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/313019 | 6 | Let $p:\mathbb{G}\_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p\_\*p^\*T$ canonically isomorphic to the cocharacter lattice $L$ of $T$?
If $\operatorname{Spec}A$ is affine scheme, it seems to... | https://mathoverflow.net/users/130207 | Cokernel of map of étale sheaves | There is a small issue with your computation related to $\mathbb G\_m$-linearity. By the $\operatorname{Gal}(k^{\text{sep}}/k)$-module view on étale sheaves on $\operatorname{Spec} k$, it suffices to understand the $k^{\text{sep}}$-points. We compute
\begin{align\*}
p\_\*p^\*T(k^{\text{sep}}) &= p^\*T(\mathbb G\_{m,k^{... | 5 | https://mathoverflow.net/users/82179 | 313070 | 136,113 |
https://mathoverflow.net/questions/313043 | 7 | Let $\mathcal{D} = \{D \in \mathbb{Z} : D \equiv 0, 1 \pmod{4}\}$ be the set of *discriminants*. It is well-known that each element in $\mathcal{D}$ is the discriminant of a primitive binary quadratic form, namely $x^2 - (D/4)y^2$ when $D \equiv 0 \pmod{4}$ and $x^2 + xy - (D-1)y^2/4$ when $D \equiv 1 \pmod{4}$.
For... | https://mathoverflow.net/users/10898 | A class number estimate | This is from Buell, Binary Quadratic Forms, pages 109-119.
Note that $h(-16) = h(-4) = 1.$
When $p$ is an odd prime, we use the Legndre symbol in
$$ h(-4p^2) = \frac{p - (-1|p)}{2} $$
After that, if $u > 1,$ while $u$ is allowed odd or even as needed; if $p$ does not divide $u,$ then
$$ h(-4u^2p^2) = h(-4 u^2) \c... | 5 | https://mathoverflow.net/users/3324 | 313073 | 136,114 |
https://mathoverflow.net/questions/313066 | 3 | The following question looks simple, but the answer is not obvious for me:
Let $S$ be a $\*$-algebra and $\left\Vert \cdot \right\Vert \_{1}$, $\left\Vert \cdot \right\Vert \_{2}$ $C^\*$-norms on $S$ with $\left\Vert \cdot \right\Vert \_{1} \leq \left\Vert \cdot \right\Vert \_{2}$. Denote by $A$, $B$ the generated $C... | https://mathoverflow.net/users/64444 | Sequence in *-algebra with different limits for two C*-norms? | The answer is no (to the main question, not the title). Consider the \*-algebra $S$ of \*-polynomials generated by one variable $z$ such that $zz^\* = z^\*z$, *i.e.* the free commutative \*-algebra on one generator. Each $a \in S$ can be considered to be a continuous function $\mathbb{C} \rightarrow \mathbb{C}$, so we ... | 5 | https://mathoverflow.net/users/61785 | 313074 | 136,115 |
https://mathoverflow.net/questions/313071 | 1 | I have a random variable $X \sim \operatorname{Gamma}(\alpha, \beta)$.
How can I compute or approximate $\mathbb{E} \sin(X)$ very quickly? Iterative quadrature would be too slow, I need some closed form expression.
One idea I considered was to use the Gamma moments and Tyler approximation, but it would take too man... | https://mathoverflow.net/users/99548 | Expected value of sin(X) for Gamma r.v. X in closed form (approximation is fine) | You can use the characteristic function. From [Wikipedia](https://en.wikipedia.org/wiki/Gamma_distribution), we have $$E[e^{itX}] =
\left(1 - \frac{it}{\beta} \right)^{-\alpha}.$$
You have to be a bit careful when $\alpha$ isn't an integer, as you have a branch cut.
This means that $E[\sin(X)] = \operatorname{Im}\... | 6 | https://mathoverflow.net/users/69870 | 313077 | 136,116 |
https://mathoverflow.net/questions/312766 | 7 | The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}\_2,\mathbb{Z}\_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}\_2,\mathbb{Z}\_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup\_1 2 Sq^1 x.
$$
The $\cup\_1$ is a higher cup product. The $Sq^1 x= x \cup\_1 x$.
It shall be true that
$$
\mathcal{P}(x) \mod 2= x \... | https://mathoverflow.net/users/106497 | Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$ | (1) No, not for any reasonable interpretation of your condition "$x \cup\_1 Sq^1 x = 0$". Consider $M= S^2 \times S^2$, let $y \in H^2(M; \mathbb{Z})$ be the sum of the two obvious generators, and let $x \in H^2(M; \mathbb{Z}\_2)$ be the mod 2 reduction of $y$. Then $P(x)$ is the mod 4 reduction of $y^2 = 2$, but certa... | 5 | https://mathoverflow.net/users/13061 | 313081 | 136,117 |
https://mathoverflow.net/questions/313062 | 7 | A Grothendieck's universe is such a set $U$ so that
>
> * $\forall x \in U, x \subseteq U$,
> * $\forall x,y \in U, \{x,y\} \in U$,
> * $\forall x \in U, \mathcal{P}(x) \in U$,
> * given a family $(X\_i)\_{i \in I}$ such that $I \in U$ and any $X\_i \in U$ we have $\bigcup\_{i \in I} X\_i$,
> * $\mathbb{N} \in U$.
... | https://mathoverflow.net/users/83143 | Enhancing Grothendieck's universes and Grothendieck's axiom: Feferman's universe | I haven't thought deeply about this question, but shooting from the hip my reaction is that one might sometimes need two or three such universes, or maybe even countably many, but proper-class many seems unlikely to be needed.
The point of Feferman's ZFC/S (and its slight modification ZMC/S) for category theory is th... | 12 | https://mathoverflow.net/users/49 | 313083 | 136,119 |
https://mathoverflow.net/questions/313052 | 3 | Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|\_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|\_{H^k(\mathbb{R}^m)}<M$
Can we say that
$$|\sum\limits\_{|\alpha| = k}\sum\limits\_{|\beta| = k}\int\_{\mathbb{R}^m}D^{\a... | https://mathoverflow.net/users/14414 | A specific problem on : Can bounding the Sobolev norm, bound a higher derivative? | This seems to be a consequence of the Cauchy–Schwarz inequality:
$$
\begin{split}
\bigg\vert
\sum\_{\vert \alpha \vert = k}
\sum\_{\vert \beta \vert = k}
\int\_{\mathbb{R}^m}
D^\alpha f
D^\beta \phi
\bigg\vert
&\le
\bigg(
\sum\_{\vert \alpha \vert = k}
\int\_{\mathbb{R}^m}
(D^\alpha f)^2
\bigg)^\frac{1}{2}
\bigg(
\sum\... | 5 | https://mathoverflow.net/users/42047 | 313085 | 136,120 |
https://mathoverflow.net/questions/313061 | 1 | This question is related to [one of previous questions.](https://mathoverflow.net/questions/312803/is-there-a-generalized-order-space-x-with-countable-tightness-which-is-not-fir)
For any generalized order space $X$, $X$ has countable tightness iff $X$ is first countable.
Since a generalized order space is monotoni... | https://mathoverflow.net/users/129635 | A question on monotonically normal spaces | There are many examples. Take one ultrafilter $u$ on $\mathbb{N}$ and consider $\mathbb{N}\cup\{u\}$ as a subspace of $\beta\mathbb{N}$ (every point of $\mathbb{N}$ is isolated and the basic open neighbourhhods for $u$ are the sets of the form $U\cup\{u\}$ with $U\in u$). The resulting space has countable tightness but... | 3 | https://mathoverflow.net/users/5903 | 313089 | 136,122 |
https://mathoverflow.net/questions/313086 | 10 | Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.
>
> Does $R$ still have the Invariant Basis Number (IBN) property?
>
>
>
Recall that a ring is said to have the IBN property if $R^m \cong R^n \Rightarrow m=n$.
All commutative rings have the IBN property, but the standard pro... | https://mathoverflow.net/users/1849 | Do commutative rings without unity have the IBN property? | (I will write my comment as an answer.)
The answer is "not necessarily" for the way IBN is defined in the problem. For a counterexample, let $R=\oplus^{\omega} \mathbb Z$
with zero multiplication.
But the definition of IBN in the problem is not the right one. It IS true that when $R$ is a commutative nonunital rin... | 16 | https://mathoverflow.net/users/75735 | 313091 | 136,124 |
https://mathoverflow.net/questions/313093 | 6 | Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality:
For any hermitian matrices $A,B$:
$$
\text{tr}(\exp{(A+B)}) \le \text{tr} \exp{(A)}\exp{(B)}.
$$
Lemmas involve the spectral pinching method which uses the eigendecomposition $A=... | https://mathoverflow.net/users/66064 | Intuitive proof of Golden-Thompson inequality | $\newcommand{\al}{\alpha}
\newcommand{\la}{\lambda}$
Let $\la\_1,\dots,\la\_n$ be the distinct eigenvalues of $A$, so that $|\text{spec}(A)|=n$. Then
\begin{multline}
\sum\_{y=1}^n U\_yXU\_y^\*=\sum\_{y=1}^n\sum\_{u,v=1}^n e^{i2\pi yu/n}P\_{\la\_u}XP\_{\la\_v}e^{-i2\pi yv/n}\\
=
\sum\_{u,v=1}^n P\_{\la\_u}XP\_{\la\_v}... | 4 | https://mathoverflow.net/users/36721 | 313095 | 136,125 |
https://mathoverflow.net/questions/312967 | 1 | Baxter & Rennie at pag. 162 state the following theorem.
Let $W$ be an $n$-dimensional $\mathbb Q$-Brownian motion and let $M\_t=(M\_1(t),...,M\_n(t))$ be an $n$-dimensional $\mathbb Q$-martingale process, which has volatility matrix $(\sigma\_{ij}(t))$, in that $dM\_j(t) = \Sigma\_i \sigma\_{ij}(t) dW\_i(t)$ and th... | https://mathoverflow.net/users/23261 | n-factor martingale representation theorem | The statement and associated definitions in the book seem to gloss over an important assumption: the process $N\_t$ needs to be a martingale with respect to the measure $\mathbb{Q}$ *and* the filtration $\{\mathcal{F}\_t^W\}$ generated by the $n$-dimensional Brownian motion $W(t)$, i.e. $\mathcal{F}\_t^W = \sigma(W(s) ... | 1 | https://mathoverflow.net/users/4832 | 313110 | 136,128 |
https://mathoverflow.net/questions/313057 | 3 | In the context of Euclidean graphs with vertices randomly embedded in either a 2D plane (for instance square with length $L$) or in 3D (similarly, cube of side $L$), where an edge between two given vertices $u,v$ is placed if their euclidean distance $d(u,v)\le \epsilon,$ with $\epsilon$ be being a given threshold dist... | https://mathoverflow.net/users/115841 | Component properties in Euclidean graphs with distance threshold | A geometric random graph $G(\mathbf{X}\_{n},r)$ is given by a threshold distance, $r$, for which vertices, $X$ and $Y$ in the vertex set $\mathbf{X}$ with $\|X-Y\|\leq r$ are connected, and a point process $\mathbf{X}\_{n}:=\bigcup\limits\_{i=1}^n X\_{i}$, where $X\_{i}$ are random variables. According to [M. Penrose](... | 1 | https://mathoverflow.net/users/118731 | 313112 | 136,129 |
https://mathoverflow.net/questions/313113 | 6 | Let $A,B$ be two uncountable sets in a group $G$ such that for any elements $x,y\in G$ the intersection $xA\cap yB$ is finite. Let $\Phi:G\to 2^G$ be a function assigning to each element $x\in G$ some finite set $\Phi(x)\subset G$.
>
> **Question.** Is it true that there exist elements $x,y\in G$ and $a\in A\setmin... | https://mathoverflow.net/users/61536 | A combinatorial property of uncountable groups | Unfortunately (for my further plans) this question has negative answer. Just take any two disjoint uncountable sets $A,B$ and consider the free group $G$ over the union $A\cup B$. Let $\Phi:G\to 2^G$ be the function assigning to each $g\in G$ the set of letters in the irreducible representation of $g$.
Now assume tha... | 2 | https://mathoverflow.net/users/61536 | 313124 | 136,132 |
https://mathoverflow.net/questions/313119 | 5 | I am looking for an English translation of "Les aspects probabilistes du contrôle stochastique" written by Nicole El Karoui, or knowledge whether it exists.
Other references with similar content on Snell envelopes are also appreciated.
Reference info: *Les aspects probabilistes du contrôle stochastique. (French) [T... | https://mathoverflow.net/users/83682 | English translation of "Les aspects probabilistes du contrôle stochastique" | There is no English translation of El Karoui's lecture notes, however her work on Snell envelopes is described in [Reflected Solutions of Backward SDE'S, and Related Obstacle Problems for PDE's](https://www.jstor.org/stable/2959608). For a text book treatment of Snell envelopes see [Methods of Mathematical Finance](htt... | 6 | https://mathoverflow.net/users/11260 | 313125 | 136,133 |
https://mathoverflow.net/questions/313109 | 3 | Let $A \in GL\_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL\_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes the first column of $A$. That is, what can we say about the set
\begin{align\*}
\mathcal E = \{G \in GL\_n(\mathbb R)... | https://mathoverflow.net/users/103704 | Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column? | One can not say that $\mathcal E$ is always connected.
For $n=3, $(and similarly $n>3$) let $A$ be a matrix with $a\_{i1}=1,\quad \forall i \in \{1,2,\ldots,n\} $
Then $\mathcal E$ contains the identity matrix whose determinant is $1$ and it also contains the following matrix with determinant $-1$:
$$\begin{pmat... | 3 | https://mathoverflow.net/users/36688 | 313126 | 136,134 |
https://mathoverflow.net/questions/313131 | 3 | Chebyshev polynomials of the second kind $V\_n(x)$ can be defined as
$$V\_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$
or through the recurrence relation
$$V\_{n+1}=xV\_n-V\_{n-1}, V\_0=1, V\_1=x.$$
First few low-order Chebyshev polynomials of the second
kind are as follows:
$$V\_0=1, V\_1=x,V\_2=x^2-1,V\_3... | https://mathoverflow.net/users/42816 | how to calculate the following integral related to Chebyshev polynomials | $$I\_{nm}=\frac{1}{\pi}\int\_0^\pi \left(\frac{\sin nx}{\sin x}\right)^m dx$$
As stated in the OP, $I\_{nm}=0$ for $n$ even and $m$ odd.
Otherwise, the nonzero $I\_{nm}$ is the *central multinomial coefficient* $c\_{nm}$, the largest coefficient of $(1+x+x^2\cdots x^{n-1})^m$.
A few examples:
$$I\_{44}=44\;\; \tex... | 5 | https://mathoverflow.net/users/11260 | 313137 | 136,135 |
https://mathoverflow.net/questions/313072 | 5 | In the paper ["Poincaré duality and commutative differential graded algebras"](https://eudml.org/doc/272242), Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are two quasi-isomoprhic CDGA, and $\hat{A}$ and $\hat{B}$ be their PD models respectively, ... | https://mathoverflow.net/users/27330 | Naturality of PD model of a CDGA | I guess it depends what you mean by "quasi-isomorphic as Poincaré duality algebras". I suppose you mean: does there exist a zigzag $\hat{A} \gets \cdot \to \cdot \gets \cdot \to \hat{B}$ such that all the intermediary algebras are Poincaré duality algebras, and each map is a quasi-isomorphism that commutes with the fix... | 4 | https://mathoverflow.net/users/36146 | 313141 | 136,137 |
https://mathoverflow.net/questions/313140 | 8 | Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out.
Let $\Omega$ be a compact domain in $\mathbb{R}^n$. For any signed Borel measure $\mu$ on $\Omega$ let $\mu\_+$ denote its positive part (obtained by Hahn-Jordan decomposition). My question is:
> ... | https://mathoverflow.net/users/9652 | Is taking the positive part of a measure a continuous operation? | If $\mu =f\cdot \lambda$ for a positive measure $\lambda$ (i.e., $\mu(A)=\int\_A fd\lambda$), isn't then $\mu\_+= f\_+ \cdot\lambda$ (where $f\_+$ is the positive part $\max\{f,0\}$ of $f$) and $\|\mu\|=\int|f|d\lambda$? Then $\|\mu\_+-\nu\_+\| \le \|\mu-\nu\|$ just follows from Radon-Nikodym (applied to $\lambda=|\mu|... | 9 | https://mathoverflow.net/users/21051 | 313144 | 136,139 |
https://mathoverflow.net/questions/313146 | 6 | It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. We know that $\phi(p^k) = p^{k-1} (p-1)$ for all primes $p$.
What about the multiplicative function $\psi$ defined on... | https://mathoverflow.net/users/10898 | Does this multiplicative function have a name? If so, what is known about it? | $\psi$ is the multiplicative convolution of $\mu^2$ and the identity function, hence its Dirichlet series is
$$\sum\_{n=1}^\infty\frac{\psi(n)}{n^s}=\frac{\zeta(s)\zeta(s-1)}{\zeta(2s)},\qquad\Re(s)>2.$$
This implies by [Perron's formula](https://en.wikipedia.org/wiki/Perron%27s_formula) and standard bounds that
$$\... | 12 | https://mathoverflow.net/users/11919 | 313150 | 136,140 |
https://mathoverflow.net/questions/313134 | 6 | Let $n\in\mathbb{N}$ be a positive integer. Two elements of $\{0,1\}^n$ form an edge if and only if their [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance) equals $1$. It is known that $\{0,1\}^n$ endowed with this graph structure possesses [Hamiltonian cycles](https://en.wikipedia.org/wiki/Hamiltonian... | https://mathoverflow.net/users/8628 | Are Gray codes in $\{0,1\}^n$ isomorphic? | If you consider the Boolean addresses of points on the hypercube as base $2$ expansions of integers, then the vertices of the $4$-dimensional cube may be labeled $0, 1, \ldots, 15$. With this labeling, no two of the following Hamiltonian paths differ by a cube automorphism:
$(0,2,6,7,15,14,10,8,12,13,9,11,3,1,5,4)$,
... | 16 | https://mathoverflow.net/users/75735 | 313156 | 136,143 |
https://mathoverflow.net/questions/313047 | 0 | Let $(X,d)$ be a metric space, and suppose $S\subseteq X$ is a finite subset in which all pairwise distances are distinct (formal definition [here](https://mathoverflow.net/questions/312656/is-there-a-set-s-subseteq-0-1-with-s-2-aleph-0-and-distinct-pairwise)).
If $x\in S$ and $k$ is a non-negative integer with $k<|S... | https://mathoverflow.net/users/8628 | Graphs represented by a subset of a metric space | It is not possible to embed any non-complete graph with a universal vertex into any metric space in this way.
A universal vertex in an $n$-vertex graph has degree $n-1$ and thus $k = n-1$ in any representation of such a graph. But if $|S| = n$ and $k = n-1$, then $G(k,S)$ is a complete graph on $n$ vertices.
| 1 | https://mathoverflow.net/users/97426 | 313160 | 136,145 |
https://mathoverflow.net/questions/313108 | 5 | I have a theoretical question about comparing two objects that I have recently come across.
For concreteness, let us work over the category $C$ of schemes over $k$. Let $G$ be an algebraic group over $k$. One can construct the stack $BG$ as a fibered category in groupoids and as a simplicial scheme $(BG)\_n=G^n$ (wit... | https://mathoverflow.net/users/130207 | $BG$ the stack, $BG$ the simplicial presheaf | The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B\_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player, $BG$, which is the presheaf of ∞-groupoids on $C$ presented by $B\_\bullet G$.
The precise relation between these t... | 16 | https://mathoverflow.net/users/20233 | 313161 | 136,146 |
https://mathoverflow.net/questions/313154 | 3 | I'm looking for a reference or answer for the following question:
Let $M$ be an (compact and orientable, if it helps) smooth manifold and $\nu$ and $\mu$ two differential forms. I'm looking for conditions that allow me to find a vector field $X$ on $M$ such that $\nu=\mathcal{L}\_X\mu$, where $\mathcal{L}$ denotes th... | https://mathoverflow.net/users/69603 | Is there a vector field such that one differential form is the Lie derivative of the other? | If $\mu$ is a volume form and $\nu$ is a top degree form, then there exists a vector field $X$ such that $L\_X\mu=\nu$ if and only if $\nu$ is exact.
You can always fix a metric $g$ on $M$ such that $\mu$ is the volume form
determined by the metric and the orientation, $\mu=dVol\_g$.
Denote by $\DeclareMathOpera... | 4 | https://mathoverflow.net/users/20302 | 313167 | 136,148 |
https://mathoverflow.net/questions/313170 | 7 | Let $ \Delta\_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-1}\_+$ with zero Dirichlet boundary conditions. Does anyhow know where I can find a reference for these?
thanks
Craig
| https://mathoverflow.net/users/66623 | Eigenvalues of Laplace-Beltrami on half sphere | Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $\mathbb{S}^{N-1}$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum of the full sphere. You are searching for the spherical harmonics which vanish on the great circle $x\_N \equiv 0$. The... | 14 | https://mathoverflow.net/users/125275 | 313173 | 136,150 |
https://mathoverflow.net/questions/313165 | 2 | There is a lot of fascination with the [Vandermonde determinant](https://en.wikipedia.org/wiki/Vandermonde_matrix) and for many good reasons and purposes. My current quest is more of number-theoretic.
>
> **QUESTION.** Let $p\equiv 3$ (mod $4$) be a prime. What is the value of
> $$\prod\_{1\leq i<j\leq\frac{p-1}2}... | https://mathoverflow.net/users/66131 | Vandermonde determinant: modulo | I saw this in [this recent arxiv paper](https://arxiv.org/abs/1809.07766) by Zhi-Wei Sun (Quadratic residues and related permutations and identities, version 6, equation (1.5)). I'll reproduce the proof in case it goes under further revisions. Every equality below is $\pmod{p}$. Start by factoring
$$\prod\_{1\le i<j\... | 6 | https://mathoverflow.net/users/2384 | 313174 | 136,151 |
https://mathoverflow.net/questions/313191 | 15 | I've got ten (projective) planes in projective 3-space:
\begin{align}
&x=0\\
&z=0\\
&t=0\\
&x+y=0\\
&x-y=0\\
&z+t=0\\
&x-y-z=0\\
&x+y+z=0\\
&x-y+t=0\\
&x+y-t=0
\end{align}
If I did not make a mistake somewhere, their intersections produce $25$ lines and $15$ points, each line containing $3$ points, each plane contain... | https://mathoverflow.net/users/41291 | Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space | This configuration has automorphisms by the symmetric group $S\_5$,
and can be identified with the planes $a\_i = a\_j$ ($0 \leq i < j \leq 4$)
in the projective 3-space $a\_0+a\_1+a\_2+a\_3+a\_4 = 0$, by setting
$$
(x,z,t,y) = (a\_0 - a\_1, 2(a\_2 - a\_3), 2(a\_4 - a\_2), a\_0 + a\_1 - 2a\_2).
$$
the ten planes then h... | 19 | https://mathoverflow.net/users/14830 | 313198 | 136,156 |
https://mathoverflow.net/questions/312937 | 9 | Let $A$ be a $k \times n$ orthogonal matrix; i.e., $AA^T = I\_{k \times k}$. For $1 \leq j \leq n$, let the squared norm of the $j$-th column of $A$ be denoted by $\alpha\_j^2$; i.e.,
$$\sum\_{i=1}^k a\_{ij}^2 = \alpha\_j^2.$$
Let $\Lambda = \operatorname{Diag}(\lambda\_1, \lambda\_2, \dotsc, \lambda\_n)$ be a positive... | https://mathoverflow.net/users/20062 | Matrix determinant inequality proof without using information theory | Here's a more general result.
From [Choi's inequality](https://www.projecteuclid.org/download/pdf_1/euclid.ijm/1256051007), we know that for a positive linear map $\Phi$ and an operator convex function $f$, we have $\Phi(f(\Lambda)) \ge f (\Phi(\Lambda))$. Let $\Phi(\Lambda)=A\Lambda A^T$, and let $f(t)=-\log t$, th... | 8 | https://mathoverflow.net/users/8430 | 313199 | 136,157 |
https://mathoverflow.net/questions/313162 | 1 | Let $k > 1,$ and $a=a\_1a\_2...a\_{2k-1}$ be a binary string, i.e. $a\_i\in \{0,1\}$. Consider contiguous substrings of $a$ of length $k (k-$substrings$): b\_i := a\_ia\_{i+1}...a\_{i+k-1}, 1\leq i\leq k$. Weight of a string $x$, $w(x)$, is the number of ones in the string. For the given string $a$ consider the multise... | https://mathoverflow.net/users/84871 | $k$-substrings of a binary string | The equivalence classes of multisets may be described by integer sequences with length $k$ satisfying the following property:
$a\_n ≤ k$
$a\_n ≥ 0$
$a\_{n} ≤ a\_{n+1} ≤ a\_{n} + 1$ for all $1≤n≤k-1$
Denote the number of such sequences $b\_k$.
We have
$b\_{k+1}= 2b\_k + 2^{k-1}$, as the $2b\_k$ part comes from... | 2 | https://mathoverflow.net/users/125498 | 313203 | 136,159 |
https://mathoverflow.net/questions/313148 | 0 | Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & 0 & 0 & 0 \\
-\frac12 & -d+1 & d & 0 & -\frac12 & 0 & 0 & 0 \\
0 & -\frac12 & 0 & d & -d+1 & -\frac12 & 0 & 0 \\
0 & 0 & -\frac12 & -d+1 & d & 0 & -\frac12 & 0 \\
0 & 0 & 0 & -... | https://mathoverflow.net/users/95387 | Matrix equation of the form $C A C^\intercal = D$ | Answered in the [crosspost](https://math.stackexchange.com/questions/2960731/matrix-equation-of-the-form-c-a-c-intercal-d).
Credit to [@amd](https://math.stackexchange.com/users/265466/amd).
| 1 | https://mathoverflow.net/users/95387 | 313207 | 136,161 |
https://mathoverflow.net/questions/312433 | 6 | I believe I once had a proof of this proposition, but it's been lost to the mists of time and old hard drives, so who knows if it was correct, and try as I might I can't seem to reproduce it.
Is it possible, in Melliès' [tensorial logic](https://www.irif.fr/~mellies/tensorial-logic.html), to give a proof of ¬(¬1 ⊗ ¬1... | https://mathoverflow.net/users/129921 | Proof of ¬(¬1 ⊗ ¬1) in tensorial logic | Your notation is a bit confusing because usually in linear logic and related systems $\top$ denotes the unit of additive conjuction (i.e., categorically, the terminal object), but then when you formulate your question in categorical terms you clearly say that by $\top$ you mean the monoidal unit, which instead is tradi... | 6 | https://mathoverflow.net/users/45027 | 313213 | 136,163 |
https://mathoverflow.net/questions/312979 | 6 | In the paper [*Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries* (page 2)](https://www.egr.msu.edu/~aviyente/elad06.pdf), the authors rewrite the minimization problem
\begin{align}
\min\_{\alpha \in \mathbb R^k} \| \alpha \|\_0 && s.t. && \|D \alpha - y \|\_2^2 \leq T,
\end{align}
... | https://mathoverflow.net/users/73571 | Adding constraints as penalty with $\| \cdot \|_0$ norm | **The claim in the paper is false.**
Since the problem is not convex, the claim does not follow from general results. However, there are some results in this direction in quite general cases:
If $x^\*$ is the unique minimizer of $\min\_x F(x) + G(x)$, then it is a solution of the constrained problem
$$
\min\_x F(x)... | 3 | https://mathoverflow.net/users/9652 | 313214 | 136,164 |
https://mathoverflow.net/questions/312948 | 4 | The notes here <https://web.eecs.umich.edu/~cpeikert/lic13/lec04.pdf> have the note 'Small decryption exponent $d$: so far the best known attack recovers $d$ if it is less than $N^{.292}$. This uses a bivariate version of Coppersmith that lacks a rigorous proof of correctness, but seems to work well in practice'. It lo... | https://mathoverflow.net/users/10035 | Is total degree version and $x,y$ degree version of Coppersmith's theorem correct? | I was asked to comment here, but this question seems more relevant to the cryptography stackexchange. A move might be appropriate?
Anyway, there are three issues here.
1. I believe that the same argument as the one we presented in our *Note on the Bivariate Coppersmith Theorem* applies to Theorem 3 as well, but I h... | 3 | https://mathoverflow.net/users/37657 | 313216 | 136,166 |
https://mathoverflow.net/questions/313206 | 4 | TituRel is a language developed in conjunction with the book Relational Mathematics by Gunther Schmidt from Universität der Bundeswehr München.
My problem is that I can't find any info related to downloading this software.
Do you know where the TituRel software can be downloaded from? The [home page](https://titure... | https://mathoverflow.net/users/130287 | What happened to TituRel? | The [swMATH description](https://www.swmath.org/software/8501) says that: ***TituRel*** has a common source with the widely known **RelView**. It is far less efficient, strictly functional, and visualization-oriented. A formal deployment has not yet taken place; it exists as a one-man endeavour.
So I guess this expla... | 6 | https://mathoverflow.net/users/11260 | 313220 | 136,168 |
https://mathoverflow.net/questions/313238 | 2 | Let $M$ be a generic $2n\times 2n$ matrix and fix $k\leq n$.
Suppose $\mathcal{F}$ is a family of submatrices under the conditions that $A\in\mathcal{F}$ provided
(a) $A$ is a $k\times k$ submatrix of $M$, and
(b) $A$ overlaps with each and every $B\in\mathcal{F}$.
>
> **QUESTION.** What is the best upper b... | https://mathoverflow.net/users/66131 | On submatrices: size bound | To each such matrix $A$ we can associate $A\_1,A\_2$, the sets of column indices and row indices respectively. The families $\mathcal F\_i=\{A\_i| A\in \mathcal F\}$ are intersecting families of subsets, therefore by [Erdos-Ko-Rado](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem) we have
$$|\m... | 5 | https://mathoverflow.net/users/2384 | 313241 | 136,173 |
https://mathoverflow.net/questions/313211 | 1 | To what extent is it possible to formally susbstantiate the following affirmation that: "In a classical first order logical universe with exactly one unique and single object, the predicate calculus practically reduces to the propositional calculus" ?
Gérard Lang
| https://mathoverflow.net/users/30395 | Reduction of the predicate calculus to the propositional calculus in the case of one sigle object in the universe? | It's quite simple: in a structure with a single object - and for simplicity, let's assume only unary relations at first - we think of each unary relation as a single atomic proposition, and of the single object as a "propositional world." (What about multiple worlds? Well, thinking in this direction can take us into pr... | 3 | https://mathoverflow.net/users/8133 | 313243 | 136,174 |
https://mathoverflow.net/questions/313240 | 4 | Let $n$ be a positive integer. The Euler $\phi$-function is defined by
$$\displaystyle \phi(n) = \# \{1 \leq a \leq n-1 : \gcd(a,n) = 1\}.$$
It is in fact a multiplicative function, and one has the formula
$$\displaystyle \phi(n) = n \prod\_{p | n} \left(1 - \frac{1}{p}\right).$$
Let $S\_\phi(X)$ be the number... | https://mathoverflow.net/users/10898 | Density of numbers $n$ which are co-prime with their $\phi$-value | Oeis has this as sequence [A003277](https://oeis.org/A003277). Erdos proved in ["Some asymptotic formulas in number theory"](https://users.renyi.hu/~p_erdos/1948-11.pdf) that
$$S\_{\phi}(X)=(1+o(1))\frac{Xe^{-\gamma}}{\log\log\log X}$$
| 8 | https://mathoverflow.net/users/2384 | 313244 | 136,175 |
https://mathoverflow.net/questions/304170 | 4 | An *ordered topological space* is a topological space $X$ equipped with a partial order $\leq$ which is closed as a subset of $X\times X$. By antisymmetry of $\leq$, it follows that the diagonal of $X$ is closed as well, so that $X$ is necessarily Hausdorff.
The [*stochastic order*](https://en.wikipedia.org/wiki/Stoc... | https://mathoverflow.net/users/27013 | Antisymmetry of the stochastic order | I have been able to answer this question in the positive [for all ordered topological spaces](https://arxiv.org/abs/1810.06771).
The proof is completely elementary and unexpectedly simple.
| 3 | https://mathoverflow.net/users/27013 | 313251 | 136,178 |
https://mathoverflow.net/questions/313249 | 3 | Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix is positive semi-definite. What if anything is known about this measure, and in particular about the distribution of the... | https://mathoverflow.net/users/105314 | Eigenvalues of random matrix conditional on positive definiteness | The average density of states in the Gaussian ensembles whose eigenvalues are restricted to lie in a given interval, or to be greater than a given minimal value, was calculated by Dean and Majumdar in [Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices](https://arxiv.org/abs/0801.1730). The eigenvalue ... | 4 | https://mathoverflow.net/users/11260 | 313264 | 136,181 |
https://mathoverflow.net/questions/312971 | 7 | Suppose that $\phi(x)$ is a $\Delta^{1}\_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the set of constructible reals, roughly because $\Delta^{1}\_{2}$ cannot distinguish between sufficiently high Cohen-gener... | https://mathoverflow.net/users/49491 | Can $\Delta^{1}_{2}$ separate degrees of constructibility? | Expanding on Douglas Ulrich's answer:
Note that this question is only meaningful when there are nonconstructible reals since otherwise there is only one constructibility degree.
If there is a nonconstructible real and $A$ is $\Delta\_2^1$ set, then either $A$ or $\mathbb{R} \setminus A$ is a $\Sigma\_2^1$ set of r... | 5 | https://mathoverflow.net/users/43354 | 313266 | 136,182 |
https://mathoverflow.net/questions/313230 | 2 | Let $A = \{ a\_i + b\_i \mathbb{N} \}\_{i=1}^{k}$, where $a\_1, \ldots, a\_k \in \mathbb{N} \cup \{0\}$ and $b\_1, \ldots, b\_k \in \mathbb{N}$ be a system of arithmetic sequences.
For a positive integer $m$, system $A$ is called a *$m$-cover of $\mathbb{N}$*, if every natural number is covered by $A$ at least $m$ ti... | https://mathoverflow.net/users/83519 | Decide if a system of arithmetic sequences is an $m$-cover of $\mathbb{N}$ | In 1973， L. J. Stockmeyer and A. R. Meyer [Proc. 5th. Ann. ACM Symp. on Theory of Computing, Assoc. for Computing Machinery] proved that the question whether a given system $A=\{a\_i+b\_i\mathbb N\}\_{i=1}^k$ is a cover of $\mathbb Z$ (i.e., $1$-cover) is co-NP-complete. Thus NP=P if and only if we can decide whether $... | 6 | https://mathoverflow.net/users/124654 | 313270 | 136,184 |
https://mathoverflow.net/questions/313229 | 4 | Suppose that $ Y $ is a scheme and $ f\colon X\to Y $ a covering of $ Y $ in some Grothendieck topology on the category of schemes (i.e. if $\{ U\_i\to Y\}$ is a covering in the topological sense, then $ X = \coprod U\_i$). Then we may consider the descent groupoid $\Gamma\_0 = X$ and $ \Gamma\_1 = X \times\_{Y} X $ wi... | https://mathoverflow.net/users/80467 | When can a scheme be recovered from its descent groupoid? | In any topos, if $Y \rightarrow X$ is an epimorphism then:
$$Y \times\_X Y \rightrightarrows Y \rightarrow X $$
is indeed a colimit diagram.
If you have a site $S$ and a cover $Y \rightarrow X$ then it is an epimorphisms in the topos of sheaves, and if the fiber product $Y \times\_X Y$ exists, then it also the f... | 5 | https://mathoverflow.net/users/22131 | 313292 | 136,193 |
https://mathoverflow.net/questions/304887 | 3 | I'm having trouble with an estimate that would be helpful in information geometry.
The background is the following. Suppose we have a smooth positive function $g:X \to \mathbb{R}^+$ where $X$ is a compact manifold without boundary with $Vol(X) = 1$. We also assume that $g$ satisfies $\int\_X g^{10} dx =1.$
I want to... | https://mathoverflow.net/users/125275 | Interpolation inequality related to the 5/3-Laplace operator | We have
$$
DF(g)=\frac{\|\nabla g \|\_{L^{5/3}(X)}}{ \|g\|\_{L^{10}(X)} -\|g\|\_{L^1(X)}},$$
which is a homogeneous expression. Hence we can drop the condition $\int\_X g^{10}dx=1$ and replace it by $\|g\|\_{L^1(X)}=\int\_X g dx=1$. If $dim(X)=2$ we have by the Sobolev and the triangle inequality
$$
DF(g)=\frac{\|\na... | 1 | https://mathoverflow.net/users/100908 | 313295 | 136,194 |
https://mathoverflow.net/questions/313293 | 3 | Let $\mathbb{N}$ denote the set of positive integers. The *Collatz function* $f:\mathbb{N}\to\mathbb{N}$ is given by $f(n) = n/2$ for $n$ even and $f(n) = 3n+1$ for $n$ odd. Given $k\in\mathbb{N}$ we associate to $k$ its *Collatz sequence* $(c^{(k)}\_n)\_{n\in\mathbb{N}}$ given inductively by $$c^{(k)}(1) = k\text{ and... | https://mathoverflow.net/users/8628 | Two reasons why the Collatz conjecture could fail | The current state of research excludes none of the two cases. The occurrence of some small cycles has been actually ruled out. You can easily find the relevant references in the [Wikipedia page](https://en.wikipedia.org/wiki/Collatz_conjecture) devoted to Collatz conjecture (the same that you linked), at the section "C... | 19 | https://mathoverflow.net/users/7460 | 313296 | 136,195 |
https://mathoverflow.net/questions/313017 | 7 | Where can I find a (readable and self-contained) proof of the following result?
>
> Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\mathrm{div}(A(x)\nabla u) = \delta\_0,$$
> $$u|\_{\partial \Omega} = 0,$$
> where $A$ is bounded, measurable and uni... | https://mathoverflow.net/users/nan | Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs | Perhaps the best self contained reference on this result is the book [2] by Stampacchia himself. It is a set of typewritten course notes in French, taken from a graduate course on elliptic equations held by Stampacchia at the [Centre de Recherches Mathématiques](http://www.crm.umontreal.ca) of the Montreal University i... | 5 | https://mathoverflow.net/users/113756 | 313297 | 136,196 |
https://mathoverflow.net/questions/313298 | 2 | Let $T(t)$ be a $C\_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dissipative operator then it generates a $C\_0$-semigroup of contractions, i.e., $$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$
My question here: is there any theorem which g... | https://mathoverflow.net/users/124904 | Generation of strict contraction semigroups | Your conditions is for contraction semigroups equivalent to have uniform exponential stability, i.e., to have growth bound less than zero, see [Proposition V.1.7](https://www.math.uni-tuebingen.de/arbeitsbereiche/agfa/members/rana/engelnagel_oneparametersemigroups.pdf). in
*Engel, Klaus-Jochen; Nagel, Rainer*, [**One... | 5 | https://mathoverflow.net/users/12898 | 313302 | 136,197 |
https://mathoverflow.net/questions/313128 | 8 |
>
> **Problem 1.** Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that
>
>
> 1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and
>
>
> 2) for any function $\Phi:G\to 2^G$ assigning to each element $g\in G$ a finite subset $\Phi(g)\subset G$ there are two elements... | https://mathoverflow.net/users/61536 | A combinatorial property of uncountable groups, II | Problems 1 and 2 both have affirmative answers (implying that the finitary ballean of any uncountable group is normal).
Two cases are possible:
I. There exists a countable subgroup $A\subset G$ and an uncountable subset $B\subset G$ such that $bAb^{-1}\cap A$ is infinite for all $b\in B$. Replacing $B$ by a smaller... | 3 | https://mathoverflow.net/users/61536 | 313308 | 136,199 |
https://mathoverflow.net/questions/313268 | 4 |
>
> Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
> When is the functor $\Omega^1 : \underline{C} \rightarrow \underline{C}$ an equivalence?
>
>
>
We can assume $C$ is a module category of a ring $R$ in... | https://mathoverflow.net/users/61949 | When is $\Omega^1$ an equivalence? | There's an elementary proof that if $\Omega$ is a self-equivalence of $\underline{C}$ then $C$ also has enough injectives, and projectives and injectives coincide. In particular, this shows that if $C$ is $\text{Mod-}R$ for a ring $R$, then $R$ is quasi-Frobenius.
For any $C$ with enough projectives, it's straightfor... | 5 | https://mathoverflow.net/users/22989 | 313311 | 136,200 |
https://mathoverflow.net/questions/313155 | 9 | Let $p$ be a prime and $\mathbb{F}$ a finite field of characteristic $p$. The theorem of Khare and Wintenberger roughly states that an irreducible, odd Galois representation $\bar{\rho}:G\_{\mathbb{Q}}\rightarrow GL\_2(\mathbb{F})$ which is ramified at finitely many primes lifts to a modular Galois representation assoc... | https://mathoverflow.net/users/nan | Is there a clear-cut analogue of the strong form of Serre's Conjecture for residually reducible Galois Representations? | Billerey and Menares have studied this question for the reducible representations $\bar{\rho} = 1 \oplus \chi\_p^{k-1} $ in <https://arxiv.org/abs/1309.3717>
In this case the prime-to-$p$ part of the Artin conductor is 1, and it is not always the case that $\bar{\rho} $ arises from a cuspidal eigenform of weight $k $ a... | 6 | https://mathoverflow.net/users/6506 | 313332 | 136,203 |
https://mathoverflow.net/questions/313347 | -2 | What is an example of a connected Hausdorff space $X$ with $|X|>1$ and a surjective continous map $f:X\to (X\times X)$?
| https://mathoverflow.net/users/8628 | Example of connected Hausdorff space $X$ and surjective continous map $f:X\to X\times X$ | $X=[0,1]$. See [Space-filling curve.](https://en.wikipedia.org/wiki/Space-filling_curve)
| 4 | https://mathoverflow.net/users/75735 | 313348 | 136,208 |
https://mathoverflow.net/questions/313343 | 6 | I have some questions about Turing machines. Is there an embedding method where you embed Turing machines, finite automata into continuous space or graphs? Or are there geometrical approaches to analyze Turing machines or automata? ( embed them into some spaces and apply geometrical tools) Thank you
| https://mathoverflow.net/users/130340 | Embedding Turing machine | Let $\mathcal{M}$ be a class of binary functions acting on strings in $\Sigma^\*$,
along with a "size" function $|\cdot|:\mathcal{M}\to\{1,2,\ldots\}$ with the property that there are only finitely many $M\in\mathcal{M}$ of a given size.
Turing machines and finite automata are of this form, where size can be taken as t... | 3 | https://mathoverflow.net/users/12518 | 313355 | 136,210 |
https://mathoverflow.net/questions/313315 | 5 | Take $S^2$ with its standard metric. The space of great circles in $S^2$ can be identified with the real projective plane $\mathbb{R}P^2$. Let $X$ be an embedded circle in $S^2$; associate to it a function $f\_X:\mathbb{R}P^2\rightarrow \mathbb{Z}\cup\{\infty\}$ which counts the number of intersection points (with mult... | https://mathoverflow.net/users/130328 | Reconstructing a curve in $S^2$ from intersections with great circles | This is related to the Funk transform. To a continuous function on the sphere it associates a map from the set of great circles to $\mathbb{R}$ that sends a great circle to the integral of the function along it. This transformation is invertible on even functions, that is one can reconstruct such a function from its in... | 2 | https://mathoverflow.net/users/98590 | 313357 | 136,212 |
https://mathoverflow.net/questions/313361 | 10 | In Johnstone's *Topos Theory* appears the following lemma.
**7.41 Lemma.** Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V\_j\to V)$ be a finite $J$-covering family in $\mathsf C$, and $v$ and element of $P(h\_V)$. There there exists a refine... | https://mathoverflow.net/users/69037 | Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem | Johnstone's proof based on this lemma follows in fact the original proof of Deligne which can be found in the appendix "Criteria for the existence of points" to SGA 4, vol. VI. Both the original and a TeX reedition from 2015 can be found [here](http://fabrice.orgogozo.perso.math.cnrs.fr/SGA4/index.html).
The idea be... | 13 | https://mathoverflow.net/users/12976 | 313372 | 136,216 |
https://mathoverflow.net/questions/284718 | 12 | Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its *flow*, i.e. the unique solution to the initial-value problem
\begin{equation}
\begin{cases}
\frac{\partial}{\partial t} \mathbf X(t,x) = \mathbf V(t,\mathbf X(t,... | https://mathoverflow.net/users/111164 | Derivative of the flow for ODEs on manifolds | I think you may be interested in this remarkable, recent [paper](http://cvgmt.sns.it/media/doc/paper/3769/Ahlfors-final.pdf) by E. Brué and D. Semola (see in particular Theorem 3.11 which answers to your question in a much more general setting).
| 3 | https://mathoverflow.net/users/nan | 313384 | 136,221 |
https://mathoverflow.net/questions/313353 | 1 | Let $\{f\_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f\_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ has points of isolated disconitnuty on a dense set of measure zero. Sobolev embedding says that $f\_n$ cannot converge to... | https://mathoverflow.net/users/14414 | Unboundedness of the Sobolev norm of a sequence : Does it follow from Sobolev embedding? | In fact, for $k > \frac{m}{2}$, any sequence of continuous functions $f\_n$ which is bounded in $H^k$ norm must have a subsequence converging uniformly on compact sets. This proves your claim, since if $f\_n$ converges pointwise to a discontinuous function $f$, then every subsequence converges pointwise to the same $f$... | 2 | https://mathoverflow.net/users/4832 | 313393 | 136,226 |
https://mathoverflow.net/questions/313374 | 35 | Lagrange's four square theorem states that every non-negative integer is a sum of squares of four non-negative integers. Suppose $X$ is a subset of non-negative integers with the same property, that is, every non-negative integer is a sum of squares of four elements of $X$.
$\bullet$ Is $X=\{0,1,2,\ldots\}$?
$\bul... | https://mathoverflow.net/users/40723 | Lagrange four squares theorem | The set $X$ doesn't have to be the set of non-negative integers. This was known already to [Härtter and Zöllner](https://zbmath.org/?q=an%3A0341.10043) in 1977, who constructed an $X$ of the form $\{ 0, 1, 2, \ldots \} \setminus S $ for an infinite $S$.
For any $\varepsilon>0$, [Erdös and Nathanson](https://www.scie... | 43 | https://mathoverflow.net/users/31469 | 313394 | 136,227 |
https://mathoverflow.net/questions/313391 | 5 | It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (extrinsic) question has a simple answer: let me consider the natural embedding of $\mathrm{SO}(n)$ into $\mathbb{R}^{n... | https://mathoverflow.net/users/69071 | Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n) | What you are asking about is the second fundamental form of the embedding. Since this is a Lie group, it's enough to know what the second fundamental form is at the identity matrix $I\_n=e$. Since the tangent space of $\mathrm{SO}(n)$ at the identity is the space of $n$-by-$n$ skew-symmetric matrices and the normal spa... | 12 | https://mathoverflow.net/users/13972 | 313403 | 136,230 |
https://mathoverflow.net/questions/313255 | 9 | On page 9 of Kauffman's Formal Knot theory, Kauffman claims
>
> The Alexander-Conway Polynomial is a true refinement of the Alexander Polynomial. Because it is defined absolutely (rather than up to sign and powers of variables) it is capable of distinguishing many links from their mirror images - a capability not a... | https://mathoverflow.net/users/101463 | Are there knots that can be distinguished by the Alexander-Conway polynomial, but not the Alexander polynomial? | Let $\overline{L}$ denote the mirror image of a link $L$. The (reduced) Alexander-Conway polynomial $D\_L(t)$ of an $n$-component link $L$ satisfies $D\_{\overline{L}}(t)=(-1)^{n+1}D\_L(t)$. More generally, the multivariable potential function $\nabla\_L(t\_1,\ldots,t\_n)$ also satisfies $\nabla\_{\overline{L}}(t\_1,\l... | 8 | https://mathoverflow.net/users/36098 | 313405 | 136,231 |
https://mathoverflow.net/questions/313392 | 6 | I encountered this small combinatorial problem and do not quite know how to solve it:
Consider a set $\mathbf N:=\left\{1,2,....,N \right\}.$ This set has $\binom{N}{2}$ many subsets of cardinality $2.$ Thus, we can introduce variables $x\_1,...,x\_{\binom{N}{2}}$ taking a value in $\mathbb R$ where each of the varia... | https://mathoverflow.net/users/130130 | Is this bound uniform in $N$? | Let $u\_i\in \mathbb{R}^N$ be a vector with two coordinates equal to 1 and other equal to 0, corresponding to the characteristic vector of the set $M\_i$. Then your inequality rewrites as $N^{-1}(\sum x\_i)^2\leqslant C(\sum x\_i u\_i)^2$. Denote $v=(1,1,\dots,1)\in \mathbb{R}^N$. Then $(u\_i,v)=2$ (here $(u,v)$ stands... | 7 | https://mathoverflow.net/users/4312 | 313409 | 136,232 |
https://mathoverflow.net/questions/313381 | 7 | Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the following condition:
$$\nu\_n(O) \to \nu(O)$$
for every open set $O\subset K$ is sufficient for weak\* convergence of $\nu\_... | https://mathoverflow.net/users/130349 | Weak*-convergence of signed measures | The answer is positive if the sequence $(\nu\_n)$ is bounded; please see Theorem IV.9.15 in Dunford/Schwartz, vol. 1. PS: I just notice that this was already observed by Christian a couple of minutes ago.
| 2 | https://mathoverflow.net/users/127871 | 313417 | 136,234 |
https://mathoverflow.net/questions/313420 | 4 | Consider a (co)bordism invariant
$$
u\_2 Sq^1 u\_2+Sq^2 Sq^1 u\_2
$$
obtained from
$$
\Omega^5\_{O}(K(\mathbb{Z}/2,2)).
$$
Here $u \in H^2(K(\mathbb{Z}/2,2),\mathbb{Z}\_2)$. The $K(\mathbb{Z}/2,2)$ is Eilenberg–MacLane space. The $\mathbb{Z}/2$ is the finite group of order 2.
>
> Question: Is it true that such a (... | https://mathoverflow.net/users/106497 | (Co)bordism invariant of Eilenberg–MacLane space becomes vanished | $\newcommand{\Pin}{\mathrm{Pin}}\newcommand{\Sq}{\mathrm{Sq}}$
To simplify notation, I'll denote $(\Pin\_n^\pm\times\mathrm{SU}\_2)/(\mathbb Z/2)$ by $\Pin\_n^{h\pm}$, since these are
analogues of the groups $\mathrm{Spin}\_n^h = (\mathrm{Spin}\_n\times\mathrm{SU}\_2)/(\mathbb Z/2)$. Analogously to how a
spin$c$-struct... | 8 | https://mathoverflow.net/users/97265 | 313429 | 136,240 |
https://mathoverflow.net/questions/313445 | 1 | Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $x$ be a positive element of $M$ with $\|x\|=1$. So, $(x^n)\_{n\in\mathbb N}$ is a decreasing sequence of positive elements and $y:=\inf\_{n\in\mathbb N}x^n$ does exist. My question is under what hypothesis we have $y\neq0$. When $H$ is finite dimensio... | https://mathoverflow.net/users/84700 | When is $\inf_{n\geq0}x^n\neq0$? | Basically, Borel functional calculus translates pointwise convergence of functions into convergence in the strong operator topology. Since the functions $(x^n)$ converge to the function $\mathbb{1}\_{\{1\}}$, $y=\inf x^n$ is equal to the projection onto the eigenspace of $x$ corresponding to the eigenvalue $1$, i.e. it... | 12 | https://mathoverflow.net/users/24953 | 313446 | 136,242 |
https://mathoverflow.net/questions/313366 | 4 | For a cardinal $\kappa$ let $[\kappa]^{<\kappa}$ denote the family of subsets of cardinality $<\kappa$ in $\kappa$. The family $[\kappa]^{<\kappa}$ is endowed with the partial order of inclusion. A simple diagonal argument shows that for any infinite cardinal $\kappa$ the poset $[\kappa]^{<\kappa}$ has cofinality $\ge\... | https://mathoverflow.net/users/61536 | The cofinality of the poset $[\kappa]^{<\kappa}$ for a singular cardinal $\kappa$ | If $\kappa$ is singular, the cofinality of the poset $[\kappa]^{<\kappa}$ is $>\kappa$.
Indeed, $\kappa$ singular means that there is a limit ordinal $\alpha<\kappa$ and an increasing family $(\lambda\_\xi)\_{\xi<\alpha}$ such that $\lambda\_\xi<\kappa$ for each $\xi$ and $\sup\_{\xi<\alpha}\lambda\_\xi=\kappa$.
S... | 3 | https://mathoverflow.net/users/14094 | 313447 | 136,243 |
https://mathoverflow.net/questions/313367 | 3 | For a second-countable space $X$, we have $${\rm Bor}(X\times X) = {\rm Bor}\, X \otimes {\rm Bor}\, X,$$ that is the Borel $\sigma$-algebra of the product is the product $\sigma$-algebra. Some [counterexamples to this statement can be produced for uncountable discrete spaces](https://mathoverflow.net/questions/39882/p... | https://mathoverflow.net/users/130349 | Borel $\sigma$-algebra in $\beta \mathbb N \times \beta \mathbb N$ | The answer is no.
Jiří Nedoma proved that if $(X,\Sigma)$ is a measurable space $|X| > 2^{\aleph\_0}$, then the diagonal is not a measurable subset of $(X\times X, \Sigma \otimes \Sigma)$. (The article is called *Note on Generalized Random Variables*, the result is Lemma 2, a proof can also be found in Schechter's *H... | 4 | https://mathoverflow.net/users/61785 | 313451 | 136,245 |
https://mathoverflow.net/questions/201984 | 18 | It is known from the general theory that, given a bialgebra (over a field $k$)
\begin{equation}
\mathcal{B}=(B,\mu,1\_B,\Delta,\epsilon)
\end{equation}
the Sweedler's dual $\mathcal{B}^0$ (called also Hopf or restricted dual, i.e. the space of linear functionals $f\in B^\*$ such that $f\circ\mu\in \mathcal{B}^\*\otime... | https://mathoverflow.net/users/25256 | Bialgebras with Hopf restricted (or Sweedler) duals | Actually your guess is true. Consider the *left* Hopf algebra $\widetilde{SL\_q(2)}$ from [Iyer, Taft, *The dual of a certain left quantum group*](https://doi.org/10.1142/S0219498816500729). This is constructed essentially as $SL\_q(2)$, but imposing just half of the relations. Namely, if $SL\_q(2)$ is given by $\Bbbk\... | 5 | https://mathoverflow.net/users/105816 | 313454 | 136,246 |
https://mathoverflow.net/questions/313452 | 10 | As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: G\rightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $H$ is non-residually finite, such that $\ker f$ is finitely generated?
| https://mathoverflow.net/users/128887 | Residually finite group surjective to nonresidually finite group with finitely generated kernel | You can obtain such a surjection for any finitely presented non-residually finite group $H$ using Daniel Wise's residually finite Rips construction, which is the main result of this paper: [A Residually Finite Version of Rips's Construction.](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024609302001406... | 11 | https://mathoverflow.net/users/95340 | 313458 | 136,248 |
https://mathoverflow.net/questions/313402 | 2 | For a cardinal $\kappa$ by $[\kappa]^{<\kappa}$ we denote the family of all subsets of cardinality $<\kappa$ in $\kappa$.
>
> **Question.** Assume that for an infinite cardinal $\kappa$ there exists a family $\mathcal B\subset[\kappa]^{<\kappa}$ of cardinality $|\mathcal B|=\kappa$ such that for any $A\in[\kappa]^... | https://mathoverflow.net/users/61536 | A possible characterization of regular cardinals? | Yes. The assumption implies that $\mathcal{B}$ is cofinal in the poset $[\kappa]^{<\kappa}$, and hence that this poset has cofinality $\le\kappa$. This implies that $\kappa$ is regular, [by this answer](https://mathoverflow.net/a/313447/14094).
| 3 | https://mathoverflow.net/users/14094 | 313459 | 136,249 |
https://mathoverflow.net/questions/313464 | 10 | In a lot of problems in linear algebra one uses the existence, for each $E$ vector space over a field $k$, and each $x\in E$, of a Hyperplane $H$ such that $E=k\cdot x \oplus H$ (Let us denote $\mathcal{P}$ this property). With Zorn's Lemma, the existence of a such $H$ is trivial. However, as this seems weaker than the... | https://mathoverflow.net/users/123269 | Relation between the Axiom of Choice and a the existence of a hyperplane not containing a vector | It is not hard to see that this statement is equivalent to "In every vector space, for every vector $v$ there is a functional $f$ such that $f(v)=1$".
If $\cal P$ holds, then the projection onto $k\cdot x$ is a linear functional which is nontrivial; if there is a nontrivial functional then choose $x$ which is mapped... | 20 | https://mathoverflow.net/users/7206 | 313469 | 136,251 |
https://mathoverflow.net/questions/313350 | 2 | $$(x + y + z)(x + y\omega\_n + z\omega\_n^{n-1})(x + y\omega\_n^2 + z\omega\_n^{n-2})....(x + y\omega\_n^{n-1} + z\omega\_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega\_n$ is an *n*th root of unity.
The question is to find the polynomial $P$.
I have tried to manually multiply the terms of LHS and then equate the co... | https://mathoverflow.net/users/130300 | $(x + y + z)....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$ | As suggested in the first comment, computationally it looks like $$P\equiv P\_n=\dfrac{x^n}{t^n}L\_n(t)-x^n=\dfrac{x^n}{t^n}(L\_n(t)-t^n), $$
where $L\_n$ is the $n$th [Lucas polynomial](http://mathworld.wolfram.com/LucasPolynomial.html) in $t:=\dfrac{{ix}}{\sqrt{yz^{\phantom l}}}$.
E.g. for $n=6$, $$P=6x^4yz-9x^2y... | 3 | https://mathoverflow.net/users/29783 | 313478 | 136,255 |
https://mathoverflow.net/questions/313000 | 6 | First we give some definitions from Section 3 of the paper [Monomials, Binomials, and Riemann-Roch](https://link.springer.com/content/pdf/10.1007%2Fs10801-012-0386-9.pdf) by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally we provide an example that seems to contradict ... | https://mathoverflow.net/users/94968 | Reflection-invariant monomial ideals and Alexander duality | **Note**: The following is the result of an email exchange with one of the coauthors of the paper cited in the OP.
In the definition of a reflection-invariant monomial ideal the requirement that the map $\phi: \mathbf{x}^{\mathbf{c}} \mapsto \mathbf{x}^{\mathbf{K}}/\mathbf{x}^{\mathbf{c}}$ be an involution on $\mathr... | 3 | https://mathoverflow.net/users/94968 | 313480 | 136,256 |
https://mathoverflow.net/questions/313436 | 3 | It looks to me that the bordism group
$$\Omega\_3^{SO} (B(O(2) \times SO(3))) \tag{1}$$
(whose Pontryagin dual for the manifold generator) contains at least a nontrivial invariant:
$$
w\_1(O(2))\big(w\_1(O(2))^2 +w\_2(SO(3))\big). \tag{2}
$$
>
> Question: Is it true that if we lift this invariant respect to a new... | https://mathoverflow.net/users/106497 | Bordism invariants vanishes in a lifted twisted $Pin^- \times Spin$-structure | Since you have the constraint
$$w\_1(O(2))^2+w\_2(O(2))=w\_2(SO(3)),$$
then
$$w\_1(O(2))(w\_1(O(2))^2+w\_2(SO(3)))=w\_1(O(2))w\_2(O(2))=Sq^1(w\_2(O(2)))=w\_1(TM)w\_2(O(2))=0$$
by Wu formula. ($w\_1(TM)=0$ since you are considering oriented bordism.)
$\frac{dw\_2(O(2))}{2}=Sq^1(w\_2(O(2)))$ by the definition of Bocks... | 4 | https://mathoverflow.net/users/102515 | 313482 | 136,257 |
https://mathoverflow.net/questions/313325 | 7 | I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken, specifically, I'm reading the following theorem:
>
> $\textbf{Theorem 2.1}$ Let $R > 0$ and $\textbf{x}\_0 \in R^{n+1}$ be arbitrary and define $\v... | https://mathoverflow.net/users/114870 | Weak parabolic maximum principle on Riemannian manifolds | Firstly, you have the wrong inequality (there is a small typo in the paper). Young's inequality is typically written for nonnegative numbers, but for any $a,b\in\Bbb R$ we have
\begin{align\*}
-ab&\le |ab|\\
&\le \frac{1}{2}a^2+\frac{1}{2}b^2.
\end{align\*}
In the context of the Ecker--Huisken,
\begin{align\*}
-6v \nab... | 7 | https://mathoverflow.net/users/90154 | 313483 | 136,258 |
https://mathoverflow.net/questions/312820 | 5 | By looking at defining relations of standard deformation of $\mathfrak{sl}\_2$, which are:
$$
[E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F,
$$
some questions come around.
For example, one can check that Jacobi identity is satisfied, but it would be also satisfied... | https://mathoverflow.net/users/130105 | Is there another quantum deformation of sl(2)? | Regarding your second question, on other possible deformations of $sl(2)$:
There have been various studies on (multi-parametric) deformations of Lie algebras -as has already been mentioned in the comments to the OP- during the last decades:
An example of a $2$-parameter deformation $sl\_{pq}(2)$ which leads to a... | 4 | https://mathoverflow.net/users/85967 | 313484 | 136,259 |
https://mathoverflow.net/questions/312050 | 8 | It seems that from [this webpage](http://www.map.mpim-bonn.mpg.de/Spin_bordism#Classification), the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}\_{p^n}$ for some $n$) $$\Omega\_d(BG)\_p.$$
**Question 1**:
Then do we have
$$\Omega\_d^{spin}(BG)\_p =... | https://mathoverflow.net/users/27004 | Spin cobordism v.s. KO theory in low or in any dimensions | First of all, let me say that the page you are quoting is a little bit
misleading if not inaccurate on the Anderson-Brown-Peterson splitting.
$ko\langle 4n(J)\rangle $ should read $\Sigma ^{4n(J)} ko$ and $ko\langle 4n(J)-2\rangle $ should read $\Sigma ^{4n(J)-4} ko\langle 2\rangle $
See, e.g. <https://pdfs.seman... | 4 | https://mathoverflow.net/users/43326 | 313485 | 136,260 |
https://mathoverflow.net/questions/313490 | 0 | For a symmetric p.s.d matrix $A \in \mathcal{R}^{n\times n}$, we can calculate its SVD as $A=USV^T$, then we can use the truncated SVD to approximate it with a low-rank matrix $\tilde{A} = \sum\_i^dU\_iS\_iV\_i^T$, my question is about the element-wise comparison between $A$ and $\tilde{A}$.
Empirically, I notice th... | https://mathoverflow.net/users/83163 | Will truncated SVD ever flip the sign of any element of the matrix? | It's not true.
Consider the $3 \times 3$ PSD matrix
$$ \pmatrix{3 & \epsilon & -3\cr
\epsilon & 5 & -1\cr
-3 & -1 & 4\cr} $$
where $\epsilon > 0$ is small. Its rank-$2$ approximation will have negative $(1,2)$ and $(2,1)$ elements.
| 4 | https://mathoverflow.net/users/13650 | 313491 | 136,261 |
https://mathoverflow.net/questions/313495 | 2 | $\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\ell]:=[k,\ell]\cap\Z$. Take then any $s\in[2;n-1]$. Let $\J:=\J\_s:=\binom{[n]}s$, the set of all $s$-sets in $[n]:=[1;... | https://mathoverflow.net/users/36721 | Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets | Various variants of the matrix
$$A\_{J,K} = |J\cap K|$$
were studied, and the spectrum was computed. A one-parameter variant is given by
$$A^{(i)}\_{J,K} = \binom{|J\cap K|}{i}$$
for some fixed $i$, and your problem corresponds to $i=1$. In total, six variants are given in Section 3 of [this friendly paper](https://onl... | 1 | https://mathoverflow.net/users/31469 | 313500 | 136,264 |
https://mathoverflow.net/questions/313518 | 6 | Mazur's conjecture on the image of Galois representations of Elliptic curves states that for $N$ large enough there is a unique elliptic curve $E$ over $\mathbb{Q}$ giving rise to a fixed mod $N$ Galois representation $\bar{\rho}: G\_{\mathbb{Q}}\rightarrow GL\_2(\mathbb{Z}/N\mathbb{Z})$.
Is there a similar expectation... | https://mathoverflow.net/users/nan | Is it expected that the mod $p$ representation determines a normalized Hecke newform of fixed weight for p large enough? | This is not true. Take $k=2$, and $p \geq 5$. Let $\ell$ be a prime such that $p$ divides $\ell-1$. Then we know (by Mazur) that there exists a newform of weight $2$ and level $\Gamma\_0(\ell)$ whose residual semi-simple representation is $\overline{\rho} = 1 \oplus \overline{\chi}\_p$ where $\overline{\chi}\_p$ is the... | 4 | https://mathoverflow.net/users/60519 | 313520 | 136,267 |
https://mathoverflow.net/questions/313512 | 3 | Let $X$ be a smooth variety and $M$ a fine moduli space of certain kind of sheaves on $X$. Let $\mathcal{E}$ be the universal family on $X\times M$. Suppose there is a derived functor $F$ from $D^b(X)$ to some derived category, say $D^b(Y)$, or $D^b(A-mod)$ for some finite algebra $A$, so that $F(\mathcal{E}\_t)$ is a ... | https://mathoverflow.net/users/48616 | Derived functor giving algebraic map between moduli | If $F$ is a Fourier-Mukai functor, it can be applied not just fiberwise, but to the whole family. Namely, if $K \in D^b(X\times Y)$ is the Fourier-Mukai kernel, then by pullback it gives an object on
$$
X \times Y \times M = (X \times M) \times\_M (Y \times M) \subset (X \times M) \times (Y \times M).
$$
The correspond... | 1 | https://mathoverflow.net/users/4428 | 313526 | 136,268 |
https://mathoverflow.net/questions/313527 | 8 | I will preface this by saying that I am not an expert on type theory. I am just a curious outsider slowly making my way through the HoTT book when I (rarely) have some spare time.
I am just curious if there has been any progress towards a computational interpretation of univalence? If not, is the general feeling that... | https://mathoverflow.net/users/56938 | Progress towards a computational interpretation of the univalence axiom? | [Cubical type theory](https://ncatlab.org/nlab/show/cubical+type+theory) is a variant of type theory which has all the usual (and some unusal) computational properties, and the Univalence Axiom is a theorem of cubical type theory. As was already pointed out in the comments, there are implementations of cubical type the... | 10 | https://mathoverflow.net/users/1176 | 313531 | 136,270 |
https://mathoverflow.net/questions/313022 | 2 | Let $X$ be a real Hilbert Space and $C \subseteq X$. Let $d\_C$ be the infimal distance function to $C$ and $P\_C(x) = C \cap S[x; d\_C(x)]$ be the metric projection. We say $C$ is *proximinal* if $P\_C(x) \neq \emptyset$ for all $x \in X$.
I'm wondering if the intersection of two proximinal subsets of $X$ must be pr... | https://mathoverflow.net/users/113997 | Are the intersection of proximinal sets in a Hilbert Space proximinal? | This answer is strongly inspired by example 3.11 in the book by Bauschke and Combettes.
Let $H = \ell^2$ and consider a sequence $\{\alpha\_n\} \in (1,\infty)$ with $\alpha\_n \searrow 1$. Define
\begin{align}
C &= \{ \alpha\_n \, e\_n \mid n \in \mathbb N\} \cup \{0\}\\
D &= \{ x \in \ell^2 \mid \exists n \in \mathb... | 1 | https://mathoverflow.net/users/32507 | 313539 | 136,275 |
https://mathoverflow.net/questions/313541 | 1 | If $(X,\tau)$ is a [paracompact](https://en.wikipedia.org/wiki/Paracompact_space), is there a topology $\tau'\supseteq \tau$ such that $(X,\tau')$ is still paracompact, and $\tau'$ is maximal with respect to $\subseteq$ and paracompactness?
| https://mathoverflow.net/users/8628 | Is every paracompact topology contained in a maximal paracompact topology? | Any [discrete](https://en.wikipedia.org/wiki/Discrete_space) space [is paracompact](https://proofwiki.org/wiki/Discrete_Space_is_Paracompact), since the family of singletons is locally finite and an open refinement of every open cover. Put $\tau'=\mathcal{P}(X)$. Then the discrete space $(X,\tau')$ is paracompact, and ... | 7 | https://mathoverflow.net/users/296 | 313546 | 136,277 |
https://mathoverflow.net/questions/313554 | 1 | Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R\_n := \mathbb E\_{\hat{P}\_n}[Z]^T\operatorname{Cov}\_{\hat{P}\_n}[Z]^{-1}\mathbb E\_{\hat{P}\_n}[Z]$ ?
Here, $\mathbb E\_{\hat{P}\_n}[Z] = (1/n)\sum\_{i=1}^nz\... | https://mathoverflow.net/users/78539 | Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$ | If $\mu\ne0$, then the distribution of $R\_n$ is asymptotically normal with asymptotic mean $\mu^T\Sigma^{-1}\mu$ and an explicit asymptotic variance $\tilde\sigma^2/n$; see e.g. [Theorem 3.9, page 1018](https://projecteuclid.org/euclid.ejs/1460463653), where a bound on the rate of convergence is also given. More speci... | 1 | https://mathoverflow.net/users/36721 | 313580 | 136,289 |
https://mathoverflow.net/questions/313321 | 1 | The de Rham dg algebra $\Omega(F)$
of a closed orientable surface $F$
is formal
(that is, weakly equivalent to its cohomology algebra).
This is a special case of the fact of formality of Kähler manifolds.
Can one prove formality of $\Omega(F)$ without using complex analysis?
Say, by explicitly constructing its S... | https://mathoverflow.net/users/9878 | Formality of surfaces | Semen:
In Felix, Halperin, Thomas: Rational Homotopy Theory II, World Scientific 2015, it is shown (see Ch. 8, section 5) that orientable surfaces are even better than formal. They are intrinsically formal. That is, any commutative cochain algebra with the same cohomology $H\_g$ as a surface (of genus $g$ here) has ... | 4 | https://mathoverflow.net/users/118986 | 313597 | 136,297 |
https://mathoverflow.net/questions/312913 | 1 | Let $M \times [0, T) \to \mathbb{R}^{n+1}$ be a mean curvature flow and let $T$ be a singular time. Let $A$ denote the second fundamental form.
We have a **type I singularity** if
$$
\max\_{p \in M} |A(p, t)| \le \frac{C}{\sqrt{2(T-t)}}
$$
for some constant $C > 0$ and a **type II singularity** otherwise.
Why are... | https://mathoverflow.net/users/86341 | Slow and fast forming singularities of the mean curvature flow | I could be wrong but I believe the answer is disappointingly uninteresting: Set $f(t)\doteqdot \sup\_{M\times[0,t]}\vert A\vert^2$. If the flow is of type I then the remaining time is bounded by $1/f(t)$, so a 'singularity' occurs sooner than if the flow is not of type I.
| 2 | https://mathoverflow.net/users/38509 | 313613 | 136,299 |
https://mathoverflow.net/questions/313610 | 9 | Fix a differentiable non-compact manifold $M$. Denote by $\mathrm{Lor}(M) := \{\text{Lorentzian metrics on $M$}\}.$ One can define a topology on this set via: fix any open covering $\mathcal{A}$ on $M$. For each $g \in \mathrm{Lor}(M)$ and for each positive continuous function $r : M\to ]0,\infty[,$ one defines:
$$\mat... | https://mathoverflow.net/users/94097 | Is the set of Lorentzian metrics metrizable? | First of all, there is a bunch of basic things that you need to write in a slightly clearer way. If you try to topologize the set of Lorentzian metrics as you did, you need first:
1. Restrict to the subset of Lorentz metrics of a given (say, $C^k$, $0\leq k\leq\infty$) regularity, otherwise your definition for the b... | 14 | https://mathoverflow.net/users/11211 | 313616 | 136,301 |
https://mathoverflow.net/questions/313313 | 4 | Computations suggest that
$$\int\_{0}^{\infty}\int\_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$
The question is how to prove this equality.
Background: this is the mean width of an ellipsoid with semiaxes $s,s,1/s^2$ and th... | https://mathoverflow.net/users/130319 | Integral equality of 1st intrinsic volume of spheroid | First of all make a replacement $x=t^2$ and go to polar coordinates
\begin{equation\*}
\int\_{0}^{\infty}\int\_{0}^{2\pi}
\cos(\phi)r^3 \cdot e^{-\frac{\frac{r^2\cos^2(\phi)}{s}+r^2\sin^2(\phi)s^2}{2}}drd\phi.
\end{equation\*}
After than notice that we can take integral by radius. Let $R=\frac{1}{2}(\cos^2(\phi)\frac... | 3 | https://mathoverflow.net/users/130319 | 313623 | 136,306 |
https://mathoverflow.net/questions/313626 | 2 | Let $X$ be a set and let ${\frak T}$ be a collection of [paracompact](https://en.wikipedia.org/wiki/Paracompact_space) topologies on $X$ such that for any $\tau, \tau'\in {\frak T}$ we have $\tau\subseteq \tau'$ or $\tau'\subseteq \tau$. Let $\sigma$ be the topology having $\bigcup {\frak T}$ as a [base](https://en.wik... | https://mathoverflow.net/users/8628 | Is the topology generated by the union of a chain of paracompact topologies paracompact? | Here is an easy counterexample under the Continuum Hypothesis, where the *union of the topologies* generates a non-normal topology.
Take the upper half plane $\mathbb{H}$ (including $x$-axis) and enumerate $\mathbb{R}$ as $\langle x\_\alpha:\alpha<\omega\_1\rangle$.
Now let topology $\mathcal{T}\_\alpha$ be generated b... | 3 | https://mathoverflow.net/users/5903 | 313628 | 136,307 |
https://mathoverflow.net/questions/310548 | 0 | Given a number n of nodes, a diameter d (d>1) and a max-degree k. Let's assume d and k are chosen such that a graph with n nodes with the desired diameter and max-degree exists.
What is the minimum number of edges necessary to create a (connected) graph with n nodes, diameter d, and max-degree k? Is there a general c... | https://mathoverflow.net/users/128867 | Minimize edge number under diameter and max-degree constraint | I have found an answer to the question in a book by Béla Bollobás: Extremal Graph Theory. In chapter 4.1, he describes a lower bound:
Given maximal diameter $d$, maximal degree $k$, number of nodes $n$, number of edges $m$, the following inequality holds:
$m \geq \frac{n(n-1)(k-2)}{2((k-1)^d-1)}$
Sadly, I have no... | 0 | https://mathoverflow.net/users/128867 | 313630 | 136,308 |
https://mathoverflow.net/questions/313023 | 6 | Let $G$ be a connected, reductive group over a $p$-adic field with parabolic $P = MN$ defined by a set of simple roots $\theta \subset \Delta$. For $(\pi,V)$ a representation of $M$, and $\nu \in \mathfrak a\_{M,\mathbb C}^{\ast}$, we have the induced representation
$$I(\nu,\pi) = \operatorname{Ind}\_P^G \pi q^{\lang... | https://mathoverflow.net/users/38145 | Convergence of the intertwining operator as a vector valued integral | A reference for this material is Waldspurger's article "La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra," [(pdf)](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1474748003000082). See section IV.1.
Here I will make a few remarks only about the definition. All seri... | 2 | https://mathoverflow.net/users/425 | 313634 | 136,309 |
https://mathoverflow.net/questions/313639 | 6 | Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ such that $1\in \text{supp}(\alpha) \cap \text{supp}(\beta)$, $|\text{supp}(\alpha)|\leq |\text{supp}(\beta)|$ and $\alpha ... | https://mathoverflow.net/users/19075 | Zero divisors in complex group algebras of residually finite groups | The conjecture does not hold.
Let $m \geq 1$ be an arbitrary integer.
Let $G = \langle x,y | x^m, y^5 , [x,y]\rangle$ be the finite abelian group isomorphic to a product of two cyclic groups $C\_m \times C\_5$.
Let $\alpha = xy +y -x -1$ and let $\beta = y^4 +y^3 +y^2 +y +1$ in $\mathbb{Z}[G]$.
Then $\alpha \beta = ... | 8 | https://mathoverflow.net/users/54441 | 313643 | 136,312 |
https://mathoverflow.net/questions/313540 | 3 | I have a question about commutative direct summands of $C$\*-algebras.
Let $A$ be a $C$\*-algebra (with unit) and suppose that its bidual $A^{\*\*}$ has a commutative direct summand, that is, $A^{\*\*}=B\oplus C$ such that $B$ is non-zero and commutative. Does this force $A$ to have a non-zero commutative direct summ... | https://mathoverflow.net/users/58366 | Commutative direct summands of C*-algebras | Let $A$ be the continuous functions $f$ from $[0,2]$ into $M\_2(\mathbb{C})$ such that $f(t)$ is diagonal for $0 \leq t \leq 1$. Then $A$ has no commutative direct summand, but the atomic part of its bidual should be equal to the bounded functions $f$ from $[0,2]$ into $M\_2(\mathbb{C})$ such that $f(t)$ is diagonal fo... | 1 | https://mathoverflow.net/users/46472 | 313647 | 136,313 |
https://mathoverflow.net/questions/313632 | 3 | Lets $\phi(n)$ is Euler's phi function. Let's define error term
$$
d(n) = {E(n) \over n} = {1 \over n} \left(\sum\_{i\le n}\phi(i) - {3 n^2 \over \pi^2}\right)
$$
Is it known that asymptotic of mean (and median) of the first N values $d(n)$ is equal to 3/\pi^2?
$$
\lim\_{N\to\infty} {\sum\_{n=1}^N d(n) \over N} = ... | https://mathoverflow.net/users/130457 | Mean and median of error term of average Euler-phi function | The limit is not $1$ but $3/\pi^2$ (in agreement with [your Mathematica code](https://i.stack.imgur.com/d1NKs.jpg)). To see this, let us use the following result of Pillai and Chowla ([On the error terms in some asymptotic formulae in the theory of numbers, (I)](https://doi.org/10.1112/jlms/s1-5.2.95), J. London Math. ... | 5 | https://mathoverflow.net/users/11919 | 313648 | 136,314 |
https://mathoverflow.net/questions/313642 | 11 | I was reading about [the appearance of Calabi-Yau manifolds in Feynman integrals](https://4gravitons.wordpress.com/2018/10/19/a-micrographia-of-beastly-feynman-diagrams/), and I thought to wonder if there is such a thing as "infinite-dimensional Hodge theory". Googling the phrase turned up references to a 2010 lecture ... | https://mathoverflow.net/users/9756 | What is Kontsevich's Hodge theory of path integrals? | The most usual Hodge theory is related to integrals, periods of algebraic differential forms on algebraic varieties, whereas the physics path integral usually involves integration of an exponential quantity. So the first thing to understand is that there is a finite dimensional Hodge theory for "exponential integrals",... | 13 | https://mathoverflow.net/users/25309 | 313649 | 136,315 |
https://mathoverflow.net/questions/313651 | 3 | Given any finite set $X$ the set $\mathcal{T}(X)=X^X$ of all functions from $X$ to $X$ clearly forms a monoid under composition. Now if we call any family of functions $\mathcal{F}\subseteq \mathcal{T}(X)$ a generating set of $\mathcal{T}(X)$ iff every function in $\mathcal{T}(X)$ can be expressed as a composition of f... | https://mathoverflow.net/users/38626 | Size of a minimum generating set for full transformation monoids | The minimal number to generate the full transformation monoids is 3 maps. You must include a generating set for the symmetric group, which requires 2, and then you can add any idempotent function collapsing exactly two elements.
Here are hints.
1. Use double transitivity of the symmetric group to show you get all... | 6 | https://mathoverflow.net/users/15934 | 313652 | 136,316 |
https://mathoverflow.net/questions/313629 | 3 | Definition: Let $X$ be a Banach space and $X^\*$ be its continuous dual of $X,$ that is, $X^\*$ contains all bounded linear functionals on $X.$
Denote
$$B\_{X^\*} = \{x^\*\in X^\*: \|x^\*\|\_{X^\*}\leq 1\}.$$
We say that $x^\*\in B\_{X^\*}$ is an **extreme point of** $B\_{X^\*}$ if whenever
$$x^\* = \frac{1}{2}(y\_1... | https://mathoverflow.net/users/42411 | Is it true that every Banach space has at least one extreme point that is normed by some point? | Take any $x$ with $\|x\|=1$. $S(x) = \{x^\* \in X^\*: x^\*(x) = \|x^\*\| = 1 \}$ is a nonempty (by Hahn-Banach) weak-\* compact convex set, so by Krein-Milman it has extreme points. Any extreme point of $S(x)$ is an extreme point of $B\_{X^\*}$.
| 7 | https://mathoverflow.net/users/13650 | 313653 | 136,317 |
https://mathoverflow.net/questions/313635 | 5 | Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)
If not, is there a known counterexample?
Similarly, does the same hold for inner anodyne extensions and inner horn inclusions?
| https://mathoverflow.net/users/1353 | Cellularity of anodyne extensions? | I believe the following seem to be a very simple counterexample without it being related to Whitehead obstruction as suggested by Tyler Lawson.
Consider the simplicial sets $D$ freely generated by:
* a $0$-cell $x$.
* a $1$-cell $t:x \rightarrow x$
* a $2$-cell $\gamma$ such that $d\_0 \gamma = d\_1 \gamma = d\_2 ... | 6 | https://mathoverflow.net/users/22131 | 313665 | 136,319 |
https://mathoverflow.net/questions/313655 | 7 | What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but when Googling I mostly found stuff for ordinary polynomial rings.
I am particularly interested in the possibile dimensions... | https://mathoverflow.net/users/15934 | Injective indecomposable modules over Laurent polynomial rings | If $R$ is a PID and $P$ is a nonzero prime ideal, then $E(R/P)=K/P\_P$, where $K$ is the fraction field of $R$ and $R/P$ is viewed as a submodule of $K/P\_P$ via the evident map. Indeed, one readily checks that $R/P\cong R\_P/P\_P$ is an essential submodule of $K/P\_P$ and that $K/P\_P$ is divisible, hence injective (i... | 4 | https://mathoverflow.net/users/86006 | 313673 | 136,322 |
https://mathoverflow.net/questions/313664 | 4 | With regards to my research (connecting character degrees' arithmetic structure with the corresponding group's structure), I find myself in the situation (when studying the symmetric group $S\_n$) of wanting the interval $(\frac 56n,n)$ to contain at least two primes. In the "Better results" section of the Wikipedia ar... | https://mathoverflow.net/users/128140 | Wanted: multiple primes in $(\frac{5n}6,n)$ | In an impressive (to me) feat of research, Jose Brox uncovered results of Molsen, Breusch, and Schur regarding the problem. Check <https://mathoverflow.net/a/289448> for details.
**Edit 2019.03.23:**
Even more impressive is the scholarship of Narkiewicz. In his book *The Development Of Prime Number Theory*, startin... | 9 | https://mathoverflow.net/users/3402 | 313674 | 136,323 |
https://mathoverflow.net/questions/313663 | 4 | As is described in the title. Is there a finitely presented group $G$, with trivial profinite completion $\widehat{G}=0$, which is not amalgamated free product?
For example, the famous example Higman groups are all amalgamated free product.
| https://mathoverflow.net/users/128887 | Infinite finitely presented simple group (or more generally with trivial profinite completion) that is not amalgamated free product | I'm going to flesh out my comment above to an answer. [Thompson's groups](https://en.wikipedia.org/wiki/Thompson_groups) $T$ and $V$ are famous examples of finitely presented infinite simple groups.
In [this](https://arxiv.org/abs/0708.1334) paper of Dan Farley, it is shown that $T$ and $V$ have Serre's property FA,... | 8 | https://mathoverflow.net/users/1463 | 313676 | 136,324 |
https://mathoverflow.net/questions/313337 | 4 | $\require{AMScd}$
**Preliminaries:** Let $\Sigma$ be a closed manifold, $X$ be a CW complex and $f:\Sigma \to X$ be a map. We say that the pair $(\Sigma,f)$ is *null-homologous* (over $\mathbb{Z}\_2$) if $f\_\*[\Sigma] = 0 \in H\_\*(X;\mathbb{Z}\_2)$, and we say that $(\Sigma,f)$ is *null-bordant* if there exists a m... | https://mathoverflow.net/users/123015 | null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes | I would like to share a careful proof of the generalized Corollary 2 from my question.
The essential idea is to factor the map $f:Z \to X$ through a map to a finite sub-complex of $X$, like I originally had in mind. However, I use Mike Miller's observation that you only need to know that $f\_\*[Z]$ is 0 in the homol... | 2 | https://mathoverflow.net/users/123015 | 313694 | 136,334 |
https://mathoverflow.net/questions/313577 | 6 | There is a well-known theorem stating that there is a bijection between diffeomorphism classes of Lefschetz fibrations over $S^2$ whose general fiber is a closed orientable surface $\Sigma\_g$ of genus $g\geq 2$ and factorizations of identity in the mapping class group $\Gamma(\Sigma\_g)$ up to Hurwitz moves and global... | https://mathoverflow.net/users/nan | Symplectic Lefschetz fibrations in terms of factorization in symplectic mapping class group | Up to homotopy, giving a Lefschetz fibration over $S^2=\mathbb C\cup\{\infty\}$ with singular fibers over $\{e^{2\pi ik/N}\}\_{0\leq k<N}$ is the same as giving the data of the fiber $F$ over $0\in\mathbb C$, the vanishing cycles $\{L\_i\}\_{0\leq i<N}$ (as marked Lagrangian spheres in $F$), and a path in $\operatornam... | 3 | https://mathoverflow.net/users/35353 | 313710 | 136,339 |
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