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https://mathoverflow.net/questions/417668 | 1 | Decks are composed of 1 copy of each of $D$ unique cards. The set of cards is $C$ ($|C|=D$), the set of people is $P$ ($|P|=n\geq k$).
Starting with a simpler case (dropping the $k-1$ restriction)
-------------------------------------------------------------
One answer is to give $n-k+1$ full decks to $n-k+1$ peopl... | https://mathoverflow.net/users/478354 | How to distribute least number of $D$ card decks amongst $n$ people so that any $k$ people have a full deck and no $k-1$ people have a full deck | For each $G\subseteq P$ define $C\_G\equiv\{$**Cards not held by any member of G**$\}$.
Notice $\forall G,H\subseteq P, C\_{G\cup H}=C\_G\cap C\_H$
Let $K\equiv\{G\subseteq P : |G|=k-1\}$. Notice $|K|=\left (\frac{n}{k-1}\right )$ [By definition of $\left (\frac{n}{r}\right )$]
Claim
=====
$\forall G\in K, C\_G... | 0 | https://mathoverflow.net/users/478354 | 417857 | 170,184 |
https://mathoverflow.net/questions/417234 | 10 | Let $G$ be a real connected Lie group. I am interested in its special homotopy properties, which distinguish it from other smooth manifolds
For example
1. $G$ is homotopy equivalent to a smooth compact orientable manifold. In particular, Poincaré duality holds for $G$.
2. $\pi\_1(G)$ is abelian, [$\pi\_2(G) = 0$](h... | https://mathoverflow.net/users/148161 | Homotopy properties of Lie groups | The problem you mention has a long history. The best homotopy characterization is probably using the notion of finite loop spaces:
A finite loop space is a space $BG$ such that $\Omega BG$ is homotopy equivalent to a finite CW-complex. There are many of those, but one can give a precise homotopy characterization of w... | 12 | https://mathoverflow.net/users/6574 | 417858 | 170,185 |
https://mathoverflow.net/questions/417856 | 8 | If $F$ is a free group then it has cohomological dimension one, which implies that the augmentation ideal $IF=\operatorname{ker}(\epsilon:\mathbb{Z}G\to \mathbb{Z})$ of its group ring is a projective $\mathbb{Z}F$-module. Hence $IF$ is a direct summand of a free $\mathbb{Z}F$-module $M$.
>
> Question: Is it possibl... | https://mathoverflow.net/users/8103 | Augmentation ideal of a free group | Let $X$ be a free basis for $F$. The Cayley graph of $F$ is a tree $T$ on which $F$ acts freely. The augmented chain complex gives a resolution
$$0\to \mathbb ZF^{(X)}\to \mathbb ZF\to \mathbb Z\to 0$$ (since $T$ is a tree) where we identify $\mathbb ZF^{(X)}$ as the free abelian group on the edges of $T$ and the image... | 10 | https://mathoverflow.net/users/15934 | 417860 | 170,187 |
https://mathoverflow.net/questions/417863 | 10 | The question is in the title.
To make the statement more precise, is is true that for any given monoidal category $(\mathcal C, I, \otimes)$ there exists at least one braiding $\beta$? In other words, does the forgetful functor from braided monoidal categories admit a section (not a left adjoint!)?
I strongly suspe... | https://mathoverflow.net/users/50376 | Does every monoidal category admit a braiding? | No, sometimes there are even $x,y$'s with no abstract isomorphism $x\otimes y \cong y\otimes x$.
Here are two families of examples:
* Monoids, viewed as discrete categories. The tensor product is just the multiplication, and the existence of a braiding would simply mean that the monoid is commutative, which it typi... | 26 | https://mathoverflow.net/users/102343 | 417865 | 170,189 |
https://mathoverflow.net/questions/417864 | 0 | Let $R\_n$ be a random variable with values in $[0,1]$ and $nR\_n$ converges to $\frac{1}{1+C} \chi\_m^2$ in distribution for some constant $C>0$ and $m\in \mathbb{N}$. Is it possible to show that $(1-R\_n)^{-\frac{n-1}{2}} =O\_P(1)$ holds?
I came across this result in the [proof of Theorem 3 of this paper](https://p... | https://mathoverflow.net/users/163533 | Random variable is Big O in probability notation | For each natural $n$, let $R\_n$ be random variable with values in $[0,1]$ such that $nR\_n$ converges in distribution, as $n\to\infty$, to a random variable $X$ with a continuous cdf.
Let $T\_n:=(1-R\_n)^{-(n-1)/2}$. Take any real $t>1$. Then for all large enough natural $n$
$$t^{-2/(n-1)}=e^{-2\ln t/(n-1)}<1-\frac{... | 2 | https://mathoverflow.net/users/36721 | 417870 | 170,192 |
https://mathoverflow.net/questions/417848 | 3 | Let $K$ be a number field and let $A\_K$ be the adele ring of $K$. Then $K$ sits in $A\_K$ via the diagonal embedding and the quotient $A\_K/K$ is compact. All this is well known. Many proofs of the above fact first reduce the case to that of $K=\mathbb{Q}$ and solves the problem in this case. The proof also provides a... | https://mathoverflow.net/users/148866 | Fundamental domain for $A_K/K$ | One does not need that the class number is $1$, the construction in Brian Conrad's notes works in general. See Propositions 6-7 in Ch.V-4 of Weil: Basic Number Theory. The class group enters for $A\_K^\times/K^\times$, not for $A\_K/K$.
| 4 | https://mathoverflow.net/users/11919 | 417874 | 170,195 |
https://mathoverflow.net/questions/417891 | 1 | I would like to find a upper bound of principal polarization of abelian variety in the following stiution:
Suppose $A$ is an abelian variety over a $char=0$ algebraically closed field. And for any two principal polarization $\lambda\_1$, $\lambda\_2$:$A \to A^t$, we we say they are eqivalent if and only if there exsi... | https://mathoverflow.net/users/478518 | The upper bound of number of the automorphism of principal polarization of abelian variety over algebraically closed field | No, not without some extra assumptions. Take for example $A = E \times E'$ where $E$ and $E'$ are generic elliptic curves connected by an isogeny $E \to E'$ with kernel $\mathbb{Z}/m\mathbb{Z}$, for some squarefree integer $m$. For each factorization $dk = m$, you can find elliptic curves $E\_k, E\_d$ on $A$ which are ... | 3 | https://mathoverflow.net/users/949 | 417897 | 170,200 |
https://mathoverflow.net/questions/417815 | 5 | In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho\_{f,\ell}$ attached to a non-CM newform form $f$ satisfies
$${\rm SL}\_2(\mathbb{F}\_\ell)\subset \overline{\rho}\_{f,\ell}(G\_\mathbb{Q}).\qquad\qquad(\*)$$
I am wondering:
(i) Why does $(\*)$ i... | https://mathoverflow.net/users/477704 | Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform | Here's an answer that's more work than Kevin Ventullo's, but works for $\ell=2$ and $\ell=3$, and gives you a method that will work in various more general cases.
There is a surjection $\pi\_1\colon G\_\mathbb Q \to \overline{\rho}\_{f,l} (G\_{\mathbb Q})$ and a surjection $\pi\_2 \colon G\_\mathbb Q \to \operatornam... | 6 | https://mathoverflow.net/users/18060 | 417900 | 170,201 |
https://mathoverflow.net/questions/417380 | 27 | For positive integers $m$ and $n$, what is the integral of the function $(-1)^{\lfloor x \rfloor + \lfloor y \rfloor}$ on the triangle with vertices $(0,0)$, $(m,0)$, and $(0,n)$?
Pictorially, we are putting a red/black checkerboard coloring on the plane and finding the signed difference between the red region enclos... | https://mathoverflow.net/users/3621 | Area-differences for lattice triangles in a checkerboard | Define
$$ h(x)=(-1)^{\lfloor x\rfloor}\, (x-\lfloor x\rfloor)(x-\lfloor x\rfloor-1) $$
Then we have
$$I(n,m)= \frac{{\rm Mod}(n,2)}{2}+\frac{n}{m} \sum\_{j=1}^{n-1} (-1)^{n-j} \, h\left(\frac{jm}{n}\right)$$
Proof: We use the two primitives
\begin{eqnarray\*}
f(y)=\int\_0^y dx\, (-1)^{\lfloor x\rfloor} &=& {\rm Mod}(y,... | 12 | https://mathoverflow.net/users/478524 | 417907 | 170,204 |
https://mathoverflow.net/questions/417894 | 1 | Say that you place an asymptotically Euclidean metric on $\mathbb{R}^3,$ e.g. $\mathbb{R}^3$ is endowed Riemannian metric $g$ such that $\text{supp}(g^{ij}-\delta^{ij})\subseteq\{|x|\leq R\}$ for some large $R>0.$ Does it follow that $(\mathbb{R}^3,g)$ is geodesically complete? This seems to be applied in PDE literatur... | https://mathoverflow.net/users/371650 | Completeness of asymptotically Euclidean manifolds | Yes, and it follows from [Hopf-Rinow](https://en.wikipedia.org/wiki/Hopf%E2%80%93Rinow_theorem) (let's assume $g$ is at least $C^2$ so that there's no ambiguity about the geodesic flow).
Since $g$ and $\delta$ differs only on a compact set, you have the the lengths defined by $g$ and $\delta$ are globally comparable,... | 1 | https://mathoverflow.net/users/3948 | 417911 | 170,205 |
https://mathoverflow.net/questions/417723 | 4 | A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$,
$$\frac{1}{n} \sum\_{i \le n} \phi \circ f^i \to 0 \text{ almost surely and in } L^1(\mu).$$
Therefore, $\max\_{n \ge N} \frac{1}{n} \sum\_{i \le n} \phi \circ f^i \to ... | https://mathoverflow.net/users/139946 | Maximal ergodic inequality | For a nice introductory discussion to the maximal ergodic inequality, see
[1]. In particular, inequality (5) there is Wiener's maximal ergodic theorem. See also Lemma 15.3 in [2]. A more advanced account, that in particular includes the $L^p$ maximal inequalities, is in the book [3], see Cor 2.2 page 8 for the Wiener i... | 5 | https://mathoverflow.net/users/7691 | 417914 | 170,207 |
https://mathoverflow.net/questions/417855 | 1 | Let $a>1$ and define $G\_a(x)=\sum\limits\_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\mathbb{R}$ thanks to a ratio test and Stirling's formula.
For $a=2$, using Legendre duplication formula $\Gamma(z)\Gamma(z+... | https://mathoverflow.net/users/294161 | Power series of ratio of Gamma functions | For arbitrary real $a > 0$ this is a special case of the generalized $\_p\Psi\_q(A;B;ζ)$ Fox-Wright function, where $A=[(a\_1,\alpha\_1),(a\_2,\alpha\_2),...,(a\_p,\alpha\_p)]$ and $B=[(b\_1,\beta\_1),(b\_2,\beta\_2),...,(b\_q,\beta\_q)]$ being $a\_j, j=1,..,p$ and $b\_k, k=1,..,q$ complex parameters and $\alpha\_j, \b... | 1 | https://mathoverflow.net/users/141375 | 417935 | 170,213 |
https://mathoverflow.net/questions/417905 | 3 | I am new to the Albanese map so am not sure about its properties.
>
> **Question** Let $k=\mathbb{C}$. Suppose $X$ smooth projective variety, let $\alpha$ be the Albanese map of $X$. Is there a
> descripition of $X$ such that $\dim(X)=\dim(α(X))$?
>
>
>
Any remarks or references would be appreciated.
Cross-P... | https://mathoverflow.net/users/99732 | Maximality of Albanese dimension | I think that the varieties you are interested in are called *of maximal Albanese dimension* or *of Albanese general type*.
As remarked in the comments, their Kodaira dimension must be non-negative; however, since the behavior of the Albanese map is related to the $1$-forms and not to the top forms (canonical divisor)... | 6 | https://mathoverflow.net/users/7460 | 417938 | 170,214 |
https://mathoverflow.net/questions/417936 | 5 | We know
$$
\sum\_{m=0}^\infty \frac{x^m}{(a-m)!m!} = \frac{1}{a!}(1+x)^m
$$
where we understand the factorial as Gamma function $\Gamma(x)$ such that it is divergent if the argument is negative integer.
We also know $$
\sum\_{m=0}^\infty \frac{x^m}{(b+m)!m!} \sim \,\_0F\_1(b,x)
$$
as hypergeometric function while this ... | https://mathoverflow.net/users/477870 | Any name for this special function? | This is a standard hypergeometric function. Note that
$$ \frac{1}{(a-m)!} = (-1)^m \frac{(-a)\_m}{a!}\quad\text{and}\quad \frac{1}{(b+m)!} = \frac{1}{b!\,(b+1)\_m}$$
in terms of the rising Pochhammer symbol $(q)\_m = q(q+1)\cdots(q+m-1)$. Hence,
$$f\_{abc} = \frac{1}{a!b!c!} \sum\_{m=0}^\infty \frac{(-a)\_m(-c)\_m}{(b+... | 17 | https://mathoverflow.net/users/47484 | 417940 | 170,215 |
https://mathoverflow.net/questions/417913 | 2 |
>
> Is it true that
> $$\sum\_{r=0}^p \sum\_{i=0}^r a\_{n,p,r,i}=0$$
> for all natural $n$ and all natural $p\ge2n$, where
> $$a\_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)!
> (p-r+i)! (n-r+i)! (p-i)!}?
> $$
>
>
>
This **is** true if $n\in\{1,\dots,10\}$ and $p\in\{2n,\dots,2n+10\}$.
(H... | https://mathoverflow.net/users/36721 | Another combinatorial identity | Subst $k = p - 2n \ge 0$ and $s = r - i$ to get the symmetric
$$\sum\_{s \ge 0,i \ge 0} [s + i \le 2n + k] \frac{(-1)^{s+i} (3n+k-s-i-1)! (2n^2 + nk - is)}{i!(n-i)!(2n + k-i)! s!(n-s)!(2n + k-s)!}$$
But then we see from the $(n-i)!(n-s)!$ in the denominator that the bounds should actually be $0 \le i, s \le n$, and... | 4 | https://mathoverflow.net/users/46140 | 417948 | 170,217 |
https://mathoverflow.net/questions/417896 | 22 | Given a connected smooth manifold $M$ of dimension $m>1$, points $p\_1,\dots,p\_n\in M$ and positive values $\{d\_{i,j};1\leq i<j\leq n\}$ satisfying the strict triangle inequalities $d\_{i,j}<d\_{i,k}+d\_{k,j}$,
Can we give $M$ a complete riemannian metric $g$ so that $d\_g(p\_i,p\_j)=d\_{i,j}$, where $d$ is the geo... | https://mathoverflow.net/users/172802 | Can we make distances in a finite subset of a manifold whatever we want? | It is not possible to find $5$ points $x\_1,\ldots,x\_5$ on a genus zero Riemannian 2-manifold (a sphere) such that $d(x\_i,x\_j)=1$ for all $i,j$.
The reason is that the complete graph $K\_5$ is not planar.
Assume by contradiction that we have $5$ points $x\_1,\ldots,x\_5$ with $d(x\_i,x\_j)=1$.
Up to permuting th... | 28 | https://mathoverflow.net/users/5690 | 417951 | 170,218 |
https://mathoverflow.net/questions/417944 | 1 | Given a [pushdown automaton](https://en.wikipedia.org/wiki/Pushdown_automaton) (PDA), we seek a shortest word accepted by it. A standard approach is to map the problem in the corresponding context-free grammar. Can we analyze and solve this problem directly in the PDA?
| https://mathoverflow.net/users/52871 | Shortest word accepted by a PDA | I think you can just solve the "obvious inequalities" to get a polynomial time algorithm. I.e. assume acceptance by empty stack, and for each pair of states $p$, $q$ and a stack symbol $t$, let $T(p, q, t)$ be the minimal word length you need to move from state $p$ to $q$ while erasing $t$ from the stack, without dippi... | 1 | https://mathoverflow.net/users/123634 | 417960 | 170,219 |
https://mathoverflow.net/questions/417736 | 1 | Can the [Schreier coset](https://en.wikipedia.org/wiki/Schreier_coset_graph) graphs can be seen as a subgraph of Cayley graph on the same groups(neglecting the loop edges) and, hence, have their chromatic numbers bounded by the chromatic numbers of the Cayley graphs on those groups with the same generating set?
Also,... | https://mathoverflow.net/users/100231 | Difference in chromatic number between Schreier coset graphs and Cayley graphs | Here's a good example.
Let $G=S\_n$, $S=\{(i,j)\mid 1\leq i<j\leq n \}$. Then $\operatorname{Cay}(G,S)$ is a bipartite graph and so its chromatic number is $2$. Let $H=\{g\in S\_n\mid g(n)=n\}$. It is easy to check that $\operatorname{Sch}(G/H,S)$ is a complete graph (if we forget about loops) and hence its chromatic... | 2 | https://mathoverflow.net/users/173068 | 417963 | 170,220 |
https://mathoverflow.net/questions/417977 | 1 | $\DeclareMathOperator\Norm{Norm}$Suppose $E/\mathbb{Q}(j(E))$ is a CM elliptic curve and $d$ is a non-square. Let $E\_d$ denote the twist of $E$ by $\mathbb{Q}(j(E))(\sqrt{d})$. I know if $d$ is relatively prime to the conductor of $E$, then we have $$N\_{E\_d} = d^2N\_{E}$$
However, by computational investigations, th... | https://mathoverflow.net/users/103423 | Primes of bad reduction for CM elliptic curves | These are exactly the primes where the Neron model of $E$ has fiber type, in [Kodaira's table](https://en.wikipedia.org/wiki/Elliptic_surface), anything other than $I\_0$ and $I\_0^\*$. Here $I\_0$ represents good reduction and $I\_0^\*$ represents a quadratic twist of good reduction by a ramified extension (quadratic ... | 4 | https://mathoverflow.net/users/18060 | 417978 | 170,225 |
https://mathoverflow.net/questions/417964 | 4 | Let $\mathcal{O}(n)$ and $\mathcal{D}(n)$ denote the set of all integer partitions of $n$ into odd parts and distinct parts, respectively. Let $o(n)=\#\mathcal{O}(n)$ and $d(n)=\#\mathcal{D}(n)$. Euler established that $o(n)=d(n)$.
Introduce the enumerations: $a(n)=$ number of parts in $\mathcal{O}(n)$ and $b(n)=$ nu... | https://mathoverflow.net/users/66131 | A refinment of Beck's conjecture | Given a partition $\lambda$ with all parts odd except for one even
part $2k$ appearing $u$ times, let $\mu$ be $\lambda$ with the parts
equal to $u$ removed. Apply your favorite bijection to transform $\mu$
into a partition $\nu$ with distinct parts. Then adjoin to $\nu$ $2k$
parts equal to $u$, obtaining a partition $... | 3 | https://mathoverflow.net/users/2807 | 417996 | 170,231 |
https://mathoverflow.net/questions/417888 | 4 | I have the following expression:
$$
\sum\_{k=0}^{\infty}\frac{n!}{(k+n)!}x^k(L\_n^k(x))^2,
$$
where
$$
L\_n^k(x)=\sum\_{j=0}^n(-1)^j\binom{n+k}{n-j}\frac{x^j}{j!}
$$
is the usual associated Laguerre polynomial and $k\in\mathbb N$. In particular $\int\_0^{\infty}e^{-x}x^k(L^k\_n(x))^2=\frac{(k+n)!}{n!}$.
I am trying t... | https://mathoverflow.net/users/205771 | Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$? | Everything becomes simpler if add some parameters and start the sum at $k=-n$ instead of $k=0$. Note that if $k$ is a negative integer with $-n\le k \le -1$ then $L\_n^k(x)$ is a polynomial in $x$ divisible by $x^{-k}$.
Then we have the formula
$$\sum\_{k=-n}^\infty \frac{n!}{(k+n)!}z^k L\_n^k(x)L\_n^k(y)=e^z\sum\_{i... | 8 | https://mathoverflow.net/users/10744 | 417997 | 170,232 |
https://mathoverflow.net/questions/417994 | 2 | Edge coloring of a graph is an assignment of “colors” to the edges of the graph so that no two adjacent edges have the same color with an optimal number of colors. Two edges are said to be adjacent if they are connected to the same vertex.
There is no known polynomial time algorithm for edge-coloring every graph with... | https://mathoverflow.net/users/148974 | Efficient algorithm for edge-coloring complete graphs | Yes, for all $n$, the edge-chromatic number of $K\_{2n}$ is $2n-1$ and the edge-chromatic number of $K\_{2n+1}$ is $2n+1$. Moreover, it is easy to construct such edge-colourings in polynomial time. For example, see this [Wikipedia](https://en.wikipedia.org/wiki/Graph_factorization) page for an easy method to construct ... | 3 | https://mathoverflow.net/users/2233 | 417998 | 170,233 |
https://mathoverflow.net/questions/417954 | 6 | $\DeclareMathOperator\cd{cd}$Are there any known examples of non-free groups with a property that $\cd(G)+1 = \cd(G \times G)$, or, less restrictive, $G, H$ with $\cd \neq 1, \infty$ such that $\cd(H)+1 = \cd(G \times H)$?
| https://mathoverflow.net/users/81055 | Groups with unusual cohomological dimension of direct product | Let $G=(\mathbb{Q},+)$. Then ${\rm cd}(G)=2$ and ${\rm cd}(G\times G)=3$.
| 6 | https://mathoverflow.net/users/124004 | 418015 | 170,236 |
https://mathoverflow.net/questions/418010 | 0 | Let $P = (V, \sqsubseteq)$ be a partial order and $\mathfrak{D}(P)$ denote the class of downward-closed subsets of the partial order $P$ (i.e, the class of $A \subseteq V$ such that $y\in A \;\&\; x \sqsubseteq y$ implies $x \in A$).
Let a partial order be called *downward complete*, if every non-empty subset has an ... | https://mathoverflow.net/users/122435 | Partial orders on downward closed sets | Conditions 4 and 5 show that $(\mathfrak{D}(P),{\subseteq})$ satisfies condition 2 in the list: because $V\in\mathfrak{D}(P)$ we have the second part of 2; and 5 says that $(\mathfrak{D}(P),{\subseteq})$ is downward complete, because $\bigcap\mathcal{B}=\inf\mathcal{B}$.
| 1 | https://mathoverflow.net/users/5903 | 418016 | 170,237 |
https://mathoverflow.net/questions/417967 | 7 | Assume that $\omega<\kappa\_1<\dotsb< \kappa\_n$ are infinite cardinals such that for each $1\le i\le n$ there is a $\kappa\_i$-complete, $\kappa\_i^+$-saturated ideal $\mathcal I\_i\subset \mathcal P(\kappa\_i)$. Can you obtain a ZFC model which contains $n$-many measurable cardinals? The natural candidate is $L[\math... | https://mathoverflow.net/users/71011 | Measurable cardinals from saturated ideals | Yes. And if the $\mathscr{I}\_k$s are normal, then the suggested candidate works (I haven't really thought about whether the candidate still works without normality). Here is the argument, which is a typical one:
First, we may assume that each $\mathscr{I}\_k$ is normal, by Jech
Lemma 22.28.
Case 1. There is an pro... | 4 | https://mathoverflow.net/users/160347 | 418025 | 170,240 |
https://mathoverflow.net/questions/416781 | 1 | Consider an $M/M/1$ queue with the arrival rate $\lambda>0$ and the service rate $\mu>\lambda$ (so that it is stable), in the stationary regime. Let $A\_t$ be the number of arrivals in the time interval $[0,t]$ and $D\_t$ be the number of departures in that time interval; then both $A\_t$ and $D\_t$ have Poisson distri... | https://mathoverflow.net/users/81488 | The input and output processes in a single-server queue | Let $\eta\_t$ be the number of customers in the system at time $t$ and $\rho=\lambda/\mu<1$ be the load. It holds that $\eta\_0+A\_t-D\_t = \eta\_t$, so $A\_t-D\_t = \eta\_t-\eta\_0$. Write
$$
A\_t D\_t = \frac{1}{2}(A\_t^2 + D\_t^2 - (A\_t-D\_t)^2),
$$
from which we obtain (recall that both $A\_t$ and $D\_t$ are Poiss... | 0 | https://mathoverflow.net/users/81488 | 418026 | 170,241 |
https://mathoverflow.net/questions/417048 | 1 | This is, in a way, a follow up question to [Unipotent orbits and intersection with Levi and pseudo-Levi subgroups](https://mathoverflow.net/questions/415890/unipotent-orbits-and-intersection-with-levi-and-pseudo-levi-subgroups#415890).
I was reading "[A generalisation of the Bala–Carter theorem for nilpotent orbits](... | https://mathoverflow.net/users/64702 | The identity connected component of centralizers of unipotent orbits | If you decompose the $E\_6$ Lie algebra over the $3A\_2$ subgroup $L$ then, in addition to the adjoint representation, there are two irreducible summands: with the right identifications these are isomorphic to $V^{\otimes 3}$ and its dual ($V$ the natural representation for ${\rm SL} \_3$). So, to describe the centrali... | 1 | https://mathoverflow.net/users/26635 | 418030 | 170,243 |
https://mathoverflow.net/questions/418019 | 8 | For each coherent category $C$, let $J\_C$ be the topology on $C$ in which a sieve $\{f\_i\colon U\_i\to X\}\_{i\in I}$ is covering if and only if there exists a *finite* set $I\_0\subseteq I$ such that $\bigcup\_{i\in I\_0} \operatorname{im}(f\_i)=X$ as subobjects of $X$. (This is a Grothendieck topology by Propositio... | https://mathoverflow.net/users/478652 | Questions about coherent topology | **Edit :** I should clarify that I've interpreted "Etale topos" to mean the petit/small étale topos everywhere. What I've said about Grothendieck-Galois duality only apply to the petit étale topos. If you are talking about the Gros topos, then these part no longer holds. I actually don't know if the Gros étale topos of... | 10 | https://mathoverflow.net/users/22131 | 418044 | 170,247 |
https://mathoverflow.net/questions/417928 | 7 | This is a question that a classmate asked me three years ago.
Let $P(x)=\sum\_{i=0}^n a\_ix^i$ be a polynomial such that each $a\_i>0$. Prove or disprove that there exists a positive integer $r$ such that $P(x)^r=\sum\_{i=0}^{nr} b\_ix^i$ and there exists $0\le j\le nr$ such that $b\_0\le b\_1\le \dots\le b\_{j}$ and... | https://mathoverflow.net/users/170895 | Prove or disprove that the power of positive term polynomial will be eventually single peak | This is answered affirmatively by Odlyzko and Richmond, *On the unimodality of high convolutions of discrete distributions*, Annals of probability (1985) 299--306: all sufficiently large powers of the polynomial (with positive coefficients and no gaps) are strongly unimodal, that is, the coefficients form a log concave... | 8 | https://mathoverflow.net/users/42278 | 418049 | 170,250 |
https://mathoverflow.net/questions/418028 | 1 | A theorem by Lebesgue, Hausdorff and Banach says the following (Kechris' *Classical Descriptive Set Theory*, p. 192):
>
> Let $X$ be a separable metrizable space and $f: X \rightarrow \mathbb{R}$ be a $\boldsymbol{\Sigma}\_2^0$-measurable function, then $f$ is the pointwise limit of a sequence of continuous functio... | https://mathoverflow.net/users/141146 | Lebesgue Hausdorff Banach theorem for Baire class $1$ functions on $\mathbb{R}^\omega$ | Is it indeed the case that the statement can be extended to functions with codomain $\mathbb{R}^\omega$.
Suppose we have $f:\mathbb{R}^\omega\rightarrow\mathbb{R}^\omega$ which is $\boldsymbol{\Sigma}\_2^0$-measurable, then in particular the functions $f^n:\mathbb{R}^\omega\rightarrow\mathbb{R}$ with $f^n(x)=f(x)(n)$... | 0 | https://mathoverflow.net/users/141146 | 418076 | 170,256 |
https://mathoverflow.net/questions/259663 | 6 | Is there a positive 128-bit integer whose square has all middle bits equal to 1?
(The "middle bits" are naturally the 65th bit through the 192nd bit, defining
the 1st bit as the least significant bit of the full integer.)
| https://mathoverflow.net/users/20757 | Mid-Square with all bits set |
>
> Is there a positive 128-bit integer whose square has all middle bits equal to 1?
>
>
>
***YES***. One is AAAAAAAAAAAAAAAB555555555555555516, which square is 71C71C71C71C71C7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8E38E38E38E38E3916.
---
It's asked if there are integers $n$ such that $0<n<2^{2k}$ and $(n^2\bmod... | 7 | https://mathoverflow.net/users/122065 | 418082 | 170,257 |
https://mathoverflow.net/questions/417588 | 6 | Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc, n\}$ with size $k$ let $e\_I$ be the function which evaluates the determinant of the $k$ by $k$ submatrix found by taking... | https://mathoverflow.net/users/38359 | Vanishing linear combinations of minors | This notion is indeed well-known and connected to the arithmetical rank of the ideal $I\_k$ of $C[V]$ generated by the maximal minors.
**Short answer** : that $\beta(k,n) = mk - k^2 +1$ and there is always a set of $mk-k^2+1$ linear combinations of maximal minors which is a *rank detecting set* (with your words).
*... | 8 | https://mathoverflow.net/users/37214 | 418086 | 170,260 |
https://mathoverflow.net/questions/418069 | 4 |
>
> Let $ 0<a<b $, $ f\in C^1\left([0,b]\right)$. Assume that $ f $ is concave on $ [0,a] $ and convex on $ [a,b] $ with $ f'(0)>f'(b) $. Please prove that there exist $ n\_0\in\mathbb{N} $ which is sufficiently large, such that for any $ n\geq n\_0 $ and $ a\_1+a\_2+...+a\_n=b $ with $ a\_i\geq 0 $ ($ i=1,2,...,n $)... | https://mathoverflow.net/users/241460 | How to prove $ \sum_{k=1}^{n}f(a_k)\leq nf\left(\frac{b}{n}\right) $ for sufficiently large $ n $ here? | Denote $b/n=s$. We want a pointwise bound $$f(x)\leqslant f(s)+(x-s)f'(s),\label{1}\tag{$\heartsuit$}$$ then summing \eqref{1} up for $x=a\_1,\ldots,a\_n$ we get the desired inequality. Note that if $s<a$ (that holds for $n>b/a$) we get \eqref{1} on $[0,a]$ by concavity. For proving \eqref{1} on $[a,b]$, by convexity i... | 3 | https://mathoverflow.net/users/4312 | 418087 | 170,261 |
https://mathoverflow.net/questions/415727 | 1 | Consider a CT Markov Process $X=(X\_t)\_{t\geq0}$ with state space $E\in\mathbb{R}^N$. Are there any general conditions under which a stationary distribution $\pi$ for $X$ is also a limiting distribution and viceversa?
Any useful reference will be much appreciated.
| https://mathoverflow.net/users/86048 | Stationary and limiting distributions | For Markov chains, a very useful condition is Harris recurrence,
see <https://en.wikipedia.org/wiki/Harris_chain>.
This has been generalized to continuous time, see
<https://www.jstor.org/stable/3690386?seq=1#metadata_info_tab_contents>
| 2 | https://mathoverflow.net/users/7691 | 418099 | 170,264 |
https://mathoverflow.net/questions/418092 | 8 | Let $P$ be a partial order on a finite set $S$ (assume that every element is related to at least one other element besides itself…this raises a few quick questions: is this implied by the definition of partial order and if not, what are the "isolated" points called and what is a partial order with no such points called... | https://mathoverflow.net/users/5090 | Smallest relation in complement of partial order that prohibits its extension | I claim that the relation $R$ is in fact unique. The uniqueness follows from the one pair extension property of finite partially ordered sets.
Proposition: Suppose that $X$ is a finite set with partial ordering $P$. Then whenever $Q$ is a partial ordering on $X$ with $P\subseteq Q,P\neq Q$, there exists an ordered pa... | 9 | https://mathoverflow.net/users/22277 | 418100 | 170,265 |
https://mathoverflow.net/questions/418093 | 1 | I am wondering about the following question: A strictly convex (concave) differentiable function $f:\mathcal{R}\to\mathcal{R}$ has the geometrical property that its graph lies completely above (below) the tangential line at any point (except at the point of contact). Now, this is a special property of such functions bu... | https://mathoverflow.net/users/19673 | Convex/concave points of a differentiable function | Let $f$ be non linear in the interval $[-1,1]$. We can suppose $f(-1)=f(1)=0$ and $f(x)>0$ for some $x\in(-1,1)$. Now let $N$ be so big that the closed ball of center $(0,-N)$ and with radius $\sqrt{N^2+1}$ doesn't contain the whole graph of $f$. Let $(x\_0,f(x\_0))$ be one of the points of the graph furthest from $(0,... | 2 | https://mathoverflow.net/users/172802 | 418101 | 170,266 |
https://mathoverflow.net/questions/418051 | 3 | Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (via the projection). So my question is the following: is the locus of points $s\in S$ such that $\cal{F}\_s$ is locally fr... | https://mathoverflow.net/users/478669 | Is local freeness open for curves? | To elaborate on Jason's comment: consider a morphism of schemes $f:\mathrm{X}\rightarrow\mathrm{S}$, and an $f$-flat coherent sheaf $\mathscr{F}$ on $\mathrm{X}$. Then
(1) the singular locus $\mathrm{Sing}(\mathscr{F})$ of $\mathscr{F}$ (given by the points $x\in\mathrm{X}$ such that $\mathscr{F}$ is not locally free... | 7 | https://mathoverflow.net/users/104669 | 418106 | 170,267 |
https://mathoverflow.net/questions/418006 | 5 | A category $\mathcal{C}$ is called $\textbf{discrete}$ if the only morphisms are identity morphisms.
Consider the following weaker notion: a category $\mathcal{C}$ is called $\textbf{totally disconnected}$ if $\text{Hom}\_\mathcal{C}(C,D)=\varnothing$ for all $C\neq D$.
Vopenka's principle ($\textbf{VP}$) states th... | https://mathoverflow.net/users/474727 | Stronger (?) form of Vopenka's principle | VP2 is equivalent to VP because every set carries a rigid binary relation. This is similar as Lemma 6.3 in my book with Adámek. In fact, VP2 is an original formulation of VP (see Jech, Set Theory).
| 6 | https://mathoverflow.net/users/73388 | 418113 | 170,270 |
https://mathoverflow.net/questions/417950 | 2 | By Reflection I mean the following schema:
**Reflection:** $$\forall X \, (\varphi \implies \exists \alpha : \varphi^{V\_\alpha})$$
where $\varphi$ is a first order formula (defined predicates and functions allowed) in which only symbol "$X$" occurs free, $\varphi^{V\_\alpha}$ is the "$\in {V\_\alpha}$" bounded for... | https://mathoverflow.net/users/95347 | Can we allow defined predicates and functions in reflection? | The first answer took care of the case where "Reflection" was a scheme of formulas in the language of ZF. Now we consider what we now think was the intent of the questioner.
We expand the language of ZF by introducing a predicate symbol P and a function symbol F
for each formula in the language of ZF. We also add add... | 2 | https://mathoverflow.net/users/133981 | 418114 | 170,271 |
https://mathoverflow.net/questions/418117 | 1 | Let $a(n)$ be [A007306](https://oeis.org/A007306), denominators of Farey tree fractions (i.e., the Stern-Brocot subtree in the range $[0,1]$).
Let $b(n)$ be [A002487](https://oeis.org/A002487), Stern's diatomic series (or Stern-Brocot sequence): $b(0) = 0, b(1) = 1$; for $n > 0$: $b(2n) = b(n), b(2n+1) = b(n) + b(n+1... | https://mathoverflow.net/users/231922 | Stern-Brocot tree and subtree | The second half is already given in the question, so really what you're asking is whether $$b(2n-1)=b(2n-3)+b(n-1)-2(b(2n-3)\bmod b(n-1))$$
But as noted in OEIS (quoted with relabelling),
>
> Moshe Newman proved that the fraction b(n+1)/b(n+2) can be generated from the previous fraction b(n)/b(n+1) = x by 1/(2\*flo... | 3 | https://mathoverflow.net/users/46140 | 418130 | 170,274 |
https://mathoverflow.net/questions/380539 | 19 | [This answer](https://mathoverflow.net/a/69248/150063) says,
>
> IIRC, the calculus of inductive constructions is equi-interpretable with ZFC plus countably many inaccessibles — see Benjamin Werner's *"Sets in types, types in sets"*. (This is because of the presence of a universe hierarchy in the CIC.)
>
>
>
B... | https://mathoverflow.net/users/150063 | Are we sure the calculus of inductive constructions and ZFC plus countably many inaccessible cardinals are equiconsistent? | The situation is a bit subtle. One can interpret CIC in any model of ZFC with infinitely many inacessibles. However, interpreting ZFC in CIC is more subtle. First one needs to assume the law of excluded middle and choice in CIC (and perhaps quotient types depending on how smooth we want things to work). These are very ... | 9 | https://mathoverflow.net/users/2000 | 418133 | 170,276 |
https://mathoverflow.net/questions/418111 | 7 | Let $\{X\_i\}\_{i=1}^{\infty}$ be i.i.d. random variables such that: i) $X\_i > 0$; and ii) $\textrm{Pr}[X\_i > x] \sim x^{-\alpha}$ at large $x$ for some $\alpha \in (0, 1)$. Define the quantities
\begin{equation}
S\_n \equiv \frac{X\_1 + \cdots + X\_n}{n^{1/\alpha}}.
\end{equation}
By Kolmogorov's zero-one law (right... | https://mathoverflow.net/users/478720 | What is the liminf of a sum of i.i.d. random variables with heavy tails? | $\newcommand{\de}{\delta}\newcommand{\ep}{\varepsilon}$Here we implement the idea proposed in mike's answer. The proof below works for all $\alpha\in(0,2]$.
Let $a:=1/\alpha$. Let
\begin{equation\*}
Z\_n:=X\_1+\cdots+X\_n,
\end{equation\*}
so that
\begin{equation\*}
S\_n=Z\_n/n^a.
\end{equation\*}
Take any real ... | 3 | https://mathoverflow.net/users/36721 | 418140 | 170,279 |
https://mathoverflow.net/questions/311958 | 5 | I found a very interesting problem in the [Open Problem Garden](http://www.openproblemgarden.org/op/a_nowhere_zero_point_in_a_linear_mapping), which I am surprised is not as well-known as I would think it would be:
Prove that If $p>3$ is prime and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb Z\_... | https://mathoverflow.net/users/7089 | A nowhere-zero point in a linear mapping conjecture | This is the Alon-Jaeger-Tarsi conjecture first stated in 1981 and [resolved](https://arxiv.org/pdf/2107.03956.pdf) very recently (for $p\ge 83$) by János Nagy and Péter Pál Pach.
| 2 | https://mathoverflow.net/users/9924 | 418150 | 170,280 |
https://mathoverflow.net/questions/418145 | 1 | Consider $n$ labeled balls, $k$ of which are red and $(n-k)$ blue. Given a permutation of these balls, we tick $n-1$ times. For the $i$-th tick, if the $i$-th ball in the permutation is red, then it paints the $(i+1)$-th ball in the permutation blue (if the latter is already blue, then it remains blue).
We secretly mar... | https://mathoverflow.net/users/478035 | Counting permutations defined by a simple process | Let us put an additional blue ball in position $0$, to the left of the $n$ balls.
The condition on the permutations of the $n$ balls is then that the marked red ball be preceded by an odd number (say $2r-1$) of red balls, which in turn must be preceded by a blue ball. Let us refer to such permutations as good.
Let $p\_... | 2 | https://mathoverflow.net/users/36721 | 418151 | 170,281 |
https://mathoverflow.net/questions/418149 | 3 | Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that
$$
\int\_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}.
$$
Does it follow that $\phi$ is a constant in a neighborhood of $t=0$?
If the answer to the above question is affirmative I have a follow up question as foll... | https://mathoverflow.net/users/50438 | On an asymptotic integral | No. E.g., let $\phi(t)=(t-1/2)^2$ (so that $\phi$ is even about $1/2$) and $a(t)=2(1/2-t)$ (so that $a$ is odd about $1/2$).
| 2 | https://mathoverflow.net/users/36721 | 418154 | 170,283 |
https://mathoverflow.net/questions/417518 | 5 | In C-H Sah's book *Hilbert's third problem: scissors congruence*, the author defines the data for *abstract scissors congruence* in order to prove Zylev's theorem by combinatorial means in great abstraction. I expect that most of the book is over my head as an undergrad but I'm looking for clarification as to wether th... | https://mathoverflow.net/users/478205 | Looking for clarification of C-H Sah's definition of abstract scissors congruence | I agree that Sah's definition is not quite correct, but it's not difficult to fix. The key point is this: when you have an intersection that contains $0$ simplices, you declare it to be empty. If you look at the definition of an assembler (which was partially designed to fix Sah's definition) it uses the categorical st... | 1 | https://mathoverflow.net/users/1378 | 418162 | 170,286 |
https://mathoverflow.net/questions/418178 | 7 | Here we choose the definition of "is a cardinal" as there is no surjective map from a smaller ordinal to it.
It's easy to prove that, if $L\_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, then $L\_\alpha \vDash ZFC$. Also it's easy to prove that if $L\_\alpha \vDash ZFC$, then α is a cardinal in $L\_{\alpha+1}$. ... | https://mathoverflow.net/users/170286 | If $L_\alpha \vDash ZFC$, then do we have $L_{\alpha+1} \vDash \alpha\text{ is inaccessible}$? | Yes. The elements of $L\_{\alpha+1}$ are exactly those subsets of $L\_\alpha$ which are definable from parameters over $L\_\alpha$. But $L\_\alpha\models\mathrm{ZFC}$, so from here we can just use the usual proof that second order ZFC implies inaccessibility. That is:
(i) every bounded subset of $L\_\alpha$ which is ... | 14 | https://mathoverflow.net/users/160347 | 418186 | 170,294 |
https://mathoverflow.net/questions/418194 | 0 | Here $B\_r$ presents the open ball of radius r in $R^3$. So I hope to know how to prove the following inequality.
$$
\int\_{B\_r} |u|\le Cr^{\frac{9}{5}}\left(\int\_{B\_r} |u|^2\right)^{\frac{1}{5}}\left(\int\_{B\_r} |u|^3\right)^{\frac{1}{5}}
$$
I did not figure out where these exponents come from. Could anyone show m... | https://mathoverflow.net/users/295572 | Using Hölder's Inequality to prove the following equation | I think that I have got the answer… it is quite a simple result from the Hölder inequality,
$u^{\frac{2}{5}}u^{\frac{3}{5}}1$ with exponents $\frac{1}{5}$, $\frac{1}{5}$ and $\frac{3}{5}$….
| 0 | https://mathoverflow.net/users/295572 | 418195 | 170,297 |
https://mathoverflow.net/questions/418185 | 4 | Let $M$ be a Ricci-flat Riemannian manifold and $N \subset M$ a totally geodesic submanifold. Is $N$ also Ricci-flat?
A partial result in that direction is that the Ricci curvature of $N$ is given by
$$\operatorname{Ric}^N(Y, Z) = \operatorname{tr}(TN \ni X \mapsto R(X, Y)Z \in TN),$$
where $R$ is the Riemmanian curv... | https://mathoverflow.net/users/409915 | Ricci curvature of totally geodesic submanifold | You can find an explicit counterexample in the Riemannian Schwarzschild solution.
Let $(z,r,\omega) \in \mathbb{R} \times (1,\infty) \times \mathbb{S}^2$, denote by $h$ the standard sphere metric. Consider the following Riemannian metric
$$ ds^2 = (1- r^{-1})~dz^2 + (1-r^{-1})^{-1} ~dr^2 + r^2 h $$
One can explic... | 8 | https://mathoverflow.net/users/3948 | 418198 | 170,299 |
https://mathoverflow.net/questions/418193 | 7 | Indirect method (associated with a certain problem of electrostatics) indicates that $$\sum\limits\_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!}=\frac{2}{3\pi}.$$ Is this result known?
| https://mathoverflow.net/users/32389 | Infinite series for $1/\pi$. Is it known? | Using the [standard power series for the complete elliptic integral of the second kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind) $$E(k) = \frac{\pi}{2} \sum\_{j=0}^\infty \left(\frac{(2j)!}{2^{2j}(j!)^2}\right)^2 \frac{k^{2j}}{1-2j},$$
we find
\begin{align\*}
\sum\l... | 19 | https://mathoverflow.net/users/47484 | 418199 | 170,300 |
https://mathoverflow.net/questions/418180 | 3 | For $\varepsilon > 0$, we say a function $f: \mathbb R^n \to \mathbb R^m$ is (pointwise) $(1 - \varepsilon)$-Hölder continuous at $x \in \mathbb R^n$ if
$$ \lim\_{ y \to 0} \frac{f(x + y) - f(x)}{\lVert y \rVert^{1 - \varepsilon}} = 0.$$
**Question:** Suppose $f$ is $(1 - \varepsilon)$-Hölder continuous at all $x \... | https://mathoverflow.net/users/173490 | Pointwise Hölder continuity of order $1-\varepsilon$ | One counter example is the Takagi function $\tau$, see Theorems 8.1 and 8.2 in the survey [2] (As noted yesterday at [4]).
More detail: Takagi [1] showed that his function $\tau$ is nowhere differentiable. On the other hand, It is easy to verify that for $0 \le x <x+h\le 1$, we have
$$|\tau(x+h)-\tau(x)| \le 2 h\log\... | 7 | https://mathoverflow.net/users/7691 | 418212 | 170,302 |
https://mathoverflow.net/questions/418211 | 7 | Is it true that every group quasi-isometric to the Heisenberg group admits a proper cocompact action by isometries on Nil?
| https://mathoverflow.net/users/159356 | Discrete cocompact group of isometries of Nil | This is true. The first step is that the group $\Gamma$ then has a subgroup of finite index that embeds as a lattice in the Heisenberg group $H$. This step is sketched [in this answer](https://mathoverflow.net/a/228717/14094) (I repeated the argument here below "original answer"). To deduce the general case, let me rel... | 7 | https://mathoverflow.net/users/14094 | 418214 | 170,303 |
https://mathoverflow.net/questions/418208 | 4 | Some statements that are true for ordinary groupoids fail for topological groupoids (by which I mean groupoids internal to the category of topological spaces): for instance, every ordinary groupoid is equivalent to the disjoint union of one-object groupoids, but [not every topological groupoid is equivalent to a disjoi... | https://mathoverflow.net/users/478652 | Topological groupoids and equivariant sheaves | First note that the definition of equivalence you are using for topological groupoids is too strong and is not the one people generally use. It is really too much to ask for a continuous quasi-inverse functor. Moerdijk has papers with the "right" notion of Morita equivalence of topological groupoids.
Here is an answe... | 6 | https://mathoverflow.net/users/15934 | 418216 | 170,304 |
https://mathoverflow.net/questions/418217 | 3 | Let $C \subset \mathbb P^3$ be a smooth complete intersection curve given by $2$ hypersurfaces of degree at least $4$ in $\mathbb P^3$. Then can it happen that:
1. $C$ is a $2$-sheeted covering of a curve of low genus (say $\leq 3$)?
2. $C$ is a $3$-sheeted covering of an elliptic curve?
Note that, in both the case... | https://mathoverflow.net/users/474635 | On $k$-sheeted covering of curves | **Edit:** As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: suppose $C$ is a $(p,q)$ complete intersection, with $p\geq q$. The maximum number $\ell$ of points of $C$ lying on a line is $\leq p\ $ — otherwise the line is contained in $C$.
By B. Basili, *Indice de Cli... | 6 | https://mathoverflow.net/users/40297 | 418219 | 170,305 |
https://mathoverflow.net/questions/323708 | 6 | Let $(V,\*)$ be an algebra and denote $A\_\*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $\*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A\_\*(a,b,c):=(a\*b)\*c-a\*(b\*c)$.
The associator $A\_\*$ is assumed to enjoy the following property:
$A\_\*(a,b,c)+A\_\*(b,c,a)-A\_\*(b,a,c)=0$.
**Qu... | https://mathoverflow.net/users/104743 | Weak associativity | Let me assume that the characteristic of the ground field is different from two. Let me start by replacing your identity by something where the existing symmetries are a bit more apparent. I claim that your identity for the associator (which I will denote simply $(a,b,c)$, dropping $A\_\*$) is equivalent to the followi... | 2 | https://mathoverflow.net/users/1306 | 418227 | 170,308 |
https://mathoverflow.net/questions/418159 | 1 | I am a PhD student. During my researches, I often have to deal with inequalities involving sums of binomial coefficients, where the sums are indexed by some set of integer compositions. For example, let $l,k$ be positive integers such that $l \leq k$ and either $l$ is odd or both $l$ and $k$ are even. One of my results... | https://mathoverflow.net/users/73667 | Inequalities between sums of products of certain binomial coefficients | Take the LHS:
$$\sum\_{\substack{\alpha \models i+k} \\ \ell(\alpha) = k \\ \alpha\_a \leq \lfloor \frac{l}{2} \rfloor} \prod\_{a=1}^k \binom{l}{2(\alpha\_a - 1)}$$
Firstly, we can simplifying by rolling the subtraction of one into the definition of $\alpha$:
$$\sum\_{\substack{\sum \alpha\_a = i \\ \ell(\alpha) = k \\... | 2 | https://mathoverflow.net/users/46140 | 418230 | 170,309 |
https://mathoverflow.net/questions/418228 | 8 | In page-479 of Visual Complex Analysis, Tristan Needham derives the flux of a vector field in Geometric form:
$$ \nabla \cdot X = \partial\_s |X| + \kappa\_p |X|$$
The $\partial\_s$ is a derivative along streamlines of the vector field $X$ an $\kappa\_p$ is the curvature of the orthogonal streamline at the point we... | https://mathoverflow.net/users/159957 | Geometric definition of divergence using curvature mentioned in Tristan Needham | Given any hypersurface of a Riemannian manifold with a unit normal vector field $\nu$, extend $\nu$ to be unit length. Then
$$\operatorname{div}(u\nu)=u\operatorname{div}\nu+\text{d}u(\nu).$$
This proves that if $X$ is any nonvanishing vector field on a neighborhood of the hypersurface, which is orthogonal to the hyper... | 10 | https://mathoverflow.net/users/156492 | 418232 | 170,310 |
https://mathoverflow.net/questions/418144 | 3 | The Hom scheme of two projective varieties over some field is constructed as an open subfunctor of the Hilbert scheme of the product of the two schemes by Grothendieck. So it is a countable union of quasi-projective varieties. For example even in the simplest case $\mathcal{Hom}(\mathbb{P}\_k^1, \mathbb{P}\_k^1)$ there... | https://mathoverflow.net/users/127776 | When Hom scheme has projective components? | Let $k$ be an algebraically closed field of characteristic zero and let $X$ be a projective variety over $k$. To try and answer your first two questions, let us try formalize the property that Hom-schemes $\mathrm{Hom}(Y,X)$ are finite unions of quasi-projective schemes.
**Definition.** We say that $X$ is *bounded ov... | 3 | https://mathoverflow.net/users/4333 | 418236 | 170,311 |
https://mathoverflow.net/questions/418222 | 4 | Let $x$ be a $n$-dimensional Gaussian random vector, i.e., $x \sim \mathcal{N}(0,\sigma^2 I\_n)$. What is its probability of falling in a cone? Say a cone $C = \{ x \in \mathbb R^n: \frac{\langle x - v, -v \rangle}{\|x-v\|\_2 \|v\|\_2} \leq \cos \theta \}$ is parameterized by $v \in \mathbb R^n$ and $\theta \in (0,\pi ... | https://mathoverflow.net/users/478836 | Probability of a Gaussian random vector in a cone | Let
\begin{equation\*}
Z=(Z\_1,\dots,Z\_n):=x/\sigma\sim N(0,I\_n),
\end{equation\*}
\begin{equation\*}
Y\_n:=Z\_2^2+\dots+Z\_n^2\sim\chi^2\_{n-1},
\end{equation\*}
\begin{equation\*}
c:=\|v\|\_2/\sigma>0,\quad t:=\cos\theta\in(0,1),\quad u:=\frac t{\sqrt{1-t^2}}=\cot\theta\in(0,\infty).
\end{equation\*}
By the sp... | 4 | https://mathoverflow.net/users/36721 | 418237 | 170,312 |
https://mathoverflow.net/questions/417637 | 1 | Consider the sparse linear regression model $y=X\theta^\*+w$, where $w\sim N(0, \sigma^2 I\_{n\times n})$ and $\theta^\*\in R^d$ is supported on a subset $S$. Suppose that the sample covariance matrix $\hat{\Sigma}=X^TX/n$ has its diagonal entries uniformly upper bounded by 1, and that for some parameter $\gamma>0$, it... | https://mathoverflow.net/users/168083 | Lasso of sparse linear regression model | The proof you are looking for should be in
>
> Theorem 1 of *Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators* by Karim Lounici (<https://arxiv.org/abs/0801.4610>).
>
>
>
Check the relationship of your curvature condition with Assumption 2 of that paper. The proof there... | 1 | https://mathoverflow.net/users/141760 | 418243 | 170,314 |
https://mathoverflow.net/questions/418229 | 2 | Consider the following: $r\_1,...,r\_t$ are iid symmetric signs taking value $\pm1$, independent of $B\sim Binomial(p, q)$ with integer $p$ and $q=p^{-1.01}$.
**Question**: Consider $t$ as a non-decreasing function of $p$. For which such non-decreasing function of $p$ does there exist a constant $\lambda>0$ independe... | https://mathoverflow.net/users/141760 | Bound on the MGF of the product of two independent binomial, one being centered | $\newcommand{\la}{\lambda}$The estimate of $t$ that you got cannot be improved. Indeed, let
\begin{equation}
S\_t:=\sum\_{u=1}^t \frac{r\_u}{\sqrt t}.
\end{equation}
Suppose, as you did, that
\begin{equation}
E\exp(\la BS\_t)\le1.05
\end{equation}
for some real $\la>0$. Then
\begin{equation}
1.05\ge\frac{\la^{2p}}... | 1 | https://mathoverflow.net/users/36721 | 418245 | 170,315 |
https://mathoverflow.net/questions/418256 | 4 | Let $\hat{D} := D \backslash \{0\}$ be a ball in $R^n$ with the origin $\{0\}$ removed. Assume that $\hat{D}$ has a structure as an orbifold (may be distinct from its standard manifold structure). Is it possible to extend the orbifold structure from $\hat{D}$ to $D$ ?
| https://mathoverflow.net/users/35716 | Extension of an orbifold structure from punctured balls to balls | No. This is not true in dimension three (nor in any higher dimension).
The three-dimensional orbifold structures allowed at a point are controlled by the list of finite subgroups of $\mathrm{SO}(3)$. In particular, there are at most three one-dimensional loci that converge at any point.
Let $S^2(2^m)$ be the two-or... | 2 | https://mathoverflow.net/users/1650 | 418258 | 170,318 |
https://mathoverflow.net/questions/418250 | 4 | Let $\mathrm{G}$ be any infinite discrete group, and $\mathrm{H}$ be any finite index subgroup of $\mathrm{G}.$ I do not know if the Frobenius reciprocity theorem is true for the infinite groups. I want to say that, given any irreducible finite-dimensional complex representation $\rho$ of $\mathrm{G}$ there exists an i... | https://mathoverflow.net/users/100578 | Frobenius reciprocity theorem for infinite groups | A semi-answer, too long for a comment.
"the Frobenius reciprocity theorem" for finite groups is just a special case of the Hom-Tensor adjunction if you phrase it as
$$\operatorname{Hom}\_{\mathbb{C}[H]}(X,\operatorname{Res}\_H^G(Y)) = \operatorname{Hom}\_{\mathbb{G}[G]}(\operatorname{Ind}\_H^G(X),Y)$$
because $\opera... | 4 | https://mathoverflow.net/users/3041 | 418262 | 170,319 |
https://mathoverflow.net/questions/418045 | 12 | It's a [standard exercise](https://math.stackexchange.com/a/3289290/28111) to show that every *countable* first-order theory has an irredundant axiomatization. For *uncountable* first-order theories, the result is much more difficult and was [proved by Reznikoff](https://arxiv.org/abs/1108.5171). For $\mathcal{L}\_{\om... | https://mathoverflow.net/users/8133 | Do second-order theories always have irredundant axiomatizations? | Here is a proof of Reznikoff’s theorem I found among my notes, not quite following Reznikoff’s proof. I stared at it for a while, and it seems to apply to second-order logic just the same; in fact, the only property of first-order logic it uses is that any sentence contains only finitely many non-logical symbols, and t... | 9 | https://mathoverflow.net/users/12705 | 418270 | 170,323 |
https://mathoverflow.net/questions/418247 | 3 | [This question is looking at the paper
* Yau, S.-T., *On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I*, Comm. Pure Appl. Math., **31** (1978) 339-411, doi:[10.1002/cpa.3160310304](https://doi.org/10.1002/cpa.3160310304), ([pdf](https://jasonpayne.webs.com/Math5339/On%20the... | https://mathoverflow.net/users/469129 | A problem of using Schauder estimate in the paper of Yau's proof of calabi conjecture | There are many non-equivalent formulations of the Schauder estimates; the version you are suggesting is the most common. For the version he needs, Yau gives a precise page & theorem reference (p.156, formula 5.5.23) to Morrey's book. In the case of the Euclidean Laplacian you can also see p.69-70 of Gilbarg & Trudinger... | 1 | https://mathoverflow.net/users/156492 | 418274 | 170,325 |
https://mathoverflow.net/questions/417810 | 0 | Define a set $C\_3(S):=\{\Delta\in R^d: \|\Delta\_{S^c}\|\_1\le 3\|\Delta\_S\|\_1\}$. Suppose we form a random design matrix $X\in R^{n\times d}$ with rows drawn iid from a $N(0,\Sigma)$ distribution and that for all $\Delta \in C\_3(S)$,
$$
\|\Sigma\Delta\|\_{\infty}\ge \gamma\|\Delta\|\_{\infty}
$$
| https://mathoverflow.net/users/168083 | Sparse linear regression model | The result is completely independent of the regression model ($y, w$ have no role and need not be involved). $\alpha$ in the definition of $C\_3(S)$ is not defined, I will assume it is $\alpha=3$.
For each $\Delta\in C\_3(S)$, there exists $k=1,...,p$ such that
$$|e\_k^TX^TX/n \Delta| - |e\_k^T\Sigma\Delta| + |e\_k^T... | 1 | https://mathoverflow.net/users/141760 | 418275 | 170,326 |
https://mathoverflow.net/questions/418210 | 4 | I'm fixing some type of structure $\Sigma$ (possibly multi-sorted, with functions symbols and relation symbols, though assuming it single sorted with only relation symbols wouldn't change anything). Let $A$ and $B$ be two $\Sigma$-structure. Is there a connection between the following to notion :
1. The locale $[A,B]... | https://mathoverflow.net/users/22131 | The locale of morphisms vs a morphism to an ultrapower? | I believe there are situations where the relevant locale is trivial, but there are many homomorphisms post-ultrapower.
For example, take $\mathcal{B}=(\mathbb{N};<)$ and let $\mathcal{A}$ be a nontrivial ultrapower of $\mathcal{B}$. There are no genuine homomorphisms from $\mathcal{A}$ to $\mathcal{B}$, and moreover ... | 2 | https://mathoverflow.net/users/8133 | 418287 | 170,330 |
https://mathoverflow.net/questions/418269 | 4 | I have a monad on an ind-category (specifically, my ind-category has a monoidal structure and I have an algebra object, so the monad is tensoring with it). It would be very useful in my work if the Eilenberg-Moore category of the monad it itself an ind-category. Is there a known criterion for the monad for this to be t... | https://mathoverflow.net/users/112314 | When is the Eilenberg-Moore category of a monad on an ind-category itself an ind-category? | Let $T$ be a monad on an [accessible category](https://ncatlab.org/nlab/show/accessible+category) (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is [finitary](https://ncatlab.org/nlab/show/finitary+monad) (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also acces... | 8 | https://mathoverflow.net/users/152679 | 418289 | 170,332 |
https://mathoverflow.net/questions/418265 | 6 | We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^\* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$$T(t) = \begin{pmatrix} \mathcal A+t & \mathcal B \\ \mathcal B^\* & \mathcal A-t\end{pmatrix}.$$
Numerical experiments seem to sh... | https://mathoverflow.net/users/119875 | Monotonicity of eigenvalues | The idea is to apply a unitary congruence $$U=\dfrac{1}{\sqrt{2}}\begin{pmatrix}I&-I\\I&I\end{pmatrix}.$$ I consider here $\mathcal{B}$ to be hermitian, $\mathcal{B}=\mathcal{B}^\*$, whereas the general case may be 'different'.
So $$R=UT(t)U^\*=\begin{pmatrix}\mathcal{A}-\mathcal{B}&tI\\tI&\mathcal{A}+\mathcal{B}\end... | 6 | https://mathoverflow.net/users/121643 | 418290 | 170,333 |
https://mathoverflow.net/questions/418244 | 2 | A ray is a continuous one-to-one image of the half-line $[0,\infty)$.
If $f:[0,\infty)\to \mathbb R ^2$ is continuous and one-to-one, then we say that the ray $X=f[0,\infty)$ limits onto itself if for each $x\in X$ there exists a sequence $(a\_n)\in [0,\infty)^\omega$ such that $a\_n\to\infty$ and $f(a\_n)\to x$. Thi... | https://mathoverflow.net/users/95718 | A plane ray which limits onto itself | It seems like there are decomposable rays which limit onto themselves.
Let $E=\bigcup\_{n\geq0}E\_n$ and $O=\bigcup\_{n\geq0}O\_n$, with $E\_n=[2n,2n+1]$ and $O\_n=[2n+1,2n+2]$. We want to construct a ray $f:[0,\infty)$ which limits onto itself and such that $f(E),f(O)$ are closed in the ray $X$ and connected.
For ... | 1 | https://mathoverflow.net/users/172802 | 418291 | 170,334 |
https://mathoverflow.net/questions/418307 | 3 | Apologies if this question is too basic for MO.
I think it should be the case that
>
> for any decreasing $f \colon [A,\infty) \to [0,\infty)$ and $k \geq 0$, if $\int\_A^\infty f(x) e^{kx} \, dx < \infty$ then $f(x)e^{kx} \to 0$ as $x \to \infty$.
>
>
>
**Proof** [I think]**.** Write $g(x)=f(x)e^{kx}$. If $... | https://mathoverflow.net/users/15570 | Is it a named result (or consequence thereof) that decreasing functions integrable against $e^{kx}$ decay faster than $e^{-kx}$? | Here is a simple proof. Since $f$ is nonnegative and decreasing, there is a limit $c:=\lim\_{x\to\infty}f(x)\ge0$, and $f\ge c$. So, $\infty>\int\_A^\infty f(x)e^{kx}\,dx
\ge\int\_A^\infty ce^{kx}\,dx=\infty$ if $c>0$. So, $c=0$ and hence
\begin{equation}
f(x+)=\int\_{(x,\infty)} \mu\_f(dt)
\end{equation}
for all $x\i... | 2 | https://mathoverflow.net/users/36721 | 418311 | 170,342 |
https://mathoverflow.net/questions/418312 | 0 | So, I am in need of an indication of literature or where to start.
I am having a problem consisting of reading a vector of characters (for example, there are n=4 possible characters {A,B,C,D}) using a moving window in the format [1 0 1].
For example:
The vector is:
>
> ABBCDE
>
>
>
and results in a reading of
... | https://mathoverflow.net/users/478901 | Probability of getting all pattern combinations in moving window over a vector of characters | The expected length needed to see all patterns of length $k$ in an alphabet of size $a$ is $k \ln(a) a^k (1+o(1))$.
See Propositions 11.9 and 11.10 in [1] for the case $a=2$.
[1] Levin, David A., and Yuval Peres. Markov chains and mixing times. Vol. 107. American Mathematical Soc. second ed., with contributions by E.... | 0 | https://mathoverflow.net/users/7691 | 418335 | 170,348 |
https://mathoverflow.net/questions/418341 | 0 | Consider the following finite version Hindman theorem:
For every sufficiently large $N\in\omega$ and 2-partition of $N=N\_0\cup N\_1$, there are $i<2,a,b,c\in N\_i$ such that $a+b=c$.
The only proof I know for this is by iteratively using Hales-Jewett theorem. What are the alternative proofs?
| https://mathoverflow.net/users/74918 | Finite Hindman theorem | This is Schur's theorem, it follows from Ramsey theorem: consider the complete graph with vertices $1,\ldots,N+1$ and color the edge between vertices $i$ and $j$ with color $s\in \{0,1\}$ iff $|i-j|\in N\_i$. A monochromatic triangles provides a monochromatic solution of $a+b=c$.
| 9 | https://mathoverflow.net/users/4312 | 418342 | 170,349 |
https://mathoverflow.net/questions/418351 | 5 | Let $f:\mathbb{CP}^n\to\mathbb{CP}^n$ be a holomorphic map; I am interested in what the subvariety of critical points could be.
More specifically, let $J=\{p\in \mathbb{CP}^n\ :\ \det\mathrm{Jac}(f)=0\}$:
1. can $J$ be a smooth submanifold of $\mathbb{CP}^n$?
2. can $J$ be an irreducible subvariety of $\mathbb{CP}^... | https://mathoverflow.net/users/17111 | Geometry of critical points of holomorphic maps in projective space | For $n=2$, the locus $J$ is smooth and irreducible for a general $f$; i.e., these $f$ form a Zariski dense subset of the parameter space of such $f$. For $n\ge3$ and for general $f$, the locus $J$ will be (mildly) singular, irreducible, and of general type. [See Theorems 14 and 15 in the linked paper.](https://arxiv.or... | 9 | https://mathoverflow.net/users/11926 | 418352 | 170,350 |
https://mathoverflow.net/questions/418367 | 4 | **Definition** A fine measure on $P\_\kappa(\lambda)$ is a non-principal ultrafilter on $P\_\kappa(\lambda)$ which contains all upper cones $\uparrow{x}=\{y\in P\_\kappa(\lambda)|x\subset y\}$, for all $x\in P\_\kappa(\lambda)$.
Bagaria and Magidor defined a cardinal $\kappa$ to be $\aleph\_1$-strongly compact if the... | https://mathoverflow.net/users/13694 | $\aleph_1$-complete fine measures on $P_\kappa(\lambda)$ | Yes, if $F$ is large enough. Let $j : V \to M$ be the embedding derived from an $\aleph\_1$-strongly compact ultrafilter $U$ on $P\_\kappa(\lambda)$. Let $[\mathrm{id}]$ be the set in $M$ represented by the identity function. Define $W \subseteq P(P\_\kappa(\lambda))$ by:
$$A \in W \Leftrightarrow [\mathrm{id}] \cap j(... | 5 | https://mathoverflow.net/users/11145 | 418368 | 170,354 |
https://mathoverflow.net/questions/418357 | 4 | The CHSH game is the standard example of a game where two cooperating players Alice and Bob who cannot communicate, but who nevertheless can get an advantage by measuring an entangled quantum state in some basis which depends on its input.
Detailed information is available here: <https://en.wikipedia.org/wiki/CHSH_in... | https://mathoverflow.net/users/143779 | Why does the CHSH game need complicated bases to show advantage? | It's not that hard to understand in terms of the [Bloch sphere](https://en.wikipedia.org/wiki/Bloch_sphere). If Alice measures along vector $\alpha$ on the Bloch sphere, and Bob measures along $\beta$, then the correlation between their answers is the dot product $\alpha \cdot \beta$. In other words, the probability th... | 5 | https://mathoverflow.net/users/297 | 418370 | 170,355 |
https://mathoverflow.net/questions/418326 | 1 | In a previous question [here](https://mathoverflow.net/questions/418265/monotonicity-of-eigenvalues), I asked the question below for block matrices and received an answer showing the question is true if $\mathcal B$ is hermitian and false, in general if $\mathcal B$ is non-hermitian. However, numerical experiments sugg... | https://mathoverflow.net/users/119875 | Monotonicity of eigenvalues II | The characteristic polynomial is even in both $X$ and $t$ : $P\_t(X)=Q(X^2,t^2)$ where
$$Q(Y,s)=(Y-s)^2-(2|a|^2+|b|^2+|c|^2)(Y-s)-4|a|^2s+|a^2-b\bar c|^2.$$
The variation of $s\mapsto Y(s)$ is given by the derivative
$$2(Y-s)(Y'-1)-(2|a|^2+|b|^2+|c|^2)(Y'-1)-4|a|^2=0.$$
The sign of $Y'$ changes when $s$ crosses the val... | 4 | https://mathoverflow.net/users/8799 | 418372 | 170,356 |
https://mathoverflow.net/questions/418359 | 7 | **Background.** (Can be skipped if you already know what is the Eguchi-Hanson metric.) The Eguchi-Hanson metric $g$ is a complete Ricci-flat Riemannian metric on the cotangent bundle of the 2-sphere, $T^\*S^2$. By removing the zero-section, we can identify it with $S^3/\mathbb{Z}\_2 \times (0, \infty) = (\mathbb{R}^4\s... | https://mathoverflow.net/users/409915 | Is the bundle map of the Eguchi-Hanson metric a Riemannian submersion? | No. The reason is basically the same why if you take flat $\mathbb R^4\setminus 0$ and quotient by the standard Hopf $\mathbb S^1$ action you get a punctured cone over $\mathbb S^2$ but the projection to that $\mathbb S^2$ is not a Riemannian submersion because the horizontal spaces scale with $r$. The Euguchi-Hanson m... | 9 | https://mathoverflow.net/users/18050 | 418379 | 170,360 |
https://mathoverflow.net/questions/418347 | 3 | Suppose $\mathcal{E} \to \mathcal{C}$ is a cocartesian fibration over (the nerve of) a classical category, and there is a section on zero simplices that sends $C$ to $s(C)$ such that, for every edge $f\colon C \to C'$ in $\mathcal{C}$, the Kan complex
$\mathrm{Hom}\_{\mathcal{E}\_{C'}}(f\_!(s(C)), s(C'))$
is discre... | https://mathoverflow.net/users/115211 | Constructing sections of a cocartesian fibration | Yes. Consider the full subcategory $\mathcal S$ of $\mathcal E$ spanned by the objects $s(C)$.
It is clear that a section extending $s$ is the same thing as a section $\mathcal C\to \mathcal S$ extending $s$.
But I claim that $\mathcal S$ is a $1$-category: indeed the fiber of $\hom\_\mathcal E(s(C),s(C'))\to \hom\... | 1 | https://mathoverflow.net/users/102343 | 418382 | 170,361 |
https://mathoverflow.net/questions/418366 | 3 | Let $\mathcal{H}\_A\otimes\mathcal{H}\_B$ be a finite-dimensional bipartite Hilbert space, $P\_A$ a positive semi-definite operator on $\mathcal{H}\_A$, $P\_B$ a positive semi-definite operator on $\mathcal{H}\_B$, and $U\_{AB}$ a unitary operator on $\mathcal{H}\_A\otimes\mathcal{H}\_B$. In some roundabout way involvi... | https://mathoverflow.net/users/98045 | Proving a bipartite trace inequality? | You can rewrite the left-hand side $\mathrm{tr}\_A\left(P\_AX P\_AX^\*\right)$,
where $X=\mathrm{tr}\_B((1\otimes P\_B)U\_{AB})$. By Hölder's inequality this is less than $\|P\_A\|\_2^2 \|X\|\_\infty^2$, so your inequality is just that $\|X\|\_\infty \leq \mathrm{tr}(P\_B)$, which is certainly "conventional". For a jus... | 4 | https://mathoverflow.net/users/10265 | 418388 | 170,363 |
https://mathoverflow.net/questions/418392 | 6 | Consider the equation
$x^5-2x^2+z=0$
How do you derive the Lagrange inversion theorem series solution for it? I know it exists because the answer is here for any trinomial <https://arxiv.org/pdf/0910.2957.pdf>
I am trying to figure out how derive the series solution.
There is this well known result, which I thi... | https://mathoverflow.net/users/470659 | Series solution for general trinomial | One pair of solutions will
be Puiseux series $ax^{1/2}+bx+cx^{3/2}+\cdots$ (where $x^{1/2}$ can
have one of two signs). Thus set
$u=x^{1/2}$ in the solution, giving a series $F(u)$ satisfying
$F(u)^5-2F(u)^2+u^2=0$. Hence $\sqrt{-F(u)^5+2F(u)^2}=u$. By ordinary
Lagrange inversion, $$ [u^n]F(u) =[u^{n-1}]\frac
1n\left(\... | 10 | https://mathoverflow.net/users/2807 | 418396 | 170,366 |
https://mathoverflow.net/questions/418394 | 1 | Suppose you have a number
$$
N = p\_1^{e\_1}p\_2^{e\_2}\cdots p\_k^{e\_k}
$$
and are looking for the largest divisor $d|N$ such that $d^2<N$ (that is, [A060775](https://oeis.org/A060775)$(N)$.) How can I efficiently find this $d$?
If $N$ has a small number of divisors, you can just iterate through them all (keeping t... | https://mathoverflow.net/users/6043 | Efficiently finding the largest divisor of N less than sqrt(N) | I believe the Schroeppel & Shamir algorithm [1] can be adapted for use here, with time something like $O\left(\sqrt{\tau(n)}\omega(n)\right)$ and space something like $O\left(\sqrt[4]{\tau(n)}\right)$. The basic idea is splitting the prime powers into four sets with roughly the same number of divisors each, listing the... | 6 | https://mathoverflow.net/users/6043 | 418399 | 170,367 |
https://mathoverflow.net/questions/418371 | 1 | I cross-post a question that has not been answered on MSE, see [here](https://math.stackexchange.com/questions/4391605/question-about-milnor-hypersurface).
Consider the Milnor hypersurface $H\_{ij}$, i.e., the smooth hypersufrace in $\mathbb CP^i \times \mathbb CP^j$ for fix pair of integers $j \ge i\ge 0$ defined by... | https://mathoverflow.net/users/170441 | Milnor hypersurface | Since the question was not answered on MSE, I will provide a short answer here.
The expression $H\_{ij}$ is a bi-homogeneous polynomial of bi-degree $(1, \, 1)$, hence it defines an effective divisor in the complete linear system $|\mathcal{O}\_{\mathbb{P}^i \times \mathbb{P}^j}(1, \, 1)|$. Moreover, a simple computa... | 5 | https://mathoverflow.net/users/7460 | 418414 | 170,371 |
https://mathoverflow.net/questions/418408 | -1 | I have a task:
Lat's take independent variables $X\_{i}$ with Poisson distribution $Poiss(a)$. Distribution of $a$ has density $p(a)=\frac{8}{3}a^3e^{-2a}, a\ge0$.
Calculate:
$E(a|X\_1=3, X\_2=2, X\_3=5, X\_4=5, X\_5=1, X\_6=4)$.
I tried to calculate this in this way:
$E(a|X\_1=3, X\_2=2, X\_3=5, X\_4=5, X\_5=1, X\... | https://mathoverflow.net/users/478987 | Poisson distribution and conditional expected value | Your problem seems to be a special case of <https://web.stanford.edu/class/stats200/Lecture20.pdf>
about prior and posterior distributions in Bayesian analysis. By assumption $X\_1,\ldots,X\_6 \sim Poisson(a)$ and $a \sim \Gamma(\alpha,\beta) = \Gamma(4,2)$, the a priori distribution. Then with $s = x\_1+\ldots+x\_n$ w... | 1 | https://mathoverflow.net/users/100904 | 418422 | 170,372 |
https://mathoverflow.net/questions/418432 | 1 | This problem, which is in the book "Probability Theory"(fourth edition) written by M. Loeve, states as follows,
Rule: In order to compute $PB$, $B = f(A\_{1}, A\_{1}^{c}, \cdots, A\_{m}, A\_{m}^{c})$, take the following steps:
1. Reduce the operations on events to complementations, intersections,
and sums;
2. Repla... | https://mathoverflow.net/users/138147 | The method of indicators for probability | For any event $B$, let $1\_B$ and $B^c$ denote the indicator and the complement of $B$, respectively. Let $A:=\bigcup\_{j=1}^{m} A\_j$.
Let $\binom{[m]}r$ denote the set of all subsets of cardinality $r$ of the set $[m]:=\{1,\dots,m\}$.
Then
$$1\_A=1-1\_{A^c}=1-\prod\_{j=1}^m 1\_{A\_j^c}
=1-\prod\_{j=1}^m (1-1\_{A\_j})... | 1 | https://mathoverflow.net/users/36721 | 418435 | 170,375 |
https://mathoverflow.net/questions/415755 | 6 | I have two questions that seem to be related.
I wonder if there is a user-friendly algorithm (starting from ribbon/slice presentation of knots/disks) for the construction of double (in general $p$-fold) coverings of $B^4$ along ribbon/slice disks.
Using the handle decomposition of ribbon/slice disks can we control ... | https://mathoverflow.net/users/475366 | Double ($p$-fold) coverings of $B^4$ along ribbon/slice disks | I would like to popularize the lecture notes of Brendan Owens about the double branched cover of knots in $S^3$ and of ribbon/slice disks in $B^4$.
He both discussed constructions and obstructions. Also, it includes the classical references listed by Professor Ruberman.
>
> Owens, Brendan. "Knots and 4-manifolds.... | 4 | https://mathoverflow.net/users/131172 | 418443 | 170,378 |
https://mathoverflow.net/questions/264622 | 5 | Given an $\omega\_1$-tree $T$ in the ground model, can Laver forcing add a cofinal branch to $T$? Assume GCH in the ground model.
Definitions:
An $\omega\_1$-tree is a well-founded tree of height $\omega\_1$ with all levels countable. A cofinal branch of $T$ is a maximal chain of type $\omega\_1$. Laver forcing $\m... | https://mathoverflow.net/users/12106 | Does Laver forcing add cofinal branches to $\omega_1$-trees? | No.
In our paper <https://arxiv.org/abs/1409.4596> with Jindra Zapletal on Y-stuff we prove that Laver forcing is Y-proper. Consequently it has the $\omega\_1$-approximation property (i.e., does not add fresh sets of size $\omega\_1$ in Joel's terminology) and hence adds no new branches to trees.
PS: Thanks to Wolf... | 4 | https://mathoverflow.net/users/33363 | 418446 | 170,381 |
https://mathoverflow.net/questions/418240 | 1 | I am trying to find an asymptotic formula for the following sum as $T \to \infty$.
$$ \sum\_{t = 1}^{T} \prod\_{\substack{p \; \textrm{prime} \\ p | t}} \rho(p) \frac{1 - \frac{1}{p^2}}{1 - \frac{\rho(p)}{p^2}}$$
where for a fixed even $c$, $\rho$ is defined on the primes as follows,
$$ \rho(p) = \begin{cases}
1 ... | https://mathoverflow.net/users/167999 | Sum of an arithmetic sequence involving Euler factors | Let $c = 2^k m$ with $m$ odd. The analysis is slightly different depending on the parity of $k$ and on whether $m\equiv 1$ or $3\ \text{mod} 4$. For simplicity, I'll assume $k$ is even, $m\equiv 1\ \text{mod 4}$, and $m$ not a perfect square. In this case, quadratic reciprocity gives
$$
\rho(p) = 1 + \chi(p)
$$
wheneve... | 2 | https://mathoverflow.net/users/307675 | 418455 | 170,383 |
https://mathoverflow.net/questions/418436 | 0 | I have a question on [Lawler – Notes on the Bessel process](http://www.math.uchicago.edu/%7Elawler/bessel18new.pdf), on page 4. Let $X\_t$ be one-dimensional Brownian motion, and we want to use $N\_t$ as a measure-changing (local) martingale, defined as $$N\_t=\left(\frac{X\_t}{X\_0}\right)^a\exp\left(-\frac{a(a-1)}{2}... | https://mathoverflow.net/users/174600 | Uniform boundedness of this SDE? And possibly a stochastic Grönwall inequality? | On the time interval $[0,\tau]$, the exponential factor in the definition of $N\_t$ is bounded below by $0$ and above by $\exp(Kt)$ for some constant $K=K(\epsilon,R)\ge 0$. Therefore $0<N\_{t\wedge\tau}\le(R/\epsilon)^a\exp(K\cdot (t\wedge \tau))$. This is "uniform" provided you ignore the dependence on $t$.
| 2 | https://mathoverflow.net/users/42851 | 418456 | 170,384 |
https://mathoverflow.net/questions/418386 | 2 | Let $V$ be a nonempty, irreducible, smooth projective variety over $\mathbf{C}$.
>
> Is there a smooth projective variety $X$ over $\mathbf{C}$, a surjective map $X\to V$ of varieties over $\mathbf{C}$, such that $X$ contains as a dense open $\mathbf{C}$-subscheme some $\mathbf{C}$-group scheme $G$ of finite type?
... | https://mathoverflow.net/users/nan | Compactifications of group varieties | Smooth projective curves of genus $>1$ are counterexamples. To see this, you can use the following lemma.
**Lemma.** Let $X$ be a proper integral variety over $\mathbb{C}$. Then the following are equivalent.
1. For every abelian variety $A$, every morphism $A\to X$ is constant.
2. For every finite type connected gr... | 3 | https://mathoverflow.net/users/4333 | 418460 | 170,387 |
https://mathoverflow.net/questions/418375 | 2 | Let $L\_1$ and $L\_2$ be two nonintersecting picewise-linear or smooth knots in $\mathbb R^3$. Suppose they are ambient isotopic. Does there exist an embedded surface $f: S^1\times[0,1]\to \mathbb R^3$ such that $L\_1=f(\cdot,0)$ and $L\_2=f(\cdot,1)$. This is similar to concordance but stronger -- this "concordance" m... | https://mathoverflow.net/users/13842 | An equivalence relation on knots similar to concordance | Take a knot, and another copy of the same knot far away. It is then a nice instructive exercise to prove that if they are not unknots they do not cobound annuli.
| 2 | https://mathoverflow.net/users/6205 | 418472 | 170,393 |
https://mathoverflow.net/questions/418490 | -1 | I am trying to understand the set
$$\mathcal{A}=\{f:f\text{ non-negative, measurable}, \text{vanishes at infinity and } f(x)=f^\*(x)\}$$
where $f:\mathbb{R}^n\to \mathbb{R}$ is vanishing at infinity if for all $t>0$ such that $\text{Vol}(\{x:f(x)>t\})<+\infty$ and $f^\*$ denotes the symmetric decreasing rearrangement.
... | https://mathoverflow.net/users/68232 | What functions are equal to their symmetric decreasing rearrangement? | It should be straightforward to verify that $\mathcal A$ consists *exactly* of the lower semi-continuous radial and decreasing functions (which are negative and vanish at $\infty$): You already know that every $f\in\mathcal A$ has these properties, so you only have to verify that if $f$ has these properties, then $f=f^... | 3 | https://mathoverflow.net/users/165275 | 418491 | 170,396 |
https://mathoverflow.net/questions/418489 | 3 | I have a question concerning certain elements having zero trace in a finite field extension and I do have the feeling that additive characters should play a role, but I am not sure how. I am stating the problem slightly more generally than what I am really interested in.
Let $q$ be prime power, let $\mathbb{F}\_{q}$ ... | https://mathoverflow.net/users/45242 | Number of certain elements in a finite field having zero trace | We have
$$ \# \{x\in \mathbb{F}\_{q^{2n}}|\mathrm{Tr}\_{\mathbb{F}\_{q^{2n}}/\mathbb{F}\_{q^2}}(x)=0 \hbox{ and } x=y^n \textrm{ for some }y\in \mathbb{F}\_{q^{2n}}\} = \frac{\#\{y\in \mathbb{F}\_{q^{2n}}|\mathrm{Tr}\_{\mathbb{F}\_{q^{2n}}/\mathbb{F}\_{q^2}}(y^n)=0 \} +n-1}{ n}$$
since $\mathbb F\_{q^n}$ contains all t... | 4 | https://mathoverflow.net/users/18060 | 418508 | 170,403 |
https://mathoverflow.net/questions/418266 | 17 | I'm interested in pairs $A=(a\_{i,j})\_{i,j=0,1,\ldots}$ and $B=(b\_{i,j})\_{i,j=0,1,\ldots}$ of infinite matrices for which:
* They are uni-lower-triangular, i.e., $a\_{i,i}=1$ for all $i$ and $a\_{i,j}=0$ for all $j>i$.
* They are inverses, i.e. their product $AB$ is the identity.
* The generating function of the $... | https://mathoverflow.net/users/25028 | Matrices of combinatorial sequences that are inverse in two ways | Following up on David's nice answer, there is a different parametrization that makes the pattern much more obvious. Namely, let $s\_1,s\_2,\dots$ be arbitrary, then you can write
$$A=\Big(h\_{i-j}(s\_1,s\_2,\dots, s\_{j+1})\Big)\_{i,j=0}^{\infty}$$
$$B=\Big((-1)^{i-j}e\_{i-j}(s\_1,s\_2,\dots, s\_{i})\Big)\_{i,j=0}^{\in... | 15 | https://mathoverflow.net/users/2384 | 418510 | 170,404 |
https://mathoverflow.net/questions/418466 | 2 | Consider the periodic Riccati equation $y'(x)=y(x)^2+q(x)$ on the real line $\mathbb{R}$, where $q\in C^\infty(\mathbb{R})$ is a periodic function with period $T=1$. Suppose $q(x)$ can take both positive and negative values but $\int\_0^1 q(x)dx<0$. Does a periodic solution to this Riccati equation exist? Any reference... | https://mathoverflow.net/users/119968 | Existence of periodic solutions to scalar Riccati equations | The answer is negative.
The MSN review says that this is the contents of
MR1466035
Tang, Fenjun
The periodic solutions of Riccati equation with periodic coefficients.
Ann. Differential Equations 13 (1997), no. 2, 165–169.
I looked at this paper and the argument there makes no sense.
Setting $y=-w'/w$ we obtain $w... | 4 | https://mathoverflow.net/users/25510 | 418513 | 170,405 |
https://mathoverflow.net/questions/418142 | 8 | Let $G = \text{GL}\_n(\mathbb{C})$ and let $N\_+$ be the subgroup of upper triangular matrices with $1$'s on the diagonal. Let $w$ be a permutation, let $B\_+ w B\_+$ be the Bruhat cell and let $\overline{B\_+ w B\_+}$ be its closure in $G$. I want to describe the ring of functions on $\overline{B\_+ w B\_+}$ which are... | https://mathoverflow.net/users/297 | Coordinates on $N_+ \backslash \overline{B_+ w B_+} / N_+$ | I have proved that the invariant ring is as I expected. As discussed in the question, $N\_+ \backslash B\_+ w B\_+ / N\_+ \cong T$, so the ring of $N\_+ \times N\_+$ invariants on $B\_+ w B\_+$ is the coordinate ring of $T$, which is the Laurent polynomial ring in the $t\_i$.
Note, let $R$ be the ring of $N\_+ \times... | 2 | https://mathoverflow.net/users/297 | 418515 | 170,406 |
https://mathoverflow.net/questions/418499 | 1 | Let $n$ and $d$ be large positive integers with $n \le d^\gamma$, for some absolute constant $\gamma>0$; i.e., $n$ is at most polynomial in $d$. Let $x\_1,\ldots,x\_n,x\_{n+1}$ be drawn iid from the uniform distribution on the sphere $S\_{d-1}(\sqrt{d})$ of radius $\sqrt{d}$ in $\mathbb R^d$, and consider the random va... | https://mathoverflow.net/users/78539 | For $x_1,...,x_n$ iid random on sphere of radius $\sqrt{d}$ in $R^d$, what is a good upper-bound on min distance of $x_{n}$ from the other $x_i$'s? | $\newcommand{\De}{\Delta}\newcommand{\R}{\mathbb R}\newcommand{\ga}{\gamma}\newcommand{\Ga}{\Gamma}$This is to provide a detalization on Will Sawin's comment. Specifically, let us show that the best upper bound on $\De:=\De(n,d)$ is $\sim\sqrt{2d}$ (as $d\to\infty$ and $d^\ga\ge n\to\infty$).
Indeed, for real $m>0$, ... | 5 | https://mathoverflow.net/users/36721 | 418520 | 170,408 |
https://mathoverflow.net/questions/418532 | 5 | For each first-order theory $T$ there is an associated [weak syntactic category](https://www.math.ias.edu/~lurie/278xnotes/Lecture2-Syntax.pdf), sometimes also called "the category of definable sets of $T$" and denoted $\mathrm{Def}(T)$.
Also, for each theory $T$ there is an associated theory $T^\mathrm{eq}$ which ca... | https://mathoverflow.net/users/478652 | Why does the category of definable sets of $T^\mathrm{eq}$ have coproducts? | In general, a theory is called proper if every sort is nonempty (in every model) and there is a sort which has (in every model) at least two elements. Then we have the following theorem: if $\mathbb{T}$ is any proper theory, the syntactic category of $\mathbb{T}^{eq}$ is a pretopos. This is a consequence of the followi... | 6 | https://mathoverflow.net/users/12976 | 418535 | 170,409 |
https://mathoverflow.net/questions/418528 | 2 | $\newcommand{\loc}{\mathrm{loc}}$Let $\Omega$ be a bounded open set (smooth as we wish if necessary) in $\mathbb{R}^n$, $(\omega\_k)$ a sequence of open subsets whose closure is contained in $\Omega$ and whose union covers $\Omega$. Let $(u\_k)\_{k\in\mathbb{N}}$ a sequence in $H^1(\Omega)$ and assume the uniform bound... | https://mathoverflow.net/users/41568 | Weak convergence in $H^1_{\mathrm{loc}}$ | The sequence $(u\_k)$ need not converge in $H^1(\Omega)$ under the given hypotheses. Indeed, recall that a weakly convergent sequence is bounded. Therefore, a sequence whose terms have $\lVert u\_k \rVert\_{1;\Omega} \to \infty$ but $\lVert u\_k \rVert\_{1;\omega\_k} \leq M$ would not converge weakly in $\Omega$.
| 2 | https://mathoverflow.net/users/103792 | 418536 | 170,410 |
https://mathoverflow.net/questions/418325 | 4 | I am a PhD student and during my research I was presented to the claim that
**For a positive definite function $f:\mathbb{R}\to \mathbb{R}$ continuous in $0$, with $0$ a stable point at $t=0$ for $x$, one has
$${\lim\inf}\_{t\to\infty} f(x(t))=0\Longrightarrow {\lim\inf}\_{t\to\infty} \|x(t)\|=0.$$**
In this contex... | https://mathoverflow.net/users/172600 | $\lim\inf_{t\to \infty} f(x(t))=0\Longrightarrow \lim\inf_{t\to\infty} \|x(t)\|=0$ | Here is my take on your question. Let $f:\mathbb R\to\mathbb R\_+$ be a function such that $\{x:f(x)=0\}=\{0\}$ and $f$ is continuous at zero. Let $\phi:\mathbb R\to\mathbb R$ be locally Lipschitz continuous, so that the differential equation $x'=\phi(x)$ admits a unique local solution for any given initial condition $... | 2 | https://mathoverflow.net/users/129074 | 418539 | 170,411 |
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