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 |
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https://mathoverflow.net/questions/345894 | 4 | Let $H$ be a separable Hilbert space. Let $A \subseteq B(H)$ be a finitely generated unital algebra. Let $M$ be the strong operator topology closure of $A.$
Let $B\_A$ be the closed ball in $A$ and $B\_M$ be the closed ball in $M.$
Is $B\_M$ the strong operator topology closure of $B\_A?$ (More likely, is there a count... | https://mathoverflow.net/users/32470 | The ball formulation of the Kaplansky density theorem in nonselfadjoint algebras | W. R. Wogen, Some counterexamples in nonselfadjoint operator algebras, Ann. Math. 126 (1987), 415-427, constructs an operator $T$ for which the WOT and weak\* closures of the algebra generated by $T$ and $I$ are distinct. Since the WOT closure equals the SOT closure (and WOT and weak\* agree on bounded sets), this give... | 5 | https://mathoverflow.net/users/23141 | 345898 | 146,583 |
https://mathoverflow.net/questions/345899 | 6 | Suppose $f$ is a weight $k$ cuspidal Hecke eigenform on $\Gamma\_0(N)$. Then $f(2z)$ is a weight $k$ cuspform on $\Gamma\_0(2N)$.
Is it possible that $f(z)$ and $f(2z)$ can be orthogonal (regarded as forms on $\Gamma\_0(2N)$)? That is, can the Petersson inner product $\langle f(z), f(2z) \rangle = 0$, where the produ... | https://mathoverflow.net/users/14508 | Can the Petersson inner product $\langle f(z), f(2z) \rangle$ be zero? | Yes, the Petersson inner product can be zero. In my paper ["Explicit bounds for sums of squares](http://users.wfu.edu/rouseja/cv/sumsofsqrs.pdf) (see Lemma 5) I show that if $f$ is a newform of level $N$ and $p$ is a prime that does not divide $N$, then
$$
\langle f(z), f(pz) \rangle = \frac{a(p)}{p^{k-1} (p+1)} \lang... | 7 | https://mathoverflow.net/users/48142 | 345902 | 146,585 |
https://mathoverflow.net/questions/345838 | 21 | Grothendieck and Dieudonné prove in $EGA\_{II}$ (Proposition 5.5.5.(ii), page 105) that if $f:X\to Y, g:Y\to Z$ are projective morphisms of schemes and **if** $Z$ is separated and quasi-compact, or **if** the underlying topological space $\operatorname {sp}(Z)$ is noetherian, then the composition $g\circ f:X\to Z$ is a... | https://mathoverflow.net/users/450 | Can you give an example of two projective morphisms of schemes whose composition is not projective? | Here is a locally Noetherian separated counterexample. I also give some motivation for this construction afterwards.
**Definition.** Let $Z$ be an infinite chain of affine lines: $Z = Z\_1 \amalg\_{p\_1} Z\_2 \amalg\_{p\_2} \ldots$, where $Z\_i \cong \mathbf A^1\_{\mathbf C}$ and $p\_i$ is the point $1$ in $Z\_i$ and... | 17 | https://mathoverflow.net/users/82179 | 345922 | 146,588 |
https://mathoverflow.net/questions/345896 | 10 | From [this](https://mathoverflow.net/a/127412/148514) answer I learned that the coefficient ring $MSO^{\*}[1/2]$ of oriented bordism with 2 inverted supports an odd formal group law and is infact the universal such ring. Is there a reference/proof for this fact?
As motivation, I should mention that I'm trying to pro... | https://mathoverflow.net/users/148514 | Formal group law for oriented bordism | One basic point is as follows. The usual inclusions $U(n)\to O(2n)$ give rise to a map $BU\to BSO$ of $E\_\infty$ spaces and then a map $MU\to MSO$ of ring spectra, which gives a complex orientation of $MSO$. Note that a map $X\to MSO(n)$ of spaces represents a class in $MSO^n(X)$. The orientation class $x\in MSO^2(\ma... | 5 | https://mathoverflow.net/users/10366 | 345932 | 146,591 |
https://mathoverflow.net/questions/345921 | -1 | Define the Volterra operator $V$ on $C\_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int\_0^{\cdot} f(s)ds.
$$
Is there an example of an ergodic and $V$-invariant Borel probability measure $\mu$ on $C\_0([0,1])$?
| https://mathoverflow.net/users/36886 | Invariant ergodic measure Volterra operator | Yes - the delta measure on the identically 0 function (and this is the only one).
| 1 | https://mathoverflow.net/users/8588 | 345939 | 146,593 |
https://mathoverflow.net/questions/345909 | 1 | Let $n$ be a positive integer. Is the function $F\_n=\frac{\Gamma^{(n)}}{\Gamma^{2n}}$ entire on $\mathbb C$? If yes, is $F\_n(0)$ a ramarkable value?
Following <https://en.wikipedia.org/wiki/Digamma_function#Infinite_product_representation>, it is the case for $n=1$.
| https://mathoverflow.net/users/33128 | higher derivatives of gamma function | Yes, it is entire: $\Gamma$ has no zeros and only simple poles. At a pole of $\Gamma$, $\Gamma^{(n)}$ has a pole of order $n+1$, while the denominator has a pole
of order $2n$. Therefore the ratio has no poles. We also have $F\_n(0)=0$ when $n\geq 2$ which is of course a remarkable value.
| 4 | https://mathoverflow.net/users/25510 | 345941 | 146,595 |
https://mathoverflow.net/questions/345942 | 3 | Let $X$ and $Y$ be two smooth complex projective varieties. Then they are also symplectic manifolds. We know Gromov-Witten (GW) invariants are symplectic invariants. That means if there exists a a bijective symplectomorphism $f: X \to Y$, then GW invariants of $X$ and $Y$ are the same.
Now suppose $f: X \to Y$ is an... | https://mathoverflow.net/users/146366 | Are Gromov-Witten invariants birational invariants? | On a complex projective variety, Gromov-Witten invariants can be interpreted as virtual counts of curves, so they are biregular invariants.
However, they are *not* birational invariant in general. The behaviour of Gromov-Witten invariants under an arbitrary birational modification is in fact rather subtle.
For mor... | 2 | https://mathoverflow.net/users/7460 | 345944 | 146,596 |
https://mathoverflow.net/questions/345905 | 4 | Let $G$ be a connected Lie group. Then by a theorem of Cartan there is a diffeomorphism
$$
G \cong K \times \mathbb{R}^n
$$
where $K$ is a maximal compact subgroup of $G$. Now, let $M$ be a homogeneous manifold. In other words, there exists a Lie group $G$ acting transitively on $M$. Is it true that $M$ deformation ret... | https://mathoverflow.net/users/148524 | Homogeneous manifold deformation retracts onto compact submanifold | Mostow-Karpelevich theorem says that if $G/G^\prime$ is a homogeneous space where $G$, $G^\prime$ are Lie groups with finitely many connected components, and maximal compact subgroups $K\supset K^\prime$, respectively, then $G/G^\prime$ is a vector bundle over $K/K^\prime$. In fact, it is a homogeneous $K$-vector bundl... | 4 | https://mathoverflow.net/users/1573 | 345945 | 146,597 |
https://mathoverflow.net/questions/345936 | 1 | A *clutter* is a pair $C=(V,E)$ where $V\neq\emptyset$ is a set, and $E\subseteq {\cal P}(V)$ such that no member of $E$ is included in another member of $E$. A *matching* in $C$ is a collection of pairwise disjoint members of $E$. Zorn's Lemma implies that every matching is contained in a maximal matching (with respec... | https://mathoverflow.net/users/8628 | Clutters with no maximum-size matchings | Yes, this is possible. For each prime $p$ and $c \in \{0,1, \dots, p-1\}$ let $A\_{c,p}=\{c+kp \mid k \in \mathbb{Z}\}$. Clearly, the set of all $A\_{c,p}$ is a clutter $\mathcal C$ with ground set $\mathbb{Z}$. If we fix $p$, then the set of $A\_{c,p}$ is a matching of size $p$. Since there are infinitely many primes,... | 5 | https://mathoverflow.net/users/2233 | 345949 | 146,598 |
https://mathoverflow.net/questions/345915 | 5 | Given an $n \times n$ random matrix $\mathbf{Z}$ with each entry i.i.d. $\mathcal{N} (0,1)$, what is $\mathbb{E} [\max\_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ as $n \to \infty$? If this is too much to ask, are there any good known upper or lower bounds?
| https://mathoverflow.net/users/148528 | What is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ for a random Gaussian matrix $Z$? | This gives a lower bound of $\frac{4}{3\sqrt{\pi}}n^{3/2}+O(\sqrt{n})$ and an upper bound of $2n^{3/2}+O(\sqrt{n})$. Note: in the arguments below, I messed up the constants somewhere, so I'm not sure what the correct constants they should give is. The arguments should still work.
To get a lower bound, we can try to b... | 5 | https://mathoverflow.net/users/47135 | 345952 | 146,599 |
https://mathoverflow.net/questions/345943 | 1 | This is a follow-up to [this question](https://mathoverflow.net/questions/345921/invariant-ergodic-measure-volterra-operator).
Define the Volterra operator $V$ on $C\_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int\_0^{\sqrt{\cdot}} f(s)ds.
$$
Is there an example of an *and locally positive* ergodi... | https://mathoverflow.net/users/36886 | Invariant ergodic measure Volterra operator on Continuous Functions | The only $V$-invariant probability measure is the delta concentrated on the origin. A quick way to see it is to look at the conjugate of $V$ with the multiplication operator $M$ by $e^x$. Indeed we have for any $f\in C\_0$ and $x\in[0,1]$
$$\big|M^{-1}VMf(x)\big|=\Big|e^{-x}\int\_0^{\sqrt x}f(s)e^s ds\Big|\le e^{-x}\... | 2 | https://mathoverflow.net/users/6101 | 345965 | 146,602 |
https://mathoverflow.net/questions/345783 | 4 | Let $T$ be a theory and let $I$ be an imaginary sort of $T$. We'll let $T\_I$ denote the induced theory on $I$, by which I mean specifically the theory of all $\varnothing$-definable relations on $I$.
Suppose that
1. for all models $\mathfrak{I} \models T\_I$ there is an $\mathfrak{M} \models T$ such that $\mathfra... | https://mathoverflow.net/users/83901 | A Beth definability style result for imaginaries? | Your question lies in the realm of relative categoricity (over a predicate), which is related to the theory of stability over a predicate. Here the predicate in question is the imaginary sort $I$.
You can read about this in Section 12.5 of Hodges's (longer) *Model Theory*. Using Hodges's terminology, your condition ... | 3 | https://mathoverflow.net/users/2126 | 345968 | 146,603 |
https://mathoverflow.net/questions/345925 | 2 | Let $A\in\mathbb{R}^{m\times n}$ be a full column rank matrix. Then there exists a left inverse $A^+$ of $A$. Let $w\in \mathbb{R}^n$ be a vector. Is there a closed-form solution for the following problem?
$$
\begin{aligned}
\min\limits\_{A^+} \ & \|{A^+}^Tw\|\_1\\
\text{s.t.} \ & A^+A= I
\end{aligned}
$$
| https://mathoverflow.net/users/148531 | Is there a closed-form solution for this problem? | If $B$ is one left inverse of $A$, then $B+X$ is a left inverse of $A$ (where $X$ is $n \times m$) iff
$X A = 0$, i.e. the restriction of $X$ to $\text{Ran}(A)$ is $0$.
Of course if $A$ is surjective, there is no choice: $X$ must be $0$, so let's suppose it is not. We may also assume $w \ne 0$.
We can choose $X$ so t... | 2 | https://mathoverflow.net/users/13650 | 345972 | 146,605 |
https://mathoverflow.net/questions/345884 | 11 | Basically I want to know whether the sum being discrete uniform effectively forces the two component random variables to also be uniform on their respective domains.
To be a bit more precise:
Suppose we know $X$ and $Y$ are independent and
$$ X+Y \sim UNIF({1, \dots , n})$$
Does this necessarily imply that bot... | https://mathoverflow.net/users/148509 | If the sum of two independent random variables is discrete uniform on $\{a, \dots,a + n\}$, what do we know about $X$ and $Y$? | 1. Proof of the claim
---------------------
The following lemma shows that $X$ and $Y$ may also be regarded as integer-valued random variables in OP's scenario.
>
> **Lemma.** Assume that $X$ and $Y$ are independent random variables. Suppose that there exists a finite set $S\subset\mathbb{R}$ satisfying
> $$ \ma... | 6 | https://mathoverflow.net/users/15602 | 345976 | 146,608 |
https://mathoverflow.net/questions/344345 | 9 | Are there any relationship between the scalar curvature and the simplicial volume?
The simplicial volume is zero (positive) on Torus (Hyperbolic manifold) and those manifolds does not admit a Riemannian metric with positive scalar curvature. What do we know about the simplicial volume of a Riemannian manifold with p... | https://mathoverflow.net/users/90512 | Does positive scalar curvature imply vanishing of the simplicial volume on a closed Riemannian manifold? | In a [preliminary version](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf) of what would become Gromov's "A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature", he writes on page 88:
>
> Neither is one able to prove (or disprove) that manifolds with positive ... | 8 | https://mathoverflow.net/users/4362 | 345986 | 146,612 |
https://mathoverflow.net/questions/345981 | 0 | Let the sequence $u\_n\in L^2(0,\infty)$ weakly converges to $u\in L^2(0,\infty)$. What can we say about the corresponding Laplace transforms $U\_n(s)$ and $U(s)$?
1. $U\_n(s)$ converges point-wise to $U(s)$ for almost all $s>0$.
2. The convergence in (1) but also uniform.
| https://mathoverflow.net/users/113264 | Weak continuity under Laplace transform | The notation in the question for Laplace transform should be improved for clarity.
1. The Laplace transforms $L(u\_n)(s)=\int\_0^\infty e^{-sx}u\_n(x)\, dx$ converge to $L(u)(s)$ pointwise for each $s>0$ by the definition of weak convergence.
2. The convergence need not be uniform. E.g. take $u(x)=x/(1+x^2) \in L^2(0... | 1 | https://mathoverflow.net/users/7691 | 345990 | 146,615 |
https://mathoverflow.net/questions/345314 | 5 | Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see <https://elibm.org/article/10011880>) says that (under certain assumptions on the base field) the homotopy fixed points of the action of $C\_2$ on the motivic algebraic $K$-theory spectrum $KGL$ give the $\eta$-completio... | https://mathoverflow.net/users/2191 | Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete? | If I understand the question correctly, you're asking if there are base fields $k$ of characteristic not $2$ and such that $k[\sqrt{-1}]$ has a finite $\mathbb{Z}/2$-cohomological dimension (the assumptions used in the above mentioned paper) for which the map $KQ \to KGL^{hC\_2}$ is not an equivalence. Note that by the... | 3 | https://mathoverflow.net/users/51164 | 345994 | 146,617 |
https://mathoverflow.net/questions/345826 | 4 | I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question.
Let $m$ be the Lebesgue measure in $\mathbb{R}$. A measurable set $E\subseteq\mathbb{R}$ has **density** $d$ at $x$ if $$\lim\_{h\to 0} \frac{m(E\cap [x-h, x+h])}{2h} $$ exists and equal... | https://mathoverflow.net/users/138770 | Question about additive subgroups of the real line and the density topology | I think that below I manage to answer the first of your questions. I will be very grateful for verification of this argument. Unless it is correct, I will delete this answer.
I denote Lebesgue outer measure by $m^\*$ and $\sigma$-algebra of Lebesgue measurable sets by $\mathcal{L}$. Let $\mathcal{B}(\mathbb{R})$ be t... | 2 | https://mathoverflow.net/users/147687 | 345997 | 146,619 |
https://mathoverflow.net/questions/156690 | 7 | Let $(X\_t, t \in \mathbb{N})$ be a martingale, and let $a \leq b \leq T \in \mathbb{N}$ be constants. Is there something like Doob's inequality for $\mathbb{E} \sup\_{a \leq t \leq b} X\_t(X\_T-X\_t)$, i.e. is it possible to bound this supremum by something involving just a deterministic variance? I am hoping for one ... | https://mathoverflow.net/users/17883 | Doob's inequality for martingale "convolution" | Assume that $X\_t$ have independent and centered increments, but not necessarily identically distributed.
Let $D\_i=X\_i-X\_{i-1}$ for $i\geqslant a+1$ and $D\_a=X\_a$. Let
$
S\_t=X\_t\left(X\_T-X\_t\right).
$
Then $$S\_t=\left(\sum\_{i=a}^tD\_i\right)\left(\sum\_{j=t+1}^TD\_j\right)$$
and
\begin{align}
S\_{t+1}-S\_... | 1 | https://mathoverflow.net/users/17118 | 346009 | 146,622 |
https://mathoverflow.net/questions/345924 | 3 | The topic of Bousfield localizations has a lot of literature which has on most of the occasion some tameness assumption on the presentability of the model category. Recently I have been trying to **avoid any tameness** assumption but I can't avoid a couple of results about localizations.
Thus, the content of this qu... | https://mathoverflow.net/users/104432 | Transfer model structures, reflective subcategories and Bousfield localizations | So far, the best results that I managed to find in the direction of my questions appear both in some paper of Boris Chorny and his collaborators.
---
**1st. Theorem, A non-functorial Bousfield-Friedlander localization**,
This result appears as **A.8** in the paper below, it is a bit too long to be cast in this ... | 1 | https://mathoverflow.net/users/104432 | 346030 | 146,627 |
https://mathoverflow.net/questions/346000 | 0 | In [this note](http://math.uchicago.edu/~amathew/pi1.pdf) of Akhil MATHEW, when he proves the fundamental group of a smooth projective curve over a algebraic closed field $k$ of characteristic $0$ admits $2g$ topological generators, there are two sentences in the proof of theorem 4.5(page 16):
*"Indeed, by “noetheri... | https://mathoverflow.net/users/nan | Fundamental group of a smooth projective curve of char $0$ | Regarding the second question, the following is true.
>
> Let $k$ be a field and let $k'\subset k$ be a subfield. Then a variety over $k'$ is smooth and proper iff its base change to $k$ is.
>
>
>
Proof. In one direction it is enough to note that an arbitrary base change of a smooth (resp. proper) morphism is ... | 0 | https://mathoverflow.net/users/nan | 346033 | 146,629 |
https://mathoverflow.net/questions/344977 | 4 | TL;DR:
Given representations $D,\Lambda$ of subgroups $K,Q$ of a Lie group $G$, is it true that every intertwining operator $T$ between the resulting induced representations of $G$ can be written
$$
(T\varphi)(g) = \int\_G t(g^{-1} g')\varphi(g')\,dg'
\tag1
$$
for some function $t$ on $G$?
---
LONG VERSION:
... | https://mathoverflow.net/users/137120 | A morphism intertwining two induced representations | I have now found a remarkably simple proof of the fact that the variables $\,g\,$ and $\,g^{\,\prime}\,$ enter $\,t(g\,,\, g^{\,\prime})\,$ through convolution:
$$
t(g\,,\, g^{\,\prime})\;=\;t(g^{-1}\, g^{\,\prime})~~.\qquad\qquad\qquad (\*)
$$
To prove this, we should use the definition of an intertwiner,
$$
U\_{g... | 1 | https://mathoverflow.net/users/137120 | 346035 | 146,630 |
https://mathoverflow.net/questions/346026 | 2 | I came across the following problem while doing a piece of research on automata theory.
Suppose we have a probability space $(\Omega, \mathcal{F}, \mu)$, where $\Omega$ is a set, $\mathcal{F}$ is a $\sigma$-algebra, and $\mu$ is a probability measure. Suppose we have **non-negative bounded** function $f:\Omega\righta... | https://mathoverflow.net/users/75549 | Does bounded integral over sequence of subsets of $X$ whose union is $X$ imply bounded integral over X? | Set $\Omega = [0,1)$, $\mu$ the Lebesgue measure, $f = \tfrac{a N}{2} \times \mathbb{1}\_{(0,1/2)}$ for an arbitrarily large integer $N$, and
$$ A\_k = \bigl([\tfrac{1}{2}, 1) \setminus [\tfrac{N+k-1}{2N}, \tfrac{N+k}{2N})\bigr) \cup [\tfrac{k-1}{2N}, \tfrac{k}{2N}) ,$$
where $k = 1, 2, \ldots N$. Then the union of $A\... | 3 | https://mathoverflow.net/users/108637 | 346040 | 146,631 |
https://mathoverflow.net/questions/346050 | 2 | It is not known whether there is a largest prime in the Fibonacci sequence, but of course there are quite a few primes.
Similarly, the Lucas sequence starting with $L\_1=1, L\_2 = 3$ comes to a prime at $L\_4 = 7$. On the other hand, for $x\_1=3, x\_2 = 11$ all the $x\_n$ are composite until $x\_7 = 103$.
It seems ... | https://mathoverflow.net/users/82067 | Existence of some prime $x_k | k > 2$ in $x_n = x_{n-1} + x_{n-2}$ whenever $x_1$ is coprime to $x_2$ | As Gerhard Paseman said in his comment, there are counterexamples. This is discussed in A3 of UPINT. According to the third edition, the smallest known counterexample as of the printing (in 2003) is $x\_1= 8983542533631$ and $x\_2 = 248272649660939$ due to John Nicol. [This article by Knuth discusses the basic idea](ht... | 4 | https://mathoverflow.net/users/127690 | 346053 | 146,634 |
https://mathoverflow.net/questions/346055 | 1 | Let $X$ be an noetherian, affine, isolated of dimension at least $2$ and $\pi:\widetilde{X} \to X$ a resolution of singularities. Let $\mathcal{E}$ be a locally-free sheaf on $\widetilde{X}$ such that $\pi\_\*\mathcal{E}$ is a free $\mathcal{O}\_X$-module. Is $\mathcal{E}$ necessarily trivial i.e., a direct sum of copi... | https://mathoverflow.net/users/58203 | Pushforward of locally free sheaves and resolution of singularities | No. If $X$ is the cone over a normal rational curve of degree $d > 1$ and $\tilde{X}$ is its blowup at the vertex with the exceptional curve $E$ then $\pi\_\*\mathcal{O}(nE) = \mathcal{O}$ for any $n \ge 0$.
| 3 | https://mathoverflow.net/users/4428 | 346059 | 146,639 |
https://mathoverflow.net/questions/346081 | 7 | Let $(X,\omega)$ be a symplectic manifold and $L\subset X$ be a Lagrangian subspace.
Let $\mu\_L:H\_2(X,L;\mathbb{Z})\to \mathbb{Z}$ be the *Maslov index*
homomorphism.
**Usual hypothesis**
Recall that $L$ is said to be *monotone*, if there exists $c>0$ such that the following identity holds for all $\beta\in \pi\... | https://mathoverflow.net/users/99042 | Lagrangian intersection Floer homology: understanding some assumptions | When you try and prove that $d^2=0$ ($d$ being the Floer differential) you need to look at the boundary of the moduli space of index 2 J-holomorphic strips with one boundary on $L\_0$, one on $L\_1$. Certainly one component of the boundary will consist of "broken strips", i.e. pairs of index 1 strips with a common asym... | 9 | https://mathoverflow.net/users/10839 | 346092 | 146,649 |
https://mathoverflow.net/questions/346069 | 5 | My question is about existing of basis of club filter club($\omega\_1$) with cardinality $c$. Does it exist?
| https://mathoverflow.net/users/118366 | Club filter basis in $\omega_1$ | That's independent of ZFC.
On the one hand, it's consistent with ZFC that $2^{\aleph\_1}=\mathfrak c$, in which case the whole club filter on $\aleph\_1$ has cardinality $\mathfrak c$.
On the other hand, the continuum hypothesis is consistent with ZFC and implies that the club filter has no basis of size $\mathfrak... | 12 | https://mathoverflow.net/users/6794 | 346096 | 146,651 |
https://mathoverflow.net/questions/346052 | 7 | Since I am from a different mathematical field and couldn't find it: Is there something which would be best called an Arzela-Ascoli version for the $L\_p$-norm, namely:
Let $X,Y$ be two nice measurable/normed spaces (i.e. compact, locally compact, hausdorff and whatever properties you might want), e.g. $X$ a closed i... | https://mathoverflow.net/users/99554 | Arzela-Ascoli for L_p-norm | For your interest in a minimal $f$, you might want to read a beginners textbook on [Sobolev-Spaces](https://en.wikipedia.org/wiki/Sobolev_space) and the calculus of variations, especially on the [direct method](https://en.wikipedia.org/wiki/Direct_method_in_the_calculus_of_variations), which is all about this. The begi... | 2 | https://mathoverflow.net/users/51695 | 346098 | 146,652 |
https://mathoverflow.net/questions/346077 | 2 | Let $\psi(x,t)$ be a solution of the Schroedinger on the line
$$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$
One assumes that $\psi(x,0)$ "behaves well" as $|x|\to \infty$, e.g. square integrable on the line. If necessary one may impose some extra assumptions.
>
> For a fix... | https://mathoverflow.net/users/16183 | Estimate of a solution of Schroedinger equation for a free particle | For the free evolution specifically, in relation to [Mateusz's comment](https://mathoverflow.net/questions/346077/estimate-of-a-solution-of-schroedinger-equation-for-a-free-particle#comment866168_346077): on the Fourier side you can write the solution as (for $m = -1/2$; you can rescale/invert time to get other scaling... | 3 | https://mathoverflow.net/users/3948 | 346102 | 146,653 |
https://mathoverflow.net/questions/346095 | 1 | What is the salient uses of Fourier analysis in machine learning? What subjects of Fourier analysis have had more effect on machine learning?
Please mention the references.
| https://mathoverflow.net/users/84390 | What subjects of Fourier analysis have had more effect on machine learning? | Here is a [summary](http://www.mathcs.duq.edu/~jackson/tutorial.html) that could provide a good entry point into the literature:
>
> Since Linial, Mansour, and Nisan introduced the use of discrete
> Fourier analysis in machine learning in 1989, it has been a powerful
> tool for proving both positive and negative ... | 1 | https://mathoverflow.net/users/11260 | 346111 | 146,654 |
https://mathoverflow.net/questions/346061 | 1 | Suppose $X\_1,\dots,X\_n$ are jointly distributed random variables such that the random $n$-tuple $(X\_1,\dots,X\_n)$ is uniformly distributed on the set of $n$-tuples of nonnegative integers summing to $k$. Clearly each $X\_i$ has expected value $k/n$. Does the published literature provide nice formulas for the expect... | https://mathoverflow.net/users/3621 | Moments of a combinatorial ensemble of random variables | (You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature,
and a brief internet search didn't reveal anything.)
A simple way is to use the joint (probability) generating
function of $X\_1,\ldots,X\_n$.
It is easy to see that it can be writen as
$$\math... | 4 | https://mathoverflow.net/users/48831 | 346125 | 146,657 |
https://mathoverflow.net/questions/346105 | 1 | Let $\mathcal{V}= 0 \subset V\_1 \subset \cdots \subset V\_{n-1}\subset V\_n=V$, $\mathcal{W}=0 \subset W\_1 \subset \cdots \subset W\_{n-1} \subset W\_n=W$ be two flags. We say that $\mathcal{V}$ and $\mathcal{W}$ are transverse if $V\_i \cap W\_{n-i}=0$ for all $i$. Now consider the following theorem:
**Theorem** (... | https://mathoverflow.net/users/148672 | Schubert cycles that intersect generically transversely | By generic transversality, there is an open dense set $U$ in $\mathrm{GL}\_n$ such that $\Sigma\_a(V)$ is transversal to $\Sigma\_b(gV)=g\Sigma\_b(V)$. There is another open dense set $V$ in $\mathrm{GL}\_n$ such that $gV$ is transversal to $V$. It follows that there is a $g$ such that $\Sigma\_a(V)$ is transverse to $... | 1 | https://mathoverflow.net/users/124862 | 346132 | 146,662 |
https://mathoverflow.net/questions/346122 | 1 | Let $M$, $N$ be $n$-dimensional real manifolds. Does $M\times N$ admits a complex structure? If not, are there known condidtions ensuring that $M\times N$ admits a complex structure?
| https://mathoverflow.net/users/147073 | Complex structure on product of two $n$-dimensional real manifolds | I'm going to take $M$ and $N$ to be connected. Note that they cannot have boundary. Furthermore, as complex manifolds are orientable, $M$ and $N$ must also be orientable.
If $n = 1$, then $M, N \in \{\mathbb{R}, S^1\}$. Note that $\mathbb{R}\times\mathbb{R}$ has a complex structure, in fact, precisely two: $\mathbb{D... | 6 | https://mathoverflow.net/users/21564 | 346137 | 146,664 |
https://mathoverflow.net/questions/346085 | 1 | Consider the multivariate polynomial
$$f(x\_1,\ldots,x\_m)=mk\sum\_{i=1}^mx\_i^2-mk(k-1)\sum\_{i=1}^mx\_i-\left(\sum\_{i=1}^mx\_i\right)^2,$$
for integers $m,k\ge2$. We are looking for integral zeros of $f$ with $0\le x\_i\le k$. If $mk$ is square free, then it is easily seen that the only zeros of $f$ under the above ... | https://mathoverflow.net/users/136842 | Integral zeros of a multivariate polynomial | First, I will consider the case of $k\geq 4$.
Let $mk=aq^2$, where $q$ is an odd prime.
Since $f(x\_1,\dots,x\_m)=0$ implies that $mk$ divides $(\sum x\_i)^2$, we will look for a zero with $\sum x\_i = aq$. Then $f(x\_1,\dots,x\_m)=0$ will follow from the two equations:
$$\begin{cases}
\sum\_{i=1}^m x\_i = aq, \\
\su... | 1 | https://mathoverflow.net/users/7076 | 346139 | 146,665 |
https://mathoverflow.net/questions/346138 | 3 | Let $n$ and $k$ be natural numbers. I will consider [North-East lattice paths](https://en.wikipedia.org/wiki/Lattice_path#North-East_lattice_paths) (NE-paths) from $(0,0)$ to $(n,n)$ and encode these as strings of length $2n$ with letters $\mathsf{N}$ and $\mathsf{E}$. A *peak* of such a lattice path $\mathsf{P} = \mat... | https://mathoverflow.net/users/125013 | NE-Lattice paths from $(0,0)$ to $(n,n)$ with $k$ peaks | The formula in 2 is a very special case of a result of Richard Stanley's, though it certainly could be older. (I wouldn't be surprised if it can be found in MacMahon's work.) See, e.g., my paper *[A historical survey of P-partitions](https://arxiv.org/abs/1506.03508)*, section 7.2, for references. Just in case the conn... | 5 | https://mathoverflow.net/users/10744 | 346144 | 146,668 |
https://mathoverflow.net/questions/346153 | 21 | The category $\text{AffSch}\_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(\text{AffSch}\_S)$ of abelian sheaves on $\text{AffSch}\_S$ with respect to some Grothendieck topology. In fact, many de... | https://mathoverflow.net/users/101861 | Surmounting set-theoretical difficulties in algebraic geometry | Let me start by discussing a bit the option of having a large class of generators. You might be interested in the notion of **locally class-presentable**.
To be precise here, I need to be a bit set-theoretical, thus, let me start with an informal comment.
**Informal comment.** Indeed your category is locally class-... | 20 | https://mathoverflow.net/users/104432 | 346162 | 146,671 |
https://mathoverflow.net/questions/346160 | 2 | Let $f \in L^1(\mathbb{R}^n)$. It's obvious that if $\int\_R f = 0$ for all rectangles $R$ then $f = 0$ $a.e.$ since every open set is union of almost disjoint rectangles and consequently with zero integration. Then I came up with a question. Can rectangles be replace by balls? The answer is yes by elenebtary version o... | https://mathoverflow.net/users/133871 | If $\int_E f = 0$ for all $E$ the translation and dilation of $E_0$ then $f = 0 \text{ } a.e.$ | The finite measure case is quite simple. Set $E\_\lambda = -\lambda E\_0$ for $\lambda > 0$, and $g\_\lambda = \mathbb{1}\_{E\_\lambda}$. By the assumption, $f \* g\_\lambda = 0$, which implies that $\hat{f} \hat{g}\_\lambda = 0$, where $\hat f$ is the Fourier transform of $f$. However, $\hat{g}\_\lambda(\xi) = \lambda... | 2 | https://mathoverflow.net/users/108637 | 346169 | 146,675 |
https://mathoverflow.net/questions/346074 | 9 | This is a [cross-post](https://math.stackexchange.com/questions/3274711/can-we-recover-all-matrix-minors-from-some-of-them).
Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" **invertible** $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of $A$).
>
> Can ... | https://mathoverflow.net/users/46290 | Can we recover all $k$-minors of a square matrix from some of them? | It depends how you frame the question, but the answer is yes in some sense. Let $A$ be the $n \times n$ generic matrix with linear entries in $\mathbb{k}[x\_1, \ldots, x\_{n^2}]$. I denote by $I\_m$ the ideal generated by the $m \times m$ minors of $A$.
It has been proved by Bruns that there exists $q=n^2-m^2+1$ hom... | 7 | https://mathoverflow.net/users/37214 | 346173 | 146,676 |
https://mathoverflow.net/questions/346171 | 0 | Let $X\sim beta(\alpha,\beta)$ be a random variable and let $\tau\in(0,1)$.
Are there any known closed-form bounds (I'm specifically interested in lower bounds) on
$$
\mathbb E[X\ | X\le \tau]?
$$
| https://mathoverflow.net/users/47499 | Are there known bounds on the expectation of the truncated Beta distribution? | Mathematica 12.0 is your friend, answering it by
```
Mean[TruncatedDistribution[{-Infinity,\[Tau]},BetaDistribution[\[Alpha], \[Beta]]]]
```
Unfortunately the $\LaTeX$ form of the result
$$\frac{
\begin{array}{cc}
\{ &
\begin{array}{cc}
\frac{\alpha }{\alpha +\beta } & \tau \geq 1 \\
\frac{B\_{\tau }(\alpha +... | 2 | https://mathoverflow.net/users/35959 | 346177 | 146,677 |
https://mathoverflow.net/questions/346088 | -1 | Let $A \in \mathbb{R}^{m\times n}$ and $p,q \in \mathbb{R}^{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$. I am interested to prove the following:
$$ \|A\|\_{p}=\|A^T\|\_q$$
I have tried using Holder Inequality for vectors $Ax$ and $A^Ty$ and thereby mapping back to the original matrix norm using basic properties but I... | https://mathoverflow.net/users/148659 | Holder inequality for a general rectangular matrix | We have
$$
\begin{eqnarray}
LHS&=& \sup\_{x\in \mathbb{R}^m\backslash \{0\},y\in \mathbb{R}^n\backslash \{0\}} \frac{|x^TAy|}{\|x\|\_q\|y\|\_p}\\
&\leq& \sup\_{x\in \mathbb{R}^m\backslash \{0\},y\in \mathbb{R}^n\backslash \{0\}} \frac{ \|x\|\_q\|Ay\|\_p}{\|x\|\_q\|y\|\_p}\\
&=& \sup\_{x\in \mathbb{R}^m\backslash \{0\},... | 0 | https://mathoverflow.net/users/100908 | 346178 | 146,678 |
https://mathoverflow.net/questions/346161 | 3 | I asked this question on MSE [here](https://math.stackexchange.com/questions/3437400/does-clairaut-schwarz-theorem-hold-when-mixed-partial-derivatives-are-of-order-g). One person gave an answer but then he deleted it because my version of Clairaut-Schwarz theorem is stronger than his. I meant my version only requires t... | https://mathoverflow.net/users/99469 | Does this version of Clairaut-Schwarz theorem hold when mixed partial derivatives are of order greater than $2$? | Assume $f:X\to F$ has the partial derivatives $\partial\_m\partial\_{m-1}\dots\partial\_1f(x)$ at any point $x$ of the open rectangle $X:=\prod\_{1\le i\le n}]b\_i,c\_i[$.
For $u\in \mathbb{R}^n$, let $\delta\_u $ denote the finite difference $\delta\_uf(x):=f(x+u)-f(x)$, defined for $x$ and $x+u\in X$, and let $(e\_... | 2 | https://mathoverflow.net/users/6101 | 346181 | 146,679 |
https://mathoverflow.net/questions/346029 | 2 | Let $G$ be a finite group, with subgroups $A \leqslant H$. Is there an isomorphism of $N\_G A$-sets (or just sets)
$$ N\_G A / N\_H A \cong (G/H)^A ?$$
This dropped out of some calculations of Mackey functors, but I would like a more direct proof (or a counterexample).
| https://mathoverflow.net/users/35150 | Quotient of normalizers is the fixed points of a homogeneous space | The statement is false, as shown by the counter examples of LSpice and Neil Strickland.
| 0 | https://mathoverflow.net/users/35150 | 346183 | 146,681 |
https://mathoverflow.net/questions/346175 | 4 | [Jeff Smith's theorem](https://ncatlab.org/nlab/show/combinatorial+model+category#SmithTheorem) gives a simple criterion for the existence of a combinatorial model category. Is there a similar theorem for combinatorial left semi-model categories? I see two problems that occur when you try to apply the theorem in this c... | https://mathoverflow.net/users/62782 | Is there Jeff Smith's theorem for left semi-model structures? | Yes, and this is one of the main results in a paper I hope to put on arxiv very soon. I wrote about this result in a previous mathoverflow answer [here](https://mathoverflow.net/questions/300879/left-bousfield-localization-without-properness-what-is-known). You are right that the way to do it is to focus on maps betwee... | 5 | https://mathoverflow.net/users/11540 | 346185 | 146,682 |
https://mathoverflow.net/questions/345041 | 1 | I have a Markov Chain of $N$ states. Such states represent the energy levels in a molecule.
The states' connectivity is as follows:
1. States $j\in\{0,\ldots,N\}$ transition to $k\in\{\max(j-M,0),...,\min(j+M,N)\}\setminus j$, with $N\gg M$. The resulting transition matrix $\mathbf{P}$ has known terms $P\_{jk}$, an... | https://mathoverflow.net/users/148012 | Stationary distribution of Markov Chain with departure | As the comments state, the resulting Markov Chain must have equilibrium distribution
$$
\tilde\pi\_j = \delta(j-J)
$$
since any random walk through the Chain will eventually lead to states $j\ge L$, and then to $J$. This is justified by the [Perron-Frobenius Theorem](https://en.wikipedia.org/wiki/Perron%E2%80%93Frobeni... | 0 | https://mathoverflow.net/users/148012 | 346186 | 146,683 |
https://mathoverflow.net/questions/346017 | 3 | Consider a Unitary Modular Tensor Category constructed from the quantum group of some compact, simple and simply-connected Lie group $U\_q(G)$ at $q=e^{2\pi i/(k+h)}$ for some integer $k$. In general, the associator ("$F$-symbols", i.e., the pentagon equations) is very hard to compute explicitly. To the best of my know... | https://mathoverflow.net/users/131264 | F-symbols for compact Lie groups | We do The $B$ series at level 2 in our paper [classifying metaplectic fusion categories](https://arxiv.org/abs/1608.03762).
| 1 | https://mathoverflow.net/users/25642 | 346199 | 146,687 |
https://mathoverflow.net/questions/346198 | 11 | Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum\_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum\_{l=0}^k\binom{ak}{al}\biggr)$$
I was able to "simplify" the above expression for $a=1$:
$$\sum\_{k=0}^\infty\frac{x^k}{k!}\cdot2^k=e^{2x}$$
I also managed t... | https://mathoverflow.net/users/144262 | How can I simplify this sum any further? | You might be able to use the fact that
$$\sum\_{k=0}^\infty b\_{ak}=\sum\_{k=0}^\infty \left(\frac{1}{a}\sum\_{j=0}^{a-1} \exp\left(2\pi ijk/a\right)\right)b\_k.$$
For example, when $a=1$, taking $b\_k = \frac{x^k}{k!}\sum\_{\ell \ge 0} \binom{k}{\ell}$ yields
$$\sum\_{k=0}^\infty b\_{k}=\sum\_{k=0}^\infty \frac{x^k}{k... | 11 | https://mathoverflow.net/users/141766 | 346201 | 146,688 |
https://mathoverflow.net/questions/346190 | 1 | Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with pointwise addition as group law; here $C(\mathbb{R}^d,\mathbb{R}^d)$ ... | https://mathoverflow.net/users/36886 | Continuous semigroup homomorphism of composition to additive structure | The new version, for semigroups, is much easier: there is no such homomorphism $\varphi$, for purely algebraic reasons. Consider just the constant functions $c$. Since $cd=c$ in the semigroup, you must map $\varphi(d)=0$. But for any $f$, $fc$ is also constant ($=f(c)$), so $\varphi(f)+\varphi(c)=0$ and thus $\varphi(f... | 5 | https://mathoverflow.net/users/48839 | 346204 | 146,689 |
https://mathoverflow.net/questions/346044 | 2 | We have $k$ blocks of integer sequences $B\_1,\dots,B\_k$ where $B\_i$ is a sequence $$a\_{i,1},\dots,a\_{i,n\_i}$$ with $a\_{i,j}\leq a\_{i,j+1}$.
Denote the permutation matrix $M\_{\ell,\ell'}$ that merges $B\_\ell$ and $B\_{\ell'}$ (there are more than one sometimes).
Assume for every pair $\ell,\ell'$ in $\{1,\... | https://mathoverflow.net/users/136553 | Minimum local permutation data needed to globally merge locally sorted sequences? | 1) *Is it possible to choose $O(k\log k)$ matrices beforehand that will allow you to construct the global matrix M (sort all the elements of the $a\_{i, j}$) regardless of the values of $a\_{i, j}$?*
**No.**
Even in the case $n\_1 = n\_2 = \ldots = n\_k = 1$ (so in the setting of the usual sorting problem) if we do... | 1 | https://mathoverflow.net/users/88291 | 346210 | 146,692 |
https://mathoverflow.net/questions/346157 | 4 | Let $X\_{(k)}$ be the $k$th order statistic out of $n$ uniform $[0,1]$ random variables. Let $q$ be the location of the $p$ quantile of $X\_{(k)}$, i.e. $\Pr[X\_{(k)}\leq q] = p$. For small $p$, Is it true that $q = O(p^{1/k} \frac{k}{n})$?
| https://mathoverflow.net/users/148689 | The behavior of a uniform order statistic near zero | Your conjecture is true. More specifically,
\begin{equation\*}
q \lesssim 2C p^{1/k} \frac kn \tag{1}
\end{equation\*}
uniformly as
\begin{equation\*}
p\to0,\quad C\_1k\ge\ln n, \quad n-k\to\infty, \tag{1a}
\end{equation\*}
where $C\_1$ is any positive real constant and $C$ is any positive real constant such that
\... | 2 | https://mathoverflow.net/users/36721 | 346211 | 146,693 |
https://mathoverflow.net/questions/346213 | 8 | Let $M$ be a compact manifold without boudary and let $X\_{1},\ldots,X\_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space:
$$ W\_{X}^{1}(M)=\{f\in L^{2}(M)|X\_{j}f\in L^2(M)， 1\leq j\leq m\}.$$
We can prove that $W\_{X}^{1}(M)$ is a Hilbert space. **My question is**: Can we claim that $... | https://mathoverflow.net/users/98451 | Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$? | The density result is true for any family of vector fields with Lipschitz coefficients.
>
> **Theorem.** Let $X\_1,\ldots,X\_k$ be a system of vector fields with Lipschitz coefficient on a compact Riemannian manifold (with or without boundary), If $f\in L^p(M)$ and $X\_j\in L^p(M)$, $j=1,2,\ldots,k$, then there is... | 9 | https://mathoverflow.net/users/121665 | 346216 | 146,694 |
https://mathoverflow.net/questions/346172 | 2 | I asked this question in Mathematics StackExchange ([link](https://math.stackexchange.com/questions/3403704/partitioning-integers-into-two-parts-and-exploring-relationships-with-positional)) about a month ago, but I have received no answer. It is about the following problem:
>
> Problem:
> Are there sets $A,B$ o... | https://mathoverflow.net/users/144695 | Partitioning integers into two parts and exploring relationships with positional numeral systems | Here is one kind of crazy example: Start with $A,B$ any two finite sets so that all the sums $a+b$ are distinct. $A=B=\emptyset$ for example. Now consider the integers in order $0,1,-1,2,-2,\cdots$ one at a time. For each, $t$, if it can already be written as $a+b$ go to the next. Otherwise pick any huge $N$ and put $ ... | 2 | https://mathoverflow.net/users/8008 | 346222 | 146,695 |
https://mathoverflow.net/questions/346218 | 4 | Is there a model of $\mathsf{ZFC}$ such that for every cardinal $\beta > \aleph\_0$ there is a cardinal $\alpha < \beta$ such that $|{\cal P}(\alpha)|
>\beta$?
| https://mathoverflow.net/users/8628 | Model in $\mathsf{ZFC}$ such that ${\cal P}(\ldots)$ has "jumping property" | There cannot be such a model. Let for instance $\beta=\beth\_\omega$, the supremum of $\beth\_0=\aleph\_0,\beth\_1=2^{\beth\_0},\beth\_2=2^{\beth\_1},\dots$, for any $\alpha<\beta$ we have $\alpha\leq\beth\_n$ for some $n$, and hence $|\mathcal P(\alpha)|\leq\beth\_{n+1}<\beta$.
The exact same argument shows that thi... | 15 | https://mathoverflow.net/users/30186 | 346223 | 146,696 |
https://mathoverflow.net/questions/346220 | 4 | Chapter 1.2 of the HoTT book says this about eta-conversion:
>
> $$
> f \equiv (\lambda x . f(x)).
> $$
> This equality is the **uniqueness principle for function types**, because it shows that $f$ is uniquely determined by its values.
>
>
>
That "$f$ is uniquely determined by its values" seems like a stronge... | https://mathoverflow.net/users/148105 | Uniqueness principle for functions types in the HoTT book | It is less confusing to first see how things work for ordered pairs. Consider the following two rules:
1. **$\eta$-equality for pairs:**
$$\frac{u : A\times B}{u \equiv (\pi\_1 u, \pi\_2 u)}$$
2. **extensionality rule for pairs:**
$$\frac{v : A\times B \quad w : A\times B \quad \pi\_1 v \equiv \pi\_1 w \quad \pi\_2 v... | 7 | https://mathoverflow.net/users/1176 | 346236 | 146,703 |
https://mathoverflow.net/questions/225819 | 21 | There are several questions here on MO about the Cantor-Bernstein-Schröder ((C)BS) theorem, but I could not find answers to what arose to me recently.
Although I don't think I need to recall it here, the theorem states that if there are embeddings $i:A\hookrightarrow B$, $j:B\hookrightarrow A$ then there is a bijecti... | https://mathoverflow.net/users/41291 | How strong is Cantor-Bernstein-Schröder? | The paper [Cantor-Bernstein implies Excluded Middle](https://arxiv.org/abs/1904.09193) by Pierre Pradic and Chad E. Brown shows that Excluded Middle is equivalent to CBS in constructive set theory.
I don't understand Toposes very well, but I believe the same proof goes through in a Topos with NNO.
| 2 | https://mathoverflow.net/users/133533 | 346241 | 146,704 |
https://mathoverflow.net/questions/346167 | 4 | Let $M$ be a subspace of $C[0,1]$ isomorphic to $c\_0$.
QUESTION: Is it possible to find a normalized disjoint sequence $(f\_n)$ in $C[0,1]$ such that the distance of $f\_n$ to $M$ tends to $0$ as $n$ goes to $\infty$?
Arguments in favor:
1. If $1\leq p<\infty$, then the result is true for subspaces of $L\_p(0,... | https://mathoverflow.net/users/39421 | Copies of $c_0$ in $C[0,1]$ and disjoint sequences | Take sequences of clopen sets $M\_i$, $N\_i$, s.t. any two of the $N\_i$ have non empty intersection but any three have empty intersection, and the $M\_i$ are pairwise disjoint and disjoint from the $N\_i$. Let $f\_i$ be the characteristic function of $M\_i \cup N\_i$. This gives a counterexample in $C(\Delta)$, $\Delt... | 5 | https://mathoverflow.net/users/2554 | 346243 | 146,705 |
https://mathoverflow.net/questions/346227 | 1 | I came across the following while doing some related proof;
It seems easy to prove. $\quad$
We are in ${\mathbb{M}}\_n(\mathbb{C})$, $n>1$:
$1$) Given a unitary $n\times n$ matrix $U$, there is some permutation of the columns such that the modulus of each entry on the diagonal of the resulting matrix is $\le \dfr... | https://mathoverflow.net/users/121643 | Unitary condition | Consider a unitary matrix $U = (u\_{i,j})$. We will show that there is some permutation $\pi:[n] \to [n]$ such that $|u\_{i,\pi(i)}| \leq \sqrt{2}/\sqrt{n+1}$ for all $i$, which is sufficient to prove the first stated conjecture when $n \geq 3$, and when $n=2$ we can use the expression of $U$ as:
$$\left(\begin{array}{... | 2 | https://mathoverflow.net/users/95129 | 346246 | 146,707 |
https://mathoverflow.net/questions/346225 | 3 | Does anybody shed light on what is A. R. D. Mathias' idea that Bourbaki's $\tau$-calculus (Logically the same as Hilbert's $\varepsilon$-calculus) is not suitable for set theory, especially because of incompleteness of set theory? I have fully read Bourbaki's Theory of Sets and Mathias' two papers on bourbaki's system ... | https://mathoverflow.net/users/148738 | Why Bourbaki's epsilon-calculus is not suitable for set theory? | One model-theoretic problem with Bourbaki's system is that, for example, the $V\_{\alpha}$'s, $H\_{\alpha}$'s and $L\_{\alpha}$'s are not closed under $\tau$ in general - it is not sure even if $1\in V\_{\alpha}$, for a given infinite ordinal $\alpha$. The usual stuff on reflection and models of set theory becomes comp... | 12 | https://mathoverflow.net/users/9825 | 346249 | 146,708 |
https://mathoverflow.net/questions/346224 | 4 | Consider a polytope with a $2$-dimensional surface and the corresponding metric on this surface (coming from the embedding in $3$-dimensional Euclidean space). Intrinsically the metric is flat everywhere apart from the vertices of the polytope, where one has cone-like singularities if the angle sum does not equal $2\pi... | https://mathoverflow.net/users/115363 | Is every conformal manifold equivalent to a flat one with cone singularities? | The answer is yes, and there are several ways to prove it. The result can be restated as "on every Riemann surface there exists a flat conformal metric with conic singularities". In fact the singularities can be prescribed, the only condition is
that Gauss Bonnet holds. References: For compact surfaces:
MR1005085
Tr... | 4 | https://mathoverflow.net/users/25510 | 346260 | 146,713 |
https://mathoverflow.net/questions/346232 | 6 | Let $X$ be a topological space, $\kappa$ be a cardinal number, such that there exists a dense subset $A\subseteq X$ of cardinality $\kappa$ but there does not exist a dense subset $A'\subseteq X$ of cardinality less than $\kappa$.
Now, suppose that $X$ is a metric space which satisfies the above property for $\kappa... | https://mathoverflow.net/users/36886 | Extension of Baire's Theorem | Such a hypothetical Baire theorem is not true: for every cardinal $\kappa$ of uncountable cofinality and any cofinal subset $C\subset \kappa$ of cardinality $|C|=\mathrm{cf}(\kappa)$, the Hilbert space $\ell\_2(\kappa)$ of density $\kappa$ is a complete metric space that can be written as the union $\bigcup\_{\alpha\in... | 12 | https://mathoverflow.net/users/61536 | 346262 | 146,714 |
https://mathoverflow.net/questions/346251 | 13 | It is well known that if $M$ is a compact orientable $n$-dimensional manifold, then $[M, \mathbb{S}^n] \cong \mathbb{Z}$, i.e the maps are classified by their degree.
What is known about $[M, \mathbb{RP^n}]$ under the same hypotheses?
| https://mathoverflow.net/users/137622 | Homotopic classification of maps $M \to \mathbb{RP}^n$ where $M$ is a compact orientable $n$-dimensional manifold | This seems to have been worked out in the 1960s by Paul Olum, see Section 1 of
*Olum, P.*, [**Cocycle formulas for homotopy classification; maps into projective and lens spaces**](http://dx.doi.org/10.2307/1993740), Trans. Am. Math. Soc. 103, 30-44 (1962). [ZBL0135.23203](https://zbmath.org/?q=an:0135.23203).
Brie... | 16 | https://mathoverflow.net/users/8103 | 346269 | 146,715 |
https://mathoverflow.net/questions/346271 | 0 | Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M\_{\ge d}:=\oplus\_{\ell \ge d} M\_\ell$ the sub-module of $M$. Is $M\_{\ge d}$ again a projective $A$-module?
| https://mathoverflow.net/users/58203 | Projective modules and gradings | A counter example would be $A = k[x, y]$, graded by the total degree, and $M = A$, $d = 1$.
The module $M\_1$ is then the ideal of $A$ generated by $x$ and $y$. It is not projective.
Because:
Consider the surjection $A^2 \rightarrow M\_1$, sending $(u, v)$ to $xu + yv$. It's an exercise to show that this map does... | 2 | https://mathoverflow.net/users/76332 | 346276 | 146,716 |
https://mathoverflow.net/questions/346242 | 6 | Let $\{X\_i^n\}$ be a sequence of smooth compact Riemannian $n$-dimensional manifolds with boundary. Assume that this sequence has uniformy bounded below sectional curvature, and each $X\_i$ is geodesically locally convex near its boundary (the latter assumption is equivalent that $X\_i$ is an Alexandrov space). Assume... | https://mathoverflow.net/users/16183 | Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense | [The statement holds for a sequence of extremal susbsets (not necessary boundary).]
According to Theorem 1.2. in my "Applications of quasigeodesics and gradient curves",
$\partial X\_i\to \partial X$ as length spaces in the sense of Gromov--Hausdorff.
Then the same argument as in Burago--Gromov--Perelman shows that $... | 3 | https://mathoverflow.net/users/1441 | 346279 | 146,717 |
https://mathoverflow.net/questions/346237 | 2 | Let $X$ be a compact $n$-dimensional Alexandrov space with curvature bounded below. Let $\partial X$ denote its boundary in the sense of the theory of Alexandrov spaces.
>
> Is it true that if $\partial X\ne \emptyset$ then it has finite and positive $(n-1)$-Hausdorff measure? (The case $n=2$ is already interesting... | https://mathoverflow.net/users/16183 | Measure of the boundary of Alexandrov space | It can be proved by induction on $n$.
Base case $n=1$.
The step follows since gradient exponent is locally Lipschitz and it maps $T\_p(\partial X)=\mathrm{Cone}[\Sigma\_p(\partial X)]$ to a neighborhood of $p$ in $\partial X$.
It proves that $\partial X$ has finite $(n-1)$-Hausdorff measure.
The differential of gradi... | 4 | https://mathoverflow.net/users/1441 | 346281 | 146,718 |
https://mathoverflow.net/questions/346273 | 4 | I'm unsure if my question is advanced enough for this site, but let's see.
Let $\mathcal{C}$ be a locally cartesian closed category, so that it always has dependent products $\Pi\_f$, i.e., right adjoints to base change functors $f^{\ast}$, along with dependent sums $\Sigma\_f$, i.e., left adjoints to base change fun... | https://mathoverflow.net/users/115055 | Checking the functoriality of an expression involving dependent sum and product | You are right that $(-)^{\mathsf \Pi}$ is not functorial on the arrow category of $\mathcal{C}$. However, it is functorial on the category whose objects are arrows in $\mathcal{C}$ and whose morphisms are *pullback squares*, and I believe that's what the authors were referring to, since in their case $\tilde{U\_0}$ is ... | 10 | https://mathoverflow.net/users/49 | 346283 | 146,719 |
https://mathoverflow.net/questions/346099 | 0 | I recently asked for the original journal citation on Littlewood's result
$$ \lim\limits\_{n\to\infty}|\gamma\_n-\gamma\_{n-1}| =0 ~~,$$
wherein $\gamma\_n$ is an increasing sequence of the imaginary parts of the zeros the RZF in the upper complex half plane. Thank you very much to this great community for providin... | https://mathoverflow.net/users/148452 | Looking for a fine exposition of a result of Littlewood | The result you want is in article 9.12 of Titchmarsh. This is on page 191 of the first edition and page 224 of the new Heath\_Brown edition. This result is unconditional,
whereas Littlewood's best result assumes R.H. so is stronger.
| 0 | https://mathoverflow.net/users/104002 | 346285 | 146,721 |
https://mathoverflow.net/questions/345380 | 4 | Here $s=\sigma+it$ denotes the complex variable. We denote the Ramanujan's zeta function
$$\varphi(s)=\sum\_{n=1}^\infty\frac{\tau(n)}{n^s}$$
for $\Re(s)>7$, where $\tau(n)$ is the Ramanujan tau function.
While I was studying a video by Harper, a video from YouTube with title *The Riemann zeta function in short int... | https://mathoverflow.net/users/142929 | A principle around the Ramanujan's zeta function in short intervals | It seems, essentially, that you want to estimate the $L$-function
$L(s,\Delta)=\sum\_{n=1}^{\infty}\tau(n) n^{-s}=\prod\_p (1-\tau(p)p^{-s}+p^{11-2s})^{-1}$
as a short Euler product, in the sense that there exists a reasonably small integer $X(s)>0$ such that
$L(s,\Delta)\approx\prod\_{p\leq X(s)} (1-\tau(p)p^{-s... | 3 | https://mathoverflow.net/users/111215 | 346289 | 146,723 |
https://mathoverflow.net/questions/346268 | 2 | Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}\_1$, a Hirzebruch surface.
My confusion is two fold, I am having difficulty spotting which two torus invariant lines of $\mathbb{F}\_1$ are ... | https://mathoverflow.net/users/148757 | Lines on a toric cubic surface with a line of nodes | The cubic scroll in $\mathbb{P}^4$ is isomorphic to $\mathbb{F}\_1$ and its torus-invariant divisor has three line components and one conic component. The linear projection $\mathbb{P}^4 \dashrightarrow \mathbb{P}^3$ from a point lying in the linear span of the conic, but not on the conic itself, gives the normalizatio... | 2 | https://mathoverflow.net/users/4428 | 346296 | 146,726 |
https://mathoverflow.net/questions/346071 | 3 | Is the following slightly generalization of Corollary 20.17 in Hewitt and Ross Book (page 296) correct?
Let $A$ be a subset of a profinite group $G$ ( compact, Hausdorff, totally disconnected topological group) such that $A=A^{-1}$ and $\lambda(A)>0$, where $\lambda$ is the normalized Haar measure of $G$. Suppose tha... | https://mathoverflow.net/users/19075 | Continuous function defined by measurable sets | The answer is yes, and the assumption of profiniteness is not needed (see comments). This can be proved by approximating in $L^2$
the indicator function $1\_A$ by a continuous function $f$.
Using the fact that such $f$ must be uniformly continuous, any small perturbation $f\_g$ of $f$ is within a small $L^\infty$ e... | 2 | https://mathoverflow.net/users/18698 | 346304 | 146,729 |
https://mathoverflow.net/questions/346291 | 5 | All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the residual representation (or make that an assumption) and then infer automorphy of the original representation using defor... | https://mathoverflow.net/users/nan | Proving automorphy of the Galois representations of number fields without considering the residual representation | The canonical answer to that question is certainly the world of so called *converse theorems*, whose basic ideas go back to Hecke's remark that an holomorphic $L$-function satisfying a suitable functional equation should be automorphic. In the legendary paper
*Über die Bestimmung Dirichletscher Reihen durch Funktiona... | 7 | https://mathoverflow.net/users/2284 | 346307 | 146,730 |
https://mathoverflow.net/questions/346259 | 0 | Let us define a power Diophantine equation by 2 algebraic functions $f,g$ *(having different degree)* and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $l$ such that-
there are infinite integer solutions for the equation $f(x)+l=g(y)$. One may add more functions of differen... | https://mathoverflow.net/users/69301 | Different solution of power Diophantine equation based on constant term | I'll assume that by "algebraic functions" is meant polynomials with integer coefficients. If both $f$ and $g$ have degree at least two, and at least one of them has degree at least three, then, trivial cases aside, it's well-known that the equation has only finitely many integer solutions.
If both polynomials have d... | 1 | https://mathoverflow.net/users/3684 | 346309 | 146,731 |
https://mathoverflow.net/questions/346264 | 29 | [This article](https://nonperele.com/three-physicists-stumbled-upon-a-striking-mathematical-discovery/) describes the discovery by three physicists, Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago, and Peter Denton of Brookhaven National Laboratory, of a striking relati... | https://mathoverflow.net/users/7113 | Consequences of eigenvector-eigenvalue formula found by studying neutrinos | The OP asks about generalisations and applications of the formula in [arXiv:1908.03795](https://arxiv.org/pdf/1908.03795.pdf).
$\bullet$ *Concerning generalisations:* I have found an older paper, from 1993, where it seems that the same result as in the 2019 paper has been derived for [*normal* matrices](https://en.wi... | 35 | https://mathoverflow.net/users/11260 | 346313 | 146,732 |
https://mathoverflow.net/questions/346238 | 6 | Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
>
> Does there exist a sequence of vector fields $V\_n \in C^\infty \cap W^{1,2}$ on $\mathbb{R}^2$, such that $V\_n \to V$ in $W^{1,2}$ and the $V\_n$ do... | https://mathoverflow.net/users/46290 | Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$? | Going from the topological index idea: let $C$ be a circle around one of the isolated zeros, and let $D$ be a disk containing $C$. By the trace theorem, your vector fields in $W^{1,2}(D)$ restricts to vector fields $W^{1/2,2}(C)$ on the circle. Since they do not vanish, you can regard them as vector fields $W^{1/2,2}(C... | 3 | https://mathoverflow.net/users/3948 | 346324 | 146,738 |
https://mathoverflow.net/questions/346329 | 3 | Consider the arithmetical theory sometimes denoted by $\mathsf{R}$. The non-logical vocabulary of $\mathsf{R}$ consists of '$0$', '$S$', '$+$' and '$\times$'. The axioms of this theory are all true instances of the following schemata:
$\overline{m}+\overline{n}=\overline{k}$
$\overline{m}\times \overline{n}=\overli... | https://mathoverflow.net/users/147858 | Models of arithmetical theory R + induction in which successor is not injective | I don't have a general classification of this kind of models, but it is rather easy to construct quite a lot of models with this property.
For example (for first-order variant of your system), consider any non-standard model $\mathfrak{M}$ of $\mathsf{PA}$ (or even $\mathsf{I}\Delta\_0$) and a non-standard number $a\... | 2 | https://mathoverflow.net/users/36385 | 346332 | 146,740 |
https://mathoverflow.net/questions/346333 | 6 | A complex weighted projective is $\mathbb{P}(k\_1, \cdots, k\_{n+1})=Proj(\mathbb{C}[x\_1, \cdots, x\_{n+1}])$ with $x\_i$ of degree $k\_i$ (sometimes people ask for each $n$ of the weights being coprime). My first question is whether all weighted projective spaces are rational.
The rationality of $\mathbb{P}(k\_1, \... | https://mathoverflow.net/users/108424 | rationality of weighted projective space | The $n$-dimensional weighted projective space $X = \mathbb{P}(k\_1,\dots,k\_{n+1})$ is a toric variety, hence is automatically rational since it contains an isomorphic copy of $\mathbb{G}\_m^n$ as a dense open subvariety.
Indeed, let $T$ be the torus $\{(t\_1,\dots,t\_{n+1}) \mid t\_i \in \mathbb{G}\_m\}/\{(t,\dots,t)... | 10 | https://mathoverflow.net/users/110362 | 346335 | 146,741 |
https://mathoverflow.net/questions/346347 | 0 | Let $\operatorname{C}$ be a category and $c\in \operatorname{C}$ an object. Consider the coslice (sometimes called slice under) category ${\operatorname c}/{\operatorname C}$. My question is whether ${\operatorname c}/{\operatorname C}$ is in general a *full* subcategory of $\operatorname{C}$?
I am particularly interes... | https://mathoverflow.net/users/142626 | Is a coslice (slice under) category a full subcategory of it ambient category? | For a counterexample, consider $R = \mathbb{C}$. The automorphism $i \colon \mathbb{C} \to \mathbb{C}$ given by complex conjugation is certainly a ring homomorphism, but it isn't $\mathbb{C}$-linear.
Edit: Another example would be the slice $p / \text{Set}$, where $p$ is a one element set. This category is the catego... | 3 | https://mathoverflow.net/users/140821 | 346352 | 146,750 |
https://mathoverflow.net/questions/346354 | 3 | Consider a closed Riemannian manifold $(M,g)$ and let $u \in C^{2,\alpha}(M)$ be a positive function on $M$.
I am interested on the existence of solution for the following problem: given a continuous function $\psi$ on $M$, when does there exists unique $v \in C^{2,\alpha}(M)$ such that
$$\Delta v + F(u,\nabla u)v + ... | https://mathoverflow.net/users/94097 | Simple existence and uniqueness for second order and linear elliptic PDE | You are dealing with a linear, second-order elliptic linear partial differential operator (LPDO) $P\_u$ with $C^{2,\alpha}$ coefficients (you have to take $\alpha\in(0,1)$) on the left hand side of the equation, whose principal part is the Laplacian $\Delta$ on $(M,g)$. The standard approach to existence of solutions t... | 5 | https://mathoverflow.net/users/11211 | 346362 | 146,755 |
https://mathoverflow.net/questions/346360 | 11 | The definition of K-theory of a scheme $X$ is defined as
$G\_i(X):=K\_i(\mathrm{Coh}(X))$ or $K\_i(X):=K\_i(\mathrm{Vec}(X))$.
But usually the schemes are required to be (at least locally) Noetherian, and usually it is said that if it is not then the $G\_i$'s are pretty bad.
But for what reasons that we really need t... | https://mathoverflow.net/users/106136 | Why does K-theory need schemes to be Noetherian? | You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least the following properties
* K-theory of an affine scheme $\mathrm{Spec}\,R$ is given by the algebraic K-theory of proj... | 21 | https://mathoverflow.net/users/43054 | 346363 | 146,756 |
https://mathoverflow.net/questions/346364 | 2 | A manifold $M$ together with a transitive $G$-action is always diffeomorphic a quotient $G/H$ for $H < G$ Lie groups. On the other hand, there might be a proper subgroup of $G$ that also acts transitively on $M$, so this representation may not be unique.
If $H$ is compact, we may choose a metric on $G$ that descends ... | https://mathoverflow.net/users/145929 | Is every homogeneous space Riemannian homogeneous? | You can take $M=\operatorname{SL}\_3 (\mathbb R)/ \mathbb R$, where we can choose any non-compact one-parameter subgroup of $\operatorname{SL}\_3$. Because the stabilizer is noncompact, $G'$ must be proper, but to act transitively, $G'$ must have codimension at most one in $\operatorname{SL}\_3(\mathbb R)$. But no codi... | 5 | https://mathoverflow.net/users/18060 | 346369 | 146,760 |
https://mathoverflow.net/questions/300545 | 6 | Consider the expansion
$$
e\_\mu(x) = \sum\_\lambda c\_{\mu\lambda}(\alpha) J\_\lambda^{(\alpha)}(x)
$$
where $J\_\lambda^{(\alpha)}(x)$ are the integral-form Jack polynomials (the ones with $n!$ as coefficient of $m\_{1^n}(x)$).
Is there some result which proves that each $c\_{\mu\lambda}(\alpha)$ is of the form $... | https://mathoverflow.net/users/1056 | Expansion of elementary symmetric function in Jack's? | Regarding my comment above, note that by induction we only need to show that $\langle J\_\lambda, J\_{1^n}J\_\nu\rangle\in\mathbb{N}[\alpha]$. This follows from the dual Pieri formula for Jack polynomials (obtained by combining Theorems 3.3 and 6.1 of <http://math.mit.edu/~rstan/pubs/pubfiles/73.pdf>), so we do get $c\... | 2 | https://mathoverflow.net/users/2807 | 346376 | 146,762 |
https://mathoverflow.net/questions/346388 | 19 | Given a space $X$, I am looking for a characterization of classes $\alpha \in H^n(X;\bf Q)$ such that there is a map $f\colon T^n \to X$ so that $f^{\ast} \alpha$ pairs non-trivially against the fundamental class of $T^n = (S^1)^n$. Let us denote the subset of such classes by $T^\*(X) \subset H^\*(X;\bf Q)$.
Here ar... | https://mathoverflow.net/users/14233 | Which cohomology classes are detected by tori? | I'd prefer to work in homology (with coefficients $\mathbb{Q}$ everywhere). I'll say that a class $x\in H\_k(X)$ is *basically toral* if there is a map $f\colon T^k\to X$ sending the fundamental class of $T^k$ to $x$. I'll say that $x$ is *toral* if it is a linear combination of basically toral classes, and I'll write ... | 17 | https://mathoverflow.net/users/10366 | 346400 | 146,766 |
https://mathoverflow.net/questions/346401 | 3 | In the Yang-Mills-Higgs (also called magnetic Ginzburg-Landau) model in the plane the energy has the form
$$
E(A,\phi)=\int\_{\mathbb{R}^2}\left(|(d-iA)\phi|^2+\frac{1}{2}F\_{jk}F\_{jk}+\frac{1}{4}(1-|\phi|^2)^2\right)d^2x
$$
where $F\_{jk}=\partial\_jA\_k-\partial\_kA\_j$, $\phi$ is a complex smooth function and $A$ i... | https://mathoverflow.net/users/148829 | A question on the nature of the vortex number | The Higgs field $\phi $ is (mostly) a red herring here. The vortex number can be equally understood in its absence and is determined by the boundary conditions on $A$, which you don't specify - they're the crucial ingredient. In particular, you cannot assume that $\int\_{\partial B\_R } A=0$. The thin vortex field $A=e... | 4 | https://mathoverflow.net/users/134299 | 346419 | 146,772 |
https://mathoverflow.net/questions/346356 | 1 | In a number of different contexts, I have wanted to estimate hitting times for a monotonic process $(T\_n)$ taking values in the reals (or sometimes a process $(T\_n,X\_n)$ taking values in $\mathbb R^2$ where the first component is monotonic). I'm assuming the step size is small, and am interested in the first time, $... | https://mathoverflow.net/users/11054 | Hitting time estimates | $\newcommand{\de}{\delta}
\newcommand{\be}{\beta}
\newcommand{\si}{\sigma}$
Consider the iid case, when $T\_n=X\_1+\dots+X\_n$, where $X\_1,X\_2,\dots$ are positive iid random variables with, say, $EX\_1=\de\to0$, $Var\,X\_1=\de^2\si^2$, $E|X\_1|^3=\de^3\be<\infty$ (scaling with $\de$) such that
\begin{equation}
\be\s... | 1 | https://mathoverflow.net/users/36721 | 346424 | 146,775 |
https://mathoverflow.net/questions/346416 | 3 | I'm trying to find the moments (or the pdf but I'm less confident there's a closed form) of $\frac X{X + Y}$ where $X$ and $Y$ are two independent random variables with a Beta distribution. There's a paper from Pham-Gia that I tried to read, and a similar (but yet different) [question posted here](https://mathoverflow.... | https://mathoverflow.net/users/148841 | Statistical moments of $\frac X{X + Y}$ when $X$ and $Y$ are two independent random variables with a Beta distribution | Suppose that $X$ and $Y$ are independent beta random variables (r.v.'s) with parameters $(a,b)$ and $(c,d)$, respectively. Let
\begin{equation\*}
V:=\frac X{X+Y}. \tag{0}
\end{equation\*}
The transformation $(x,v)\mapsto(x,x\frac{1-v}v)$ transforms $(X,V)$ to $(X,Y)$.
The Jacobian determinant of this transformation ... | 6 | https://mathoverflow.net/users/36721 | 346430 | 146,776 |
https://mathoverflow.net/questions/346427 | 6 | Consider the vertical strip of angle $\alpha=\frac{\pi}{2}$
In this case, the harmonic function which is $0$ on the left line and $1$ on the right line is given by $$f(a+ib)=\frac{a}{T}.$$
Now, when the angle $0<\alpha <\frac{\pi}{2}$ the strip becomes bent.
My question is: can we determine explicitly the harmonic ... | https://mathoverflow.net/users/nan | A harmonic function | Yes, it can be found explicitly, though not in elementary functions but in terms of a combination of elementary and hypergeometric functions.
The problem is (almost) equivalent to finding a conformal map
from your straight strip onto the broken strip. For this it is sufficient to find
a conformal map from the upper ha... | 13 | https://mathoverflow.net/users/25510 | 346433 | 146,777 |
https://mathoverflow.net/questions/339634 | 13 | For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum\_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function *A004125* from the [OEIS](https://oeis.org).
**Example.** We've that for $n=6$ $$S(6)=0+0+0+6\text{ mod }4+6\text{ mod }5+0=2+1=3.$$
This arithmetic fu... | https://mathoverflow.net/users/142929 | Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function? | Well, let me elaborate Ilya Bogdanov's argument. First of all, $$S(n)=\sum\_{k=1}^n \left(n-k\lfloor n/k\rfloor\right)=n^2-\sum\_{k,d:kd\leqslant n} k=
n^2-\sum\_{d=1}^n (1+2+\ldots+\lfloor n/d\rfloor).
$$
We have $1+2+\ldots+\lfloor n/d\rfloor=\frac1{2d^2}n^2+O(n/d)$, thus
$$S(n)=\beta n^2+O\left(n\sum\_{i=1}^n \frac... | 6 | https://mathoverflow.net/users/4312 | 346448 | 146,781 |
https://mathoverflow.net/questions/346385 | 14 | I'm a computer scientist, not a mathematician, so apologies if I've messed up a lot of things greatly.
I've been reading about [synthetic differential geometry](https://ncatlab.org/nlab/show/synthetic+differential+geometry), and [trying to formalize it in Coq](https://github.com/bollu/diffgeo/blob/master/diffgeo.v). ... | https://mathoverflow.net/users/123769 | Constructing computable synthetic differential geometry? | I still don't quite understand what OP wants, but let me just cite a few papers that I think might be relevant to such questions. First, there are a lot of literature that describe how to work with real numbers in a computationally meaningful way. To give a few examples:
* [Andrej Bauer, Iztok Kavkler, A constructive... | 3 | https://mathoverflow.net/users/62782 | 346458 | 146,783 |
https://mathoverflow.net/questions/346461 | 8 | Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to [this one](https://mathoverflow.net/questions/178654/characterizing-the-newforms-s-t-the-associated-symmetric-square-l-function-ha) :
>
> Can $L(1, \pi, \mathrm{sym}^2)$ be zer... | https://mathoverflow.net/users/128718 | Does the symmetric square L-function vanish at one? | For $GL(3)$, the exterior square $L$-function $L(s,\wedge^2\pi)$ is entire as it agrees with $L(s,\tilde\pi\otimes\omega)$, where $\omega$ is the central character of $\pi$. Therefore, $L(1,\mathrm{sym}^2\pi)=0$ would imply that $L(1,\pi\otimes\pi)=0$, contradicting a result of [Shahidi (1980)](https://projecteuclid.or... | 10 | https://mathoverflow.net/users/11919 | 346462 | 146,785 |
https://mathoverflow.net/questions/346245 | 8 | Let $X$ be a smooth projective variety. A *fibration* is a surjective map with connected fibers between projective varieties. Is it true that there's a finite number of birational equivalence classes of projective varieties (say, with representatives $Y\_1,\ldots, Y\_n$) such that if $X\to Y$ is a fibration then $Y$ is... | https://mathoverflow.net/users/116075 | Finiteness of birational types for targets of algebraic fibrations | As Jorge already pointed out this is too much to hope for. On the other hand, if you can put some restriction on $Y$, then there are results in this direction.
1. A theorem of Severi implies that if you restrict $Y$ to be a curve of genus at least $2$, then this is true. (Severi's theorem is slightly more general, re... | 6 | https://mathoverflow.net/users/10076 | 346464 | 146,787 |
https://mathoverflow.net/questions/337311 | 5 | In 1939 H. Weyl proved the following non-trivial theorem. Let $(M^n, g)$ be a **closed** smooth Riemannian manifold. Fix an isometric imbedding $\iota\colon M\to \mathbb{R}^N$ into a Euclidean space (now such an imbedding is known to exist e.g. by Nash Theorem; Weyl actually used weaker results on local imbeddings). Co... | https://mathoverflow.net/users/16183 | Weyl tube formula for manifolds with boundary | I was shown that the positive answer to my question (in fact even for manifolds not only with boundary but even with corners) follows from ther recent stronger result: Theorem 3.11 in the paper by J. Fu and T. Wannerer
<https://arxiv.org/abs/1711.02155>
| 1 | https://mathoverflow.net/users/16183 | 346476 | 146,791 |
https://mathoverflow.net/questions/346470 | 5 | Landweber exactness gives a criterion for when a complex oriented cohomology theory $E$ can be recovered from the formal group law over $E\_{\*}$ determined by the complex orientation. That is it gives a criterion for the following to hold (see for example [here](http://www.math.harvard.edu/~lurie/252xnotes/Lecture18.p... | https://mathoverflow.net/users/148857 | Example of a space X exhibiting the Landweber non-exactness of the additive formal group over the integers? | Firstly, you ask for a space $X$. I will instead talk about finite spectra, but they become spaces if you suspend them enough times, so that does not really make a difference.
There is a kind of tautological answer to your question as follows. By the nilpotence technology of Hopkins, Devinatz and Smith, for suitable ... | 8 | https://mathoverflow.net/users/10366 | 346477 | 146,792 |
https://mathoverflow.net/questions/346420 | 4 | Gromov's nonsqueezing theorem famously says that there does not exist any symplectic embedding $B^{2n}\_R\hookrightarrow Z^{2n}\_r:=B^2\_r\times\mathbb R^{2n-2}$ when $R>r$, but can we make this non-existence quantitative? I suspect we should have a result like follows:
*Conjecture:* For any symplectomorphism $f:\mat... | https://mathoverflow.net/users/94022 | How much of a ball $B_R^{2n}$ can be symplectically embedded in the cylinder $B_r^2\times\mathbb R^{2n-2}$? | I had thought about a quantitative version of this for a while, so let me say what I know.
The first point to make is that the conjecture as stated is incorrect. In fact, $\operatorname{Vol}(f(B\_R) \cap Z\_r)$ can be arbitrarily close to $\operatorname{Vol}(B\_R)$. This is due to Anatole Katok in his paper *Ergodic ... | 5 | https://mathoverflow.net/users/66405 | 346502 | 146,797 |
https://mathoverflow.net/questions/346506 | 11 | I remember coming across some article of Bill Thurston’s where he describes a 3-manifold (with boundary?) as being like an egg. In my recollection the interior of the egg, the shell, and even the shell after being cracked all had meaning. As you can imagine, Googling “like an egg three-manifold Thurston” is not very fr... | https://mathoverflow.net/users/135175 | Searching for a Thurston paper with egg / 3-manifold analogy? | I believe you're looking for page 211 of Hyperbolic Structures on 3-Manifolds I: Deformation of Acylindrical Manifolds (Annals of Math., 1986), which includes the following paragraph:
>
> A complete description of the three spaces $AH(M)$, $GH(M)$, and $QH(M)$ is
> certainly not rigorously known, but here is a con... | 15 | https://mathoverflow.net/users/385 | 346507 | 146,798 |
https://mathoverflow.net/questions/346498 | 0 | Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ the algebra of all bounded linear operators on $E$.
On $\mathcal{L}(E)^n$, we have two equivalent norms:
\begin{eqnarray\*}
N\_1({\bf A})
&=&\sup\left\{\bigg(\displaystyle\sum\_{k=1}^n\|A\_kx\|^2\bigg)^{\frac{1}{2}},\;x\in E,\;\|x\|=1\;\right\},
\end{eqnarray\*}... | https://mathoverflow.net/users/113054 | Equality between two norms on $\mathcal{L}(E)^n$ | As quite an obvious counterexample, take $A\_k$ to be the orthoprojector $E\ni x:=(x\_1,\dots,x\_n)\mapsto x\_ke\_k\in E$ on $E:=\mathbb{C}^n$. Then $A\_iA\_j=A\_jA\_i=\delta\_{ij}A\_i$ but $N\_1(A)=1$ and $N\_2(A)=\sqrt{n}$.
On the positive side, I think $A\_k:=f\_k(A)$ with $A$ self-adjoint, and *increasing* funct... | 1 | https://mathoverflow.net/users/6101 | 346510 | 146,799 |
https://mathoverflow.net/questions/346519 | 6 | Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum of all simple submodules of $M$.
We know that both rad and soc define covariant subfunctors of $Id:Mod\_R\rightarrow ... | https://mathoverflow.net/users/137269 | Adjoints for radical and socle functors | While it does not work for general rings, for Artin algebras one has that the left adjoint of the socle functor is the functor $M \rightarrow M/rad(M)$. I would think that for general rings that is the only choice in case a left adjoint exists.
| 7 | https://mathoverflow.net/users/61949 | 346526 | 146,803 |
https://mathoverflow.net/questions/346532 | 1 | If one add to ZF the rule that all sets are parameter free definable. Would that prove the axiom of choice?
More specifically. IF we add the following omega rule to inference rules of the language of ZF.
if $\phi\_0, \phi\_1, \phi\_2,...,$ are all formulas in the first order language of set theory, in which only th... | https://mathoverflow.net/users/95347 | Does choice always hold in a model of ZF with point-wise parameter-free definable sets? | *The following fleshes out the comments above by Asaf and Andreas.*
First, note that the idea you outline at the end **will not** work: it implicitly assumes that the relation "$\varphi$ defines $a$" is definable, which is not the case. (Indeed, if that did work it would imply that there are no pointwise-definable mo... | 4 | https://mathoverflow.net/users/8133 | 346540 | 146,808 |
https://mathoverflow.net/questions/346538 | 3 | Let $S=\{1,2,3,...,n\}$ be the set of integers up to $n$ and $p\_k(a\_1,...,a\_k)=a\_1\cdots a\_k$ the product of $k$ distinct integers $a\_1,...,a\_k \in S$. There are $\binom{n}{k}$ possibilities to construct such a product $p\_k$. I was wondering if it is anyhow possible to estimate the sum of all such $k$-products ... | https://mathoverflow.net/users/140124 | Sum of all products of k distinct integers in [1,n] | The sum of all $k$-products of numbers in the interval $\{1,2,\ldots,n\}$, it is, the number you are referring to, is known as the Stirling number of the first kind: ${n+1 \brack {n+1-k}}$. There are plenty of articles that face the problem of giving estimations of these numbers out there.
| 10 | https://mathoverflow.net/users/147861 | 346541 | 146,809 |
https://mathoverflow.net/questions/346546 | 9 | I'm reading a paper that translates between formal geometry and rigid geometry.
In particular, this paper begins with two rigid analytic spaces $A$ and $C$ (each coming from a scheme over $\mathbb{Z}\_p$), both defined over $B$, and finds the formal completion of each, calling them $\mathfrak{A}$, $\mathfrak{C}$, and... | https://mathoverflow.net/users/141571 | Translation between formal geometry and rigid geometry | No, these are not the same thing. Formal schemes are to rigid-analytic spaces as $\mathbf{Z}\_p$-schemes are to $\mathbf{Q}\_p$-schemes.
The book *[Lectures in Formal and Rigid Geometry](https://www.springer.com/gp/book/9783319044163)* by Bosch is an excellent and friendly reference on this subject - take a look esp... | 14 | https://mathoverflow.net/users/56878 | 346553 | 146,812 |
https://mathoverflow.net/questions/47882 | 9 | I posted this question at [math.stackexchange.com](https://math.stackexchange.com/questions/12487/characterizations-of-euclidean-space) but didn't get an answer. Is it a dumb question, eventually?
There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
1. axioma... | https://mathoverflow.net/users/2672 | Characterizations of Euclidean space | Herbert Busemann provided many metric characterizations of the elementary spaces in his 1955 book *The Geometry of Geodesics*. To characterize $E^n$, we can then further restrict to cases with zero curvature, or to non-compact cases with non-negative curvature, or probably in several other ways.
For a first pass at t... | 10 | https://mathoverflow.net/users/nan | 346560 | 146,815 |
https://mathoverflow.net/questions/346542 | 6 | I'm reading the Dijkgraaf–Witten paper *Topological gauge theories and group cohomology* (Comm. Math. Phys. **129** (1990) pp 393–429, doi:[10.1007/BF02096988](https://doi.org/10.1007/BF02096988)) and on page 395, 2nd paragraph they write
>
> Suppose we choose a discrete group $G$. Every principal $G$-bundle has a ... | https://mathoverflow.net/users/146495 | Why does a principal G-bundle with a discrete structure group G have a unique flat connection? | Tsemo's answer addresses the first part of your question. To answer the second, for $G$ discrete, a $G$-bundle $p:P\to M$ is in particular a covering space for $M$, and so every path in $M$ has a unique lift (which happens to be the horizontal lift for the flat connection discussed in Tsemo's answer). So a closed path ... | 4 | https://mathoverflow.net/users/17945 | 346561 | 146,816 |
https://mathoverflow.net/questions/346529 | 9 | $\def\Cat{\mathbf{Cat}}\def\Set{\mathbf{Set}}\def\A{\mathcal{A}}$I stumbled into the following statement:
Let $\Cat(\Set,\Set)\_s$ be the category of small functors[¹] $F : \Set \to\Set$ and let $F,G$ be two comonads on this category; then, the Day convolution $F \* G$ is itself a comonad.
I find this both interest... | https://mathoverflow.net/users/7952 | The convolution of comonads is a comonad | This is an instance of a standard fact about duoidal categories.
**Proposition.**
Let $(\mathcal{V},\ast,J,\circ,I,\ldots)$ be a duoidal category (see e.g. Section 4.2 of [Street's lecture notes](https://arxiv.org/abs/1201.2991)), and let $(G,\delta,\varepsilon)$ and $(H,\delta,\varepsilon)$ be $\circ$-comonoids in ... | 9 | https://mathoverflow.net/users/57405 | 346562 | 146,817 |
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