3))$ and $S=F\cap H=U(1)$. In particular, it is a connected manifold.
This can be seen as follows: Let $\xi=1\_3\otimes\sigma^3$ und $U\in\mathfrak{su}(6)$ with $U\xi U^{-1}\in\mathfrak{su}(2)$. Since $\xi$ and $U\x... | 5 | https://mathoverflow.net/users/89948 | 407507 | 166,949 |
https://mathoverflow.net/questions/407505 | -1 | If $n\in\mathbb{N}$ is a non-negative integer, we consider it as a cardinal, so $n = \{0, \ldots, n-1\}$. If $X$ is a set, and $\kappa$ is a cardinal, we let $[X]^\kappa$ be the collection of subsets of $X$ having cardinality $\kappa$.
If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) and $\kap... | https://mathoverflow.net/users/8628 | Chromatic number of $(n, [n]^k)$ | It is $\lceil \frac{n}{k-1}\rceil$, simply since every color class must contain at most $k-1$ elements.
| 2 | https://mathoverflow.net/users/4312 | 407510 | 166,950 |
https://mathoverflow.net/questions/407521 | 1 | Let $\{E\_j\}$ be measurable subsets in $B\_1\subset\mathbb{R}^n$ and $\exists$ $A>0$, such that
$\int\_{B\_1}\chi\_{E\_j}(x)dx\geq A$ for any $j=1,2,3,...$. Can we select a subsequence of functions $\{\chi\_{E\_j}(x)\}$ such that $\chi\_{E\_j}(x)\rightarrow\chi\_E(x)$ a.e. $x\in B\_1$, for some measurable set $E$ in $... | https://mathoverflow.net/users/99411 | a.e. convergence of characteristic functions | The answer is no. E.g., let $n=1$ and let $B\_1$ be the set of all irrational numbers in the interval $[0,1]$. For each $x\in B\_1$, let $b\_j(x)$ be the $j$th binary digit of $x$. For each natural $j$, let $E\_j:=\{x\in B\_1\colon b\_j(x)=1\}$, so that $\chi\_{E\_j}=b\_j$ on $B\_1$.
Then no subsequence of the sequen... | 1 | https://mathoverflow.net/users/36721 | 407525 | 166,953 |
https://mathoverflow.net/questions/407488 | 3 | Let $K$ be a number field and consider a finite Galois extension $L|K$. Moreover let $X$ be a projective, regular, integral variety over $K$. After a base change we obtain a morphism of varieties $f:X\_L\to X$. Assume that $\mathcal L$ is a $\operatorname{Gal}(L|K)$ invariant line bundle on $X\_L$; why is it true that ... | https://mathoverflow.net/users/65980 | Galois invariant line bundle and base change | I am posting my comment as an answer. This result is discussed in many sources. I do not have Serre's "Galois cohomology" with me at this moment, but I am certain that it is discussed there. It should be discussed also in "Dix exposes sur le Groupe de Brauer".
In fact the reference where I first learned this is Igor ... | 4 | https://mathoverflow.net/users/13265 | 407531 | 166,955 |
https://mathoverflow.net/questions/407516 | 3 | I try to calculate the following series
\begin{align\*}
S\_{n,m}(z)=\sum\_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k},
\end{align\*}
where $\Gamma(z)$ is the Gamma function and where $n,m\in \mathbb Z^+$ (positive integers).
More precisely I w... | https://mathoverflow.net/users/84558 | $\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function | As Carlo noted, for $n$ an even integer, $S\_{n,m}(z)$ is an elementary function of $z$.
What about $n$ odd?
When $n,m$ are both odd, I get something in terms of arcsinh, also elementary.
---
But for $n$ odd and $m$ even, Maple gets complete elliptic integrals, which are not elementary... Examples
$$
S\_{1,... | 6 | https://mathoverflow.net/users/454 | 407536 | 166,957 |
https://mathoverflow.net/questions/407535 | 1 | This question can be seen as a variant of the post [Bounded density for diffusions with diffusion coefficients bounded away from $0$](https://mathoverflow.net/questions/405624/bounded-density-for-diffusions-with-diffusion-coefficients-bounded-away-from-0) by Iosif Pinelis. Namely, consider the diffusion
$$X\_t=\int\_... | https://mathoverflow.net/users/261243 | On the marginal distributions of an absorbed diffusion | The answer is yes. Indeed, let
$$Y\_t=\int\_0^t a(s,Y\_s)\,dW\_s\quad \forall t\ge 0.$$
Then $X\_t=Y\_t$ on the event $\{\tau>t\}$. So, for any Borel set $A\subseteq(-1,1)$, we have
$$P(X\_t\in A)=P(Y\_t\in A,\tau>t)\le P(Y\_t\in A).$$
So, the distribution of $X\_t$ is absolutely continuous with respect to the distribu... | 1 | https://mathoverflow.net/users/36721 | 407538 | 166,958 |
https://mathoverflow.net/questions/407540 | 4 | Let $t$ be a natural number. For a unitary matrix $U$ let $U\_{1,1}$ be the top left matrix element of $U$. I am trying to figure out the value of $\int |U\_{1,1}|^{2t} dH(U)$ where $H$ is the Haar measure over the unitary group.
I know that this integral could be expressed as $$\sum\_{\sigma,\tau\in S\_t} \mbox{Wg}\... | https://mathoverflow.net/users/42993 | Expected even power of absolute value of an element of a random unitary matrix | We have
$$\mathbb{E}\_{U(N)} |U\_{1,1}|^{2t} = \binom{t+N-1}{t}^{-1}$$
for all integers $t \ge 1$. A reference is Corollary 1.2 of J. Novak, ``Truncations of random unitary matrices and Young tableaux'', Electron. J. Combin. 14 (2007), no. 1, Research Paper 21. He indeed uses a Weingarten-type approach, as part of a mu... | 5 | https://mathoverflow.net/users/31469 | 407545 | 166,960 |
https://mathoverflow.net/questions/407517 | 5 | For two functions $x,y:\omega\to\mathbb R$ let $xy:\omega\to\mathbb R$, $xy:n\mapsto x(n)y(n)$, be their pointwise product.
It is clear that for any elements $x,y\in\ell\_2$ their pointwise product $xy$ is an element of $\ell\_2$.
>
> **Problem.** Let $\sum\_i x\_i$ be an unconditionally convergent series in $\el... | https://mathoverflow.net/users/61536 | A perturbation of an unconditionally convergent series in $\ell_2$ | Volodymyr Kadets kindly informed me that the answer to this problem is affirmative. His argument easily generalizes to prove the following
>
> **Theorem.** For any $p\in[1,\infty)$, any unconditionally convergent series $\sum\_{n\in\omega}x\_n$ in the Banach space $\ell\_p$ and any sequence $\{y\_n\}\_{n\in\omega}\... | 3 | https://mathoverflow.net/users/61536 | 407551 | 166,962 |
https://mathoverflow.net/questions/407553 | 26 | Differential equations are at the heart of applied mathematics - they are used to great success in fields from physics to economics. Certainly, they are very useful in modelling a wide range of phenomena.
Integral equations, on the other hand, do not receive such attention. While I have seen some integral equations c... | https://mathoverflow.net/users/114143 | Importance of integral equations | One important point is that differential equations encode *local* behaviour of a system, while integral equations typically endcode *global* behaviour. Local behaviour is often easier to model and to grasp intuitively. In many cases, it can also be described by much simpler formulae.
More specifically:
* Let us con... | 24 | https://mathoverflow.net/users/102946 | 407557 | 166,965 |
https://mathoverflow.net/questions/406831 | 18 | This question was motivated by [a recent MO post.](https://mathoverflow.net/q/406753/11260) You know $n$ elements of the $N\times N$ matrix $M$ and you do *not* know $n$ elements of the inverse $M^{-1}$ (but you know the other $N^2-n$ elements of $M^{-1}$). Equating $(M^{-1})^{-1}=M$ gives $n$ nonlinear equations in $n... | https://mathoverflow.net/users/11260 | Uniquely reconstruct a matrix $M$ from its inverse $M^{-1}$ if $n$ elements of $M^{-1}$ are unknown and $n$ elements of $M$ are given | The conjecture is true. More precisely, here is what I will prove:
**Theorem** Partition $\{ (i,j) : 1 \leq i \leq j \leq n \}$ into two disjoint sets, $A \sqcup B$. Let $X$ and $Y$ be positive definite $n \times n$ matrices and suppose that $X\_{ij} = Y\_{ij}$ for $(i,j) \in A$ and $(X^{-1})\_{ij} = (Y^{-1})\_{ij}$ ... | 16 | https://mathoverflow.net/users/297 | 407563 | 166,967 |
https://mathoverflow.net/questions/405552 | 2 | Let $X(t)$ be a $C^1$ (continuously differentiable) path in the Lie algebra (actually I just need finite-dimensional matrices). It is well-known (from [Wikipedia page of Derivative of the exponential map](https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map), also in many Lie algebras/groups textbooks) that
... | https://mathoverflow.net/users/13838 | Derivative of adjoint action of exponential map | Long story short, here is the answer:
$$
\begin{aligned}
\frac{d}{dt}\exp{X(t)}
&=
\exp{X}\frac{1 - \exp(-\mathrm{ad}(X))}{\mathrm{ad}(X)}\frac{dX}{dt}
\\
\frac{d}{dt} \mathrm{Ad}(\exp{X(t)})
&=
\mathrm{Ad}(\exp{X}) \mathrm{ad} \left(\frac{1 - \exp(-\mathrm{ad}(X))}{\mathrm{ad}(X)}\frac{dX}{dt}\right)
\end{aligned}
$$
... | 0 | https://mathoverflow.net/users/13838 | 407573 | 166,969 |
https://mathoverflow.net/questions/407562 | 3 | What is an example of a real solvable simply-connected Lie group $G$ whose nilradical does not have a complement (that is, $G$ is not a semidirect product of the nilradical and another subgroup)? Is it possible to produce such a group if $G$ is supersolvable (that is, $G$ can be embedded in a group of upper triangular ... | https://mathoverflow.net/users/442553 | Example of a supersolvable Lie group/algebra whose nilradical does not have a complement | $\newcommand{\mk}{\mathfrak}$Let $\mk{h}\_{2n+1}$ be the $(2n+1)$-dimensional Lie algebra (basis $x\_1,\dotsc,x\_n,y\_1,\dotsc,y\_n,z$, nonzero brackets $[x\_i,y\_i]=z$).
The $n$-dimensional abelian Lie algebra $\mk{a}\_n$ (basis $a\_1,\dotsc,a\_n$) acts on it by $a\_i\cdot x\_i=x\_i$, $a\_i\cdot y\_i=-y\_i$, rest of... | 1 | https://mathoverflow.net/users/14094 | 407576 | 166,971 |
https://mathoverflow.net/questions/407492 | 8 | By constructive mathematics in this matter we mean [intuitionistic ZF](https://en.wikipedia.org/wiki/Constructive_set_theory#Intuitionistic_Zermelo%E2%80%93Fraenkel) (\*).
In the language of [locales](https://ncatlab.org/nlab/show/locale), the Jordan curve can be defined as $f\colon S^1 \to X$ such that "if $U \cap V... | https://mathoverflow.net/users/148161 | What is the status of Jordan's theorem in constructive mathematics in the language of locales? | Let me first clarify some confusion in the comments to the original question. To be clear : I'm not at all saying the persons making them were confused, as far as I can tell all the comments were correct, my point is rather than the mixture of the "topological" and "localic" point of view created some confusion (which ... | 4 | https://mathoverflow.net/users/22131 | 407589 | 166,977 |
https://mathoverflow.net/questions/407552 | 5 | Let $f(n)$ denote the proposition "There exists some $k>1$ such that
$$
\sum\_{m=k}^{k+n-1}\tau(m) < \sum\_{m=1}^n\tau(m)
$$
where $\tau(m)$ is the number of the divisors of $m$." (This is like the so-called second Hardy-Littlewood conjecture on $\pi(x)+\pi(y)$ vs. $\pi(x+y)$ in testing the initial interval, which has ... | https://mathoverflow.net/users/6043 | Analogue of the second Hardy-Littlewood conjecture for numbers of divisors? | This is false, as you suspect.
By definition, $$ \sum\_{m=k}^{k+n-1} \tau(m)$$ is the sum of the number of divisors of integers $m$ in $[k, k+n)$. Equivalently, it is the sum over $d$ of the number of $m\in [k, k+n)$ such that $d$ divides $m$.
For each $d$, the number of $m\in [k, k+n)$ such that $d$ divides $m$ is... | 7 | https://mathoverflow.net/users/18060 | 407590 | 166,978 |
https://mathoverflow.net/questions/407592 | 2 | If $Y\_n=\sum\_{i=1}^n X\_i$ is a martingale, where $X\_i$ is a martingale difference sequence, $\mathbb{E}[X\_n\mid \mathcal{F}\_{n-1}]=0$ for all $n$, we know that
$$ \mathbb{E}\big[Y\_n^2-Y\_{n-1}^2\big]=\mathbb{E}X\_n^2.$$
A similar property, but now as an inequality, holds if we replace the square with the absolut... | https://mathoverflow.net/users/443327 | Inequality for increments of $r$th absolute moments of martingales, $1<r<2$ | Yes, this inequality, with the best possible $C$ ($\le 2$), was proved in [this paper](https://projecteuclid.org/journals/annals-of-functional-analysis/volume-6/issue-4/Best-possible-bounds-of-the-von-Bahr--Esseen-type/10.15352/afa/06-4-1.full); see e.g. inequality (1.11) there.
Indeed, that inequality implies that
$... | 3 | https://mathoverflow.net/users/36721 | 407595 | 166,979 |
https://mathoverflow.net/questions/407584 | 2 | We consider any non-negative integer $n$ as a cardinal, so $0 = \emptyset$, and $n=\{0,\ldots,n-1\}$ for positive $n$. Given $n,k\in \mathbb{N}$, let $[n]^k$ denote the collection of $k$-element subsets of $n$. We say ${\cal L}\subseteq [n]^k$ is *linear* if $|a\cap b|\leq 1$ whenever $a\neq b\in {\cal L}$, and we call... | https://mathoverflow.net/users/8628 | Cardinalities of maximal linear $k$-subsets of $n = \{0,\ldots,n-1\}$ | For the minimum question, this is answered in Theorem 4 of
*Erdős, Paul; Füredi, Zoltán; Tuza, Zsolt*, [**Saturated (r)-uniform hypergraphs**](http://dx.doi.org/10.1016/0012-365X(91)90035-Z), Discrete Math. 98, No. 2, 95-104 (1991). [ZBL0766.05060](https://zbmath.org/?q=an:0766.05060).
The answer is $\frac{n^2}{k(k-1... | 4 | https://mathoverflow.net/users/17798 | 407608 | 166,983 |
https://mathoverflow.net/questions/407603 | 7 | There is a lot of important recent work on the construction of Calabi-Yau metrics on non-compact complex manifolds, such as $\mathbb{C}^n$. For example:
[1] Li, Y. *A new complete Calabi-Yau metric on $\mathbb{C}^3$.* Invent. Math. 217 (2019), no. 1, 1–34.
[2] Székelyhidi, G. *Degenerations of $\mathbb{C}^n$ and Ca... | https://mathoverflow.net/users/392184 | What is the definition of a Calabi-Yau metric on a non-compact manifold? | There are two slightly different definitions. The first is that it is a Kähler metric that is Ricci-flat, and the second is that it is a Kähler metric on a (usually connected) complex $n$-manifold with holonomy in $\mathrm{SU}(n)$. They are equivalent in the simply-connected case, but not always in the non-simply conne... | 6 | https://mathoverflow.net/users/13972 | 407610 | 166,985 |
https://mathoverflow.net/questions/407612 | 1 | Let $k=(k\_1,k\_2) \in \mathbb{Z}^2$. Let $\lambda=(\lambda\_1,\lambda\_2)\in [0,2\pi]^2$ and $F(\lambda)$ be a bounded real function of $\lambda\in [0,2\pi]^2$.
I am interested in the following equation:
$$
\frac{1}{(2\pi)}\int\_{[0,2\pi]}\sum\_{l=-\infty}^\infty \lvert l\rvert \bigg\lvert\frac{1}{2\pi}\int\_{[0,2\p... | https://mathoverflow.net/users/151758 | 2-dimensional Fourier transform | $\newcommand\Z{\mathbb Z}$Suppose that $F$ is a bounded measurable complex-valued function on $[0,2\pi]^2$, so that $F\in L^2([0,2\pi]^2)$. Then
$$F(s,t)=\sum\_{(m,n)\in\Z^2}c\_{m,n}e^{i(ms+nt)} \tag{1}$$
for some complex $c\_{m,n}$'s such that $\sum\_{(m,n)\in\Z^2}|c\_{m,n}|^2<\infty$ (the above double series converge... | 2 | https://mathoverflow.net/users/36721 | 407618 | 166,988 |
https://mathoverflow.net/questions/407646 | 9 | What is an example of a ring $R$ for which the abelian category of left $R$-modules is not isomorphic to the category of right $R$-modules?
| https://mathoverflow.net/users/371382 | A ring for which the category of left and right modules are distinct | Let $Q$ be a finite acyclic quiver and $K$ a field. Let $Q^{op}$ be the opposite quiver (where all the arrows are reversed). Let $KQ$ be the path algebra. Then the category of left $KQ$-modules is equivalent to the category of right $KQ^{op}$-modules. It is well known that if $Q,Q'$ are acyclic quivers, then the catego... | 19 | https://mathoverflow.net/users/15934 | 407648 | 166,994 |
https://mathoverflow.net/questions/407649 | -1 | Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $
\delta\_n := \{x \in [0,1]^n:\,\Sigma\_k x\_n =1, (\forall i)\,x\_i>0\}
$ are homeomorphic. What I'm looking for is much weaker, however. Namely, does anyone know of an explicit continuous surjection:
$$
f:\mathbb{R}^{n-1}\r... | https://mathoverflow.net/users/36886 | Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse | Let $\Delta\subset \mathbb R^{n-1}$ be an isometric copy of $\delta\_n$ with barycenter in the origin. Let $j:\mathbb R^{n-1}\to[0,+\infty)$ be the associated Minkowski functional, defined by $x\in j(x) \partial\Delta $ for all $x\in \mathbb R^{n-1}$. Then $x\mapsto \frac {j(x)}{j(x)+1}x$ is a homeo $\mathbb R^{n-1}\to... | 2 | https://mathoverflow.net/users/6101 | 407650 | 166,995 |
https://mathoverflow.net/questions/377843 | 4 | Let $s>0$, $1<p<\infty$ and let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. Set $H^{s,p}(\Omega)=\{u\in L^p(\Omega):\exists\tilde u\in L^p(\mathbb R^n),\tilde u|\_{\Omega}=u,(I-\Delta)^\frac s2\tilde u\in L^p(\mathbb R^n)\}$ and $\|u\|\_{W^{s,p}(\Omega)}=\inf\limits\_{\tilde u}\|(I-\Delta)^\frac s2\tilde ... | https://mathoverflow.net/users/118469 | Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain | The result is mentioned in Triebel's *Function Spaces and Wavelets on Domains* Proposition 4.21. Also see Theorem 1.4 for my paper <https://arxiv.org/abs/2110.14477>
| 0 | https://mathoverflow.net/users/118469 | 407659 | 166,997 |
https://mathoverflow.net/questions/407641 | 0 | It is well known that the Riemann Hypothesis implies the following:
$|\theta(x) - x| = O(x^{1/2 + \epsilon})$ for all $\epsilon > 0$.
where $\theta$ is the first Chebyshev function; that is, $\theta(x)$ is the sum of the logs of all the primes up to $x$. Is it known whether the reverse implication holds? If so, cou... | https://mathoverflow.net/users/130113 | Question about $\theta$ and the Riemann Hypthesis - reference request | More is true. Let $c\in [1/2,1)$ be a constant. It is well-known that the Riemann zeta function $\zeta(s)$ has no zeros in the half plane $\Re(s)> c$ if and only if $|\theta(x)-x|=O(x^{c+\epsilon})$ for all $\epsilon>0$.
You can find this proved in most texts on the Riemann zeta function, such as Titchmarsh's book.
... | 5 | https://mathoverflow.net/users/3199 | 407664 | 167,000 |
https://mathoverflow.net/questions/394037 | 8 | Fix even $n$ and consider the boolean function $f : \{0, 1\}^n \rightarrow \{0, 1\}$, $f : (x\_0, \ldots , x\_{n - 1}) \mapsto (x\_0 \vee x\_1) \wedge (x\_2 \vee x\_3) \wedge \cdots \wedge (x\_{n - 2} \vee x\_{n - 1})$. Fix a field $K$ and any affine hyperplane $A \subset K^n$.
**Conjecture.** If $A \cap \{0, 1\}^n \... | https://mathoverflow.net/users/92003 | a Littlewood–Offord-type problem concerning the "cubical lattice" | An asymptotic reformulation of this conjecture has now been solved by Diamond and Yehudayoff in the paper *Explicit Exponential Lower Bounds for Exact Hyperplane Covers*. Preprint is available [here](https://eccc.weizmann.ac.il/report/2021/148/). The sharp form of the conjecture is still open.
| 0 | https://mathoverflow.net/users/92003 | 407665 | 167,001 |
https://mathoverflow.net/questions/407667 | 1 | Hellow.
I don't understand why the following formula is valid.
Can you please tell me the proof?
Let $f(x,y)$ be a function on star-shaped domain of $\mathbb{R}^2$, and let $c(x,y):=\int^1\_0tf(tx,ty)dt$.
Then
$f(x,y)=\partial\_x(xc)+\partial\_y(yc)$ holds.
| https://mathoverflow.net/users/152099 | An integral representation of a two-variable function | $$f(x,y) = \int\_0^1 \frac{d}{dt}[t^2f(tx,ty)]\, dt$$
$$=^{\ast} 2\int\_0^1 tf(tx,ty)\, dt+x\frac{\partial}{\partial x}\int\_0^1 tf(tx,ty)\, dt+y\frac{\partial}{\partial y}\int\_0^1 tf(tx,ty)\, dt$$
$$=\partial\_x(xc)+\partial\_y(yc),\;\;\text{for}\;\;c(x,y)=\int^1\_0tf(tx,ty)\,dt.$$
In the $^\ast$ equation I used that... | 2 | https://mathoverflow.net/users/11260 | 407670 | 167,002 |
https://mathoverflow.net/questions/407653 | 3 | Let $T$ be a measure preserving bijection of a probability space $(X,\nu)$. Consider the Koopman representation of $\mathbb{Z}$ on $L^2(X,\nu)$ given by $[z.f](x) = f(T^{-z}(x))$. The question is: can I tell from the representation whether $(X,\nu,T)$ has zero entropy?
| https://mathoverflow.net/users/23661 | Zero entropy and the Koopman representation | There is an example (due to [Newton - Parry](https://mathscinet.ams.org/mathscinet-getitem?mr=206209) and also attributed by [Rokhlin](http://www.mathnet.ru/links/8b6eec06952e8e0c9d023e3caa73e40b/rm5788.pdf) to Girsanov) of a zero entropy measure preserving transformation with a countable Lebesgue spectrum (and mixing ... | 3 | https://mathoverflow.net/users/8588 | 407683 | 167,007 |
https://mathoverflow.net/questions/407677 | 7 | Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and these problems has been solved in a number of special cases. The "next simplest" case are Picard curves, which can be desc... | https://mathoverflow.net/users/89064 | $y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves | The below Magma code determines the size of $J\_C(\mathbb{F}\_p)$ for various primes $p$, and finally compute the GCD of their orders, which gives you a bound on the size of $J\_C(\mathbb{Q})$. For a discussion of this, see Section 4.1 of [this paper](https://arxiv.org/pdf/2007.13929.pdf). To determine $J\_C(\mathbb{F}... | 4 | https://mathoverflow.net/users/56667 | 407686 | 167,009 |
https://mathoverflow.net/questions/407655 | 8 | This question is related to [Monotone version of one-dimensional Whitney extension theorem](https://mathoverflow.net/q/215153/121665).
Let $m$ be a positive integer or $m=\infty$.
>
> Suppose that $E\subset\mathbb{R}$ is a closed set and $f:E\to\mathbb{R}$ is a non-decreasing (strictly increasing) function such tha... | https://mathoverflow.net/users/121665 | Whitney extension theorem preserving monotonicity | No also in the strictly increasing case.
Let $E = [0,1]$ and $f: x\mapsto x^2$ is strictly increasing, and is the restriction of a $C^\infty$ function. Any $C^2$ extension of this function must have $F'(0) = 0$ and $F''(0) = 2$, and so for some $\epsilon > 0$ must have $F'(-\epsilon) < 0$.
---
On the other hand... | 9 | https://mathoverflow.net/users/3948 | 407688 | 167,010 |
https://mathoverflow.net/questions/407593 | 4 | After a long story of dancing around the effective topos $ \mathcal{Eff}$, I finally resolved to get to the bottom of it. To this effect, working as it were backward, I ended up revisiting Kleene's original definition of realizability for Heyting Arithmetic (1949), the "*Mother of all realizabilities*" .
To my surpri... | https://mathoverflow.net/users/15293 | Homotopical realizability | The reference to "truth in $\mathbb{N}$" is a mirage. As Andreas Blass points out, there is a computable procedure that determines whether $s = t$ holds for closed term $s$ and $t$ of HA. One needs only run a certain algorithm to find out whether $s = t$, there is no "faith in the model".
And because equality is deci... | 4 | https://mathoverflow.net/users/1176 | 407690 | 167,011 |
https://mathoverflow.net/questions/407691 | 22 | First of all sorry for this non-research post.
I was watching [*Jeffrey Blitz* Lucky](https://rads.stackoverflow.com/amzn/click/com/B004JM5S9I) documentary movie and it was interesting to me that a winner of Lottery was a math Ph.D. from Berkeley.
In the movie he said:
> ... | https://mathoverflow.net/users/90655 | What is so special about Chern's way of teaching? | [Louis Auslander](https://en.wikipedia.org/wiki/Louis_Auslander) has described his experiences on [S.S. Chern as a teacher:](https://academic.hep.com.cn/skld/EN/chapter/9781571462411/chapter14)
>
> Somehow Chern conveyed the philosophy that making mistakes was normal
> and that passing from mistake to mistake to tr... | 37 | https://mathoverflow.net/users/11260 | 407692 | 167,012 |
https://mathoverflow.net/questions/407700 | 15 | In teaching my linear algebra students about Gram-Schmidt orthogonalization, I found a curious sequence of polynomials. They are closely related to Legendre polynomials, but they also appear to be related to Catalan numbers. (Several of the statements below are conjectural and I am not an expert on orthogonal polynomia... | https://mathoverflow.net/users/11264 | Do you recognize this sequence of polynomials? | Let $b\_n(t)$ be the Morgan-Voyce polynomial defined by $$\begin{eqnarray}b\_0(t) &=& 1 \\
b\_1(t) &=& t + 1 \\
b\_n(t) &=& (t+2) b\_{n-1}(t) - b\_{n-2}(t)
\end{eqnarray}$$
Then $f\_n(t) = (-1)^n b\_n(-t)$ fits your recurrence.
See [Rising diagonal polynomials associated with Morgan-Voyce polynomials](https://cites... | 11 | https://mathoverflow.net/users/46140 | 407706 | 167,015 |
https://mathoverflow.net/questions/407676 | 0 | I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures $\mu$ and $\nu$ on a metric space $X$ by
$$
W\_p(\mu, \nu) = \left(\inf\_\gamma \int\_{X\times X} \mathrm{dist}(x,y)^p \mathrm{d}\gamma(x,y)\right)^{1/p},
$$
where the ... | https://mathoverflow.net/users/445256 | Couplings as generalized functions | Since you refer to transportation metrics, it seems to be fair to assume that your probability spaces are Lebesgue (=standard). Then it is the same to talk either about "lifting" your measure $\mu$ to $\gamma$ or about the **family of the conditional measures** of the projection $\gamma\to\mu$ (or, which is also the sa... | 1 | https://mathoverflow.net/users/8588 | 407714 | 167,019 |
https://mathoverflow.net/questions/407727 | 0 | This is a related question to an [older one](https://mathoverflow.net/questions/362115/are-complete-regular-linear-hypergraphs-on-omega-isomorphic).
If $H\_i = (V\_i, E\_i)$ are [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) for $i=1,2$ then we say they are *isomorphic* if there is a bijection $f: V\_1 \to V... | https://mathoverflow.net/users/8628 | Are strongly complete regular linear hypergraphs on $\omega$ isomorphic? | If $H\_1= P^2(\mathbb{Q})$ (the projective plane defined on $\mathbb{Q}$ with the points as vertices and lines as hyper-edges), and $H\_2$ the [Moulton plane](https://en.wikipedia.org/wiki/Moulton_plane) defined on $\mathbb{Q}$, then $H\_1$ and $H\_2$ are both strongly complete regular linear hypergraphs but they are n... | 3 | https://mathoverflow.net/users/125498 | 407744 | 167,030 |
https://mathoverflow.net/questions/407740 | 1 | I am looking for a reference/a hint to the following problem:
We are given $f\_1(x),f\_2(x)$ convex functions (say, on $\mathbb R^d$) such that $f\_1(x) \to\infty$ for $\|x\|\to\infty$. Also there is an $\alpha > 0$ such that $f\_1(x) - \alpha f\_2(x) \geq 0$.
A first consequence is that for any $\epsilon > 0$,
$$f... | https://mathoverflow.net/users/88505 | Convexification of difference of convex functions | $\newcommand\R{\mathbb R}$In general, the answer is no.
E.g., let $d=1$,
\begin{equation}
f\_1(x):=\sum\_{j=1}^\infty(|x|-2j)\_+,\quad f\_2(x):=\sum\_{j=1}^\infty(|x|-2j-1)\_+,
\end{equation}
where $u\_+:=\max(0,u)$ --
so that $f\_1$ and $f\_2$ are convex functions, $f\_1(x)\to\infty$ as $|x|\to\infty$, and $f\_1\ge... | 2 | https://mathoverflow.net/users/36721 | 407746 | 167,031 |
https://mathoverflow.net/questions/407704 | 1 | For $m\in \mathbb{N}$ and $a=(a\_0,a\_1,\ldots,a\_{m}) \in \mathbb{R}^{m+1}$, consider the polynomial $P\_{a}$ defined by
$$
P\_{a} (x):= a\_0 + a\_1 x^2 + \ldots + a\_{m}x^{2m}\text{, for $x \in \mathbb{R}$.}
$$
Then there is $C\_m >0$ such that for every $a \in \mathbb{R}^{m+1}$, we have
$$
\int\_{[0,1]} P\_{a}(x)^2 ... | https://mathoverflow.net/users/118316 | Comparing different norms of a polynomial | (Note: there is no need to consider only polynomials of even degree $n=2m$, since the same works for any degree $n\in\mathbb N$).
The map $Q\_{n+1}:\mathbb R^{n+1}\ni a\mapsto \int\_0^1 \big(\sum\_{k=0}^na\_kx^k\big)^2dx= \sum\_{h,k}\frac1{k+h+1}a\_ka\_h=(Ha\cdot a)$ is a positive definite quadratic form on $\mathbb ... | 1 | https://mathoverflow.net/users/6101 | 407754 | 167,034 |
https://mathoverflow.net/questions/407767 | -1 | We define an embedding of the set of prim numbers into the Cantor set as follows:
First we recall that the cantor set $\mathcal{C}$ is homeomorphic to $(\mathbb{Z}/10\mathbb{Z})^\omega $ since the latter is a compact metrizable space without any isolated point. So according to [topological characterization of the Can... | https://mathoverflow.net/users/36688 | The set of prime numbers as a subspace of the Cantor set | Your set is a countable dense subset of $(\mathbb{Z}/10\mathbb{Z})^\omega$; cf. Lucia's response [here](https://mathoverflow.net/questions/292044/distribution-of-square-roots-mod-1). Hence it is neither open, nor compact.
| 2 | https://mathoverflow.net/users/11919 | 407770 | 167,038 |
https://mathoverflow.net/questions/407760 | 8 | This seems such a simple question that I fear I must have missed some elementary maths.
I am looking for a way to solve $x+x^a = y$ by reference to an already defined function, $a,x,y > 0$ are real.
Failing that a reasonable approximation for $a$ in $(0,1)$.
Many thanks!
| https://mathoverflow.net/users/446964 | Is there a specific named function that is the inverse of $x+x^a$ for $x$ real? | The answer is yes indeed. It is a special case of Fox-H function, a variation of the confluent Fox-Wright $\_{1}\Psi\_{1}$ function (a generalization of the confluent hypergeometric function $\_{1}F\_{1}$) providing the inverse function. See a previous answer [here](https://mathoverflow.net/questions/406250/transcenden... | 16 | https://mathoverflow.net/users/141375 | 407777 | 167,039 |
https://mathoverflow.net/questions/407784 | 1 | Let $\Omega \subset \mathbb{R}^n$ be convex. We write points of $\mathbb{R}^n$ as $(x\_1, x\_2, \dots, x\_n)$. Set $p(x) = m(\Omega \cap \{x\_1 = x\})$, where $m$ is the $n-1$ dimensional Lebesgue measure and $\{x\_1 = x\}$ is the hyperplane $\{(x\_1, x\_2, \dots, x\_n) \in \mathbb{R}^n : x\_1 = x\}$.
Geometrically, ... | https://mathoverflow.net/users/447371 | Measure of intersection of convex set with hyperplane is concave function | It is not true in general that the function $p$ is concave. For instance, let $\Omega$ be the conical hull of the ball of radius $1$ centered at the point $(2,0,\dots,0)\in\mathbb R^n$. Then $p(x)=c\_nx^{n-1}$ for some real $c\_n>0$ depending only on $n$ and for all real $x\ge0$. So, $p$ is not concave even on the inte... | 0 | https://mathoverflow.net/users/36721 | 407787 | 167,042 |
https://mathoverflow.net/questions/407732 | 6 | Suppose $S$ a noetherian base scheme, $X \to S$ is projective and $F, G$ are coherent $\mathcal O\_X$-modules. Then by EGA (7.7.8) and (7.7.9) there exists a scheme $H = \underline{\operatorname{Hom}}\_X(F, G)$, affine over $S$, which represents the functor
$$(f: T \to S) \mapsto \operatorname{Hom}\_{X\_T}(f^\*F, f^\*G... | https://mathoverflow.net/users/111897 | Why is the scheme of isomorphisms of sheaves affine over the base? | So I realized Jason Starr and Johan de Jong only claim that $H = \underline{\operatorname{Hom}}\_S(F, G)$ is affine if $F$ and $G$ are locally free. In that case, if $U = \operatorname{Spec}(A) \subset H$ is such that $F$ and $G$ are free of rank $n$ on $U$, we get
$$H\_U = Gl\_n(A) = \operatorname{Spec}A[X\_{ij}|i,j =... | 5 | https://mathoverflow.net/users/111897 | 407802 | 167,044 |
https://mathoverflow.net/questions/407779 | 4 | I am learning the idea of "gradient" of a functional in Otto's calculus. It is defined as follows.
Suppose the space we are thinking about is $(\mathcal{P}\_{2,AC}(\mathbb{R}^d),W\_2)$, the space of probability measures with finite second moment that is absolutely continuous w.r.t. Lebesgue measure, and equipped with... | https://mathoverflow.net/users/174600 | Gradient of Wasserstein distance in the sense of Otto's calculus | Yes this is true, formally this follows by the envelope theorem. In an abstract and very smooth setting, the envelope theorem says that for an objective functional depending on a parameter $t$
$$
F(t)=\max\limits\_z f(t,z),
$$
then the derivative of the optimal value can be computed as
$$
\frac{dF}{dt}(t)=\partial\_t f... | 5 | https://mathoverflow.net/users/33741 | 407806 | 167,046 |
https://mathoverflow.net/questions/407795 | 4 | Suppose $\pi:U(n)\rightarrow GL(V)$ is a positive-dimensional irreducible representation of the unitary group. Given $\varepsilon>0$, how could one rigorously show that the probability that $|\det(I-\pi(g))|<\varepsilon$ as $g$ is chosen in $U(n)$ (with respect to the Haar probability measure) is small (in a quantifiab... | https://mathoverflow.net/users/447643 | Irreducible representations of U(n) and probability of being close to having fixed points | This isn't true. Let $V$ be the standard two dimensional representation of $U(2)$ and let $W = (\det V)^{-1} \otimes \mathrm{Sym}^2(V)$. If the eigenvalues of $g$ acting on $V$ are $z\_1$ and $z\_2$, then the eigenvalues of $g$ acting on $W$ are $z\_1 z\_2^{-1}$, $1$ and $z\_1^{-1} z\_2$. So $\det (\mathrm{Id} - \pi\_W... | 10 | https://mathoverflow.net/users/297 | 407816 | 167,049 |
https://mathoverflow.net/questions/407788 | 0 | I have a question that occurred to me and has been bothering me, because maybe graphically it seems obvious but I don't know how to get there. It has to do with the distribution function and monotone rearrangment.
Given a bounded function $f\colon [a,b] \rightarrow \mathbb{R}$, the (right-continuous) distribution of ... | https://mathoverflow.net/users/447433 | Evaluating a limit at a discontinuity of a monotone rearrangment (distribution function) | $\newcommand{\ep}{\varepsilon}$Let
\begin{equation\*}
c:=\inf f,\quad d:=\sup f,\quad I:=[a,b],\quad F:=D\_f,
\end{equation\*}
so that
\begin{equation\*}
F(y)=\mu(\{t\in I\colon f(t)\le y \})\quad \forall y\in[c,d].
\end{equation\*}
Also introduce the set
\begin{equation\*}
E\_t:=\{y\in[c,d]\colon F(y)\ge t\}\quad ... | 0 | https://mathoverflow.net/users/36721 | 407824 | 167,052 |
https://mathoverflow.net/questions/353340 | 4 | I'm confused by the definition of a "cusp" as found in
>
> V.S. Guba, *Conditions for the embeddability of semigroups into groups*, Math. Notes **56** (1994), Nos. 1-2, 763-769 ([link](https://link.springer.com/article/10.1007/BF02110736)).
>
>
>
In the words of Mark Sapir (from an answer that has meanwhile b... | https://mathoverflow.net/users/16537 | What is a "cusp" ("кусок") in relation to Guba's embedding theorem? | **Update:** I had an email exchange with Victor Guba. He has kindly confirmed that there is indeed a typo in (the Russian and English versions of) his paper: a "кусок" (as per his paper) and an "$s$-piece" (as per Kashintsev's paper) are meant to be one and the same thing.
The part below was written before hearing fr... | 1 | https://mathoverflow.net/users/16537 | 407827 | 167,053 |
https://mathoverflow.net/questions/407803 | 3 | Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$.
Is it true that there exists $\alpha>0$ **independent of $n$** such that, for all $x \in \mathbb{R}^n$:
\begin{equation}
\label{prop}
\tag{P}\qquad \alpha \lVert x-x^\*\rVert\_{\infty} \leq \lVert\nabla f(x)\rVert\_{\infty},
\... | https://mathoverflow.net/users/447802 | Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$ | I think it is not possible to do much better than $\frac{1}{\sqrt n}$. More precisely, I believe the best $\alpha$ is $\frac{1}{\sqrt n}$ whenever $n$ is a power of $2$, and therefore (since $\alpha$ is non-decreasing in $n$) at most $\frac{\sqrt{2}}{\sqrt n}$ in general.
Indeed, consider $f(x)=\frac 1 2 \|A^{-1} x\|... | 3 | https://mathoverflow.net/users/10265 | 407830 | 167,055 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.