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https://mathoverflow.net/questions/435659 | 10 | Assume that $\frac{1}{p}+\frac{1}{q}=1$ for two positive real numbers $p,q$.
For what kind of $C^\*$ algebras $A$ does the following hold:
$$\frac{ab+ba}{2}\leq \frac{a^p}{p}+\frac{b^q}{q}$$ $\forall a,b\in A^+$?
As a second question we pose the trace version of this question:
Let $A$ be a $C^\*$ algebra with a... | https://mathoverflow.net/users/36688 | For what kind of $C^*$ algebras does the inequality $\frac{(ab+ba)}{2}\leq\frac{ a^p}{p} +\frac{b^q }{q}$ hold for $a,b>0$? | Let me expand slightly on the comments I made above, and give the most general solution.
Clearly the inequality $\frac{ab + ba}{2} \leq \frac{a^2}{2} + \frac{b^2}{2}$ holds for all positive elements $a,b$ in a $C^\ast$-algebra since $a-b$ is self-adjoint and therefore $0\leq (a-b)^2 = a^2 + b^2 - ab - ba$.
Also (ju... | 13 | https://mathoverflow.net/users/126109 | 435780 | 176,141 |
https://mathoverflow.net/questions/435751 | 4 | Let $n\geq 3$ be a positive integer and $\kappa=(k\_1, \dots, k\_n)\in \mathbb{Z}^n$. Denote by $B\_n$ the braid group on $n$ strings. Consider the braid on $n+1$ strings $\sigma\_\kappa:=\sigma\_1^{k\_1}\dots \sigma\_n^{k\_n}$, where $\sigma\_i=\sigma\_i^+$ is the generator taking the $i$-th string to the $(i+1)$-st s... | https://mathoverflow.net/users/470091 | Alexander polynomials for a certain family of closed braids | The closure of the braid $\sigma\_\kappa$ is a connected sum of torus links $T(2,k\_i)$ (which are closures of 2-braids). Since the Alexander polynomial is multiplicative with respect to connected sums, $\Delta\_\kappa = \prod\_i \Delta\_{T(2,k\_i)}$.
You can see more generally that if you have a braid $\beta = \beta... | 3 | https://mathoverflow.net/users/13119 | 435781 | 176,142 |
https://mathoverflow.net/questions/435784 | 7 | A function $f \in L^1(\mathbb R^n)$ is said to be of *bounded variation* if there exists a constant $C \geq 0$ such that
$$
\int\_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx
\leq
C \sup\_{ x \in \mathbb R^n } \lvert\phi(x)\rvert
$$
for all compactly supported differentiable vector fields $\phi : \mathbb R^n \r... | https://mathoverflow.net/users/2082 | What happens if we consider functions of bounded variation that are not in $L^1$? | The main historical reason for which the requirement $f\in L^1$ enters in the definition of $BV$ is that functions of bounded variation (*tout court*) of several variables were introduced by Lamberto Cesari building upon previous work by Leonida Tonelli in order to solve the problem of characterization of (hyper)surfac... | 7 | https://mathoverflow.net/users/113756 | 435792 | 176,145 |
https://mathoverflow.net/questions/435749 | 1 | Let $(X,\tau)$ be a topological space. A *retraction* is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a *retract* of $X$ if there is a retraction $r:X\to X$ such that $\text{im}(r) = S$.
We say that $(X,\tau)$ is rc if all retracts are closed. It turns out that a... | https://mathoverflow.net/users/8628 | Is the class of rc-spaces closed under products? | Take $X$ to be an [RC space which isn't $T\_2$](https://topology.pi-base.org/spaces?q=rc%2B%7E%24T_2%24) such as the [one-point compactification of the rationals](https://topology.pi-base.org/spaces/S000029). We will show $X^2$ is not RC. Note that it is not $T\_2$ as its factors are not $T\_2$.
First we will note th... | 3 | https://mathoverflow.net/users/73785 | 435793 | 176,146 |
https://mathoverflow.net/questions/434947 | 0 | I'm asking here instead of the economics stackexchange because I'm interested more in the applied mathematics part, instead of just the economics; I'm interested in seeing what new research is being published, and what are the ongoing trends and currents in this area. In that sense, would anyone have interesting journa... | https://mathoverflow.net/users/204581 | Journals of applied mathematics with an economics bent? | <https://www.resurchify.com/impact/category/Applied-Mathematics>
Journal of Econometrics is a journal covering the categories related to Applied Mathematics (Q1); Economics and Econometrics (Q1); History and Philosophy of Science (Q1). It is published by Elsevier BV. The overall rank of Journal of Econometrics is 462... | 1 | https://mathoverflow.net/users/493315 | 435802 | 176,148 |
https://mathoverflow.net/questions/435767 | 3 | I start with a thesis: the natural notion of equality is additional data (paths/morphisms), not a binary relation (the fact that they exist). So, in particular, with such a constructivization (replacing property $\to$ structure):
* sets $\to$ $\infty$-groupoids
* categories $\to$ $\infty$-categories
At the same tim... | https://mathoverflow.net/users/148161 | Does the concept of a $\infty$-category have a natural definition in the $\infty$-world? | Homotopy type theory (HoTT) gives a natural internal language for studying $\infty$-groupoids. [Riehl and Shulman](https://arxiv.org/abs/1705.07442) give an extension of HoTT which gives an analogous internal language for studying $\infty$-categories. Essentially, the Riehl-Shulman framework is able to work because a b... | 6 | https://mathoverflow.net/users/2362 | 435805 | 176,149 |
https://mathoverflow.net/questions/435758 | 9 | The [nLab page on $\infty$-categories](https://ncatlab.org/nlab/show/infinity-category) splits the known definitions of $\infty$-categories into two types:
* [Algebraic $\infty$-categories](https://ncatlab.org/nlab/show/algebraic+definition+of+higher+categories), in which composition is expressed "externally", e.g. a... | https://mathoverflow.net/users/130058 | Is there a "geometric definition" of globular $\infty$-groupoids/categories? | In short there isn't: the problem is that if you just have globular sets - and if you want $k$-cells to model $k$-arrows following the globular structure - then globular sets have no way of expressing the idea that some cell $f$ is the composite of two cells $g$ and $h$. You can only express that two cells $g$ and $h$ ... | 5 | https://mathoverflow.net/users/22131 | 435807 | 176,150 |
https://mathoverflow.net/questions/435801 | 11 | Given an initial integer $x\_0>0$, one can consider the first prime of the recursive sequence $x\_i=1+2x\_{i-1}$.
Naïvely such a prime should exist for $x\_0$ arbitrary since the sequence $\log(x\_i)$ is asymptotically
an arithmetic progression. Sometimes it takes however some
time: for $x\_0=147$ my Maple algorithm ... | https://mathoverflow.net/users/4556 | First prime of the form $x_i$ for $x_0=658$ and $x_i=1+2x_{i-1}$ | The $n$th term of your sequence is $x\_{n} = 659 \cdot 2^{n} - 1$. People have long searched for prime values of numbers $k \cdot 2^{n} - 1$ for small $k$, with the goal of proving that $k = 509203$ is the *smallest* positive integer for which $k \cdot 2^{n} - 1$ is composite for all $n$. (This is called the Riesel pro... | 23 | https://mathoverflow.net/users/48142 | 435808 | 176,151 |
https://mathoverflow.net/questions/428627 | 1 | Here is the definition of a space $X$ to be Rothberger, if for each sequence $(\mathcal{U}\_{n})\_{n\in\mathbb{N}}$ of open covers of $X$, there exists a sequence $(U\_{n})\_{n\in\mathbb{N}}$ where $U\_{n}\in\mathcal{U}\_{n}$ and $X=\bigcup\_{n\in\mathbb{N}} U\_{n}$.
Definition: A subset $A$ of $X$ is said to be semi... | https://mathoverflow.net/users/489732 | Rothberger property and semi-open sets | First let's identify the semi-open sets. We first note all open sets are open. A simpler definition of the open sets are (assuming $p=0$),
$$\tau=\{U\subseteq\mathbb R:0\in U\Rightarrow\mathbb R\setminus U\text{ finite}\}.$$
If $0\not\in S$, then $S$ is open and thus semi-open.
If $0\in S$ and $S$ is open, then $... | 0 | https://mathoverflow.net/users/73785 | 435813 | 176,153 |
https://mathoverflow.net/questions/435797 | 1 | In a commutative ring $R$, when does the assumption $r\_i\mid r$ for $1\le i\le n$ imply $\prod\_{1\le i\le n} r\_i\mid r$ (when $r\_i$ are fixed)?
Does there exist any criterion for this implication that is related to regular sequences?
| https://mathoverflow.net/users/2191 | When an element of a ring that is divisible by a finite set of elements is necessarily divisible by their product? | This is just an expansion of my comment. Question is local, so we may assume that the ring is local. If $r\_1,\ldots, r\_n$ is a regular sequence, then they are so in any order and any subset forms a regular sequence. So, assume $r\_i|r$ for all $i$ and assume we have shown $r=ar\_1r\_2\cdots r\_i$ for some $i<n$, the ... | 2 | https://mathoverflow.net/users/9502 | 435815 | 176,154 |
https://mathoverflow.net/questions/435803 | 3 | Recently, [Ching and Salvatore](https://arxiv.org/abs/2002.03878 "Koszul duality for topological E_n-operads") have proven that the $E\_n$ operad is Koszul self dual. While thinking about the analogous question for the framed $E\_n$ operad, I realized there is an obvious first question: does the spectrum $\Sigma^\infty... | https://mathoverflow.net/users/134512 | Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\infty$-ring spectrum? | The question as stated probably requires clarification. If
$X$ is a space, then the S-dual $D\_+(X)$ (i.e., functions from $X\_+$ to the sphere) is always an $E\_\infty$-ring spectrum. In particular, it will also be an $A\_\infty$-ring.
Perhaps what is being asked is whether $D\_+(G)$ is an **$A\_\infty$-coalgebra*... | 6 | https://mathoverflow.net/users/8032 | 435825 | 176,156 |
https://mathoverflow.net/questions/435841 | 5 | Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$.
Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$.
What is the known generalization of this statement to abelian varieties of arbitrary dimension?
| https://mathoverflow.net/users/144623 | Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension | The $j$ invariant gives an isomorphism over $\mathbb Z$,
$$j:\mathcal A\_1\to\mathbb A^1,$$
of the moduli space of elliptic curves. So $j(E)\in\overline{\mathbb Z}$ can be interpreted as saying that $\langle E\rangle\in \mathcal A\_1(\overline{\mathbb Z})$, where I've written $\langle E\rangle$ for the isomorphism clas... | 7 | https://mathoverflow.net/users/11926 | 435854 | 176,162 |
https://mathoverflow.net/questions/435853 | -1 | Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).
Empirically, I find that the distribution of $s \cdot v$ seems to be $\mathcal{N}(0, \frac{||v||\_2^2}{d})$, or at least I can' find an exa... | https://mathoverflow.net/users/131785 | The distribution of the sum of a non-zero vector with random signs | Of course, your empirically motivated conjecture will not hold in general. E.g., it will not hold if $v\_1^2$ is much greater than $\sum\_{i\ge2}v\_i^2$, where the $v\_i$'s are the coordinates of $v$.
On the other hand, by (say) the [Berry--Esseen inequality](https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem... | 3 | https://mathoverflow.net/users/36721 | 435856 | 176,163 |
https://mathoverflow.net/questions/435429 | 9 | *[Cross-posted from MSE.](https://math.stackexchange.com/questions/4580731)*
Following [Bankston - The total negation of a topological property](https://doi.org/10.1215/ijm/1256048236), a topological space is called *anticompact* if all its compact subsets are finite. The linked MSE post above has two examples:
**E... | https://mathoverflow.net/users/458159 | Is there a connected Hausdorff anticompact space that is countably infinite? | Such an example has been constructed by [Banakh and Stelmakh](https://arxiv.org/abs/2211.12579).
More precisely, they constructed an anticompact countable connected Hausdorff space which is Brown and strongly rigid.
| 3 | https://mathoverflow.net/users/61536 | 435864 | 176,167 |
https://mathoverflow.net/questions/435523 | 1 | In the proof of Lemma 2.1 in
*Ershov, Mikhail; He, Sue*, [**On finiteness properties of the Johnson filtrations**](https://doi.org/10.1215/00127094-2018-0005), [ZBL06904638](https://zbmath.org/?q=an:06904638),
the authors claim the following (without proof).
*Let $G$ be a finitely generated group, and let $g\_1,\... | https://mathoverflow.net/users/47274 | A group-theoretic lemma in a paper by Ershov and He | Here is @AndyPutman's [comment](https://mathoverflow.net/questions/435523/a-group-theoretic-lemma-in-a-paper-by-ershov-and-he#comment1122032_435523) as an answer (so that it can be accepted), made CW to avoid reputation. If @AndyPutman prefers to post the answer, then I will delete this.
>
> This only needs the fac... | 1 | https://mathoverflow.net/users/2383 | 435867 | 176,168 |
https://mathoverflow.net/questions/435754 | 5 | Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp\_p : T\_p M \to M$ that sends a tangent vector $v \in T\_p M$ to the endpoint of the unique geodesic $\gamma$ satisfying $\gamma(0)=p... | https://mathoverflow.net/users/105103 | Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)? | Exponential map in your definition is closely related to the smooth family of smooth curves smoothly depending on the position such that in every point in every direction there exists precisely one curve at this point in this direction. In literature, such a family is sometimes called path structure.
There are two di... | 7 | https://mathoverflow.net/users/14515 | 435870 | 176,169 |
https://mathoverflow.net/questions/435863 | 3 | I'm looking at the linear PDE in 3+1 dimensions,
$$
\left[ -(\partial\_t - \xi)^2 - \partial\_k \partial\_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}
$$
Where $\xi$ is generally a complex parameter and $k=1,2,3$. So I'm after a particular Green's function $\phi(t,x) = \phi(t,x,\xi)$ and would like... | https://mathoverflow.net/users/41312 | Linear PDE, analytic continuation, Green's function and boundary conditions | **Q:** *Do I have to consider both problems (real $\xi$ or imaginary $\xi$) totally independently and work hard twice?.*
**A:** A single calculation suffices, you could just do the inverse Fourier transform of $[(\omega+i\xi)^2+k^2]^{-1}$ for complex $\xi=i\alpha+\beta$ to arrive at the solution that works for both r... | 8 | https://mathoverflow.net/users/11260 | 435876 | 176,172 |
https://mathoverflow.net/questions/435826 | 9 | Let $g$ be a piecewise smooth, zero average, function over $[0,1]$ such that $\min g^2>0$. I would like to show that
$$
\int\_0^1 g\sqrt{1-r/g^2}\int\_0^1 \frac{1}{g\sqrt{1-r/g^2}} \leq 1
$$
for all $r \in \mathopen[-1,\min g^2\mathclose[$. I don't know that this is true but I am persuaded that it is, based on intuitio... | https://mathoverflow.net/users/486571 | Integral inequality: Prove $\int_0^1 f\int_0^1 1/f \leq 1$ for a certain function $f$ | Matematika, shmatematika: any minimally decent CAS should immediately detect and tell the human operator that, for $r=1$, the inequality reads
$$
(\int\_X F-\int\_Y G)(\int\_X 1/F-\int\_Y 1/G)\le 1
$$
where $F=\sqrt{f^2-1}, G=\sqrt{g^2-1}$, $f,g> 1$, $\mu(X)+\mu(Y)=1$, and $\int\_X f=\int\_Y g$. Now it becomes obvious ... | 6 | https://mathoverflow.net/users/1131 | 435877 | 176,173 |
https://mathoverflow.net/questions/435868 | 23 | This question is inspired by [On moments of inertia of planar and 3D convex bodies](https://mathoverflow.net/questions/435361/on-moments-of-inertia-of-planar-and-3d-convex-bodies).
Let $f:{\mathbb R}^3\setminus\{0\}\to{\mathbb R}$ be an even homogeneous ($f(kx)=f(x)$ for all real $k\neq 0$) continuous function.
Can one... | https://mathoverflow.net/users/25510 | A property of even continuous functions on the sphere | It seems there aren't any counterexamples, even if $f$ is homogeneous but not even.
If there is some counterexample $f$ to the question, and letting $X=\mathbb{R}^3\setminus\{x\_1=x\_2=x\_3\}$, we can consider the map $F:SO(3)\to X$, given by $F(u,v,w)=(f(u),f(v),f(w))$ (we will sometimes represent elements $r\in SO(... | 21 | https://mathoverflow.net/users/172802 | 435882 | 176,174 |
https://mathoverflow.net/questions/435855 | 6 | The following is true:
1. The counit components $\epsilon : X^A\times A\to X$ of the cartesian closed structure of $Set$ are the components of the initial cowedge; in other words, $X$ is the coend $\int^A X^A\times A$. Size issues apart, the reason is that $\int^A X^A\times A$ is isomorphic to the value at $X$ of $\t... | https://mathoverflow.net/users/7952 | The coend of a parametric counit | Unfortunately statement 2 is not true in general. First, notice that
$$\hom\Bigl(\int^A X^A \times A,Y\Bigr) = \int\_A \hom(X^A,Y^A) = \hom(X^{(-)},Y^{(-)}).$$
So the question is if the functor
$$\mathcal{C} \to [\mathcal{C}^{\mathrm{op}},\mathcal{C}], ~ X \mapsto X^{(-)}$$ is fully faithful. The canonical map
$$\hom(X... | 3 | https://mathoverflow.net/users/2841 | 435886 | 176,175 |
https://mathoverflow.net/questions/435859 | 9 | Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the following true?
If $i\_\*(\pi\_1(\Sigma\setminus \partial \Sigma))=0$, then $\Sigma$ is orientable.
| https://mathoverflow.net/users/64231 | Links and non-orientable surfaces | Yes, the surface is orientable. To simplify the LaTex and the exposition, I will change the notation and setting a small amount.
---
Suppose that $F$ is a compact connected embedded surface in three-sphere. Suppose that the image of $\pi\_1(F)$ in $\pi\_1(S^3 - \partial F)$ is trivial. We must show that $F$ is or... | 4 | https://mathoverflow.net/users/1650 | 435887 | 176,176 |
https://mathoverflow.net/questions/435837 | 4 | To each pair $(S,\mathcal{X})$ where $S=(s\_i)\_{i\in\mathbb{N}}$ is a decreasing sequence of positive real numbers and $\mathcal{X}\subseteq\mathbb{R}$, we can associate the **alternation game** $A\_S(\mathcal{X})$ as follows:
* Players $1$ and $2$ jointly build an increasing sequence of natural numbers $$a\_0<b\_0<... | https://mathoverflow.net/users/8133 | Can these alternating series games be undetermined? | If we take $s\_i = 2^{-i}$, we should even get that $\mathrm{ZFC}$ proves the existence of some $\mathcal{X}$ with undetermined $A\_{(2^{-i})\_{i \in \mathbb{N}}}(\mathcal{X})$. The key parts are that different plays yield different reals (as nothing in the tail can overcome a difference in a prefix), and that each str... | 5 | https://mathoverflow.net/users/15002 | 435906 | 176,180 |
https://mathoverflow.net/questions/435676 | 4 | Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d\_i)\_{i \in I}$, with $I$ a finite set and $d\_i \in \mathbb{A} \cap \mathbb{R}\_{\ge 1}$, such that $$d\_i d\_j = \sum\_{k \in I} n\_{i,j,k} d\_k,$$ for all $i,j \in I$, with $n\_{i,j,k} \in \mathbb{Z}\_{\ge 0}$.
A subset $J \subset I$ is ca... | https://mathoverflow.net/users/34538 | Positive system of algebraic integers | No, here are infinitely many counter-examples for $I = \{1,2\}$.
Let $n,m \in \mathbb{Z}\_{\ge 1}$ such that $m | n^2$, $n | 2m^2$ and $n \neq 2m$ (e.g. $n=m=1$).
Take $d\_1 = n(1+\sqrt{2})$, $d\_2 = m(3+2\sqrt{2})$. Here are the matrices $M\_i = (n\_{i,j,k})\_{j,k \in I}$:
$$\left(\begin{matrix}0&\frac{n^2}{m}\\... | 2 | https://mathoverflow.net/users/34538 | 435917 | 176,183 |
https://mathoverflow.net/questions/434790 | 5 | Kechris in his *Classical Descriptive Set Theory* book gives the following definition (Definition 24.1) and characterization (Theorem 24.15) of Baire class $1$ functions:
>
> **Definition.** Let $X,Y$ be metrizable spaces. A function $f:X\rightarrow Y$ is of Baire class $1$ if $f^{-1}(U)$ is $F\_\sigma$ for every o... | https://mathoverflow.net/users/141146 | Baire class $1$ functions and Baire's characterization theorem | I had a similar problem some time ago and I had the impression that a general review on Baire classes is in fact missing, and would be useful.
Probably you already saw it, but in [this](https://www.tau.ac.il/%7Etsirel/Courses/MeasCategory/lect5.pdf) chapter there is at least a discussion of how the definition through $... | 3 | https://mathoverflow.net/users/167834 | 435918 | 176,184 |
https://mathoverflow.net/questions/435919 | 31 | In 2002, the discovery of the AKS algorithm proved that it is possible to determine whether an integer is prime in polynomial time deterministically. However, it is still not known whether there is an algorithm for factoring an integer in polynomial time.
To me, this is the most counter-intuitive observation in mathe... | https://mathoverflow.net/users/7089 | Why is integer factoring hard while determining whether an integer is prime easy? | What I think you're asking for are examples of **search** problems that seem to be hard, while a corresponding **decision** problem is solvable in polynomial time (but not totally trivial). It is true that such problems do not arise in practice very often; typically, an efficient decision procedure can be turned into a... | 53 | https://mathoverflow.net/users/3106 | 435925 | 176,188 |
https://mathoverflow.net/questions/434800 | 3 | Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine formal model $\mathfrak{X}=Spf(A^{\circ})$ is normal (i.e. $A^{\circ}$ is a normal integral domain). My question is whethe... | https://mathoverflow.net/users/476832 | On the stability of having a normal formal model under finite extensions of the base field | As for your first question, $X\_L$ indeed admits a normal formal model by virtue of normalisation. Whether $A^{\circ}\otimes\_R R\_L$ is already normal (which is equivalent to $A^{\circ}\otimes\_R R\_L=(A\_L)^{\circ}$, as the latter is normal) is less transparant and is related to issues of wild ramification.
Here is... | 1 | https://mathoverflow.net/users/93776 | 435938 | 176,191 |
https://mathoverflow.net/questions/435322 | 1 | $G$ is a finite solvable group. Let $\{P\_{1}, P\_{2}, \dotsc , P\_{s}\}$ be a Sylow basis of $G$. We have that $G=P\_{1}P\_{2}\dotsm P\_{s}$. Set
\begin{equation}
\begin{aligned}
%% The alignment is never used ….
T=\prod\limits\_{t=1}^{s-1}P\_t,
H=\prod\limits\_{k\neq3}^sP\_k,
K=\prod\limits\_{r\neq2}^sP\_r.\nonumber... | https://mathoverflow.net/users/478453 | The property of self-normalizing subgroup | Yes. Nilpotency of $T$ is not needed. Assume that $N\_G(P\_s)>P\_s$. Then for some $i<s$, $N\_G(P\_s)$ has a
Sylow $p\_i$-subgroup $Q\_i\ne 1$. Assume without loss that $p\_i$
divides $|H|$. Then $P\_iP\_s$ is a Hall $\{p\_i,p\_s\}$-subgroup of $H$
and $G$. By Hall's extension of Sylow's theorems to solvable groups,
$(... | 2 | https://mathoverflow.net/users/99221 | 435943 | 176,193 |
https://mathoverflow.net/questions/435941 | 1 | I am very confused by a sum I have been trying to solve analytically/ numerically for a long time. It comes from the idea of a physical problem where the observation is made that has a combined response of a number of entities. For example, I want to evaluate the mathematical sum at a observation point $\omega$ that lo... | https://mathoverflow.net/users/489481 | A double sum with complex numbers having stochastic variables | Your final integral can be readily evaluated by expanding the fraction of sines into sums of exponentials $e^{ikx/2}$ with integer $k$, and integrating term by term with the Gaussian weight, to arrive at
$$|L(0)|^2 \equiv\frac{M}{\sqrt{2\pi}} \int\_{-\infty}^{+\infty} e^{-x^2/2} \frac{\sin^2\left(Nx/2 \right)}{\sin^2\l... | 1 | https://mathoverflow.net/users/11260 | 435951 | 176,196 |
https://mathoverflow.net/questions/435947 | 7 | I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this [thesis](https://trace.tennessee.edu/cgi/viewcontent.cgi?article=8804&context=utk_graddiss).
Corollary 4.1.15. on page 63 says that if the number of ends (... | https://mathoverflow.net/users/363264 | If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set? | Of course, yourself and YCor already answered this in the comments, but since I see you tagged geometric group theory, maybe you will be interested in this explicit answer about bi-Holder homeomorphisms : The end boundary of an accessible infinitely-ended group is bi-Hölder equivalent to the standard Cantor ternary set... | 2 | https://mathoverflow.net/users/111917 | 435961 | 176,197 |
https://mathoverflow.net/questions/435858 | 4 | The context for this question comes from [this arxiv preprint](https://arxiv.org/abs/2210.07753). Specifically, a remark in the final proof of the paper. To make the question more self-contained, I'll phrase this question in a slightly more general setting.
Fix a category $A$ and a combinatorial model category $\math... | https://mathoverflow.net/users/76636 | Decomposing a $\mathcal{M}$-valued presheaf into a homotopy colimit of representables | The diagram $G$ is a projectively cofibrant diagram because
nondegenerate simplices split off as a coproduct summand in every degree
and every simplicial level is a coproduct of (enriched) representables.
The homotopy colimit of a projectively cofibrant diagram can be computed as its colimit.
Therefore, the homotopy ... | 2 | https://mathoverflow.net/users/402 | 435965 | 176,199 |
https://mathoverflow.net/questions/434451 | 23 | I want to follow the discussion from [here](https://mathoverflow.net/questions/191016/all-retracts-are-closed-as-separation-axiom) concerning about the strength of the separation "all retract subspaces are closed".
(A retract subspace of a topological space $X$ is a subspace $A$ where there exists continuous $f: X\to... | https://mathoverflow.net/users/494268 | "All retracts are closed" and "all compacts are closed" | RC does not imply KC: in [this paper](https://arxiv.org/abs/2211.12579) Banakh and Stelmakh construct a semi-Hausdorff countable Brown space $X$ which is strongly rigid (and hence $X$ has $RC$) and contains a non-closed compact subset (so, $X$ fails to have $KC$).
This example also shows that $KC$ does not follow fro... | 6 | https://mathoverflow.net/users/61536 | 435973 | 176,204 |
https://mathoverflow.net/questions/435893 | 3 | Let $\varphi := \frac{\sqrt{5}-1}{2}$ be the golden ratio, and $H(x):=-x\log\_2 x -(1-x) \log\_2(1-x)$ be the binary entropy function for a Bernoulli random variable.
Show that for all $\delta > 0$, one can choose sufficiently small $\gamma > 0$, such that if $\mu$ is a probability measure on $[0, 1]$, with
\begin{al... | https://mathoverflow.net/users/4923 | A variational estimate related to the union closed set conjecture | Actually I don't see what your trouble is then. Switching to natural $\log$ (which is just scaling of $H$), you have the key inequality
$$
\varphi H(x^2)\ge xH(x)
$$
(ask Zachary Chase if you want to see a reasonably short and almost computation-free complex analysis proof of it) with equality only for $x=0,\varphi,1$.... | 5 | https://mathoverflow.net/users/1131 | 435977 | 176,208 |
https://mathoverflow.net/questions/435572 | 5 | The following question comes from a typo in an old notebook of mine (I changed what I was calling my forcing notion partway through writing the definition of properness):
Say that a forcing $\mathbb{P}$ is **pseudoproper** iff for all sufficiently large $\theta$, all countable $M\prec H\_\theta$, and all $\mathbb{P}$... | https://mathoverflow.net/users/8133 | Highly improper forcings | If we allow only for separative forcings $\hat{\mathbb{P}}$ in the definition of rudeness then all non-trivial separative $\sigma$-closed forcings are rude. This does not answer your question literally, but hopefully in spirit.
EDIT: This restriction is not necessary after all, see below.
Suppose that $\mathbb P$ i... | 4 | https://mathoverflow.net/users/125703 | 435998 | 176,213 |
https://mathoverflow.net/questions/435798 | 2 | Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).
1. Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?
2. (Here we may assume $L/K$ is cyclic.)... | https://mathoverflow.net/users/149579 | Ramification of primes and order of $\smash{\hat{H}}^0$ in ray class fields with one finite prime divisor | The answer to Question 1 is negative. Take $K = {\mathbb Q}(\sqrt{-6})$
and let ${\mathfrak p} = (2,\sqrt{-6})$ denote the prime ideal above $2$.
The class number of $K$ is $2$, its maximal unramified (abelian) extension
is $L = {\mathbb Q}(\sqrt{-3},\sqrt{2})$. The ray class number formula shows that $h\{\mathfrak m\}... | 2 | https://mathoverflow.net/users/3503 | 436003 | 176,216 |
https://mathoverflow.net/questions/435849 | 4 | Is it true that
$$(v-u)^2+(w-u)^2+(w-v)^2 \\
+\left(\sqrt{\frac{1+u^2}{1+v^2}}
+\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\
-\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-v)(v-u) \\
-\left(\sqrt{\frac{1+w^2}{1+v^2}}+\sqrt{\frac{1+v^2}{1+w^2}}\right) (v-u) (w-u)>0$$
for any real $u,v,w$ su... | https://mathoverflow.net/users/36721 | An algebraic inequality in three real variables | Rewrite the inequality in terms of $a,b,c>1$ via $u=(\frac{1}{a}-a)/2$, $v=(b-\frac{1}{b})/2$, and $w=(c-\frac{1}{c})/2$. Note that $\sqrt{1+u^2}=\frac{1+a^2}{2a}$ and so on. Then the conjectured inequality is equivalent to
\begin{equation}
((c - b)(a + c)(a + b))^2(a^2b^2c^2-a^2bc+ab^2c+abc^2+ab+ac-bc+1)>0.
\end{equa... | 4 | https://mathoverflow.net/users/18739 | 436004 | 176,217 |
https://mathoverflow.net/questions/436009 | 3 | Let $T(n,k)$ be a triangle of coefficients such that $T(n,k)\geqslant0$ for $n>0$, $0<k\leqslant n$, $0$ otherwise. Also
$$T(2n+1,1)=\frac{1}{2n+1}, T(2n,1)=0$$
$$T(n,k)=\frac{1}{n}(T(n-1,k-1)+(n-2)(T(n-2,k)+\frac{T(n-2,k-1)}{n-1}))$$
Let
$$P(n,m)=m\sum\limits\_{k=1}^{m}n^{k-1}T(m,k)(-1)^{m+k}$$
I conjecture that
$$P(n... | https://mathoverflow.net/users/231922 | Powers of $2$ up to $2^{m-1}$ from a polynomial of degree $m-1$ | This is a fun problem! You start out by describing the conjectured coefficients of $P(n,m)$, but presumably you started out by computing the polynomial which interpolates $2^{n-1}$ and then noticed a pattern in the coefficients. So I'll start in that order: Let $Q(n,m)$ be the unique degree $m-1$ polynomial in $n$ whic... | 6 | https://mathoverflow.net/users/297 | 436012 | 176,220 |
https://mathoverflow.net/questions/435971 | 8 | Can we prove the consistency of $\mathsf{ZF+SVC}$ + "There is a Reinhardt cardinal?" (Preferably from the consistency of $\mathsf{ZF}$ with a Reinhardt cardinal, but using a stronger assumption is also okay.)
Here $\mathsf{SVC}$ means the *Small Violation of Choice*, claiming the axiom of choice is forcible by a set ... | https://mathoverflow.net/users/48041 | Compatibility of $\mathsf{SVC}$ and Reinhardtness | No, a Reinhardt cardinal implies SVC is false.
First, if there is a Reinhardt cardinal, then by Woodin's proof of the Kunen inconsistency theorem, for sufficiently large regular cardinals $\delta$, the set $S^\delta\_\omega$ of ordinals of cofinality $\omega$ cannot be partitioned into $\delta$-many disjoint stationa... | 9 | https://mathoverflow.net/users/102684 | 436013 | 176,221 |
https://mathoverflow.net/questions/435949 | 9 | Let $\mathcal{C}$ be a category with finite limits and a (parameterized) natural numbers object $(N,0,s)$. Let $1$ denote the terminal object of the category. It's easy to show that the following is a coproduct diagram.
$$ 1 \xrightarrow{0} N \xleftarrow{s} N $$
**My question is:** is this coproduct diagram necessarily... | https://mathoverflow.net/users/128139 | Is the coproduct $N=1+N$ universal? | The answer is no in general - but this is a fairly subtle issue. First, let's go over why this is "almost true".
Given $e:X \to N$ I denote by $X\_0$ and $X\_{>0}$ the pullback of $\{0\}$ and $N\_{>0}$ along $e$.
So, given $N$ a parametrized NNO, and $f,g :X \to Y$ two functions, what you can easily do using the NN... | 4 | https://mathoverflow.net/users/22131 | 436017 | 176,222 |
https://mathoverflow.net/questions/435997 | 2 | $\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Let $V, W$ be finite dimensional complex vector spaces and $M\in \Hom(V, W)$ a full rank linear map. I want to see if there exists a Lie group $G$ and representations $\pi: G \to \GL(V)$ and $\rho: G \to \GL(W)$ s.t. $\rho(g) M = M \pi(g)$. The problem is that I... | https://mathoverflow.net/users/265376 | Algorithm for finding the symmetries of a linear operator | Note that the identity $\rho M = M \pi$ (I'm omitting the $g$ argument here) implies that $\rho$ has to preserve the image of $M$ and can do anything it likes the cokernel of $M$, while $\pi$ has to preserve the kernel of $M$ and it can do anything it likes on the complement. Writing $V = \ker M \oplus Z$ and $W = Z \o... | 3 | https://mathoverflow.net/users/2622 | 436022 | 176,225 |
https://mathoverflow.net/questions/425214 | 6 | It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}\_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators $E\_i,F\_i$, $i=0, \ldots q-1$ on the basis of Schur polynomials $s\_\lambda$ is given by
$$E\_i s\_\lambda= \sum\_{\l... | https://mathoverflow.net/users/160416 | Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients | I ended up finding a proof of this phenomenon using the vertex operator description of the basic representation of $\widehat{\mathfrak{sl\_2}}$. It can be found in the following preprint: <https://arxiv.org/abs/2211.10584>.
| 2 | https://mathoverflow.net/users/160416 | 436023 | 176,226 |
https://mathoverflow.net/questions/436029 | 1 | Let $X$ be a complex smooth cubic threefold and $C$ be a smooth twisted cubic, then $C\subset Y\subset X$ for a unique cubic surface $Y$ in $X$ (or equivalently a hyperplane section of $X$). When $Y$ is smooth, we have $27$ lines on $Y$.
What can we say about the position of the a line $L$ and the twisted cubic $C$ (... | https://mathoverflow.net/users/nan | twisted cubic in a smooth hyperplane section of a cubic threefold | Although this is not necessarily, let me assume that $Y$ is a smooth cubic surface. Then the linear system of a twisted cubic curve $C \subset Y$ generates a linear system that induces a morphism
$$
\pi \colon Y \to \mathbb{P}^2
$$
which is a blowup of 6 points, say $P\_1,\dots,P\_6$, and so that $\mathcal{O}\_Y(C)$ is... | 1 | https://mathoverflow.net/users/4428 | 436030 | 176,227 |
https://mathoverflow.net/questions/436020 | 3 | For prime $p$ let $E\_p[\dots]$ and $P\_p[\dots]$ be the external and polynomial $\mathbb{Z}\_p$--algebras.
It is known that for $n\geqslant 1$ and odd $p$ where is an isomorphism
of primitively generated Hopf algebras
$H\_\*(\Omega^2S^{2n+1};\mathbb{Z}\_p)=E\_p[x\_0,x\_1,x\_2,\dots]\otimes P\_p[y\_1,y\_2,\dots]$
whe... | https://mathoverflow.net/users/494199 | Homology of iterated loop spaces on odd--dimensional spheres | For the case where $n=dp$, the calculation of $H\_\*(\Omega^3S^{2n+1};\mathbb{Z}/p)$ is reviewed in Section 4.2 of the paper [The triple loop space approach to the telescope conjecture](https://people.math.rochester.edu/faculty/doug/mypapers/obit.pdf) by Mahowald, Ravenel and Shick. I don't think that the assumption th... | 5 | https://mathoverflow.net/users/10366 | 436035 | 176,230 |
https://mathoverflow.net/questions/435890 | 3 | Let $B$ denote the $n$-dimensional unit ball. Assume $u\in\bigcap\_{1\le p<2} W\_0^{1,p}(B)$ satisfies $$
\int a\_{ij}\partial\_j u \partial\_iv =0,$$ for any $v\in C\_c^{\infty}(B)$, where we assume $a\_{ij}(x)$ are just uniformly elliptic, i.e. there exists $\lambda,\Lambda>0$ such that $$\lambda \lvert\xi\rvert^2\le... | https://mathoverflow.net/users/442224 | $W^{1,p}$ ($1\le p<2$) uniqueness of elliptic equations | The answer is **yes** if and only if you have optimal elliptic regularity in $W^{-1,p'}(B)$ for some $p>2$ for the adjoint differential operator, i.e., the one given by the transpose matrix $A^\top$.
Consider your question from a functional-analytic point of view: You are essentially asking whether the (bounded linea... | 4 | https://mathoverflow.net/users/85906 | 436037 | 176,231 |
https://mathoverflow.net/questions/436028 | -1 | In this [paper](https://arxiv.org/abs/1206.2251), if we denote the $k$ largest eigenvalues by $\lambda\_N,\lambda\_{n-1},··· ,\lambda\_{N-k+1}, $ then for Gaussian ensembles the joint distribution function of rescaled eigenvalues has the limit:
$$
\lim\_{N\to\infty}P(N^{2/3}(\lambda\_N-2)\le s\_1,\dots, N^{2/3}(\lamb... | https://mathoverflow.net/users/168083 | Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law? | $\newcommand\la\lambda$The answer is: yes, of course.
Indeed, let $X\_{N,i}:=N^{2/3}(\la\_i-2)$. By the limit theorem you cited and (say) [Example 2.3, p. 18](https://archive.org/details/convergenceofpro0000bill), the $k$-tuple $(X\_{N,N},\dots,X\_{N,N-k+1})$ converges in distribution to some $k$-tuple $(Y\_1,\dots,Y... | 0 | https://mathoverflow.net/users/36721 | 436038 | 176,232 |
https://mathoverflow.net/questions/436036 | 2 | I found this question: [Chernoff style concentration bound for ratio of variables](https://mathoverflow.net/questions/420837/chernoff-style-concentration-bound-for-ratio-of-variables).
I want to ask if we get similar thing for the ratio of the sum and the one Gaussian variable.
Given i.i.d. Gaussian random variables ... | https://mathoverflow.net/users/168083 | Can we find such $k$ so that the following inequality holds? | $\newcommand\ep\epsilon $In the clever answer by Fedor Petrov, it was shown that
\begin{equation\*}
Q:=P\Big(\frac{X\_1^2+\dots+X\_k^2}{X\_1^2}<C\Big)\le C/k, \tag{1}\label{1}
\end{equation\*}
where $C:=1/\ep^2>0$ and the $X\_i$'s are any iid random variables.
Let us show that for the standard normal $X\_i$'s as in ... | 2 | https://mathoverflow.net/users/36721 | 436051 | 176,237 |
https://mathoverflow.net/questions/436032 | 13 | *Below, all sentences/formulas are first-order and in the language of arithmetic. For simplicity, we conflate numbers and numerals, and conflate sentences/formulas and their Godel numbers.*
Given a formula $\varphi(x)$ and a sentence $\theta$, say that $\theta$ **asserts its own $\varphi$-ness** iff $\mathsf{PA}\vdas... | https://mathoverflow.net/users/8133 | Are there different "levels" of self-referentiality in arithmetic? | If $\varphi$ is not required to behave the same way on Gödel codes of equivalent sentences or any such thing, then $\mathsf{SR}(\theta)$ is always equivalent to the preorder on all arbitrary formulas, by defining in PA a bijection $f$ between $\mathbb{N} - \{\theta\}$ and $\mathbb{N}$, and noting that any formula $\var... | 8 | https://mathoverflow.net/users/3902 | 436053 | 176,238 |
https://mathoverflow.net/questions/436052 | 9 | We believe there is always a prime in the interval $[x,x+\sqrt{x})$, for $x$ sufficiently large, but proving this is inaccessible, even under RH.
What if we just wanted a sequence of integers free of multiples of previous elements? These sequences are called primitive. Can we find a primitive sequence with this prope... | https://mathoverflow.net/users/495982 | Primitive sequences with elements in every interval $[x, x + \sqrt x)$ | One can obtain such a sequence after some finite starting point. Let $k\_0$ be large, and for each $k \geq k\_0$ we can partition the interval $[2^k,2^{k+1})$ into about $100 \cdot 2^{k/2}$ disjoint intervals of length about $\frac{1}{100} \cdot 2^{k/2}$. In each such interval $I$, use sieve theory to select a number $... | 6 | https://mathoverflow.net/users/766 | 436064 | 176,240 |
https://mathoverflow.net/questions/436060 | 1 | It is easy to see that for any $x$ and $y$ on the unit sphere of a Hilbert space $H$ there exists a surjective isometry $U$ such that $Ux=y$. Does something more general also hold? That is, given two pairs $(x\_1, y\_1)$ and $(x\_2, y\_2)$ of vectors on the unit sphere such that $||x\_1-x\_2||=||y\_1-y\_2||$, can we fi... | https://mathoverflow.net/users/69275 | Isometries of Hilbert space | For complex scalars, this fails in $\mathbb{C}^2$. Take $x\_1 = y\_1 = (1,0)$, $x\_2 = (i,0)$, $y\_2 = (0,1)$. No (complex linear) isometry can take the one-dimensional subspace spanned by $x\_1$ and $x\_2$ onto the two-dimensional subspace spanned by $y\_1$ and $y\_2$.
For real scalars, it is true. It suffices to fi... | 3 | https://mathoverflow.net/users/23141 | 436066 | 176,241 |
https://mathoverflow.net/questions/436071 | 11 | Let $M\_1,M\_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that $\mathfrak{X}(M\_1)\cong \mathfrak{X}(M\_2)$ as Lie algebras.
>
> **Question.** Are $M\_1$ and $M\_2$ diffeomorphic?
>
>
>
This seems like a basic question but I did not find any... | https://mathoverflow.net/users/32135 | Does the Lie algebra of vector fields $\mathfrak{X}(M)$ determine the diffeomorphism class of a manifold $M$? | The answer is *yes*, and the assumptions "simply connected" and "compact" are actually unnecessary.
In fact, it is possible to reconstruct any smooth manifold $M$, up to diffeomorphisms, by using the subalgebra $\mathfrak{X}\_0(M)$ of vector fields with compact support. See
*Shanks, M. E.; Pursell, Lyle E.*, [**The... | 11 | https://mathoverflow.net/users/7460 | 436078 | 176,246 |
https://mathoverflow.net/questions/436080 | 13 | In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their regularity. The best-case scenario would be to have a color scale (let's say from red to blue) where regions in red wou... | https://mathoverflow.net/users/168737 | How to characterize the regularity of a polygon? | Internal angles are not enough to determine the regularity of a polygon. E.g., angles of $2\pi/3$ between sides of length $1,1,4,1,1,4$ make an irregular hexagon.
For a metric of regularity, I suggest $$A \ / \ \sum d\_i^2$$ where $A$ is the area of the hexagon and the $d\_i$ are the side lengths.
Using this metric... | 13 | https://mathoverflow.net/users/nan | 436088 | 176,250 |
https://mathoverflow.net/questions/425280 | 3 | I'm interested in studying a certain generalization of determinental varieties as defined here:
<https://en.wikipedia.org/wiki/Determinantal_variety>
To be more specific, I must first lay out a few definitions.
Consider the variety
$X=Mat\_{n\_1, n\_2}(\mathbb{C})\times Mat\_{n\_2, n\_3}(\mathbb{C})\times... Mat\_... | https://mathoverflow.net/users/88516 | Searching for resolutions of generalized determinental varieties | These are "type A quiver cycles" (a name chosen so as not to collide with type A quiver varieties, which involve taking quotients; here the quotient would be a point). Your guess for the closure is correct. We compute the equivariant cohomology classes of these varieties in [Four positive formulae for type A quiver pol... | 2 | https://mathoverflow.net/users/391 | 436092 | 176,252 |
https://mathoverflow.net/questions/436102 | 3 | Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}\_\text{qc}(\mathcal{O}\_X)$. (Feel free to assume $M^\bullet$ bounded if needed. But I would rather not suppose it coh... | https://mathoverflow.net/users/131975 | What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$? | No to both questions:
1. Take $M = K(X)$, the field of rational functions on $X$.
2. Take $M$ to be the structure sheaf of any regular point on $X$ (assuming $\dim(X) > 0$).
| 4 | https://mathoverflow.net/users/4428 | 436103 | 176,256 |
https://mathoverflow.net/questions/435840 | 0 | von-Neumann entropy
===================
I know von-Neumann entropy on density matrix $S=-{\rm Tr}(\rho \ln\rho)$ is similar to Shannon entropy $S=-\sum\_i p\_i\ln p\_i$ in classical mechanics. And I want to get Bose-Einstein distribution and Fermi-Dirac distribution with the principle of maximum entropy. I work with ... | https://mathoverflow.net/users/495865 | Dose density matrix with off-diagonal elements equal to zero has maximum von-Neumann entropy? | The following observation allows to reduce the problem to the classical, commuting case.
I will assume finite dimensionality although generalizations are possible.
Let the Hamiltonian have the following spectral decomposition, $H = \sum\_k E\_k \Pi\_k $, where $E\_k, \Pi\_k$ are respectively the eigenvalues, spectral... | 0 | https://mathoverflow.net/users/74539 | 436105 | 176,257 |
https://mathoverflow.net/questions/436082 | 0 | I take $F$ from $\Omega\subset \mathbb C^n$ to $\mathbb C^n$ to be a holomorphic function such that
$$| \det(J\_F)|\leq 1,$$
where $J\_F$ is the Jacobian matrix of $F$.
My question: Is there any classification of functions of this type?
| https://mathoverflow.net/users/151216 | Holomorphic function on $\mathbb C^n$ | This function has constant Jacobian by Liouville. Then it is
map of Jacobian 1 composed with a homothety or its differential is degenerate everywhere. The constant Jacobian biholomorphisms are subject of much research, see for example
*Rosay, Jean-Pierre,
Automorphisms of Cn, a survey of Andersén-Lempert theory and a... | 0 | https://mathoverflow.net/users/3377 | 436113 | 176,260 |
https://mathoverflow.net/questions/435649 | 8 | Let $k$ be a local field of characteristic $p$ and $\omega \in k$ a uniformiser. Consider the Artin-Schreier extension $L\_n = k[x]/(x^p - x - \omega^n)$ for each $n \in \mathbb{Z}$.
Is there an explicit description for the image of the norm map $N\_{L\_n/k}: L\_n^\times \to k^\times$?
If $n < 0$ it seems that the ... | https://mathoverflow.net/users/5101 | Image of the norm map for Artin-Schreier extensions | Local class field theory has something to say about the norm groups.
To set up notation, let's put $k = \kappa((t))$ with uniformiser $\omega = t$.
I'll assume $n < 0$ is not divisible by $p$ in the following (without loss).
The element $x \in L\_n$ satisfying $x^p - x = t^n$ has valuation $v(x) = n/p$ (where I alway... | 4 | https://mathoverflow.net/users/496038 | 436126 | 176,264 |
https://mathoverflow.net/questions/435426 | 2 | ### Background
Throughout, let $X$ be a smooth complex manifold.
1. It is a classical fact that a coherent analytic sheaf admits a **local** resolution by locally free sheaves (also known as a local syzygy). Griffiths and Harris' *Principles of Algebraic Geometry* (p. 696) gives a nice proof of this: by definition,... | https://mathoverflow.net/users/73622 | Resolving complexes of coherent analytic sheaves | If I interpret your question correctly, then I believe there is indeed such a construction.
The construction relies first of all on the existence of local resolutions as in your point 1. Secondly, it relies on the fact that vector bundles are projective objects over Stein domains, for example over any ball in some lo... | 2 | https://mathoverflow.net/users/49151 | 436128 | 176,265 |
https://mathoverflow.net/questions/436134 | 6 | For a positive integer $n$, the prime omega function value $\Omega(n):=\sum\_{p\mid n}{\nu\_p(n)}$ counts the number of prime divisors of $n$ with multiplicities. A result of Hardy and Wright, [1, Theorem 430 on p. 472], implies that $\frac{1}{x}\sum\_{n\leq x}{\Omega(n)}\sim\log\log{x}$ as $x\to\infty$.
**Question:*... | https://mathoverflow.net/users/57975 | Average value of the prime omega function $\Omega$ on predecessors of prime powers | Yes, this is true.
First, let us observe that replacing prime powers with primes cannot make a difference, and the same goes to replacing $\Omega$ with $\omega$.
Erdős, in "On the normal number of prime factors of $p-1$ and some related problems concerning Euler’s $\phi$-function (Quart. Journ. of Math. 6, 205-213 ... | 10 | https://mathoverflow.net/users/31469 | 436137 | 176,271 |
https://mathoverflow.net/questions/436122 | 0 | Following this question:[Can we find such $k$ so that the following inequality holds?](https://mathoverflow.net/questions/436036/can-we-find-such-k-so-that-the-following-inequality-holds/436040#436040).
Consider a sequence of independent $n-$dimensional random vectors $u, v\_1, v\_2,\dots, v\_k$ uniformly distributed... | https://mathoverflow.net/users/168083 | Can we find the following $k$ so that the following inequality holds for asymptotic normal? | $\newcommand\ep\epsilon\newcommand{\R}{\mathbb R}$By the spherical symmetry, conditionally on $u$, the $X\_i$'s are iid random variables (r.v.'s) each with a conditional distribution not depending on $u$. So, even unconditionally, the $X\_i$'s are iid r.v.'s each equal $X:=\sqrt n\,e\_1\cdot v$ in distribution, where $... | 2 | https://mathoverflow.net/users/36721 | 436140 | 176,272 |
https://mathoverflow.net/questions/436104 | 2 | If stumbled accross [self-avoiding walk](https://en.wikipedia.org/wiki/Self-avoiding_walk)s. They seem to be deterministically generated, but from a quick google search term seem to be randomly generated variants.
However, as far as I can see they are only defined on a lattice (like $\mathbb Z^d$) (which is clearly n... | https://mathoverflow.net/users/91890 | Is there something like a "self-avoiding Markov chain" on a continuous space? | As the question is asked, the answer is "no": if a continuous curve $\gamma:\mathbb{R}\_{\geq 0}\to [0,1)^2$ is self-avoiding, i.e., injective, then the image $\gamma(\mathbb{R}\_{\geq 0})$ is nowhere dense in $[0,1)^2$. Indeed, the images $\gamma([0,T])$ are compact, hence closed. Also, compactness implies the inverse... | 2 | https://mathoverflow.net/users/56624 | 436149 | 176,274 |
https://mathoverflow.net/questions/436130 | 2 | Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$?
I know it's related to the decomposition of a complex Lie algebra $\frak{g}\_{\mathbb{C}}=\frak{t}\oplus\frak{p}^{+} \oplus \frak {p} ^{-}$ but I'm having trouble finding what in Shimura's varieties theory needs this assumption.
Does anyone hav... | https://mathoverflow.net/users/169282 | Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$? | This is the condition which gives the Shimura variety the (almost) complex structure, which is obviously necessary if you want to view it as a variety over $\mathbb C$. This is explained in Theorem 1.21 of [Milne's notes](https://www.jmilne.org/math/xnotes/svi.pdf#X.1.21) (in slightly different language, but all the eq... | 1 | https://mathoverflow.net/users/30186 | 436151 | 176,275 |
https://mathoverflow.net/questions/436124 | 2 | Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}\_2F\_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) \alpha B(2,\alpha +1)}
$$
and
$$
{}\_2F\_1(2\alpha+1,\alpha+1;\alpha+3;-1)
=\frac{1}{2^{2\alpha}}\frac{\alpha B(1/2,\a... | https://mathoverflow.net/users/147732 | Ask for references or proofs of two explicit formulas for special Gauss hypergeometric functions | These formulas are special cases of what I like to call
extendable evaluations: set for simplicity $F={}\_2F\_1$. If one
has an explicit formula for $F(a,b;c;z)$ and (for example) also for
$F(a+1,b;c;z)$, then using the contiguity relations, it is immediate
to find explicit formulas for $F(a+k,b+l;c+m;z)$ for any
$(k,l... | 4 | https://mathoverflow.net/users/81776 | 436154 | 176,277 |
https://mathoverflow.net/questions/436157 | 6 | $\DeclareMathOperator\Spec{Spec}\newcommand\Ring{\mathrm{Ring}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}$In the category of schemes the objects of the form $\Spec(K)$ with $K$ a field can be characterized as follows. They are precisely the non-empty schemes which have no proper ... | https://mathoverflow.net/users/219922 | Subsheaves of Spec K, K a field | There is no hope for this in any subcanonical topology coarser than the fppf topology, or more generally, any subcanonical topology in which morphisms $\operatorname{Spec} C \to \operatorname{Spec} K$ are not automatically covers when $K$ is a field and $C$ is non-trivial.
But it is true with the fpqc topology.
Indee... | 8 | https://mathoverflow.net/users/11640 | 436161 | 176,279 |
https://mathoverflow.net/questions/419383 | 16 | I study algebraic geometry / number theory and from time to time I stumble upon 2-categorical (co)limits. I have two main examples in mind:
>
> Example 1) In étale cohomology, the (triangulated) derived category of $\overline{\mathbb{Q}\_\ell}$-sheaves is defined as a 2-colimit of the derived categories of sheaves ... | https://mathoverflow.net/users/131975 | 2-categories for the working algebraic geometer | A standard reference for category theory is *Categories for the Working Mathematician* (which I assume the OP knew about based on the question title). The closest reference I know to "2-categories for the working mathematician" is Steve Lack's *[A 2-categories companion](https://arxiv.org/abs/math/0702535v1)*, which pr... | 4 | https://mathoverflow.net/users/11540 | 436163 | 176,280 |
https://mathoverflow.net/questions/434699 | 10 | In a survey article [Algebraic geometry in mixed characteristic](http://arxiv.org/abs/2112.12010v1), B. Bhatt writes
For instance, given a commutative ring $R$ with a finitely generated ideal $I$,
* the assignment carrying $R$ to the $\infty$-category $D\_\text{$I$−comp}(R)$ of derived $I$-complete $R$-complexes fo... | https://mathoverflow.net/users/80739 | Reference request: infinity categories for the commutive algebraist/algebraic geometer | I don't want this question to hang around forever on the "unanswered queue," so let me add an answer, even though I think the comments largely answer it. My motivation here is to advertise a few other sources beyond those mentioned in the comments.
First, as the comments point out, Lurie has written a ton about this ... | 3 | https://mathoverflow.net/users/11540 | 436166 | 176,282 |
https://mathoverflow.net/questions/436108 | 4 | $\newcommand\Spt{\mathit{Spt}}\newcommand\GrAb{\mathit{GrAb}}$Let $A$ be a ring spectrum. Suppose that $A$ has a Künneth theorem — i.e. the homology theory $A\_\ast : \Spt \to \GrAb$ is a strong monoidal functor [1].
**Question:** Does it follow that $A$ is a module over Morava $K$-theory $K(h)$ for some prime $p$ an... | https://mathoverflow.net/users/2362 | If $A_\ast$ has a Künneth theorem, then is $A$ a module over Morava $K$-theory? | If you want $K(h)\_\*$ to be strong monoidal, then you need the target category to be the category of graded $K(h)\_\*$-modules, not the category of graded abelian groups. Thus, I assume you are really asking for conditions under which $A\_\*(-)$ gives a strong symmetric monoidal functor to the category of graded $A\_\... | 7 | https://mathoverflow.net/users/10366 | 436168 | 176,284 |
https://mathoverflow.net/questions/433532 | 4 | Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0?
I understand that there is a technical problem with the original proof due to Quillen given in ["Rational Homotopy Theory" (Remark on p.223)](https://people.math.rochester.edu/faculty/doug/otherpapers/q... | https://mathoverflow.net/users/143549 | Monoidal Dold–Kan correspondence for non-connected CDGA | The answer to the question:
>
> Why can't the monoidal Dold–Kan correspondence be extended to non-connected CDGAs over a field of characteristic 0?
>
>
>
is that in fact the Dold-Kan correspondence *can* be extended to this setting.
The classical Dold-Kan correspondence is an equivalence of categories betwee... | 2 | https://mathoverflow.net/users/11540 | 436171 | 176,285 |
https://mathoverflow.net/questions/435524 | 1 | Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook *Toric Varieties* by Cox, Little and Schenck says that the vector space of global sections of the invertible sheaf $\mathcal{O}\_X(D)$ associated to $D$ i... | https://mathoverflow.net/users/136356 | Explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety | Yes, since the complex global sections are isomorphic to the tensor product of the real global sections with $\mathbb C$ by flat base change, and this isomorphism is equivalent for the action of the real torus, so when we write the complex global sections as a sum of eigenspaces this also writes the real global section... | 1 | https://mathoverflow.net/users/18060 | 436176 | 176,286 |
https://mathoverflow.net/questions/417735 | 2 | Levy’s characterisation theorem for Brownian motion states that for a local martingale $X$ with $X\_0 = 0$, $X$ is a Brownian motion if and only if it has quadratic variation $\langle X, X \rangle\_t = t$.
The usual proofs of this fact use the characteristic function of the normal distribution. I am seeking an altern... | https://mathoverflow.net/users/173490 | Alternate proof of Levy’s characterisation of Brownian motion | Fix $A>0$; we will be proving that $X\_t\stackrel{\mathcal D}= B\_t$ on $[0;A].$ Let $T^{(m)}$ be an a.s. increasing sequence of stopping times such that $T^{(m)}\to\infty$ almost surely, and for each $m$, $X\_{n\wedge T^{(m)}}$ is a martingale. Then, $Y\_n=X\_{\epsilon n\wedge T^{(m)}}$, $n=0,1,\dots,$ is a discrete-t... | 1 | https://mathoverflow.net/users/56624 | 436187 | 176,289 |
https://mathoverflow.net/questions/436045 | 3 | Let $A$ be a (unital) $C^\*$-algebra and $X,Y$ right Hilbert $A$-modules which are finitely generated and projective. It seems to be well-known that if $T: X \to Y$ is an $A$-linear map, then $T$ is necessarily adjointable. I could not find a proof though. Can someone give a reference or proof of this little fact?
| https://mathoverflow.net/users/216007 | Linear map between projective finitely generated Hilbert modules is adjointable | Looking at the proof of Lemma 6.21 in the notes of de Commer that you're reading (<https://arxiv.org/pdf/1604.00159.pdf>, per the comments), it seems like the relevant property of the modules is that of having compact identity operator. Suppose that a Hilbert $A$-module $X$ has this property. Since finite sums of the f... | 5 | https://mathoverflow.net/users/85913 | 436195 | 176,291 |
https://mathoverflow.net/questions/436194 | 5 | Let $C\_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C\_{lsc}(\mathbb{R}^n)$ is:
$$ f^\*(y) := \sup\_{x \in \mathbb{R}^n} (\langle x, y \rangle - f(x)) $$
It is known that the LF transform is continuous and an i... | https://mathoverflow.net/users/136732 | Is the Legendre transform as an operator Lipschitz? | This is basically true for sup norm by Fenchel's inequality. Indeed, for all $y$,
$$
f^\*(y) = \sup\_x\left( \langle x,y\rangle - f(x) \right) \leq \sup\_x\left( g(x)+g^\*(y) - f(x) \right) \leq \|f-g\|\_{\infty} + g^\*(y).
$$
The same is true when the roles of $f,g$ are reversed, giving essentially what you want. ... | 6 | https://mathoverflow.net/users/99418 | 436199 | 176,293 |
https://mathoverflow.net/questions/436173 | 2 | Following this question: [Can we apply the continuous mapping theorem for the limiting joint distribution of the Tracy-Widom law?](https://mathoverflow.net/questions/436028/can-we-apply-the-continuous-mapping-theorem-for-the-limiting-joint-distribution).
We know that
>
> $$
> \lim\_{N\to\infty}P(N^{2/3}(\lambda\_... | https://mathoverflow.net/users/168083 | Can we get that $ P(N^{2/3}(\lambda_N-\lambda_{N-1})\le c)\ge 1-\epsilon$? | The probability distribution of the spacing $\delta\_N=\lambda\_N-\lambda\_{N-1}$ of the eigenvalues $\lambda\_N$ and $\lambda\_{N-1}$ at the edge of the spectrum decays exponentially for $\delta\_N\gg N^{-2/3}$, with a decay rate that is independent of $N$. So no matter how small $\epsilon$, you can always find a $c$ ... | 3 | https://mathoverflow.net/users/11260 | 436211 | 176,297 |
https://mathoverflow.net/questions/436209 | 3 | Suppose a function $f(x): \mathbb R^d \mapsto \mathbb R^D$, and its stochastic approximator, $g(x; W): \mathbb R^d \mapsto \mathbb R^D$. Here $W$ is some random variable. Then $g(x; W)$ is unbiased in the sense that
$$\mathbb E\_W [g(x;W)] = f(x),$$
for any $x$.
I think the following two are not equal, but how to pro... | https://mathoverflow.net/users/34972 | Expected gradient vs. gradient of expectation | Indeed, the equality will not hold in general. For counterexamples, see [this](https://mathoverflow.net/a/105773/36721) or [this](https://mathoverflow.net/a/172898/36721).
For sufficient conditions for the equality when $d=1$, see e.g. [Folland, Theorem 2.27](http://home.ustc.edu.cn/%7Eluke2001/pdf/realfolland.pdf) o... | 5 | https://mathoverflow.net/users/36721 | 436224 | 176,304 |
https://mathoverflow.net/questions/428448 | 2 | Assume that $B$ is a complete boolean algebra endowed with a Hausdorff topology, with respect to which all operations on $B$ are continuous, $0$ has a base of full sets (recall that $A\subset B$ is full if $b\le a\in A\Rightarrow b\in A$), and for any net $(b\_i)$ in $B$ which decreases to $0$ we have $b\_i\to 0$.
>... | https://mathoverflow.net/users/53155 | Is a Boolean algebra with an order continuous topology a measure algebra? | It is not true that $B$ is necessarily a measure algebra. The counterexample is due to Michel Talagrand, who constructed a Maharam algebra that is not a measure algebra.
>
> *Maharam, D.*, [**An algebraic characterization of measure algebras**](http://dx.doi.org/10.2307/1969222), Ann. Math. (2) 48, 154-167 (1947). ... | 1 | https://mathoverflow.net/users/61785 | 436249 | 176,308 |
https://mathoverflow.net/questions/436261 | 1 | I’m looking for a version of the Cameron Martin theorem for the Brownian motion under random shifts. Here is the precise statement:
Let $\mathbb P$ be Wiener measure on $\Omega := C[0, 1]$. Given a $C[0, 1] $ valued random variable $F$, define the translation map $T\_F: \Omega \to \Omega$ by $T\_F (\omega) = \omega +... | https://mathoverflow.net/users/173490 | Full version of Cameron Martin theorem for Brownian motion | Let $\varphi$ be a smooth function such that $\varphi(x)=0$ for $x\leq 0$ and $\varphi(x)=1$ for $x\geq 1$, and $0\leq\varphi(x)\leq 1$ else. If $\tau\_1$ and $\tau\_2$ are is the first times $B\_t$ hits $1,2$ respectively, and $M$ is a maximum of the $B\_t$ on $[0,1]$, put $F\_t=-M\varphi(\frac{t-\tau\_1}{\tau\_2-\tau... | 3 | https://mathoverflow.net/users/56624 | 436265 | 176,314 |
https://mathoverflow.net/questions/436244 | 5 | First, some motivation. Let $X$ be a complex manifold, and $A$ a Hermitian connection on some complex vector bundle $E$ over $X$. It is known that the existence of $A$ such that the $(0,2)$-part of the curvature form $F\_A$ vanishes implies the existence of a holomorphic structure on $E$.
Now let $M$ be an almost com... | https://mathoverflow.net/users/143629 | Does $F_{A}^{0,2}=0$ for a connection $A$ on $TM$ almost complex give a complex structure? | Here is an example to think about: Let $S^6 = \mathrm{G}\_2/\mathrm{SU}(3)$ be the $6$-sphere endowed with its $\mathrm{G}\_2$-invariant almost Hermitian structure. There is a $\mathrm{G}\_2$-invariant (special) Hermitian connection $A$ on the tangent bundle of $S^6$ whose curvature is of type $(1,1)$, but the only $\m... | 6 | https://mathoverflow.net/users/13972 | 436274 | 176,317 |
https://mathoverflow.net/questions/436275 | 2 | For $X$ compact metric spaces and $f:X\to X$ continuous, is there a nice characterization of the systems $(X,f)$ for which, for every pair of points $x,y\in X$ with disjoint orbits, we have $\omega(x)\cap\omega(y)=\emptyset$?
| https://mathoverflow.net/users/167834 | Dynamical systems with disjoint $\omega$-limits of single points | This seems to be true if and only if every $x \in X$ is eventually periodic.
The reverse direction is obvious; if all points of $X$ are eventually periodic and $x, y$ have disjoint orbits, then each has $\omega$-limit set equal to a different periodic orbit, which must be disjoint.
For the forward direction, assume... | 6 | https://mathoverflow.net/users/116357 | 436277 | 176,318 |
https://mathoverflow.net/questions/436278 | 8 | I would like to know if there is a special name for the following concept, papers that feature something similar or a general reference. Let $\mathcal{C}$ be a category and $\mathcal{D}$ a subcategory (or more generally a subclass of objects and morphisms). For a diagram $F:J\to\mathcal{C}$, say that the $\mathcal{D}$-... | https://mathoverflow.net/users/494777 | Reference for certain categorical limits | This is such a natural notion, that I feel it must be written about somewhere, but I don't think it's in any of my usual sources like Borceux's books, Mac Lane's *Categories for the working mathematician*, or Emily Riehl's book *Category Theory in Context*.
It seems to me that such a notion, if previously invented, m... | 4 | https://mathoverflow.net/users/11540 | 436288 | 176,320 |
https://mathoverflow.net/questions/436290 | 6 | (This is in a sense a follow-up to my [earlier question on a geometric definition of globular $\infty$-groupoids](https://mathoverflow.net/questions/435758/is-there-a-geometric-definition-of-globular-infty-groupoids-categories))
---
We know by [Scholie 8.4.14 of Cisinski's thesis](http://www.numdam.org/item/?id=A... | https://mathoverflow.net/users/130058 | Inexistence of a Kan–Quillen model structure on globular sets | Zhen Lin points out below that I've been way too cavalier with transferring model structures along a reflection. So the following answer is not clearly correct. I will leave this up as community wiki because I still think it addresses the spirit of the question, showing that spaces can be "modeled" in some sense by glo... | 1 | https://mathoverflow.net/users/2362 | 436295 | 176,321 |
https://mathoverflow.net/questions/435698 | 4 | For an algebraic number field $K$, let $\zeta\_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta\_K'(s)$ be its derivative.
I believe that the following statement is true:
$$\zeta\_K\left(\dfrac12\right)\neq 0 \implies \zeta\_K'\left(\dfrac12\right)\neq0.$$
I maganged to prove this for quadratic a... | https://mathoverflow.net/users/175440 | Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$ | If $\zeta\_K(1/2) \neq 0$ then $\zeta'\_K(1/2) = 0$ if and only if
$$
\log |D\_K| = (\log(8\pi) + \gamma) n + \frac\pi2 r\_1,
$$
where $D\_K$ is the discriminant of $K$, and
$\gamma = 0.5772156649\ldots$ is Euler's constant.
We expect that this is impossible because such an equality would give
a closed form for $\gam... | 17 | https://mathoverflow.net/users/14830 | 436297 | 176,322 |
https://mathoverflow.net/questions/436296 | 1 | I was reading an article on Probabilistic Number Theory by M.Kac where I am not able to understand why a particular equation mentioned [here](https://projecteuclid.org/journalArticle/Download?urlId=bams%2F1183513946) in page $657$ equation $(7.7)$ is true?
I do understand that $\frac{(\nu(m)-\log\log n)^2}{\log\log n... | https://mathoverflow.net/users/483436 | Why is $\sum_{m=1}^{n}\frac{(\nu(m)-\log\log n)^2}{n\log\log n}=\int_{-\infty}^{\infty}\omega^2\, \mathrm{d}\sigma_n(\omega)$? | Denote $f\_m=(\nu(m)-\log\log n)(\log\log n)^{-1/2}$ and define $\rho(\omega)=\sum\_{m=1}^n\delta(\omega-f\_m)$, with $\delta(x)$ the [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function). Because of the identity $\int g(\omega)\delta(\omega-\omega\_0)\,d\omega=g(\omega\_0)$, one has
$$n^{-1}\sum\_... | 4 | https://mathoverflow.net/users/11260 | 436299 | 176,323 |
https://mathoverflow.net/questions/436310 | 4 | Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$? That is, if there are infinitely many different $a$ and for which at least one value of $b$ can be found for a given $a$. Some $a$ values are $0,2,3,7,11,17....$
I think that the answer is yes but I have no ... | https://mathoverflow.net/users/265714 | Division problem | If $6a+1$ is a prime and a quadratic residue modulo $17$ (which is true for infinitely many values of $a$), then there are infinitely many positive integers $b$ with the required property.
First observe that $b$ is good if and only if $$f(a,b):=6b^2+6ab+b-6a^2-2a-3$$ is even and divisible by $6a+1$. Hence $b$ must be... | 11 | https://mathoverflow.net/users/11919 | 436319 | 176,331 |
https://mathoverflow.net/questions/436321 | 15 | There is a (most likely folklore) theorem - if in a category every morphism has a right inverse then that category is a groupoid. The proof is an honest oneliner: for $x:A\to B$ find $x':B\to A$ with $xx'=\operatorname{id}\_B$; now for $x'$ find $x'':A\to B$ with $x'x''=\operatorname{id}\_A$. Then $x''=xx'x''=x$, so $x... | https://mathoverflow.net/users/41291 | Is there a higher analog of "category with all same side inverses is a groupoid"? | Yes, this is possible. The following is a classical result of the theory of quasi-categories (You'll find it in the early part of Lurie's Higher topos theory or in Joyal notes on quasi-categories - where they discuss equivalences in quasi-categories):
**Proposition:** A quasi-category $X$ is a Kan complex if and only... | 21 | https://mathoverflow.net/users/22131 | 436322 | 176,332 |
https://mathoverflow.net/questions/436316 | 11 | Let $f\colon\mathbb R^2\to\mathbb R$ be a measurable function such that
\begin{equation\*}
F(t):=\int\_{\mathbb R}dx\,f(t,x)
\end{equation\*}
exists and is finite for all real $t$. Suppose that
\begin{equation\*}
f\_t(t,x):=\frac{\partial f(t,x)}{\partial t}
\end{equation\*}
exists and is finite for all real $t,x$, ... | https://mathoverflow.net/users/36721 | Counterexamples to differentiation under integral sign, revisited | A simple example is given by
$$
f(t,x)=\cases{\exp(-(x-t^{-2})^2),&$t\neq 0$,\\0,&$t=0.$}
$$
For each fixed $t\neq 0$, $\int f(t,x)\,dx$ is a Gaussian integral equal to $\sqrt{\pi}$, while for $t=0$, the integral equals to zero. Therefore, $F(t)$ is not even continuous at $0$, let alone differentiable.
On the other h... | 10 | https://mathoverflow.net/users/56624 | 436333 | 176,336 |
https://mathoverflow.net/questions/436329 | 8 | Consider the following martingale: $X\_1 \sim \mathcal{N}(0, 1)$, and for any $n > 1$, $X\_n \sim \mathcal{N}(X\_{n-1}, X\_{n-1}^2)$ (notice, this is a conditional distribution given $X\_{n-1}$).
I am looking for concentration bounds for $X\_n$, which I suspect exist, based on numerical simulation.
**Additional inf... | https://mathoverflow.net/users/496182 | Concentration bounds for martingales with adaptive Gaussian steps | Observe that $X\_n=X\_{n-1}(1+Z\_n)$ where $\{Z\_k\}\_{k \ge 1}$ are i.i.d. standard normal. Hence to analyze the asymptotic distribution of $|X\_n|$, pass to logarithms, to get $$\log(|X\_n|)= \log(|X\_1|) +\sum\_{k=2}^n \log(|1+Z\_k|) \tag{$\*$} \,.$$
According to Wolfram alpha,
\begin{align}
& \mu:=E\bigl[ \log(|1... | 11 | https://mathoverflow.net/users/7691 | 436338 | 176,338 |
https://mathoverflow.net/questions/436326 | 9 | Let $V$ be an irreducible finite dimensional complex representation of the product of groups $G\times H$. Is it necessarily isomorphic to a tensor product of irreducible representation of $G$ and $H$? If not what is a counter-example, and under what extra assumptions this is known to be true?
Remark. I think for cont... | https://mathoverflow.net/users/16183 | Irreducible representations of a product of two groups | If your groups are finite, then Andy’s [answer](https://mathoverflow.net/a/436339/2383) is perfectly fine. If you want to do topological groups you must be slightly (but not much) more careful. First since we are dealing with finite dimensional representations, algebraic and topological reducibility are the same.
It ... | 6 | https://mathoverflow.net/users/15934 | 436343 | 176,341 |
https://mathoverflow.net/questions/436346 | 47 | It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from [this list](https://en.wikipedia.org/wiki/List_of_lemmas).
When I was first learning algebra, one of the key lemmas we were ... | https://mathoverflow.net/users/3199 | Zorn's lemma: old friend or historical relic? | I agree with almost everything in your post. But still, I believe I know why people use Zorn's lemma.
**My answer.** Zorn's lemma encapsulates succinctly many of the consequences of AC via transfinite recursion, but without requiring any involvement of the ordinals or knowledge of transfinite recursion to be used.
... | 57 | https://mathoverflow.net/users/1946 | 436348 | 176,343 |
https://mathoverflow.net/questions/390681 | 2 | The Routh-Hurwitz criterion explicitly specifies a finite set of inequalities on the coefficients of a polynomial, necessary and sufficient that all zeros lie in the unit circle or in the left half complex plane.
Is there a similar set of explicit inequalities on the coefficients of a (real or complex) matrix, necess... | https://mathoverflow.net/users/56920 | Routh-Hurwitz criterion for matrices | Hmm, a "similar set". Is the following similar enough? I do not know. But it has to be pointed out that there are of course very nice and largely tractable numerical tests for stability of matrices (and much better than attempting to compute the eigenvalues). Let $I$ denote the identity matrix. The spectrum of a (real ... | 2 | https://mathoverflow.net/users/85570 | 436360 | 176,350 |
https://mathoverflow.net/questions/436369 | 8 | This question is naive, but I didn't get an answer at [MSE](https://math.stackexchange.com/questions/4587217/is-there-a-measure-theory-for-proper-classes): Is it straightforward to extend measure theory to proper classes?
Of course when one tries to define measures on "large sets" problems of non-measurability arise ... | https://mathoverflow.net/users/96899 | Is there a measure theory for proper classes? | As a [sledgehammer](https://www.collinsdictionary.com/us/dictionary/english/a-sledgehammer-to-crack-a-nut), working in Higher Order Set Theory (HOST) as proposed in
* *Rhea, Alec*, An Axiomatic Approach to Higher Order Set Theory. (2022)
<https://doi.org/10.48550/arXiv.2206.10060>
you can just go ahead and collect ... | 6 | https://mathoverflow.net/users/92164 | 436373 | 176,354 |
https://mathoverflow.net/questions/436372 | 8 | Let $u$ and $v$ be two vectors in $\mathbb{C}^n$.
Define a permutation of a vector $v':=\sigma(v)$ by $v'\_j = v\_{\sigma(j)}$ for any $\sigma \in S\_n$.
It is easy to show the following for $u,v \in \mathbb{C}^n$ by considering $\sigma$ being a swap:
>
> $u = c\vec{1}$ or $v = c\vec{1}$ for some $c \in \mathbb{C... | https://mathoverflow.net/users/22954 | Use random inner product to test if at least one vector is uniform | This (and specifically the $\Leftarrow$) is false: let
$$u = (-1,0,1)$$
$$v = (1, \omega, \omega^2)$$
where $\omega$ is a cube root of unity.
| 5 | https://mathoverflow.net/users/nan | 436376 | 176,355 |
https://mathoverflow.net/questions/436374 | 2 | Recently I have read many papers which focusing on the image enhancement. The description,
>
> $D$ are the Toeplitz matrices from the discrete gradient with forward difference
>
>
>
occurs many times.
An example is shown below:
To minimize an object function,
$$ F(T)=\sum\_x\left((T(x)-L(x))^2 + \lambda\... | https://mathoverflow.net/users/475095 | What is the "Toeplitz matrices from the discrete gradient operators with forward difference" | A Toeplitz matrix is a band matrix in which each descending diagonal from left to right has constant elements. The derivative of a function can be represented in a finite-difference calculation as the product of a Toeplitz matrix and equally spaced values of $f$. One distinguishes forward differences and backwards diff... | 2 | https://mathoverflow.net/users/11260 | 436378 | 176,356 |
https://mathoverflow.net/questions/436366 | 4 | In Walter Michaelis' paper [*Lie Coalgebras*](https://doi.org/10.1016/0001-8708(80)90056-0), he gives on page 9 an explicit example of a Lie coalgebra which is not the union of its finite-dimensional Lie subcoalgebras. In fact, Michaelis' example has exactly two finite-dimensional Lie subcoalgebras: the zero coalgebra ... | https://mathoverflow.net/users/126183 | Lie coalgebra with no finite-dimensional subcoalgebras | Consider the Lie algebra of vector fields on the formal disk, $\mathfrak g=k[[t]]d/dt$, where $k$ is a field of characteristic zero and $k[[t]]$ is the $k$-algebra of formal Taylor power series in the variable $t$. Then $k[[t]]$ is naturally a topological $k$-algebra with a pro-finite-dimensional ( = pseudocompact = li... | 2 | https://mathoverflow.net/users/2106 | 436385 | 176,359 |
https://mathoverflow.net/questions/436381 | 18 | It may seem silly to ask *"Why are there three types of non-Archimedean geometry?"*, that would be like asking why there are three (and even more) different Weil cohomologies. So I have to clarify my question.
Let $X$ be a scheme on a $p$-adic ring (i.e. an extension of $\mathbb{Z}\_{p}$).
**-** What are the relati... | https://mathoverflow.net/users/169282 | Why are there three kinds of non-archimedean geometry? | Tate's rigid-analytic geometry was the first theory of (global) nonarchimedean geometry to have been devised, and in some sense could be seen as a "proof of concept" that such a theory can exist, despite some general skepticism which was present at the time, including from Grothendieck. For a while this theory was the ... | 19 | https://mathoverflow.net/users/30186 | 436389 | 176,361 |
https://mathoverflow.net/questions/436391 | 1 | I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^\* & 0 \end{pmatrix}.$
The matrix I have is of the form
$$ C = \begin{pmatrix} 0 & a & b & 0 & 0 & 0 \\
\bar a & 0 & 0 &b & 0& 0\\
\bar b & 0 & 0 & a & f & 0 \\
0 & \bar b & \bar a & 0 & 0 &f \\
0 & 0 & \bar f & 0 & 0 & a\\
0... | https://mathoverflow.net/users/119875 | Transforming matrix to off-diagonal form | The general recipe to accomplish the block off-diagonalization is as follows. The matrix $C$ has eigenvalues $\pm\lambda\_1,\pm\lambda\_2,\ldots \pm\lambda\_3$. Define $\Lambda=\text{diag}\,(\lambda\_1,\lambda\_2,\lambda\_3)$, and decompose
$$C=U\begin{pmatrix}\Lambda&0\\ 0&-\Lambda\end{pmatrix}U^\ast,$$
with $U$ the u... | 2 | https://mathoverflow.net/users/11260 | 436397 | 176,366 |
https://mathoverflow.net/questions/436353 | 8 | Let $X\_0$ be a compact complex algebraic surface with an isolated singularity and let $X\_t$ be a smoothing of $X\_0$ over the disc. How can we compute the fundamental group of $X\_t$ say in terms of the topology of a minimal resolution and some local information of the singularity? If it helps we can assume the singu... | https://mathoverflow.net/users/12402 | Fundamental group of a smoothing of a complex surface | Non surprisingly, this usually involves Seifert-Van Kampen theorem, but the actual computation can be a tricky one. However, since you have just one singularity, life will be probably easier.
For an example where the fundamental group turns out to be trivial, you can look at the celebrated paper by Lee and Park, in w... | 5 | https://mathoverflow.net/users/7460 | 436398 | 176,367 |
https://mathoverflow.net/questions/436405 | 7 | One model of Robinson arithmetic which is obviously not our usual integers is $\mathbb{Z}[X]^+$, that is the set containing 0 and also all polynomials with coefficients in $\mathbb{Z}$ with positive lead coefficient. This ring also satisfies a bit more, since it satisfies distributivity, commutativity, and associativit... | https://mathoverflow.net/users/127690 | Systems intermediate in strengthen between Robinson arithmetic and PA | Yes. For example, you can take the nonnegative part of the ring of polynomials in $\mathbb Q[X]$ with integer constant coefficient (i.e., $\mathbb Q[X]X+\mathbb Z$).
For a more sophisticated example due to Shepherdson [2], the nonnegative part of the ring of *Puiseux polynomials* $\sum\_{i\le n}a\_iX^{i/k}$ over a re... | 16 | https://mathoverflow.net/users/12705 | 436406 | 176,368 |
https://mathoverflow.net/questions/436393 | 7 | Let $F : \mathbb C \to \mathbb C$ be an entire function of finite order. Since the zeros of $F$ are countable there exists a constant $c \in \mathbb R$ such that $F$ is zero-free on the line $e^{ic} \mathbb R$. I'm wondering of the following stronger statement holds true: there exists a $c \in \mathbb R$ and a $d>0$ su... | https://mathoverflow.net/users/170539 | Are entire functions uniformly bounded from below on a line through the origin? | The answer is negative, and here are two ways to construct counterexamples.
1. Let $G$ be a bounded simply connected region whose closure does not contain zero, but $G$ intersects every ray from the origin. For example, $G$ can be a neighborhood of a sufficiently long compact piece of a logarithmic spiral.
Consider t... | 9 | https://mathoverflow.net/users/25510 | 436410 | 176,369 |
https://mathoverflow.net/questions/436411 | 1 | Let $X$ be a smooth Fano threefold with a finite group $G$ action. Assume that the orbit space $X/G$ is smooth. Is it true that $J(X/G)\cong J(X)^G$ As an abelian variety? Here, $J(X)^G$ is the $G$-invariant part of $J(X)$.
I am particular interested in the case that $G$ is $\mathbb{Z}\_2=\langle 1,\tau\rangle$ and $... | https://mathoverflow.net/users/41650 | Intermediate Jacobian under group action | No, because there is no reason for $J(X)^G$ to be connected. Here is a silly example: consider a smooth elliptic curve $C\subset \mathbb{P}^3$ given by $\sum x\_i^2=\sum a\_ix\_i^2=0$; take for $X$ $\mathbb{P}^3$ with $C$ blown up, and let $\tau $ be the involution of $\mathbb{P}^3$ which changes the sign of one coordi... | 2 | https://mathoverflow.net/users/40297 | 436419 | 176,372 |
https://mathoverflow.net/questions/436218 | 11 | $\newcommand{\Z}{\mathbb Z/p\mathbb Z}$
Can one partition a group of prime order as $A\cup(A-A)$ where $A$ is a subset of the group, $A-A$ is the set of all differences $a'-a''$ with $a',a''\in A$, and the union is disjoint?
As stated, the answer is "yes", at least if the order of the group is $p\equiv 2\pmod 3$, in ... | https://mathoverflow.net/users/9924 | $\mathbb Z/p\mathbb Z=A\cup(A-A)$? | *This is a previous comment which was moved to [chat](https://chat.stackexchange.com/rooms/141236/discussion-on-question-by-seva-mathbb-z-p-mathbb-za-cupa-a) by Ben Webster:* In fact for every prime $p$ if $A=[-(2m-1),-m] \cup [m, 2m-1]$ for $(p+3)/8\le m<(p+3)/6$, then $\mathbb Z/p\mathbb Z$ is a disjoint union of $A$... | 5 | https://mathoverflow.net/users/18739 | 436422 | 176,373 |
https://mathoverflow.net/questions/436428 | 4 | Call a topological space $X$ standard Borel if $X$ is standard Borel as a measurable space (equipped with its Borel $\sigma$-algebra), i.e. if there is a Borel isomorphism between $X$ and a Polish space.
Clearly all Polish spaces are standard Borel by definition, but the converse is not true (any Borel subset of a Po... | https://mathoverflow.net/users/160416 | Which topological spaces have a standard Borel $\sigma$-algebra? | Here are two examples showing that none of your candidate notions work.
First, we can observe that every Quasi-Polish space (<https://doi.org/10.1016/j.apal.2012.11.001>) admits a Baire class 1 isomorphism to a Polish space, and thus has a standard Borel $\sigma$-algebra. However, take e.g. the Scott domain $\mathcal... | 5 | https://mathoverflow.net/users/15002 | 436430 | 176,375 |
https://mathoverflow.net/questions/435800 | 16 | Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod\_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \mathbb{R}$ to the family $(x \mod a) \_ {a>0}$; depending on your inclination, you may prefer to view it as $\Phi \colon ... | https://mathoverflow.net/users/17064 | Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$? | Short answer: $S$ is known as the *Bohr compactification* of $\mathbf R$ (often denoted $S=b\mathbf R$, see [1], 26.11), and $R$ is known as $\mathbf R$ with the *Bohr topology* (often denoted $R=\mathbf R^+$, see e.g. [2, 3, 4]).
For a longer answer, write $\chi\_b(x)=\exp(2\pi ibx)$ so that your $\Phi(x)=(\chi\_b(x... | 16 | https://mathoverflow.net/users/19276 | 436436 | 176,379 |
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