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/325671 | 10 | Let $f:X\rightarrow Y $ be a fibration (with fiber $F$) between simply connected spaces such that
$H\_{\ast}(f):H\_{\ast}(X,\mathbb{Z})\rightarrow H\_{\ast}(Y,\mathbb{Z})$ is an isomorphism for $\ast\leq n$
Is it true that the reduced homology of the fiber is $\tilde{H}\_{\ast}(F,\mathbb{Z})=0$ for $\ast\leq n$?
| https://mathoverflow.net/users/136128 | Homology of the fiber | As usual, there's no loss of generality in assuming that $f$ is the inclusion of a subspace $X\subset Y$, replacing $Y$ with the homotopy equivalent mapping cylinder of $f$ if necessary. By your assumptions and the five lemma, $H\_\*(Y,X)=0$ for $\*\leq n$, and the pair $(Y,X)$ is simply connected, therefore by the Hur... | 23 | https://mathoverflow.net/users/12166 | 325675 | 140,260 |
https://mathoverflow.net/questions/325655 | -2 | Are there $2$ non-adjacent points in the icosahedron graph $G$ such that contracting them leaves the [Hadwiger number](https://en.wikipedia.org/wiki/Hadwiger_number) unchanged?
| https://mathoverflow.net/users/8628 | Contracting non-adjacent points in the icosahedron | No.
The icosahedron graph is distance transitive, meaning that for any two pairs $(a,b)$ and $(c,d)$ of vertices of the icosahedral graph such that $\text{dist}(a,b) = \text{dist}(c,d)$, there is an automorphism $\sigma$ such that $\sigma(a) = c$ and $\sigma(b) = d$. Since the icosahedral graph has diameter 3, there ... | 4 | https://mathoverflow.net/users/18606 | 325681 | 140,261 |
https://mathoverflow.net/questions/325686 | 2 | Let $x \in (0,L)$, $t \in (0,T)$, and let $f\_1 = f\_1(x,t) \in \mathbb{R}$, $f\_2 = f\_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions.
My question is:
>
> Can we find a function $u = u(x,t) \in \mathbb{R}$ that satisfies
> \begin{equation}
> \partial\... | https://mathoverflow.net/users/137201 | Two PDE for one unknown? | A solution exists if and only if the following compatibility conditions are satisfied:
$$\partial\_xf\_1=\partial\_tf\_2,\quad g'(t)=g(t)f\_1(0,t),\quad u\_0'(x)=u\_0(x)f\_2(x,0).$$
For the existence, you can solve the Cauchy problem in $x$, then that in $t$, and verify that both solutions coincide.
| 4 | https://mathoverflow.net/users/8799 | 325689 | 140,263 |
https://mathoverflow.net/questions/324691 | 2 | The paper I am referencing is "Normal Subgroups of the Cremona Group." <https://arxiv.org/abs/1007.0895>. In theorem 5.14, at the bottom of page 52, the author stated for the abelian surface $Y= \mathbb{C}/\mathbb{Z}[i] \times \mathbb{C}/\mathbb{Z}[i]$, we consider the quotient $X= Y/ \Gamma$ where $\Gamma$ is the grou... | https://mathoverflow.net/users/134269 | Lifting of automorphism of rational surface to that on abelian variety | Denote $X\backslash\text{Sing}(X)$ by $X\_0$ and its preimage in $Y$ as $Y\_0$. Note that $Y\_0$ is the Galois cover of $X\_0$ corresponding to the normal subgroup $\mathbb{Z}[i]\times\mathbb{Z}[i]$ inside $\Lambda.$ For any automorphism $f$ of $X\_0$, the pullback of $Y\_0$ along $f$ is the cover corresponding to the ... | 1 | https://mathoverflow.net/users/51424 | 325690 | 140,264 |
https://mathoverflow.net/questions/325678 | 3 | Let $\mathcal{U} = \{ U\_i \; |\; i\in I \}$ be an open covering of $X$. Spanier defined $\pi (\mathcal{U}, x)$ to be the subgroup of $\pi\_1 (X, x)$ which contains all homotopy classes having representatives of the following type: $
\prod\_{j=1}^{n}u\_j \*v\_j \* u^{-1}\_{j},
$
where $u\_j$'s are paths (star... | https://mathoverflow.net/users/114476 | Concerning the Spanier group relative to an open cover | No. Let $X\_1=X\_2=X\_3=S^1$ be copies of the unit circle and consider $X\_1\vee X\_2\vee X\_3$ with wedge basepoint $x$. Let $\gamma\_i$ be a loop traversing $X\_i$ whose homotopy class generates $\pi\_1(X\_i,x)$. Construct $X$ by attaching a 2-cell to $X\_1\vee X\_2\vee X\_3$ by the attaching loop $\gamma\_1\ast\gamm... | 1 | https://mathoverflow.net/users/5801 | 325693 | 140,266 |
https://mathoverflow.net/questions/324290 | 10 | Assume that
$$
\iota\_1:\mathbb{S}^k\to\mathbb{R}^n,
\quad
\iota\_2:\mathbb{S}^\ell\to\mathbb{R}^n,
\quad
k+\ell=n-1,
$$
are two embeddings of spheres with disjoint images $\iota\_1(\mathbb{S}^k)\cap\iota\_2(\mathbb{S}^\ell)=\emptyset$. Then, the linking number $Lk(\iota\_1,\iota\_2)$ of the embedded spheres can be def... | https://mathoverflow.net/users/121665 | Different definitions of the linking number | Proposition 3.3 in [De Turck and Gluck's "Linking Integrals in the n-sphere"](http://mat.unb.br/~matcont/34_10.pdf) states:
>
> Let $K^k$ and $L^\ell$ be disjoint closed oriented smooth submanifolds of $S^n$ with $k+\ell=n-1$ and let $f:K^k\ast L^\ell \rightarrow S^n$ be [the map sending the line segment $\{(\mathb... | 5 | https://mathoverflow.net/users/353 | 325695 | 140,267 |
https://mathoverflow.net/questions/325698 | 6 | Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$.
Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$.
>
> **Question 1:** Is F the identity map?
>
>
>
If the answer is negative: Is $F$ an isometry?
| https://mathoverflow.net/users/137210 | $SO(m+1)$-equivariant maps from $S^m$ to $S^m$ |
>
> **Theorem.** $F:\mathbb{S}^m\to \mathbb{S}^m$, $m\geq 2$, is $SO(m+1)$ equivariant if and only if $F=\operatorname{Id}$ or $F=-\operatorname{Id}$.
>
>
>
Let me write a **very detailed proof** that only requires a basic knowledge of linear algebra.
**Proof.**
It is easy to see that both $F=\operatorname{Id... | 6 | https://mathoverflow.net/users/121665 | 325705 | 140,270 |
https://mathoverflow.net/questions/325706 | 8 | My question is on a seemingly-natural extension of classical logic that I've been playing around with a bit in the context of generalized recursion theory. I'm sure it's been treated extensively already, but my literature search skills have failed me.
>
> What is a good source on what happens when we extend first-o... | https://mathoverflow.net/users/8133 | Logic with "co-relations" - sources? | Let us first distinguish between two kinds of operations that "take formulas to terms":
1. We might ask for *reflection* of syntax into the theory. A typical example is a quoting operator which takes a formula and returns its Gödel number.
2. We might ask for the ability to mix formulas and terms. A typical example i... | 8 | https://mathoverflow.net/users/1176 | 325711 | 140,272 |
https://mathoverflow.net/questions/325697 | 0 | I would like to perform the integration ($u \in [0,1]$),
\begin{equation}
\int\_{a=-1}^1 \int\_{b=-\sqrt{1-a^2}}^{\sqrt{1-a^2}} u^2 \sqrt{-a^2 u^2-b^2+1} \tan ^{-1}\left(\frac{\left| a\right|
}{\sqrt{-a^2-b^2+1}}\right) db da .
\end{equation}
I also have a companion problem
\begin{equation}
\int\_{a=-1}^1 \int\_{b=-... | https://mathoverflow.net/users/47134 | A pair of integrals involving square roots and inverse trigonometric functions over the unit disk | Further to my comment, one can set $y = \frac{z}{\sqrt{1-z^2}}$ to obtain $$\frac{8}{3} \, u^2 \int\_0^1 \! \sqrt{1 - u^2 z^2} \sin^{-1}(z) \, dz \;,$$ which *Mathematica* evaluates to
$$
\frac{2}{3} \, u \, \bigl(\operatorname{Li}\_2(-u) - \operatorname{Li}\_2(u) + \pi \sqrt{1 - u^2} \, u - (1 - u^2) \tanh^{-1}(u) - u... | 3 | https://mathoverflow.net/users/nan | 325715 | 140,274 |
https://mathoverflow.net/questions/325724 | 3 | Assume the positive integers $\mathbb{N}$ are partitioned as
$$\mathbb{N} = \cup\_{i = 1}^n (a\_i + b\_i \mathbb{N})$$
where $a\_i, b\_i \in \mathbb{N}$. Prove that all such partitions are obtained by the following algorithm: partition $\mathbb{N}$ into a finite number of a.p.'s with equal steps, then partition one of ... | https://mathoverflow.net/users/84963 | Covering integers by finitely many arithmetic progressions structure | The Schnabel paper mentioned in the comments gives the example,
$2(6),4(6),1(10),3(10),7(10),9(10),0(15),5(30),6(30),12(30),18(30),24(30),25(30)$
where $a(b)$ stands for $a+b{\bf N}$. This is said to follow from the impossibility of splitting the vanishing sum $$\zeta^5+\zeta^6+\zeta^{12}+\zeta^{18}+\zeta^{24}+\ze... | 4 | https://mathoverflow.net/users/3684 | 325726 | 140,278 |
https://mathoverflow.net/questions/325725 | 5 | A unital $C^\*$ algebra $A$ is said a Tietze algebra if it satisfies the following:
For every ideal $I$ of $A$ and every unital morphism $\phi: C[0,1] \to A/I$ there is a unital morphism $\tilde{\phi}:C[0,1] \to A$ with $\phi=q\circ \tilde{\phi}$ where $q:A\to A/I$ is the quotion map.
Obviousely every commutative a... | https://mathoverflow.net/users/36688 | (Noncommutative) Tietze $C^*$ algebras | Every unital C\*-algebra has this property.
Let $f \in C[0,1]$ be the function $f(t) = t$. If $\phi: C[0,1] \to \mathcal{A}$ is a unital $\*$-homomorphism then $\phi(f)$ is a positive element $x$ of $\mathcal{A}$ whose norm is at most $1$. Conversely, given any positive element $x$ of $\mathcal{A}$ whose norm is at m... | 7 | https://mathoverflow.net/users/23141 | 325732 | 140,281 |
https://mathoverflow.net/questions/324994 | 7 | For $p \in \mathbb{R}$, consider the function
$$F\_p(\lambda\_1, \dots, \lambda\_n) = \lambda\_1^p + \dots + \lambda\_n^p.$$
My goal is to maximize this function under the constraints that
$$ \lambda\_1^2 + \dots + \lambda\_n^2 = 1, ~~~\text{and}~~~ \lambda\_1 + \dots + \lambda\_n = 0.$$
I am mainly interested in the c... | https://mathoverflow.net/users/16702 | Maximize $L^p$ norm over sphere | Proof for the case $p$ odd:
The Lagrange-multiplier equation yields
$$
p\lambda\_i^{p-1}+a\lambda\_i+b=0
$$
for some $a,b \in \mathbb{R}$. If $p>1$ is odd the LHS is a strictly convex function in $\lambda\_i$ and can be zero at at most two different real values.
Assume now we have $n\_1, n\_2 \in \mathbb{N}$ times ... | 7 | https://mathoverflow.net/users/35593 | 325743 | 140,283 |
https://mathoverflow.net/questions/325740 | 2 | Let $H$ be a hypeplane in $\mathbb{P}^3$ containing a point $p$ and $I\_p$ be the ideal sheaf corresponding to $p$. Consider the natural exact sequence :
$0 \to \mathcal{O} \to \mathcal{O}(H) \to \mathcal{O}(H) \mid\_H \to 0$.
Is it true that the tensoring the exact sequence by $I\_p$ remains exact ?
I guess not, ... | https://mathoverflow.net/users/130022 | Restricting sheaves in projective space | Essentially, you are asking whether the map $I\_p \to I\_p(1)$ induced by the multiplication with the equation of a hyperplane is injective. Since $I\_p$ is a torsion-free sheaf, it is enough for this to check the map at the generic point of $\mathbb{P}^3$. But there $I\_p$ agrees with $\mathcal{O}$, hence injectivity ... | 3 | https://mathoverflow.net/users/4428 | 325744 | 140,284 |
https://mathoverflow.net/questions/325738 | 1 | Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation
$$u\_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$
$$u(x,0)=f, \ \ u\_t(x,0)=g, \ \ x\in \mathbb{R}^n,$$
where $c(x)>c\_0>0$ is also a bounded smooth functions on $\mathbb{R}^n.$
... | https://mathoverflow.net/users/42326 | Global solutions of the wave equation with bounded initial condition | The answer is no, already for the wave equation $c(x)=1$.
Let me be more precise.
If the initial data belong to $H^s\times H^{s-1}$ with $s$ strictly larger than $n/2$, then of course energy estimates give you that the $H^s$ norm remains bounded and hence the solution remains bounded in $L^\infty$ for all finite time... | 1 | https://mathoverflow.net/users/7294 | 325753 | 140,286 |
https://mathoverflow.net/questions/325752 | 10 | Let $X$ be an irreducible surface such that $X \times \mathbb{P}^1$ is rational. Is it true that $X$ is rational?
If the field is not algebraically closed, the answer is no in general (see A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc et Sir Peter Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles... | https://mathoverflow.net/users/23758 | Are stably rational surfaces all rational? | The result is true in all characteristics. See O. Zariski, Illinois J. Math. 2(1958), 303-315.
| 13 | https://mathoverflow.net/users/7666 | 325755 | 140,288 |
https://mathoverflow.net/questions/325667 | 10 | Let $Log(n) = \sum\_{i=1}^r \alpha\_i \cdot e\_i$, where $n = \prod\_{i=1}^r p\_i^{\alpha\_i}$ and $p\_i$ is the $i$-th prime, $\alpha\_i \ge 0$, $e\_i$ is the $i$-th standard basis vector. For example $6 = 2\cdot3$, hence:
$$Log(6) = (1,1,0,0,0,\cdots)$$
$$Log(15) = (0,1,1,0,0,\cdots)$$
$$Log(7) = (0,0,0,1,0,\cdots)$$... | https://mathoverflow.net/users/nan | Is there a pattern for the irreducible factors and degrees of a characteristic polynomial? | I made hand calculations for $n\le4$. It turns out that even with $V\_n=1\_n$, the factor $U\_n$ is not unique. In other words, the factorisation problem is underdetermined. This is due to the fact that the few last columns of $A\_n$ vanish. For instance, if $n=3$, the general $U\_3$ is
$$\begin{pmatrix} a & 1 & 0 \\ b... | 6 | https://mathoverflow.net/users/8799 | 325760 | 140,289 |
https://mathoverflow.net/questions/325273 | 16 | Consider the set $\mathbb{R}^{m \times n}$ of $m \times n$ matrices. I am particularly interested in properties of polytope $P$ defined as a convex hull of all $\{-1,1\}$ matrices of rank $1$, that is,
$$ P = \mbox{conv} \{ uv^T : u \in \{-1,1\}^m, v \in \{-1,1\}^n \}. $$
In particular, I would like to know how all... | https://mathoverflow.net/users/58990 | Convex hull of all rank-$1$ $\{-1, 1\}$-matrices? | That polytope $P\_{m,n}$ appears under many names : correlation polytope, Bell polytope, local hidden variable polytope, local polytope (sometimes these names refer to slightly different polytopes), and probably more. It is the unit ball for the projective norm on $\ell\_{\infty}^m \otimes \ell\_{\infty}^n$.
The faci... | 12 | https://mathoverflow.net/users/908 | 325761 | 140,290 |
https://mathoverflow.net/questions/325734 | 3 | I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.
| https://mathoverflow.net/users/136078 | A clean upper bound for the expectation of a function of a binomial random variable | This is the mean absolute deviation (MAD) for a binomial distribution, divided by $n$. The expectation is hence $$2 \, (1-p)^{n+1-\lceil np \rceil} \, p^{\lceil np \rceil} \, \binom{n-1}{\lceil np \rceil-1} \;.$$
See [this paper](https://www.cs.bgu.ac.il/~karyeh/sharp-mad.pdf) (Berend & Kontorovich 2013, doi: 10.1016/j... | 2 | https://mathoverflow.net/users/nan | 325766 | 140,291 |
https://mathoverflow.net/questions/325768 | 5 | Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?
| https://mathoverflow.net/users/137242 | Connection Between Knot Theory and Number Theory | The question seems very general, but the first book to come to mind is this:
*The Arithmetic of Hyperbolic 3-Manifolds*, with C. Maclachlan, Graduate Text in Math. 219, Springer-Verlag (2003)
| 9 | https://mathoverflow.net/users/12218 | 325769 | 140,292 |
https://mathoverflow.net/questions/325672 | 3 | For $X$ a *paracompact* space, I am trying to classify all locally trivial fibration with base the suspension $SX = X \times [-1,1]\, /\, (X \times \{-1\} \cup X \times \{1\})$, and fiber-type a space $F$ *such that $G\_F = Homeo(F)$ with the C.O. topology is a topological group*.
I duplicate the reasonning for clas... | https://mathoverflow.net/users/74372 | Locally trivial fibration over a suspension | It is independent of the choice of base point.
Let $Map((X,x\_0), (G\_F,id))$ be the based mapping space (based at $x\_0$).
Let $Map(X, G\_F)$ be the free mapping space. Then we have a split short exact sequence of topological groups:
$$Map((X,x\_0), (G\_F,id)) \to Map(X, G\_F) \stackrel{ev\_{x\_0}}{\to} G\_F$$
... | 1 | https://mathoverflow.net/users/184 | 325772 | 140,294 |
https://mathoverflow.net/questions/325359 | 10 | Consider the higher Cantor space $2^\kappa$ with the ${<}\kappa$-box topology ($\kappa$ at least inaccessible). This canonically defines the notion of higher Borel sets.
A higher Borel code $\mathbf{B}$ is a wellfounded tree of size $\kappa$ consisting of finite sequences, such that terminal nodes are label with basi... | https://mathoverflow.net/users/134910 | On the absoluteness of higher Borel sets? | In the case of forcing extensions by ${<}\kappa$-complete posets (which you probably meant), we have $\Sigma^1\_1$-absoluteness. (This is well known / folklore, see e.g. [Friedman Khomskii Kulikov](http://www.logic.univie.ac.at/~ykhomski/papers/Regprop.pdf), Lem 2.7). Strategic closure is sufficient. And ``Borelcode ev... | 5 | https://mathoverflow.net/users/26862 | 325773 | 140,295 |
https://mathoverflow.net/questions/325546 | 8 | **Background:** It is currently unknown whether $e$ is [normal](https://en.wikipedia.org/wiki/Normal_number). A natural way to approach this question is to find a class to which $e$ belongs, and prove all members of that class are normal. For example, if we want to know whether $\sqrt{2}$ is normal, it makes sense to c... | https://mathoverflow.net/users/45707 | Non-normal numbers definable without parameters in the langauge of differential rings with composition | We can define Greg Martin's absolutely abnormal number in this language, based on Martin's paper as in the [arxiv](https://arxiv.org/abs/math/0006089) or the [American Mathematical Monthly](https://www.jstor.org/stable/2695618).
Our number which is abnormal in all bases is $\alpha=M(1)$, where we will define the func... | 7 | https://mathoverflow.net/users/nan | 325783 | 140,302 |
https://mathoverflow.net/questions/325742 | 3 | Let $S$ be a scoring rule for probability functions. Define
$EXP\_{S}(Q|P) = \sum \limits\_{w} P(w)S(Q, w)$.
Say that $S$ is striclty proper if and only if $P$ always minimises $EXP\_{S}(Q|P)$ as a function of $Q$. Define
$D\_{S}(P, Q) = EXP\_{S}(Q|P) - EXP\_{S}(P|P)$.
If $S$ is the logarithmic scoring rule de... | https://mathoverflow.net/users/45570 | Strictly Proper Scoring Rules and f-Divergences | In a word, yes, KL is the only one. You're correct that $S$ is strictly proper if and only if $D\_S$ is a Bregman divergence of some strictly convex function[1] (note you should swap the terms in your definition of $D\_S$). You're also apparently right (going from the abstract) that the only f-divergence on the simplex... | 1 | https://mathoverflow.net/users/29697 | 325788 | 140,304 |
https://mathoverflow.net/questions/325584 | 6 | Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}\_0\}$. Sorted, this is OEIS sequence [A088915](https://oeis.org/search?q=2%2C6%2C12%2C16%2C20%2C30%2C42%2C48&language=english&go=Search). What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.
I'm actually even more interested in the asymptotic... | https://mathoverflow.net/users/2480 | Asympotic density of a very simple sequence | It can be shown that the cardinality of
$$R(X) = \{n \in \mathbb{N} : \exists u,v \in \mathbb{Z} \text{ s.t. } n = uv(u+v), n \leq X\}$$
satisfies
$$\displaystyle R(X) = C X^{2/3}(1 + o(1))$$
for some explicit constant $C$. Indeed, by the main result of this [paper](https://arxiv.org/abs/1605.03427) and the fa... | 2 | https://mathoverflow.net/users/10898 | 325795 | 140,308 |
https://mathoverflow.net/questions/325796 | 5 | Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself ([Dedekind infinite](https://en.wikipedia.org/wiki/Dedekind-infinite_set)). Without Choice it is consistent that there are infinite sets which can't be partitioned into two infinite subsets ([amorphous sets](https://en.wikip... | https://mathoverflow.net/users/92164 | Amorphous proper classes in MK | Unless I'm missing something, the answer is **no**: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and this lets us partition the original class into two proper classes, for example $s^{-1}(limits)$ versus $s^{-1}(succ... | 7 | https://mathoverflow.net/users/8133 | 325800 | 140,309 |
https://mathoverflow.net/questions/325722 | 1 | Is it true that the definition of [approximate differentiability presented here](https://www.encyclopediaofmath.org/index.php/Approximate_differentiability) of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?
>
> $$\lim\_{r \to 0} \rlap{-}\!\!\int\_{B\_r(x)} \min \left\{\frac{f(y)-f(x... | https://mathoverflow.net/users/122620 | Equivalent notion of approximate differentiability | I think the answer is no.
First of all consider a related question: is [the standard definition of approximate continuity](https://www.encyclopediaofmath.org/index.php/Approximate_continuity) of $g\colon \mathbb R \to \mathbb R$ equivalent to the following one
$$
\lim\_{r \to 0} \rlap{-}\!\!\int\_{B\_r(x)} \min \le... | 1 | https://mathoverflow.net/users/44463 | 325838 | 140,315 |
https://mathoverflow.net/questions/325793 | 9 | **Edit**: The formulation of my question was incorrect, for several reasons. Here is what I hope to be the correct formulation:
Let $\mathbb{P}$ be a projective space, and $V$ a *general* linear subspace of $H^0(\mathcal{O}\_{\mathbb{P}}(d))$ (that is, a general point in the corresponding Grassmannian). Then for $p<d... | https://mathoverflow.net/users/40297 | Relations between homogeneous polynomials | I am posting this as an answer since the comment thread is already long. The question is a special case of Fröberg's Conjecture.
MR0813632 (87f:13022)
Fröberg, Ralf(S-STOC)
An inequality for Hilbert series of graded algebras.
Math. Scand. 56 (1985), no. 2, 117–144.
13H15 (13D03 13H10)
This spe... | 7 | https://mathoverflow.net/users/13265 | 325840 | 140,316 |
https://mathoverflow.net/questions/325733 | 22 | To be explicit, by Zermelo set theory with Choice, ZC, I mean the theory with the same language and axioms as ZFC except not Foundation (also called Regularity) and with the axiom scheme of Separation instead of Replacement. By Global Choice I mean adding a new function symbol $F$ and an axiom:$\forall v[v\neq\emptyset... | https://mathoverflow.net/users/38783 | Is Global Choice conservative over Zermelo with Choice? | The known proofs of conservativity of ZF + GC (Global Choice) over ZFC make significant use of replacement, and as far as I know the problem of conservativity of Z + GC over ZC is wide open.
Let me add that I have discussed the problem with a number of experts over the past two decades, and also posed it on FOM in 2... | 15 | https://mathoverflow.net/users/9269 | 325843 | 140,318 |
https://mathoverflow.net/questions/325837 | 4 | Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$.
What are the curves of positive curvature in $\textit{F}$ minimizing the Euclidean distance between the starting and the ending point? Is the arc of circle of proper radius one of t... | https://mathoverflow.net/users/132140 | What curve of positive curvature minimizes distance from the origin, given length and total curvature? | The minimizer is not smooth, it is formed by two sides of an isosceles triangle with angle $\pi-\bar k$ at its vertex. You can approximate it by a smooth curve with large curvature around the vertex and nearly zero curvature elsewhere.
| 5 | https://mathoverflow.net/users/1441 | 325870 | 140,323 |
https://mathoverflow.net/questions/325862 | 10 | Suppose $X$ is a random vector denoted as $(X\_1,\cdots,X\_n)$, where $X\_1,\cdots,X\_n$ are iid random variables with sub-Gaussian distributions. For all $i$, suppose $E[X\_i^2]=1$ for simplicity and $\|X\_i\|\_{\psi\_2}=K$ where$\|\cdot\|\_{\psi\_2}$ is the sub-Gaussian norm.
Let $Y=\|X\|$ be the 2-norm of $X$. A k... | https://mathoverflow.net/users/120302 | Expectation of the norm of a random vector | $\newcommand{\si}{\sigma}$
Let us prove a stronger estimate of $EY$, and let us do that under less restrictive conditions. Namely, let us prove that
\begin{equation\*}
EY-\sqrt n=O(1/\sqrt n) \tag{1}
\end{equation\*}
assuming only that $EX\_1^4<\infty$ (instead of the $X\_i$'s being sub-Gaussian).
Substituting
$$... | 14 | https://mathoverflow.net/users/36721 | 325881 | 140,328 |
https://mathoverflow.net/questions/325806 | 3 | Let $F$ be a finite set equipped the discrete topology. Let $X = F \times F \times ...$ be the countably infinite product space equipped with the product topology. Let $\mathcal A$ be any field of subsets of $X$ that contains the open subsets of $X$. Let $\mu$ be a finitely additive, finite measure with domain $\mathca... | https://mathoverflow.net/users/96899 | If the finitely additive measure of an open set is approximable by clopen sets, is it approximable from within? | The answer to this question is negative. To construct a counterexample, fix any free ultrafilter $\mathcal U$ on $X$ containing a discrete subspace $D$ of $X$. The characteristic function $u:\mathcal P(X)\to \{0,1\}$ of this ultrafilter determines a $\{0,1\}$-valued finitely additive measure $u$ defined on the algebra ... | 1 | https://mathoverflow.net/users/61536 | 325890 | 140,331 |
https://mathoverflow.net/questions/325901 | 3 | Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a\_1, a\_2$ be smooth positive functions such that $a\_1-a\_2$ is compactly supported in $\Omega$, and $a\_i>c>0$, for some constant $c$. Suppose there exists $\lambda>0$ and $u\_i\in H^1\_0(\Omega)$ such that
$$-a\_i \Delta u\_i=\lambda u\_i,$$
and $\frac{\par... | https://mathoverflow.net/users/42326 | Eigenfunctions of elliptic equations | In general the answer is "no." Let $\Omega = (-\pi/2,\,\pi/2) \subset \mathbb{R}$, and for $\varphi \in C^{\infty}\_0(\Omega)$ take
$$u\_1 = \cos(x), \quad u\_2 = \cos(x) + \epsilon \varphi$$
$$a\_1 = 1, \quad a\_2 = \frac{\cos(x) + \epsilon \varphi}{\cos(x) - \epsilon \varphi''},$$
$$\lambda = 1.$$
For $\epsilon > 0$ ... | 3 | https://mathoverflow.net/users/16659 | 325903 | 140,333 |
https://mathoverflow.net/questions/325805 | 4 | A topological space $X$ is defined to have *countable discrete cellularity* if each discrete family of open subsets of $X$ is at most countable.
A family $\mathcal F$ of subsets of a topological space $X$ is called *discrete* if each point $x\in X$ has a neighborhood $O\_x\subset X$ that intersects at most one set $... | https://mathoverflow.net/users/61536 | Does each $\omega$-narrow topological group have countable discrete cellularity? | Mikhail Tkachenko informed me that the problem has a counterexample, constructed in [Example 8.2.1](https://books.google.com/books?id=hIEnzrOBbW0C&pg=PA526) of [his book](https://books.google.com/books/about/Topological_Groups_and_Related_Structure.html?id=hIEnzrOBbW0C&redir_esc=y) with Arhangelskii.
This example lo... | 4 | https://mathoverflow.net/users/61536 | 325906 | 140,335 |
https://mathoverflow.net/questions/291089 | 4 | Is there a formula to compute the determinant of a structurally symmetric $n$-banded matrix? I am specifically interested in the 5-banded matrix:
$$ \left[\begin{matrix}
c\_{0} & s\_{0} & 0 & w\_{0} & 0 & 0 & 0 & 0 & 0\\
n\_{0} & c\_{1} & s\_{1} & 0 & w\_{1} & 0 & 0 & 0 & 0\\
0 & n\_{1} & c\_{2} & 0 & 0 & w\_{2} & 0 ... | https://mathoverflow.net/users/102586 | Determinant of structurally symmetric $n$-banded matrix? | This determinant can be calculated using the block transfer matrix method of Molinari (Linear Algebra and its Applications 429 (2008) 2221-2226, <https://arxiv.org/abs/0712.0681>), if $w\_i \neq 0$. You have $3 \times 3$ blocks that read, after renumbering,
$$
\mathbf A\_k =
\begin{pmatrix}
c\_{k,0} & s\_{k,0} & 0 \\
... | 3 | https://mathoverflow.net/users/90413 | 325908 | 140,336 |
https://mathoverflow.net/questions/325859 | 2 | A boolean algebra is *rigid* if it has no nontrivial automorphisms. Call it *semi-rigid* if none of its nontrivial automorphisms has any fixed points other than 0 and 1.\* The four-element algebra $\{0, b, \neg b, 1\}$ is a simple example of a semi-rigidity. Preliminary question: Are there semi-rigid complete atomless ... | https://mathoverflow.net/users/100571 | Semi-rigid boolean algebras | There is no such algebra. In fact, suppose that $B$ satisfies the indicated condition. Choose $a$ in $B$ with $a$ not equal to $0$ or $1$. Let $f$ be a nontrivial automorphism of the principal ideal determined by $a$. Define $g(x)=f(x \wedge a)\vee(x \wedge-a)$. Then $g$ is a nontrivial automorphism of $B$ with fixed p... | 9 | https://mathoverflow.net/users/90095 | 325913 | 140,337 |
https://mathoverflow.net/questions/325907 | 1 | Let $(F,\nu)$ be a Thurston's foliation on a surface $S$ with a non-zero transverse measure $\nu.$ Assume that $F$ has no closed leaves nor compact separatrices. Did anyone study such foliations?
More specifically, I believe that any other transverse measure on $F$ is a scalar multiple of $\nu$ and I am looking for a... | https://mathoverflow.net/users/23935 | Is transverse measure on a foliation without closed leaves unique? | Such foliations were studied rather intensely in the early works on measured foliations that introduced them to the mathematical world. See for example "Thurston's work on surfaces" aka "Travaux de Thurston sur les Surface" by Fathi, Laudenbach, Poenaru et. al.
Your statement about scalar multiples of $\nu$ is false ... | 3 | https://mathoverflow.net/users/20787 | 325918 | 140,339 |
https://mathoverflow.net/questions/325939 | 3 | I am wondering if the following rings are catenary:
1. If $k$ is a field, is the ring of formal power series $k[[X\_1,\dots,X\_n]]$ catenary?
2. Is the ring of complex power series with a non-zero radius of convergence $\Bbb C\{X\_1,\dots,X\_n\}$ (*id est* the ring of germs of holomorphic functions at zero) a catenar... | https://mathoverflow.net/users/83945 | Are the ring of power series and the ring of germs of holomorphic functions catenary? | Yes, they are regular (the maximal ideal is generated by a number of elements equal to its dimension) and therefore Cohen-Macaulay (Matsumura, Theorem 17.8). And a Cohen Macaulay ring is catenary (Matsumura, Theorem 17.4).
Matsumura, Commutative Ring Theory, CUP, 1986
| 10 | https://mathoverflow.net/users/6348 | 325940 | 140,344 |
https://mathoverflow.net/questions/300399 | 0 | Let $M$ be a **positive semidefinite** operator on a complex Hilbert space $(F,(\cdot,\cdot))$.
On the quotient space $F/\text{Ker}(M)$ we have the following inner product
$$\langle \overline{x},\overline{y}\rangle = (Mx,y),$$
for all $\overline{x},\overline{y}\in F/\text{Ker}(M)$.
According to some papers, the fol... | https://mathoverflow.net/users/116483 | The completion of $F/\text{Ker}(M)$ is isomorphic to the closure of the range of $M$ | I think what is going on is the following.
As Meisam Soleimani Malekan says, for any Hilbert space $H$ with inner product $(\cdot|\cdot)$, and $\newcommand{\mc}{\mathcal}T\in\mc B(H)$, we have that $\ker T^\*T=\ker T$, because $T^\*T\xi=0\implies (T^\*T\xi|\xi)=0 \implies \|T\xi\|^2=0\implies T\xi=0$. Also, $\newcomm... | 4 | https://mathoverflow.net/users/406 | 325942 | 140,345 |
https://mathoverflow.net/questions/325938 | 4 | I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the homomorphism $Hom\_R(B,I)\to Hom\_R(A,I)$ to be surjective. Is this condition strictly weaker than the injectivity of $I$; ... | https://mathoverflow.net/users/2191 | On definitions and explicit examples of pure-injective modules | This is not pure-injectivity. The relevant concept is that of an fp-injective ("finitely presented-injective") module, otherwise known as an "absolutely pure" module. A left $R$-module $J$ is said to be fp-injective if for any finitely presented left $R$-module $M$ one has $Ext^1\_R(M,J)=0$.
The notion of an fp-inje... | 6 | https://mathoverflow.net/users/2106 | 325943 | 140,346 |
https://mathoverflow.net/questions/325924 | 2 | I have iid random variables $X\_1, \dots, X\_n$ with $X\_i \geq 0$, $E[X\_i]=1$ and $V[X\_i] = \sigma^2$.
Let $S\_n = \frac{\sum\_{i=1}^n X\_i}{n}$.
I'd like to say that $E[\sqrt{S\_n}] = 1-O(1/n)$.
My first approach was to write $E[\sqrt{S\_n}] = \sqrt{E[S\_n] - V[\sqrt{S\_n}]} = \sqrt{1-V[\sqrt{S\_n}]}$.
I'm th... | https://mathoverflow.net/users/57321 | Asymptotic rate for the expected value of the square root of sample average | Substituting $S\_n$ for $u$ in the inequalities
$$\frac{1+u-(u-1)^2}2\le\sqrt u\le\frac{1+u}2$$
for $u\ge0$, taking the expectations, and using that $ES\_n=1$ and $E(S\_n-1)^2=V(S\_n)=\sigma^2/n$, we have
$$1-\frac{\sigma^2}{2n}\le E\sqrt{S\_n}\le1,$$
so that $E\sqrt{S\_n}=1-O(1/n)$, as desired.
| 2 | https://mathoverflow.net/users/36721 | 325947 | 140,350 |
https://mathoverflow.net/questions/325912 | 4 | I am wondering if there is a simple example of a manifold such that, given a value for the scalar curvature $R$, I can find a manifold such that the Ricci tensor has all zero components except for one component which takes the value $R$.
I feel like this can be achieved using a warped product of two metrics to separa... | https://mathoverflow.net/users/119114 | Example of a Manifold which has One Non-zero Component of Ric corresponding to Scalar Curvature | Your intuition is correct that warped product metrics will do the trick.
Maybe try to redo the computations yourself, but if I'm not mistaken the Ricci curvature of $g=dt^2+f(t)^2dx^2$ (where $x\in\mathbb{R}^n$ and $dx^2$ denotes the euclidean metric on $\mathbb{R}^n$) is given by :
$$\text{Ric}\_g=-(n-1)\frac{f''(t)... | 4 | https://mathoverflow.net/users/8887 | 325959 | 140,354 |
https://mathoverflow.net/questions/325949 | 6 | Both the ultraweak and ultrastrong topologies are intrinsic topologies in the sense that the image of a continuous (unital) $\*$-homomorphism between von Neumann algebras (in either topology) is a von Neumann subalgebra of the target von Neumann algebra, unlike say just norm-continuous $\*$-homomorphisms. One may then ... | https://mathoverflow.net/users/127523 | Why are ultraweak *-homomorphisms the `right' morphisms for von Neumann algebras (and say, not ultrastrong)? | A $\*$-homomorphism between two von Neumann algebras is weak\* to weak\* continuous if and only if it is ultrastrong to ultrastrong continuous. See Proposition III.2.2.2 of [Blackadar's book](https://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf) (which, basically, answers all questions of this type that you might have).
| 8 | https://mathoverflow.net/users/23141 | 325960 | 140,355 |
https://mathoverflow.net/questions/325880 | 3 | Let $Log(n) = \sum\_{i=1}^r \alpha\_i \cdot e\_i$, where $n = \prod\_{i=1}^r p\_i^{\alpha\_i}$ and $p\_i$ is the $i$-th prime, $\alpha\_i \ge 0$, $e\_i$ is the $i$-th standard basis vector. For example $6 = 2\cdot3$, hence:
$$Log(6) = (1,1,0,0,0,\cdots)$$
$$Log(15) = (0,1,1,0,0,\cdots)$$
$$Log(7) = (0,0,0,1,0,\cdots)$$... | https://mathoverflow.net/users/nan | Is this line of thought (using linear algebra to get number theoretic results) already being pursued in the literature? | I don't know of such a reference, but I think I can give you a short proof for your intuition and conjecture, hope that helps:
One basis for $1,...,n$ is given by $e\_1,\ldots, e\_k$, where $k = \Pi(n)$. Now assume we have any basis $v\_1, \ldots , v\_k$ and fix an $1 \leq i \leq k$. Then there exists a $j$ such that... | 2 | https://mathoverflow.net/users/109932 | 325962 | 140,356 |
https://mathoverflow.net/questions/325973 | 3 | Let $S^n/\Gamma\_i\,(i=1,2)$ be a $n$-dimensional spherical space form, where $\Gamma\_i \subset SO(n+1)$ is a finite subgroup acting freely on $S^n$.
Suppose $S^n/\Gamma\_1$ is diffeomorphic to $S^n/\Gamma\_2$, can we show they are isometric?
| https://mathoverflow.net/users/105900 | Isometries between spherical space forms | Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G.
Complexes à automorphismes et homéomorphie différentiable.
Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W.... | 10 | https://mathoverflow.net/users/1573 | 325975 | 140,361 |
https://mathoverflow.net/questions/325963 | 42 | I have noticed that there is a huge amount of work which has been done on numerically verifying the [Riemann hypothesis](https://en.wikipedia.org/wiki/Riemann_hypothesis) for larger and larger non-trivial zeroes.
I don't mean to ask a stupid question, but is there some particular reason that numerical verifications g... | https://mathoverflow.net/users/119114 | Why is so much work done on numerical verification of the Riemann Hypothesis? | People are interested in computing the zeros of $\zeta(s)$ and related functions not only as numerical support for RH. Going beyond RH, there are conjectures about the vertical distribution of the nontrivial zeros (after "unfolding" them to have average spacing 1, assuming they are on a vertical line to begin with).
... | 49 | https://mathoverflow.net/users/137323 | 325981 | 140,363 |
https://mathoverflow.net/questions/325988 | 6 | Let $X$ and $Y$ be two copies of $S^2$, and let $A\_5$ act on each of them (as a group of rotations). Call these actions $\theta\_X$ and $\theta\_Y$.
Moreover, let $g \in A\_5$ be a fixed element of order $5$ (without loss of generality, the cycle $(12345)$) and suppose that $\theta\_X(g)$ is a rotation by $\frac{2 \... | https://mathoverflow.net/users/39521 | Bijection from $S^2$ to itself interchanging actions of $A_5$ | Yes.
You need to describe $X$ and $Y$ as $A\_5$-sets. One is obtained from the other by twisting the action by a non-inner automorphism $\sigma$ of $A\_5$, say $Y=X^\sigma$. But one observation is that two subgroups of $A\_5$ are conjugate in $A\_5$ iff they're conjugate in $S\_5$. So any $A\_5$-set $X$ is isomorphic... | 11 | https://mathoverflow.net/users/14094 | 325991 | 140,368 |
https://mathoverflow.net/questions/325985 | 10 |
>
> **I. Klein**
>
>
>
In "*[On the Order-Seven Transformation of Elliptic Functions](http://library.msri.org/books/Book35/files/klein.pdf)*" (pp. 287-331), he discusses in p. 298 what we now call the *Klein quartic*,
$$\lambda^3\mu+\mu^3\nu+\nu^3\lambda= 0\tag1$$
and in p. 313 introduces what we can call th... | https://mathoverflow.net/users/12905 | Can we use Ramanujan's parameterization of Klein's quartic to solve Klein's septic? | There seems to be a problem with Klein's septic equation $(2)$ combined with the purported roots in $(3)$. Let $\,k\,$ be any integer. Define
$$ P\_1(k) := \gamma^{k}\mu^2 + \gamma^{2k}\lambda^2 + \gamma^{4k}\nu^2 \tag{1} $$
and
$$ P\_2(k) := \gamma^{3k}\lambda\nu + \gamma^{6k}\mu\nu +
\gamma^{5k}\lambda\mu. \tag{2} ... | 8 | https://mathoverflow.net/users/113409 | 326001 | 140,370 |
https://mathoverflow.net/questions/325997 | 3 | Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let
$\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\mathbf{d}$ on the diagonal,
and $$M\_\mathbf{d}:=\operatorname{diag}(\mathbf{d})+Z+Z^2$$.
Notation: call a subset of $[n]... | https://mathoverflow.net/users/48831 | Determinant of an "almost cyclic" matrix | For a length $n-1$ vector $a=(a\_1,\dots,a\_{n-1})$, define the $n$-by-$n$ tridiagonal matrix $T(a)$ with $1$'s on its main diagonal and subdiagonal and $a$ on its superdiagonal. The first two determinants are $\det T(\emptyset)=1$ and $\det T(a\_1)=1-a\_1$. By cofactor expansion on the last row we have $\det T(a\_1,\d... | 2 | https://mathoverflow.net/users/112641 | 326003 | 140,371 |
https://mathoverflow.net/questions/325822 | 3 | $(M,\omega)$ is a compact Kaehler manifold and $f\_{t,s}$ are 1-parameter group generated by holomorphic vector fields $V\_s$. My question is whether the function $\frac{f\_{t,s}^\* \omega^n}{\omega^n}$ is a bounded independent of $t$ and $s$. If not, can we control the $L^p$($p>1$) norm of $\frac{f\_{t,s}^\* \omega^n}... | https://mathoverflow.net/users/41065 | Volume form under holomorphic automorphisms | There is no such bound. As a counterexample take $(M,\omega)=(\mathbb{CP}^1, \omega\_{FS})$. Identify $\mathbb{CP}^1$ with $\mathbb{C}\cup \infty$; in such co-ordinates
$$\omega\_{FS}=\frac{i dz\wedge d\overline z}{(1+|z|^2)^2}.$$
The radial vector field $V=z\partial/\partial z$ on $\mathbb{C}$ extends smoothly to a ho... | 3 | https://mathoverflow.net/users/2819 | 326007 | 140,373 |
https://mathoverflow.net/questions/325572 | 3 | This question is related to [Lifting points via étale morphism of adic spaces](https://mathoverflow.net/questions/242430/lifting-points-via-%C3%A9tale-morphism-of-adic-spaces).
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\circ)$. Let $f\colon A\to B$... | https://mathoverflow.net/users/137002 | Extending section of étale morphism of adic spaces | The statement is correct and here is one way to prove it. The slogan is that "finite etale extensions of $k(x)$ and $\widehat{k(x)}$ are the same for "quasi-complete" fields $k(x)$" (this is a toy example of Gabber's Approximation Lemma).
I decided to write a proof in complete details mostly to be sure that there is... | 3 | https://mathoverflow.net/users/115211 | 326013 | 140,375 |
https://mathoverflow.net/questions/135680 | 5 | A recent project has forced me to consider a rather special object in a rather nasty category. Consider any category $\mathcal{C}$ which has
* **image objects**, meaning for each morphism $f: x \to y$ there exists an object $\text{im }f$ along with morphisms $$x \stackrel{s}{\twoheadrightarrow}\text{im }f \stackrel{i... | https://mathoverflow.net/users/18263 | What is the image of the intial object inside the final object called? | One context where an object like your $d$ has been studied at least a bit is in [quasitopos](https://ncatlab.org/nlab/show/quasitopos) theory. I doubt this will be very helpful to you since your "nasty" category is probably not a quasitopos, but perhaps it is interesting and may give some ideas.
Unlike in a topos, in... | 3 | https://mathoverflow.net/users/49 | 326014 | 140,376 |
https://mathoverflow.net/questions/325951 | 3 | If a semigroup S has no proper ideals can it have both regular and non-regular members? My guess would be 'yes' but in that case does anyone know of an example in the literature?
| https://mathoverflow.net/users/83019 | Is a J-simple semigroup with an idempotent necessarily regular? | Yes. In section 8.5 of volume 2 of Clifford and Preston they give a construction due to Bruck that embeds any monoid $M$ is a simple monoid $C(M)$. In Thm 8.48 they say $C(M)$ is regular iff $M$ is regular. So taking any nonregular monoid $M$ gives you $C(M)$ as the answer.
| 2 | https://mathoverflow.net/users/15934 | 326020 | 140,379 |
https://mathoverflow.net/questions/326033 | 7 | Let $X$ be an isotropic random vector (i.e. $E[XX^T]=I\_n$) and $X$ takes value in a finite set $S \subset\mathbb R^n$. If $X$ is a sub-Gaussian random vector and the norm $\|X\|\_{\psi\_2}\le C$ where $C$ is a constant, it will imply that $S$ is expoentially large with dimension $n$.
Note that $X$ is said to be sub-... | https://mathoverflow.net/users/120302 | Prove that a sub-Gaussian random vector over a finite set $S \subset\mathbb R^n$ implies that $|S|$ is exponentially large | If $(Z\_i)\_{1 \leq i \leq N}$ are scalar random variables with $\|Z\_i\|\_{\Psi\_2} \leq C$, then $\mathbf{E} \max Z\_i \leq CC'\sqrt{\log N}$ by the usual union bound argument.
Now let $X$ be an isotropic random vector in $\mathbf{R}^n$ such that $\|\langle X,\theta \rangle \|\_{\Psi\_2} \leq C $ for every unit ve... | 10 | https://mathoverflow.net/users/908 | 326050 | 140,388 |
https://mathoverflow.net/questions/326035 | 7 | I'm studying the proof of Thm 1.5.1. in Laumon's "Cohomology of Drinfeld Modular Varieties". Notation: $\mathfrak{m}$ is a square zero ideal of $\mathcal{O}$ and $k=\mathcal{O}/\mathfrak{m}$. Laumon shows that the obstruction to the existence of a lift of a Drinfeld module $\phi: A \rightarrow k[\tau]$ to a Drinfeld mo... | https://mathoverflow.net/users/137354 | Smoothness of the moduli space of Drinfeld modules | At this point of the proof, Laumon assumes that $\phi$ is standard—I'll do so as well. Let $a\in A$ be nonzero. Then
$$ \phi\_a = \sum\_i \phi\_{a,i} \tau^i$$
with $\phi\_{a,-d\deg(\infty)\infty(a)} \in k^{\times}$ and $\phi\_{a,i} = 0$ in higher degrees. Note that an element $x$ of $\mathcal{O}$ is a unit if and ... | 4 | https://mathoverflow.net/users/nan | 326051 | 140,389 |
https://mathoverflow.net/questions/326017 | 1 | We consider the equation
$$ \sum\_{j=1}^n \frac{\lambda\_j}{x-x\_j} =i$$
where $\lambda\_j>0$ and $x\_j$ are real distinct numbers.
I want to show that if $\lambda\_k$ is small compared to the distance of all $x\_j$ from $x\_k$ then there exists a solution $x\approx x\_k- i y\_k$ to this equation in the neighbour... | https://mathoverflow.net/users/nan | Existence of solution to linear fractional equation | Here's a "real" method proof:
Without loss of generality, assume $k = 1$. You are equivalently looking for the roots of
$$ P(x, \eta) = \eta \prod\_{j > 1} (x - x\_j) + \sum\_{\ell > 1} \lambda\_\ell \prod\_{j \neq \ell} (x - x\_j) - i \prod\_{j} (x - x\_j) $$
where we wrote $\eta$ for $\lambda\_1$. (We consider... | 3 | https://mathoverflow.net/users/3948 | 326060 | 140,391 |
https://mathoverflow.net/questions/326061 | 4 | Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}\_n$. Does $G$ have finitely many connected components? I heard it's true when $\mathbf G$ is semisimple, by a theorem of Cartan. ... | https://mathoverflow.net/users/38145 | Real points of reductive groups and connected components | For every $\mathbb{R}$-scheme $X$ of finite type, $\pi\_0(X(\mathbb{R}))$ is finite. This follows e.g. from Theorem 2.3.6 in
Bochnak, Coste, Roy, *Real Algebraic Geometry*
(basic structure theorem for semi-algebraic sets).
| 7 | https://mathoverflow.net/users/7666 | 326067 | 140,393 |
https://mathoverflow.net/questions/326070 | 5 | Is there a bicategory $V$ and a definition of *monoid in a bicategory* so that $\text{Monoids}(V)$ is the category of groups and homomorphisms?
EDIT: For example, is there a bicategory $V$ so that Monad(V) is the category of groups and group homomorphisms?
| https://mathoverflow.net/users/2536 | Can groups be recovered as "monoids" in a bicategory? | There is a definition of monoids in a monoidal category and every monoidal category is a bicategory. You can try to extend this definition to a general bicategory. For example, we can say that a monoid in a bicategory $\mathcal{C}$ is a monoid in the monoidal category $\mathrm{Hom}\_\mathcal{C}(X,X)$ for every object $... | 5 | https://mathoverflow.net/users/62782 | 326073 | 140,396 |
https://mathoverflow.net/questions/326074 | 5 | If $X$ is a separable Banach space and $(\epsilon\_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon\_n$ for all $n$? I suspect the answer is positive, but cannot come up with an example.
| https://mathoverflow.net/users/69275 | Quasinilpotent , non-compact operators | On the Argyros-Haydon space every operator is a compact perturbation of a scalar multiple of the identity, and hence every quasinilpotent operator is compact.
| 7 | https://mathoverflow.net/users/2554 | 326076 | 140,397 |
https://mathoverflow.net/questions/326075 | 3 | Suppose I have a random variable $\theta=(\theta\_1,\dotsc,\theta\_n)$; where the $\theta\_i$ might have pairwise correlations. I decompose it into $\theta=\hat\theta(\phi\_1,\dotsc,\phi\_k)$, where $\hat\theta$ is a deterministic function, and the $\phi\_i$ are random variables such that each $\phi\_i$ is independent ... | https://mathoverflow.net/users/120465 | Is there a name for "splitting a probability distribution into independent components"? | I think this particular property is unlikely to have a generally accepted name.
However, a more general concept is commonly called decoupling, when the joint distribution of a function of dependent random variables (r.v.'s) is represented as a mixture of distributions of this function of independent r.v.'s. See e.g.... | 2 | https://mathoverflow.net/users/36721 | 326078 | 140,398 |
https://mathoverflow.net/questions/326066 | 3 | Let $f:\mathbb{Z}^+\to \mathbb{C}$ be bounded. Say we are interested in studying how $f$ behaves in short three-term arithmetic progressions. It is very well-known that we can bound
$$\sum\_{h\leq H} \sum\_{n\leq N} f(n) f(n+h) f(n+2 h)$$
by $o(H N)$ (to set ourselves a low bar...) provided that we can bound the *Gower... | https://mathoverflow.net/users/136932 | Gowers norms and three-term arithmetic progressions in the mean | If you expand square as a double sum, you get a sum over copies of some configuration of 6 points. By the usual Cauchy--Schwarz argument, this sum is going to be controlled by some $U^d$ norm. The smallest $d$ such that this is controlled by $U^{d+1}$ is called ``true complexity'' of the system. For your case of 6 line... | 3 | https://mathoverflow.net/users/806 | 326082 | 140,400 |
https://mathoverflow.net/questions/326094 | 4 | Suppose $(f\_n)\_n$ is a countable family of entire, surjective functions, each $f\_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a\_n)\_n$, not all zero, such that $\sum\_{n=1}^{\infty} a\_n f\_n$ is entire but not-surjective? In fact, I am interested in this question under the additional assumptio... | https://mathoverflow.net/users/137377 | Sums of entire surjective functions | One expects there to be no such $a\_n$ in general, because the
"typical" entire functions is surjective (those that aren't are of the
special form $z \mapsto c + \exp g(z)$). An explicit example is
$f\_n(z) = \cos z/n$: any convergent linear combination $f = \sum\_n a\_n f\_n$
is of order $1$, so if $f$ is not surjec... | 4 | https://mathoverflow.net/users/14830 | 326103 | 140,406 |
https://mathoverflow.net/questions/326041 | 4 | When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, it is also known that the Riemann $\zeta$ function does not vanish on (0, 1).
Are there any result showing that some Di... | https://mathoverflow.net/users/133679 | Real non trivial zeros of Dirichlet L-functions | Perhaps the most comprehensive result in this direction is due to Conrey and Soundararajan (<https://arxiv.org/abs/math/0111013> or <https://link.springer.com/article/10.1007/s00222-002-0227-x>). They prove (among other things) that the density of odd positive squarefree integers $d\leq x$ such that $L(s,\chi\_{8d})>0$... | 3 | https://mathoverflow.net/users/111215 | 326111 | 140,409 |
https://mathoverflow.net/questions/326110 | 5 | Let A be a non-commutative algebra and let X be some geometric space (such as a topological space or an algebraic variety or scheme). Is there a notion of cohomology ring of X with coefficients in A? What is the correct set up to consider cohomologies with non-commutative coefficients?
If a topological group $G$ act... | https://mathoverflow.net/users/21491 | Cohomology ring with non-commutative coefficient ring |
>
> Is there a notion of cohomology ring of X with coefficients in A?
>
>
>
Yes, and nothing new is needed. The underlying additive group of $A$ is abelian so you take cohomology with coefficients in that abelian group; then the multiplication on $A$ is a bilinear map $A \times A \to A$ which induces a map
$... | 10 | https://mathoverflow.net/users/290 | 326120 | 140,410 |
https://mathoverflow.net/questions/326091 | 3 | Let $\pi:\mathcal{A}\rightarrow C$ be a semi-abelian scheme, i.e. $\mathcal{A}$ is a smooth separated commutative group scheme over $C$ via $\pi$ with geometrically connected fibres, such that each fibre $\mathcal{A}\_v$, where $v\in C$, is an extension of an abelian variety $\mathcal{B}\_v$ by a torus $T\_v$ over the ... | https://mathoverflow.net/users/110471 | Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open? | I am posting my comments as an answer.
Let $k$ be a field. Let $$(G,m:G\times\_{\text{Spec}\ k}G \to G)$$ be a locally finitely presented group scheme over $\text{Spec}\ k$. For every open $U$, denote by $m\_U$ the restriction of $m$, $$m\_U:U\times\_{\text{Spec}\ k}U \to G.$$
**Lemma 1.** If $G$ is connected, the... | 2 | https://mathoverflow.net/users/13265 | 326136 | 140,413 |
https://mathoverflow.net/questions/326139 | 12 | Is there any references on the tensor product of (locally) presentable categories ?
All I know about this is Lurie's book that deals with the $\infty$-categorical version, and a few references that deals with special cases (Grothendieck abelian categories, toposes etc...)
Is there any references that defines it pro... | https://mathoverflow.net/users/22131 | Is there any references on the tensor product of presentable (1-)categories? | The canonical reference is Chapter 5 of [Greg Bird's thesis](http://maths.mq.edu.au/~street/BirdPhD.pdf).
| 14 | https://mathoverflow.net/users/57405 | 326145 | 140,415 |
https://mathoverflow.net/questions/326132 | 0 | Suppose I have two independent Brownian motions $B^1\_t, B^2\_t$ and $\mathbb F\_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q\_t$ be a $[-1,1]$ valued $\mathbb{F}\_t$ martingale that the analyst controls.
Let $\mathcal Q$ be the set of $[-1,1]$ valued $\mathbb F\_t$ ma... | https://mathoverflow.net/users/78761 | Conditioning on an irrelevant variable in a martingale control problem | This is true. For any $q \in \mathcal{Q}$, you may define $\tilde{q}$ as the *optional projection* of $q$ onto the filtration generated by $B^1$. This $\tilde{q}$ remains a martingale and achieves the same value thanks to linearity of $f$. More generally, using Jensen’s inequality, your conjecture is true as long as $f... | 1 | https://mathoverflow.net/users/44169 | 326149 | 140,417 |
https://mathoverflow.net/questions/326105 | 5 | Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w\_n)\_{n=1}^\infty\in c\_0\setminus\ell\_1$ is a normalized decreasing weight.
Is there very much known about the complemented subspaces of $d(\textbf{w},p)$? In general (i.e., without any restrictions on $\textbf{w}$ or $... | https://mathoverflow.net/users/73784 | Complemented subspaces of Lorentz sequence spaces? | Question 2 has a negative answer. Suppose $\ell\_p$ contains uniformly complemented copies $E\_N$ of $\text{span}(d\_n)\_{n=1}^N$. Take an ultra power to get a copy $E$ of the completion of $\text{span}(d\_n)\_{n=1}^\infty$ in an $L\_p$ space. The corresponding copies of $E\_n$ in the $L\_p$ space are uniformly complem... | 6 | https://mathoverflow.net/users/2554 | 326154 | 140,419 |
https://mathoverflow.net/questions/326156 | 17 | $\mathrm G$ is Catalan's constant.
I recently found the product
$$
\alpha=\prod\_{n=1}^{\infty}\frac{E\_n(\frac12)E\_n(\frac7{12})E\_n(\frac1{20})E\_n(\frac{13}{20})}{E\_n(\frac14)E\_n(\frac1{12})E\_n(\frac3{20})E\_n(\frac{11}{20})}=\\
\exp\left[\frac{47\mathrm G}{30\pi}+\frac34\right]\sqrt{\frac{33}{91\pi}\sqrt{\fr... | https://mathoverflow.net/users/130062 | Closed-form expression for certain product | the OP asks for some numerical evidence: plotted below is the constant $\alpha$ minus the $\prod\_{n=1}^N$ of the expression in OP, as a function of $N$; so at least within 1 part in 1000 the infinite product does seem to converge from above to the stated constant.
=\lim\_{n\to\infty}\frac{|A\cap n|}{n}$ is the density of subset $A\subset\omega$ if the limit exists. Let us define the filter $\mathcal{F\_1}=\{A\subset\omega~|~\rho(A)=1\}$.
**Question:** Does there exist (in ZFC & CH) selective ultrafilter $\mathcal{U}$ and a bijection $\varphi:\omega\to\omeg... | https://mathoverflow.net/users/118366 | Dense filter and selective ultrafilter | No, there are no such $\mathcal U$ and $\varphi$.
Let me start the proof with two simplifying observations. First, $\varphi$ is irrelevant, because bijections of $\omega$ to itself preserve selectivity of ultrafilters.
Second, every selective ultrafilter $\mathcal U$ is a P-point, which means that, given any counta... | 4 | https://mathoverflow.net/users/6794 | 326178 | 140,425 |
https://mathoverflow.net/questions/326150 | 6 | It is a student exercise that no group can be represented as a set-theoretic union of its two proper subgroups. The same also can be shown for Boolean algebras. On the other hand, it's not hard to show that any infinite Boolean algebra $\mathcal{A}$ can be covered by its $k$ proper subalgebras $\mathcal{A}\_0,\ldots,\m... | https://mathoverflow.net/users/15860 | Finite covers of Boolean algebras by their subalgebras | It's a good idea for such questions to think about the finite case, since it indeed allows to solve the general case.
>
> Proposition: every unital Boolean algebra $A$ of (possibly infinite) cardinal $\ge 16$ (i.e., whose spectrum has cardinal $\ge 4$) is a non-redundant union of 4 unital subalgebras.
>
>
>
Pr... | 4 | https://mathoverflow.net/users/14094 | 326186 | 140,429 |
https://mathoverflow.net/questions/325834 | 11 | We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$.
Let $f$ be measurable. For every $e$ in $(0, 1]$, by Lusin’s theorem, we can write our measurable function as continuous on $[0, 1]-H$, and horrid on a set $H$... | https://mathoverflow.net/users/132446 | A question concerning Lusin’s Theorem | A counterexample to this problem can be constructed as follows. Take a sequence $(K\_n)\_{n\in\omega}$ of pairwise disjoint nowhere dense compact sets $K\_n\subset[0,1]$ of positive Lebesgue measure $\lambda(K\_n)>0$ such that $\sum\_{n=0}^\infty\lambda(K\_n)=1$. Consider the function $f:[0,1]\to [0,1]$ defined by $$
f... | 7 | https://mathoverflow.net/users/61536 | 326187 | 140,430 |
https://mathoverflow.net/questions/326098 | 2 | A (commutative unitary) Noetherian ring $R$ of finite dimension is said to be catenary if for every prime ideal $\mathfrak{p}$ of $R$ one has $\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\mathrm{dim}R$.
Let $A$ be a catenary ring and $B$ a finite extension of $A$ (*id est* $A\subseteq B$ and $B$ is a finit... | https://mathoverflow.net/users/83945 | Are integral extensions of a catenary ring still catenary? | No. Nagata's famous family of examples of non-catenary rings yields a non-catenary finite extension of a catenary noetherian local domain.
**Reference:** M. Nagata, [*On the chain problem of prime ideals*](https://projecteuclid.org/euclid.nmj/1118799769), Nagoya Math. J. 10 (1956), 51-64.
| 2 | https://mathoverflow.net/users/11025 | 326192 | 140,433 |
https://mathoverflow.net/questions/326190 | 4 | A $P$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ there is $x\in {\scr U}$ such that the restriction $f|\_x$ is either constant, or finite-to-one.
A $Q$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ with the property that ... | https://mathoverflow.net/users/8628 | Models of $\mathsf{ZFC}$ with neither $P$- nor $Q$-points | Your definition of P-point should end with "finite-to-one" rather than "injective". As it stands, it defines selective ultrafilters, not P-points.
As far as I know, the consistency of the statement "neither P-points nor Q-points exist" is still an open problem. It is known that this statement implies $2^{\aleph\_0}\g... | 12 | https://mathoverflow.net/users/6794 | 326201 | 140,436 |
https://mathoverflow.net/questions/326183 | 3 | Suppose $X$ is a isotropic sub-Gaussian $n$-dimensional random vector (i.e. $EXX^T=I\_n$, and for any unit vector $u$,$\|\left<X,u\right>\|\_{\psi\_2}\le K$). It is said that $\|X\|\_2-\sqrt n$ may not be sub-Gaussian with bounded norm $CK$ which does not depend on $n$. But I havn't found a counter example.
When $X$... | https://mathoverflow.net/users/120302 | The norm of isotropic sub-Gaussian random vector may not be sub-Gaussian | Let
\begin{equation}
\mu\_X=\tfrac12\,\mu\_{aZ}+\tfrac12\,\mu\_{bZ},
\end{equation}
where $\mu\_U$ denotes the probability distribution of a random vector $U$, $Z\sim N(0,I\_n)$,
and $a,b$ are constants such that
\begin{equation}
0<a<1<b\quad\text{and}\quad \tfrac12\,a^2+\tfrac12\,b^2=1.
\end{equation}
Then... | 4 | https://mathoverflow.net/users/36721 | 326210 | 140,441 |
https://mathoverflow.net/questions/326212 | 16 | I recently had a paper accepted by a journal. When I looked it up on the AMS’ *[Mathematical Reviews](https://mathscinet.ams.org/mathscinet/index.html)*, I noticed that it was previously indexed by the service but, at present, is it not. The journal is not pay-for-play, and one of the previous editors was a very famous... | https://mathoverflow.net/users/104633 | Journal losing indexing services | To answer the first question: There are three main reasons why a journal would be removed from indexing lists, at least these are the three used for the [Impact Factor list:](https://clarivate.com/essays/journal-selection-process/)
*A journal may be removed if it encourages self-citation (e.g. if authors are strongly... | 13 | https://mathoverflow.net/users/11260 | 326213 | 140,442 |
https://mathoverflow.net/questions/326199 | 10 | What are the open problems concerning *all* the finite groups?
The references will be appreciated. Here are two examples:
* **Aschbacher-Guralnick conjecture** ([AG1984](https://doi.org/10.1016/0021-8693(84)90183-2) p.447): the number of conjugacy classes of maximal subgroups of a finite group is at most its class... | https://mathoverflow.net/users/34538 | Open problems concerning all the finite groups | There are plenty of examples; the below are taken from the [Kourovka Notebook](https://kourovka-notebook.org/), and include a fairly broad range of topics.
* (4.55) Let $G$ be a finite group and $\mathbb{Z}\_p$ the localization at $p$. Does the Krull-Schmidt theorem hold for projective $\mathbb{Z}\_pG$-modules?
* (8.... | 6 | https://mathoverflow.net/users/120914 | 326217 | 140,443 |
https://mathoverflow.net/questions/326214 | 5 | I hope this question fits here.
Let $H/k$ be a genus $2$ curve and $J$ its Jacobian variety. Since $J(k)\cong \text{Pic}^0(H)(k)$ we have that its generic point looks like $[(x\_1,y\_1)+(x\_2,y\_2)-2\infty]\in J$. In Mumford coordinates we can see it as $g:=\langle x^2
-Ax + B,Cx+D\rangle:=\langle u(x),v(x)\rangle\i... | https://mathoverflow.net/users/91023 | Identifying elements in the kernel of an explicit endomorphism of a Jacobian variety | I would suggest that you work with the Kummer surface $K$ of $J$
instead of using Mumford coordinates. The advantage is that $K$ is
a quartic surface in $\mathbb P^3$; in the case you are
considering when the curve has a unique point at infinity,
the vanishing of the first coordinate means that the point is
in the thet... | 2 | https://mathoverflow.net/users/21146 | 326225 | 140,445 |
https://mathoverflow.net/questions/325925 | 4 | Let us define the density of subset $A\subset\omega$ :
$$\rho(A)=\lim\_{n\to\infty}\frac{|A\cap n|}{n}$$
if the limit exists. Let $\mathcal{F\_1}=\{A\subset\omega~|~\rho(A)=1\}$. $\mathcal{F\_1}$ is the filter and for the Frechet filter we have $\mathcal{N}\subset\mathcal{F\_1}$. For arbitrary selective ultrafilter $\m... | https://mathoverflow.net/users/118366 | The property of the dense subfilter of a selective ultrafilter | No, there do not exist such selective $\mathcal U$ and bijection $\varphi$. Since selectivity is preserved by bijections and since the Fréchet filter $\mathcal N$ is included in $\mathcal F\_I\cap\mathcal U$, it suffices to show that no selective ultrafilter $\mathcal U$ on $\omega\times\omega$ includes $\mathcal N\oti... | 4 | https://mathoverflow.net/users/6794 | 326227 | 140,447 |
https://mathoverflow.net/questions/326221 | 2 | Given selective ultrafilter $\mathcal{U}$ on $\omega$ and dense filter $\mathcal{F\_1}=\{A\subset\omega~|~\rho(A)=1\}$, where $\rho(A)=\lim\_{n\to\infty}\frac{|A\cap n|}{n}$ if the limit exists. Let $\mathcal{F}=\mathcal{F\_1}\cap\mathcal{U}$.
**Question:** Does there exist a family $\{A\_i\subset\omega\}\_{i<\omega}... | https://mathoverflow.net/users/118366 | Dense subfilter of selective ultrafilter | The answer is negative. The equivalent question was answered [here](https://mathoverflow.net/questions/325925/the-property-of-the-dense-subfilter-of-a-selective-ultrafilter) by Andreas Blass
| 0 | https://mathoverflow.net/users/118366 | 326231 | 140,450 |
https://mathoverflow.net/questions/326159 | 5 | I have a problem with understanding what is a neighbourhood of a singular fiber in a Seifert fibered space coming from the zero surgery. For me a 3-manifold $Y$ is a SFS if it has a decomposition into a disjoint sum of circles s.t. every circle has a neighbourhood of form $V(p,q)$, where $V(p,q)$ is obtained from $D^2 ... | https://mathoverflow.net/users/42750 | Zero surgery on a Seifert fiber space | What you call a *0-surgery* on a Seifert manifold is a move that typically does not produce a Seifert manifold. It is a move that "kills the fiber" and it produces a graph manifold, homeomorphic to a connected sum of lens spaces.
Since this may be a potential source of confusion, let me mention that a a 0-surgery on... | 3 | https://mathoverflow.net/users/6205 | 326238 | 140,451 |
https://mathoverflow.net/questions/326244 | 3 | I am doing some exercises in the analytic and there is a problem as following:
``Let $\{f\_n\}\_{n \in \mathbb{ N}}$'' to be a positive sequence such that:
1. $\sum\limits\_{n=1}^{+\infty} f\_n = 1$.
2. $\gcd(i : f\_i > 0) = 1$.
Suppose that $\mu = \sum\limits\_{n =1}^{+ \infty} nf\_n < + \infty$.
If the sequenc... | https://mathoverflow.net/users/101222 | Where to find the proof of this property? | In probabilistic terms, the result you want is precisely the so-called [local renewal theorem, lattice version](https://projecteuclid.org/download/pdf_1/euclid.pjm/1103051394), due to Kolmogorov '36, Erdös--Feller--Pollard '49, and Chung--Wolfowitz '52. Indeed, let $S\_1,S\_2,\dots$ be independent random variables such... | 5 | https://mathoverflow.net/users/36721 | 326250 | 140,454 |
https://mathoverflow.net/questions/326129 | 4 | We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire function. It is easy to see that a mapping theorem for the numerical range in the sense of $W(f(A)) = f(W(A))$ does not hold, ... | https://mathoverflow.net/users/102946 | Mapping inclusion theorem for the numerical range | Let $N=[\begin{smallmatrix} 0&1 \\ 0&0 \end{smallmatrix}]$ and $f(z)=z+z^2+\cdots+z^m$. On the one hand, $f(N)=N$ and $W(f(N))=W(N)=B(0,1/2)$, the closed ball of center $0$ and radius $1/2$. On the other hand, since $f(z) \approx \sum\_{k\geq1} z^k = z(1-z)^{-1}=(1-z)^{-1}-1$ to within $2^{-m}$ for $z \in B(0,1/2)$ and... | 2 | https://mathoverflow.net/users/7591 | 326251 | 140,455 |
https://mathoverflow.net/questions/237499 | 12 | It is known that there are finitely many finite groups with a given number of conjugacy classes. How can one construct (or get a character table for) the groups $G$ that realize the maximum possible order among groups with $k$ conjugacy classes? Help even with $k=4$ or 5 would be very useful.
| https://mathoverflow.net/users/33089 | Constructing the largest finite group with a fixed number of conjugacy classes | Following @verret comment, here are the list of the largest finite groups $G\_k$ with a fixed class number $k\le 14$ (i.e. the number of conjugacy classes of elements, or the number of irreducible complex representations up to equiv.), coming from the papers of Vera-López and Sangroniz [MR804489](https://mathscinet.ams... | 7 | https://mathoverflow.net/users/34538 | 326258 | 140,457 |
https://mathoverflow.net/questions/323822 | 2 | I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis. That is, *defining* the integral of a function $f$ (with some restrictions) as $$\int\_{-\infty}^{+\infty} f(x) dx := ... | https://mathoverflow.net/users/136155 | Defining integrals by residue theorem | Defining integrals using certain integration formula is sometimes done to regularize divergent integrals. Now, as pointed out by Alexandre Eremenko in the comments, the use of the residue theorem would limit the applicability of the definition of an integral over the real line to a rather small class of functions. Howe... | 1 | https://mathoverflow.net/users/52954 | 326259 | 140,458 |
https://mathoverflow.net/questions/326155 | 9 | In Steenrod's ``[Products of Cocycles and Extensions of Mappings (1947)](https://www.jstor.org/stable/1969172?seq=1#metadata_info_tab_contents),'' which derives [Theorem 5.1]
>
> $$
> \delta(u\cup\_{i} v)=(-1)^{p+q-i}u\cup\_{i-1}v+(-1)^{pq+p+q}v\cup\_{i-1}u+\delta u\cup\_{i}v+(-1)^pu\cup\_{i}\delta v
> $$
> where $... | https://mathoverflow.net/users/106497 | Use of Steenrod's higher cup product and the graded-commutativity | For $(\delta w) \cup v - v \cup (\delta w)$ to be a coboundary, it would need to also be a cocycle: so we would have to have
$$
0 = \delta((\delta w) \cup v - v \cup (\delta w)) = (\delta w) \cup (\delta v) - (\delta v) \cup (\delta w)
$$
Let $X$ be the standard 6-simplex $[v\_0,v\_1,\dots,v\_6]$. Define the followin... | 2 | https://mathoverflow.net/users/360 | 326271 | 140,459 |
https://mathoverflow.net/questions/326262 | 1 | An integral has been pushed me over the edge for several weeks. It reads as:
$$\displaystyle\int\_{\mathbb{R}\_y^3}\int\_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$
I tried to calculate the surface integral inside using spherical coordinates, but it seems that I couldn't do any further ca... | https://mathoverflow.net/users/137463 | About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$ | Without loss of generality you can orient the axes so that the vector $\mathbf{x}$ is along the $x\_3$ axis. Using also that $|\mathbf{\omega}|=1$, one has
$$|\mathbf{x}-[(\mathbf{x}-\mathbf{y})\cdot\mathbf{\omega}]\mathbf{\omega}|^2=(\mathbf{\omega}\cdot\mathbf{y})^2-x\_3^2(\omega\_3^2-1).$$
Subtitution into the integ... | 3 | https://mathoverflow.net/users/11260 | 326272 | 140,460 |
https://mathoverflow.net/questions/326176 | 6 | I was wondering how to compute the eta invariant $\eta(T^3)$ of a flat torus $T^3$, with respect to the signature operator.
In general, how can we compute the $\eta(T^3/\Gamma)$ of a finite quotient of a flat torus?
| https://mathoverflow.net/users/105900 | How to compute the eta invariant of torus | The general formula can be found in [Ouyang - Geometric Invariants For Seifert Fibred 3-Manifolds](https://www.ams.org/journals/tran/1994-346-02/S0002-9947-1994-1257644-9/).
In particular, for $\Gamma \cong 1, \mathbb{Z}\_2, \mathbb{Z}\_3, \mathbb{Z}\_4, \mathbb{Z}\_6, \mathbb{Z}\_2^2$, we have $\eta(T^3/\Gamma) = 0,... | 6 | https://mathoverflow.net/users/126206 | 326273 | 140,461 |
https://mathoverflow.net/questions/315466 | 6 | Im not sure whether this question is appropriate for MO, but I do not have much experience with bimodules (or I forgot many things).
Let $A$ be a finite dimensional (connected) algebra over a field $k$ and $M$ an indecomposable $A$-bimodule (finite dimensional prefered). For simplicity we can assume that $A$ is even ... | https://mathoverflow.net/users/61949 | Tensor product of bimodules | Let $Q=( 2\_{\circlearrowleft c}\xleftarrow a 1 \xrightarrow b 3\_{\circlearrowleft d})$, that is, 3 vertices with 4 arrows, one loop at vertex 2 and 3 and two arrows from vertex 1 to 2 and 3. Consider the relations $\{ac, c^2, d^2, bd\}$ on $Q$. Define $A = \mathbb{Z}\_7Q/\langle ac, c^2, d^2, bd\rangle$.
Let $D(A)... | 3 | https://mathoverflow.net/users/130741 | 326278 | 140,462 |
https://mathoverflow.net/questions/326008 | 1 | The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma\_n\}\_{n>0}$ in groups. Berger-Fresse defined [here](https://arxiv.org/pdf/math/0109158.pdf) an operad morphism from the normalized chains of $E$ onto the Surjection operad, anoth... | https://mathoverflow.net/users/43574 | The table reduction morphism of operads from Barratt-Eccles to Surjection | As written in the paper that you cite, this construction originates from two papers of McClure–Smith, and a geometric interpretation of the table reduction morphism is given in:
>
> Clemens Berger and Benoit Fresse. "Une décomposition prismatique de l'opérade de Barratt-Eccles." (French) *C. R. Math. Acad. Sci. Par... | 2 | https://mathoverflow.net/users/36146 | 326281 | 140,463 |
https://mathoverflow.net/questions/326290 | 5 | Let $\Gamma\_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures.
It is clear that in general, presentations for these groups are hard. I only need for the general case a set of generators (I want to show that two morphisms $... | https://mathoverflow.net/users/124042 | Generators of the mapping class group for surfaces with punctures and boundaries | See the paper
B. Wajnryb, "An elementary approach to the mapping class group of a surface,"
Geometry & Topology 3 (1999) 405–466.
See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: <https://arxiv.org/pdf/math/9811162.pdf>.
| 5 | https://mathoverflow.net/users/1335 | 326297 | 140,468 |
https://mathoverflow.net/questions/326274 | 5 | If ${\scr U}$ and ${\scr V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the *tensor product* ${\scr U}\otimes{\scr V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\subseteq A\times B: \{a\in A:\{b\in B: (a,b)\in X\}\in {\scr V}\}\in {\scr U}\big\}.$$
Is there a non-principal ultrafilter... | https://mathoverflow.net/users/8628 | Relation between ultrafilters ${\scr U}$ and ${\scr U} \otimes {\scr U}$ | As Will Brian noted in a comment, either projection $p:\omega^2\to\omega$ satisfies $p(\mathcal U\otimes\mathcal U)=\mathcal U$. If you also had a function $f$ as in the question, with $f(\mathcal U)=\mathcal U\otimes\mathcal U$, then the composite function $f\circ p$ would send $\mathcal U\otimes\mathcal U$ to itself.... | 5 | https://mathoverflow.net/users/6794 | 326303 | 140,470 |
https://mathoverflow.net/questions/326301 | 2 | The title is the question.
Given a locally compact completely $T\_{4}$ space $X$ (every subspace is $T\_{4}$) and a (Hausdorff) compactification $\overline{X}$ of $X$, is $\overline{X}$ also completely $T\_{4}$?
I have been unable to think of an obvious counterexample, but I suspect that there is one.
The only ki... | https://mathoverflow.net/users/115694 | Are compactifications of completely $T_{4}$ spaces completely $T_{4}$? | The obvious (to me) counterexamples are $\beta \mathbb{N}$ and $\beta \mathbb{R}$ ( the Čech-Stone compactifications) which are non-completely normal compactifications (classic fact, see Engelking or Gilman and Jerrison's book) of $\mathbb{N}$ and $\mathbb{R}$ resp., which fit your bill (locally compact and even metris... | 4 | https://mathoverflow.net/users/2060 | 326306 | 140,472 |
https://mathoverflow.net/questions/326211 | 9 | Let me preface this by saying I'm not someone who has every studied mathematical logic or philosophy of math, so I may be mangling terminology here (and the title is a little tongue in cheek).
I (and probably most of us) would define a linear subspace, $W$ of $\mathbb{Q}^n$ to be a non-empty set of vectors that is cl... | https://mathoverflow.net/users/127803 | Constructivist defininition of linear subspaces of $\mathbb{Q}^n$? | Your question almost answers itself, but this may not be so obvious. So I will try to give a short and clear answer.
Regarding 1) of your question:
As a mathematician well-schooled in intuitionistic and constructive mathematics, I would always consider more than one constructive notion of 'linear subspace'.
Your ... | 10 | https://mathoverflow.net/users/101577 | 326316 | 140,475 |
https://mathoverflow.net/questions/326313 | 3 | Let $A$ be an $n$-by-$n$ matrix with integer entries whose absolute values are bounded by a constant $C$. It is well-known that the entries of the inverse $A^{-1}$ can grow exponentially on $n$. (See the replies to <https://math.stackexchange.com/questions/1146929/estimations-for-the-size-of-the-biggest-entry-in-an-inv... | https://mathoverflow.net/users/136932 | Bounding entries of the inverse of a matrix with bounded entries | Christian Remling's answer to this question: [Smallest non-zero eigenvalue of a (0,1) matrix](https://mathoverflow.net/questions/157472/smallest-non-zero-eigenvalue-of-a-0-1-matrix) indicates that the answer is that the entry can grow super-exponentially.
| 4 | https://mathoverflow.net/users/11142 | 326328 | 140,479 |
https://mathoverflow.net/questions/322181 | 27 | If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x\_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x\_o$. It is not particularly difficult to prove the following counterpart of this result for Lipschitz mappings (very nice exercise). Recall that by the Rademache... | https://mathoverflow.net/users/121665 | Rademacher theorem | While I don't have a reference, let me add another perspective. The following result is proved e.g. in the book *Functions of Bounded Variation and Free Discontinuity Problems* by Ambrosio, Fusco and Pallara (see the proof of Theorem 2.16).
>
> **Theorem.** Let $n\ge k$. If $f\_j\to f\_\infty$ is a converging seque... | 6 | https://mathoverflow.net/users/36952 | 326333 | 140,481 |
https://mathoverflow.net/questions/326334 | 6 | Are there good books that cover the history of math and mathematical science (ex. physics, chemistry, computer science) PhD programs in the Occident? My primary motivation is to figure out how the PhD system developed into what it is today.
**Note:** I will probably start a PhD focused on theoretical neuroscience in... | https://mathoverflow.net/users/56328 | Books on the History of math research at European universities | The classic reference is [The History of Mathematics in Europe from the Fall of Greek Science to the Rise of the Conception of Mathematical Rigour](https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.19260060413) by Sullivan (1925).
Do note that Ph.D.'s in mathematics were late arrivals in some countries. In the UK... | 5 | https://mathoverflow.net/users/11260 | 326339 | 140,482 |
https://mathoverflow.net/questions/326338 | 1 | Is there an algorightm generating all digraphs with $n$ edges up to isomorphism whose underlying graph is not a tree? For example, for $n=3$, there are only two such digraphs, representable as $\text{Digraph}({[1,2],[2,3],[3,1]})$ and $\text{Digraph}({[1,2],[2,3],[1,3]})$.
| https://mathoverflow.net/users/132052 | Algorithm generating digraphs | There are algorithms that will directly enumerate non-isomorphic graphs, without duplicates, e.g. `geng` available as a part of [nauty](http://users.cecs.anu.edu.au/~bdm/nauty/) and it also have procedures to generate non-isomorphic digraphs with a fixed underlying graph,
one is `directg` (see chapter 15 of [nauty man... | 3 | https://mathoverflow.net/users/11100 | 326341 | 140,483 |
https://mathoverflow.net/questions/326104 | 4 | Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.
| https://mathoverflow.net/users/126532 | Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises? | These are the two oldest US geometry texts with unsolved exercises:
 
[Elements of geometry; with practical applications to mensuration](https://archive.org/details/elementsofgeomet00greerich/page/n6) Greenleaf, ... | 3 | https://mathoverflow.net/users/11260 | 326342 | 140,484 |
https://mathoverflow.net/questions/326232 | 2 | Let $f(x)=x \psi(x+1)$, where $\psi$ is the digamma function. Define
$$g(x)=(f(ax)+f((1-a)x)-f(x))-(f(ax+by)+f((1-a)x+(1-b)y)-f(x+y)),$$
where $0\le a,b\le 1$ and $x,y\ge 0$.
How to show that $g(x)$ is increasing in $x$ on $[0,\infty)$? Thank you!
---
Here is what I have tried. Let $h(x,y)=f(ax+by)+f((1-a)x+(1... | https://mathoverflow.net/users/33278 | A challenging inequality that involves the digamma function and polygamma functions | Max reduced the problem to proving that $(xf'')'\geqslant 0$. We have $f=x\psi(x+1)=1+x\psi(x)$, so $f''=(x\psi(x))''=x\psi''+2\psi'$, $f'''=x\psi'''+3\psi''$,$(xf'')'=xf'''+f''=x^2\psi'''+4x\psi''+2\psi'$. Exactly this guy is proved to be non-negative in the recent clever [MO answer](https://mathoverflow.net/questions... | 3 | https://mathoverflow.net/users/4312 | 326359 | 140,490 |
https://mathoverflow.net/questions/326361 | 2 | I was wondering if the reverse Loomis-Whitney inequality holds for general functions:
Let $n\geq 2$. Let $(X\_i,\mu\_i)$, $1\leq i\leq n$ be measure spaces. Write $x=(x\_1,\dots,x\_n)$ and for each $1\leq i\leq n$, write $$
\pi\_i(x)=x\_i'=(x\_1,\dots,x\_{i-1},x\_{i+1},\dots,x\_n)\in X'\_i:=X\_1\times \cdots X\_{i-1... | https://mathoverflow.net/users/111012 | Reverse Loomis-Whitney Inequality for functions | This conjecture is trivially true for $n=2$, but false for any $n\ge3$. Indeed, take any $n\ge3$ and for all $i$ let $X\_i=\mathbb R$ and $\mu\_i=\lambda$, the Lebesgue measure. Next, let
$$f\_i(x'\_i):=\prod\_{j:\ j\ne i}1\_{i<x\_j<i+1}.
$$
Then
$$\prod\_i f\_i(x'\_i)=\prod\_i\prod\_{j:\ j\ne i}1\_{i<x\_j<i+1}
=\pro... | 1 | https://mathoverflow.net/users/36721 | 326393 | 140,497 |
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