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https://mathoverflow.net/questions/333850 | 4 | It is known that any subgroup of the symmetric group on $n$ elements can be generated by a linear number (in $n$) of elements. Can we choose a small set of generators such that the resulting maximal word length is also small? For example, both linear, or perhaps polynomial?
| https://mathoverflow.net/users/112954 | Number of generators and word length for subgroups of symmetric group | For a subgroup $G \le S\_n$ we can choose generating sets $X$ that satisfy
any of the following three bounds for the
maximum lengths of elements of $G$ as words over $X$.
1. $|X| \le n(n-1)/2$, maximum word length $n-1$.
2. $|X| \le n-1$, maximum word length $n(n-1)/2$.
3. $|X| \le n\log n$, maximum word length $2n\l... | 6 | https://mathoverflow.net/users/35840 | 333982 | 142,789 |
https://mathoverflow.net/questions/333990 | 1 | ### Motivation:
At the present time it really isn't clear to me why this question might be inappropriate for the MathOverflow. However, it appears that some people are down-voting this question even if the notation we use is a key part of doing mathematics. Mathematicians of the contrary opinion may read Terrence Tao... | https://mathoverflow.net/users/56328 | Symbol for monotone relationship between two probability distributions | See, for example, [this paper](http://www.ams.org/journals/proc/1998-126-05/S0002-9939-98-04702-9/S0002-9939-98-04702-9.pdf), and apply the definition to $X=\mathbb{R}^n$:
**Definition 1**. Two functions $f : X → \mathbb{R},$ and $g : X → \mathbb{R},$ are said to be *similarly
ordered*, in short $f$ s.o. $g,$ if
$$(f... | 3 | https://mathoverflow.net/users/17773 | 333991 | 142,791 |
https://mathoverflow.net/questions/333985 | 2 | I already asked this question on MSE here <https://math.stackexchange.com/questions/3254184/cute-striking-applications-of-snake-lemma-outside-homological-algebra>, but still received no answer. I hope I will be more lucky here.
When you teach algebra to students, it's often easy to find cute/direct applications of "... | https://mathoverflow.net/users/36683 | Cute/striking application(s) of snake lemma outside homological algebra | The Snake Lemma is used in the Iwasawa theory of elliptic curves. This is a branch of number theory.
Control theorems in Iwasawa theory following Mazur's control theorem all rely on repeated use of the Snake Lemma. See Theorem 1.2 of Greenberg's article <https://arxiv.org/pdf/math/9809206.pdf> for a statement of this... | 2 | https://mathoverflow.net/users/nan | 334004 | 142,796 |
https://mathoverflow.net/questions/334036 | 3 | Consider a simple regular graph $G$ with $n$ vertices and $E$ edges. Then, can we say that the edges can be colored equitably in $\Delta+1$ colors? By equitability is meant that a proper $\Delta+1$ coloring has either $\left\lceil\frac{E}{\Delta+1}\right\rceil$ or $\left\lfloor\frac{E}{\Delta+1}\right\rfloor$ independe... | https://mathoverflow.net/users/100231 | Equitable edge coloring of graphs | Yes. Choose any proper edge coloring with $\Delta+1$ colors (it exists by Vizing theorem). If we have two color classes with $a$ and $b$ edges respectively, $a\geqslant b+2$ (say, $a$ red eges and $b$ blue edges), consider the graph formed by these $a+b$ red or blue edges. It contains a component with more red edges th... | 9 | https://mathoverflow.net/users/4312 | 334041 | 142,809 |
https://mathoverflow.net/questions/330324 | 5 | As is mentioned in the introduction of [this](https://arxiv.org/abs/1807.07486v1) paper of Spodzieja there is a lack of 'natural' examples of differentially closed fields. The immediate naive guesses, namely the field of germs of meromorphic functions at some point, only partially work in that these structures contain ... | https://mathoverflow.net/users/83901 | Constructing a model of $\mathrm{DCF}_0$ via forcing | I had a conversation about this with James Freitag at a conference recently and he pointed me to a fundamental result of Seidenberg that seems to resolves this problem in the positive.
The relevant result is from Seidenberg's paper 'Abstract Differential Algebra and the Analytic Case II.' Here it is restated slightly... | 4 | https://mathoverflow.net/users/83901 | 334046 | 142,811 |
https://mathoverflow.net/questions/333978 | 7 | (This is a follow-up to [this question](https://mathoverflow.net/questions/333846/shimura-varieties-and-connected-components) of mine.)
Is there an example of a connected reductive algebraic group $G$ over $\mathbb{R}$ such that:
* $G$ is not isomorphic to a product $G\_1 \times G\_2$ of smaller groups (*isogenous*... | https://mathoverflow.net/users/2481 | Connected components of real Lie groups | There is NO such example.
Note that any semisimple algebraic ${\mathbb{R}}$-group $H$ of Hermitian type has a *compact* (anisotropic) maximal torus.
Indeed, by a definition of a group of Hermitian type
(see, e.g., [Deligne, Travaux de Shimura](http://www.numdam.org/item/SB_1970-1971__13__123_0/), condition (1.5.3)... | 6 | https://mathoverflow.net/users/4149 | 334048 | 142,812 |
https://mathoverflow.net/questions/332821 | 5 | With great pleasure I read the [recent paper of Griffin, Ono, Rolen and Zagier](https://www.pnas.org/content/early/2019/05/20/1902572116) proving the surprising result that the Jensen polynomials $J^{d, n}\_\alpha$ for a sequence $\alpha = \{\alpha(0), \alpha(1), \ldots \}$ of real numbers whose growth (?) is controlle... | https://mathoverflow.net/users/41139 | Intuition behind the growth condition in the result of Griffin, Ono, Rolen and Zagier on Jensen polynomials | The condition does restrict the rate of growth of the functions considered, and $E$, $C\_i$, and $\delta$ do have a certain amount of freedom, however they are meant to encode specific information. The $E$ term is meant to account for the exponential part of the growth of $\alpha(n)$. Once $E$ has been fixed, the right... | 5 | https://mathoverflow.net/users/61910 | 334050 | 142,813 |
https://mathoverflow.net/questions/334053 | 1 | How wild can the topology of a length space be? That is,
**Questions:**
1. Let $X$ be a metric space where the distance between two points $x,y \in X$ is the infinum of lengths of rectifiable paths from $x$ to $y$. What can be said about the topology of $X$? Can spaces of this form be completely characterized?
2. ... | https://mathoverflow.net/users/2362 | Topology of length spaces | 1 Look at 3.
3 It is almost obvious from the definition that length spaces are locally path connected, and if you assume local connectedness, 2 is true, but nontrivial:
Bing, R. H. A convex metric for a locally connected continuum. Bull. Amer. Math. Soc. 55 (1949), no. 8, 812--819. <https://projecteuclid.org/euclid... | 2 | https://mathoverflow.net/users/113851 | 334054 | 142,814 |
https://mathoverflow.net/questions/333988 | 3 | For Compact pmp actions of a group $G$ on measure space $(X,\mu)$, if a subalgebra $B$ is such that $L(G)\subset B \subset L^{\infty}(X)\rtimes G$, is it always true that $B$ is also of the form crossed product with action of $G$.
| https://mathoverflow.net/users/136400 | On crossed product subalgebra | You tagged your question with "c-star-algebras" and "von-neumann-algebras" but you obviously seem to be mostly interested in the W\*-crossed product, rather than its C\*-version. Nevertheless let me give a counter example in the C\*-case, hoping that it may be modified to fit your interest.
Let $X$ be a set with two ... | 2 | https://mathoverflow.net/users/97532 | 334067 | 142,817 |
https://mathoverflow.net/questions/314834 | 10 | Recently, I am interested in the graph polynomial of the product of cycles.
Let $G = (V , E)$ be an undirected multi-graph with vertex set $\{1,\cdots,n\}$. The graph polynomial of $G$
is defined by
$$f\_G(x\_1,x\_2,\cdots,x\_n)=\prod\_{1\leq i<j\leq n, (i,j)\in E}(x\_i-x\_j).$$
**Conjecture:** Let $G $ be the Cartes... | https://mathoverflow.net/users/42816 | A conjecture on the coefficient of a special term in the expansion of the graph polynomial? | The following proof is obtained jointly with Alexey Gordeev.
We start with a general
**Lemma.** Let $H=(X,E)$ be a $2d$-regular graph on the vertex set $X$, $G=H\square C\_k$ be the product of $H$ and a cycle of length $k$. Fix a field $\mathbb{F}$ and a subset $A\subset \mathbb{F}$, $|A|=d+2$. Let $\mathcal{U}$ de... | 9 | https://mathoverflow.net/users/4312 | 334071 | 142,819 |
https://mathoverflow.net/questions/334064 | 6 | Let $G$ be a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$.
Is there some sort of formula for $\tau(S\_n)$, for the symmetric group $S\_n$?
Here $XXX$ stands for $\{abc| a, b, c \in X\}$.
Similar problems for some different c... | https://mathoverflow.net/users/110691 | Is there some sort of formula for $\tau(S_n)$? | Here is a lower bound: let $H$ be a subgroup of $S\_{n}$ containing $(12)$ and let $\sigma$ be the $n$-cycle $(12 \ldots n)$. Let $X = H \cup \{\sigma \}$ and note that $\langle X \rangle = S\_{n}$ since we already have $S\_{n} = \langle (12),\sigma \rangle.$
Suppose that we have chosen $H$ so that $|X| > \tau(S\_{n... | 5 | https://mathoverflow.net/users/14450 | 334073 | 142,820 |
https://mathoverflow.net/questions/334074 | 7 | There are informative and easily accessible images and videos that illustrate the Hopf fibration $S^3\to S^2$ by describing what happens to the fibers in the unit cube $(0,1)^3\approx S^3\backslash \ast$, e.g. [those by Niles Johnson](https://nilesjohnson.net/hopf.html). Is there a similar way to visualize the map $S^3... | https://mathoverflow.net/users/39609 | Visualizing a Whitehead product: the attaching map $S^3\to S^2\vee S^2$ | The natural generality is as follows. For any finite-dimensional inner-product spaces $U$ we have a unit sphere $S(U)\subset U$ and a one-point compactification $S^U\simeq S(U\oplus\mathbb{R})$. If $V$ is another finite-dimensional inner-product space, then we can ask about the attaching map of the top cell in the prod... | 6 | https://mathoverflow.net/users/10366 | 334080 | 142,821 |
https://mathoverflow.net/questions/334105 | 5 | Does every compact metric space continuously embed into a Hilbert space (possibly with large distortion)?
| https://mathoverflow.net/users/12518 | Embedding metric spaces into Hilbert ones | You may also send $X$ into a space $L\_2(X,\mu)$ via the Fréchet-Kuratowski isometry $x\mapsto d(\cdot,x)\in C^0(X),\|\cdot\|\_\infty$, followed by the bounded linear inclusion $C^0(X)\to L\_2(X,\mu)$, where $\mu$ is a probability measure on $X$. If $\text{supp}(\mu)=X$ (e.g. $\mu$ is a series of deltas $\sum\_{k=1}^\i... | 5 | https://mathoverflow.net/users/6101 | 334106 | 142,827 |
https://mathoverflow.net/questions/334034 | 2 | Fix $f: \mathbb{R}\times \mathbb{R}^n \mapsto \mathbb{R}^n$ and $v\_0:\mathbb{R}^n \mapsto \mathbb{R}^n$. Let $X\_t$ be the solution to the second-order ODE
$$\frac{d^2}{dt^2}X\_t = f(t,X\_t), \quad X\_0 = id, \frac{d}{dt}X\_t|\_{t = 0}=v\_0.$$
Under what conditions on $f$ and $v\_0$ is $X\_t$ a diffeomorphism?
| https://mathoverflow.net/users/57697 | Diffeomorphisms as solutions to second-order ODEs | First, I just want to consider the requirement that $X\_t$ be a *local* diffeomorphism, which by the inverse function theorem is equivalent to requiring the Jacobian of $X\_t$ to never vanish. We have
$$ X\_t = id + \int\_0^t V\_s\,ds = id + \int\_0^t \left(v\_0 + \int\_0^s f\_r(X\_r)\,dr\right)\, ds \\= id + v\_0 t + ... | 1 | https://mathoverflow.net/users/5279 | 334114 | 142,828 |
https://mathoverflow.net/questions/334103 | 7 | How large can the chromatic number of an $n$-vertex $C\_4$-free graph be? If the maximum degree of the graph $G$ is $\Delta$, is there a bound of the form
$\chi(G) \leq O(\Delta/\log(\Delta))$ as in the case of triangles? What happens if $e(G)$ is close to $ex(n,C\_4)$, say $e(G) \geq n^{3/2-\alpha}$; is there a bette... | https://mathoverflow.net/users/141963 | Chromatic number of $C_4$-free graphs | For $G$ an $n$ vertex graph which is $C\_4$-free, $\chi(G)=O(\sqrt{n})$, follows from Kővári–Sós–Turán by the argument found [here](https://math.stackexchange.com/questions/2419506/chromatic-number-of-a-graph-with-no-4-cycles) for instance.
Before Johannson proved the chromatic number bound for triangle free graphs, ... | 4 | https://mathoverflow.net/users/118731 | 334120 | 142,832 |
https://mathoverflow.net/questions/334013 | 8 | Fix a prime $p$. I have a sketch of a proof that if $X$ is a finite simply-connected CW complex with $\mathrm{dim}(X) < p$ then for some $t\in \mathbb{N}$, the $p$-localization $\Sigma^t X\_{(p)}$ is a wedge of Moore spaces.
(Basically, the idea is that all the interesting attaching maps are Whitehead products, henc... | https://mathoverflow.net/users/3634 | Splitting low-dimensional $p$-local CW complexes for large $p$ | This result appears in the PhD thesis of Hans-Werner Henn, and in this paper:
```
@article {MR884630,
AUTHOR = {Henn, Hans-Werner},
TITLE = {Classification of {$p$}-local low-dimensional spectra},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {45},
... | 6 | https://mathoverflow.net/users/10366 | 334124 | 142,833 |
https://mathoverflow.net/questions/334143 | 2 | Question:
----------
Let's suppose that $S \subset \mathbb{R}^n$ is convex and symmetric so:
\begin{equation}
x \in S \iff -x \in S \tag{1}
\end{equation}
Now, if we define the radius of $S$ as $R$ such that:
\begin{equation}
R = \sup\_{x \in S} \lVert x \rVert \tag{2}
\end{equation}
and use (2) to define:
... | https://mathoverflow.net/users/56328 | A conjecture concerning symmetric convex sets | Consider the set
$$S:= \{ (x,y) \in \mathbb R^2 : x^2+4y^2 \leq 4 \}$$
This is convex and symmetric, and $R=2$.
But $V= \{ (2,0), (-2,0) \}$ and $\mbox{conv}(V)= \{ (x, 0) : -2 \leq x \leq 2 \} \neq S$.
**P.S.** About the new question.
Let $A=(0,1), C=(0,-1)$ and $B,D$ be the intersection between the circle $x... | 6 | https://mathoverflow.net/users/11552 | 334144 | 142,840 |
https://mathoverflow.net/questions/334075 | 5 | Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]$. Two obvious solutions for $F$ and $H$ are
* $F = I$, the identity function, and $H(u,x,y,z) = \alpha u x + \beta... | https://mathoverflow.net/users/110301 | Does there exist another form of the derivative for polynomials? | It's not hard to see that $H$ must be of the form $\alpha ux + \beta uz + \gamma yx + \delta yz$ by $\mathbb{R}$-linearity of $F$ (see Jan-Cristoph Schlage-Puchta's answer for a fuller explanation).
By using symmetry of multiplication, we can assume without loss of generality that $\beta = \gamma$.
We have the equ... | 8 | https://mathoverflow.net/users/44191 | 334151 | 142,844 |
https://mathoverflow.net/questions/334111 | 2 | Assume we have a connected CW-complex $Y$ and $X\hookrightarrow Y$ a connected sub-complex. We know that the inclusion induces injection on all homotopy groups. Is it true (or under what conditions it can be true) that we can attach cells to $Y$ so that the inclusion induces isomorphism on homotopy groups? (This will s... | https://mathoverflow.net/users/127776 | Turning injection of homotopy groups to an isomorphism | Your question is equivalent to the following:
*Given a cellular inclusion $i : X\to Y$, when is there a retraction $r:Y \to X$?*
(Being a retraction means that $r\circ i: X\to X$ is the identity.)
The answer is usually phrased in terms of obstruction theory.
For simplicity, let's assume that $Y$ is a finite c... | 7 | https://mathoverflow.net/users/8032 | 334155 | 142,846 |
https://mathoverflow.net/questions/334158 | 1 | <http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf> gives a general method to solve quadratic bivariate diophantine equation while [Coppersmith introduced a method to solve bivariate polynomials](https://en.wikipedia.org/wiki/Coppersmith_method) which work provably and have been shown to break $RSA$ syste... | https://mathoverflow.net/users/10035 | Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver? | Partial answer about "regular diophantine solver".
Finding points on conics in general require integer factorization.
Several papers deal with points on $a x^2+b y^2=c z^2$
and they require factorization of $a,b,c$.
Another example is $x^2 - a y^2=n z^2$. Solving it
will compute the square root of $a$ modulo $n$.... | 0 | https://mathoverflow.net/users/12481 | 334167 | 142,848 |
https://mathoverflow.net/questions/334164 | 2 | Suppose $E$ is an elliptic curve over a number field with good ordinary reduction at the primes above a fixed odd prime $p$. We are interested in the Iwasawa theory over the cyclotomic $\mathbb{Z}\_p$ extension under the additional assumption that $E$ has a point of infinite order and also that $L(E,s)$ has a simple ze... | https://mathoverflow.net/users/nan | Euler characteristics in the rank one case | Let $E$ be an elliptic curve over a number field $F$ and $p$ a prime such that $E$ has good ordinary reduction at all places above $p$. Suppose we know that the dual $X$ of the usual Selmer group over the cyclotomic $\mathbb{Z}\_p$-extension is torsion, e.g. if $F=\mathbb{Q}$.
Then we can compute the order of vanishi... | 2 | https://mathoverflow.net/users/5015 | 334169 | 142,849 |
https://mathoverflow.net/questions/334096 | 7 | A well known model theory fact is that for any first-order theory $T$ the collection of universal consequences of $T$, written $T\_\forall$, is a precise axiomatization of the class of substructures of models of $T$. This is related to the fact that a sentence is preserved under passing to substructures if and only if ... | https://mathoverflow.net/users/83901 | A mixed preservation theorem for two-sorted structures? | $\let\fii\varphi\def\fk{\mathfrak}\def\cL{\mathcal L}\def\aeba{A\exists B\forall}\let\TO\Rightarrow$Here is a quick and dirty proof that Q1 is true for countable languages, using an approach to preservation theorems suggested by §1.5 of Barwise & Schlipf, *An introduction to recursively saturated and resplendent models... | 2 | https://mathoverflow.net/users/12705 | 334179 | 142,850 |
https://mathoverflow.net/questions/273764 | 21 | The [McKay conjecture](https://en.wikipedia.org/wiki/McKay_conjecture) and related ([Alperin](https://groupprops.subwiki.org/wiki/Alperin-McKay_conjecture), [Issacs-Navarro](https://groupprops.subwiki.org/wiki/Isaacs-Navarro_conjecture)) are one of the "main problems in the representation theory of finite groups" ([G.N... | https://mathoverflow.net/users/10446 | McKay conjecture for finite groups in the simplest case G=GL(2,F_p) ( puzzle: Borel knows about cuspidals) | Here is an answer of sorts. In this case, by Brauer's First Main Theorem, there is a bijection between $p$-blocks of $G = {\rm GL}(2,p)$ with defect group $P$ and $p$-blocks of $B = N\_{G}(P)$ with defect group $P$. Choose a block $\beta$ of $G$ with defect group $P$ and let $\gamma$ be the unique Brauer correspondent ... | 6 | https://mathoverflow.net/users/14450 | 334195 | 142,857 |
https://mathoverflow.net/questions/334201 | 7 | In some work on QFT the following identity has come up:
$$
\sum\_{\sigma \in S\_n}\sum\_{j=1}^n \left(\sum\_{l=1}^j w\_{\sigma\_l}\right)\prod\_{i=1,i\neq j}^{n}\frac{1}{\sum\_{l=1}^j z\_{\sigma\_l}-\sum\_{l=1}^i z\_{\sigma\_l}}=0
$$
for $2<n$ where $w\_1,...,w\_n$ are arbitrary numbers and $z\_1,...,z\_n$ are such tha... | https://mathoverflow.net/users/120080 | Identity involving sum over permutations | **June 18, 2019 Edit: The identity is now proved and generalized. See below.**
I did not have time to look at your identity for very long but I am pretty sure it follows from Lemma II.2 in my article ["Trees, forests and jungles: a botanical garden for cluster expansions"](https://arxiv.org/abs/hep-th/9409094) with V... | 4 | https://mathoverflow.net/users/7410 | 334204 | 142,859 |
https://mathoverflow.net/questions/334122 | 3 | I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, let $\mathcal{E}$ be a quasicoherent sheaf on a scheme $S$ and we define the functor $F: Sch/S \rightarrow \text{Set}$ ... | https://mathoverflow.net/users/115048 | Representability of Grassmannian functor by a scheme | First of all, being quasicompact is not a "strong finiteness assumption", come on :). For what you're doing you're actually free to restrict $F$ to quasi-compact quasi-separated schemes, or even just affine schemes over $S$, because the inclusion of sites $Aff/S \to Sch/S$ induces an equivalence of categories of sheave... | 3 | https://mathoverflow.net/users/85136 | 334210 | 142,861 |
https://mathoverflow.net/questions/332942 | 5 | Let $\mathsf{M}$ be a **simplicial model category** presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-categorical sense) with the compact objects in $\mathsf{M}$. Here's roughly what I expect to be true but if i'm missing some... | https://mathoverflow.net/users/22810 | Compact objects in the $\infty$-category presented by a simplicial model category | If $X$ is such that $X \times \Delta^n$ is compact for every $n$ then yes. This happens, for example, if the cotensor functor $(-)^{\Delta^n}$ preserves filtered colimits, a condition which is quite common. Sufficient conditions of a similar nature are described in Proposition 5.3.1 of [this paper](https://arxiv.org/ab... | 3 | https://mathoverflow.net/users/51164 | 334218 | 142,862 |
https://mathoverflow.net/questions/334199 | 2 | Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum\_{k=1}^d \lambda\_k^2$, where $\lambda\_1,\dots,\lambda\_d$ are the eigenvalues of $S$.
**What I tried:**
* We know that (see e.g. Corollary 5.35 in Ve... | https://mathoverflow.net/users/127150 | Sum of Square of the Eigenvalues of Wishart Matrix | I will assume $m=\alpha\_d d$ with $\alpha\_d\to \alpha \in [1,\infty)$ independent of $d$. The case $\alpha\to\infty$ is actually easier.
Define $Z=d^{-1} m^{-2} \sum\_{i=1}^d \lambda\_i^2$. Then $Z$ converges a.s. to $\int x^2 d\mu\_\alpha(x)$ where $\mu\_\alpha$ is the Pastur-Marchenko distribution of parameter
$\... | 1 | https://mathoverflow.net/users/35520 | 334219 | 142,863 |
https://mathoverflow.net/questions/334185 | 1 | What is the/a main reference book for spaces with curvature bounded from below (CBB spaces/spaces with curvature $\geq \kappa$ in the sense of Alexandrov)? Looking for an up to date reference.
| https://mathoverflow.net/users/54495 | Reference request: metric spaces with curvature bounded from below (CBB) spaces | You may check preliminary version of [our book/3](https://arxiv.org/abs/1903.08539).
Otherwise do "Alexandrov spaces with curvature bounded below" by Burago, Gromov, and Perelman and ["Alexandrov's space with curvatures bounded from below II" by Perelman](https://anton-petrunin.github.io/papers/alexandrov/perelmanASWCB... | 2 | https://mathoverflow.net/users/1441 | 334224 | 142,866 |
https://mathoverflow.net/questions/334221 | 6 | If $\mu$ is the normalized counting measure on a finite group $G$, then $\mu(G)=1$ and $\mu(C)=1/n$ for every coset $C$ of a subgroup of index $n$. Let's ask for the same for infinite groups:
**Question:** When $G$ is any group, is there a finitely additive measure $\mu$ on $G$ such that for every positive integer $n... | https://mathoverflow.net/users/95282 | Measure on cosets in a group? | This should be possible for any group, assuming I have facts about extending measures correct. The idea is to first construct such a measure on a smaller Boolean algebra of subsets of $G$, and then use general measure theory facts to extend this measure to $\mathcal{P}(G)$.
So first, let $\mathcal{B}\subseteq\mathcal... | 10 | https://mathoverflow.net/users/38253 | 334225 | 142,867 |
https://mathoverflow.net/questions/334102 | 1 | For every large enough $T$ given four integers $a,b,c,d$ with absolute value less than $T^2$ are there integers $w\_1,x\_1,y\_1,z\_1,w\_2,x\_2,y\_2,z\_2$ with absolute value less than $T$ such that
$$w\_1x\_1+y\_1z\_1=a$$
$$w\_2x\_1+y\_2z\_1=b$$
$$w\_1x\_2+y\_1z\_2=c$$
$$w\_2x\_2+y\_2z\_2=d$$
holds?
If not for what f... | https://mathoverflow.net/users/10035 | Representing integers efficiently with quadratic polynomials | Here is an answer. If it inspires you to change the question, please don't. Just ask a new one.
No, you can't even say that the $8$ unknowns are bounded by $T^2-1.$
To recast, given an integer matrix $A=\begin{pmatrix}
a & c \\
b & d
\end{pmatrix}$ with entries bounded in absolute value by $T^2,$ you wonder abou... | 5 | https://mathoverflow.net/users/8008 | 334227 | 142,869 |
https://mathoverflow.net/questions/334236 | 8 | A paper [see here](https://arxiv.org/abs/1609.03435) on arXiv claims that Chowla's conjecture (applied to the Liouville function instead of the Mobius function), i.e., that
$$
\lim\_{N\rightarrow \infty} \sum\_{n\leq N}
\lambda(n+a\_1) \lambda(n+a\_2) \cdots \lambda(n+a\_k)=o(N),
$$
implies the Riemann hypothesis. I ... | https://mathoverflow.net/users/17773 | Does Chowla's conjecture on the Liouville function imply the Riemann hypothesis? | D Karagulyan, [On certain aspects of the Mobius randomness principle](https://pdfs.semanticscholar.org/4ae8/e59b56bac940430de06c72b7caa7500d056a.pdf), writes (Remark 1, page 9), "We remark, that the result proved above contradicts with what is claimed in [Reference 1]. There it is stated, that for the Liouville functio... | 10 | https://mathoverflow.net/users/3684 | 334238 | 142,872 |
https://mathoverflow.net/questions/334244 | 4 | Let $S$ be the set of $n!$ permutations of the first $n$ integers. Let $p\in(0,1)$. Consider the Markov Process defined on the elements of $S$.
1. Let $x\in S$. Choose $1\le i <i+1 < n$ uniformly at random.
2. If $x\_i < x\_{i+1}$, swap $x\_i$ and $x\_{i+1}$ with probability $p$, otherwise do nothing. If $x\_i > x\_{... | https://mathoverflow.net/users/9118 | Stationary distribution of a Markov process defined on the space of permutations | Your conjecture is correct. In fact, provided $0 < p < 1$, the Markov process is recurrent and reversible with unique stationary distribution proportional to
$$ \pi\_\sigma = \Bigl(\frac{p}{1-p}\Bigr)^{\ell(\sigma)},$$
where $\ell(\sigma)$ is the Coxeter length of $\sigma$. (This is the number of 'misrankings' in your... | 5 | https://mathoverflow.net/users/7709 | 334256 | 142,876 |
https://mathoverflow.net/questions/334213 | 2 | I am reading Thaddeus' paper on GIT and flips (<https://arxiv.org/pdf/alg-geom/9405004.pdf>), and I am confused with a claim in the begining.
Let $R$ be a finitely generated integral algebra over an algebraically closed field $k$, and $X = Spec~R$. Choose a $\mathbb{Z}$-grading on $R$. (It is possible to find such g... | https://mathoverflow.net/users/43027 | Relation between the Spec and the Proj of a ring | Clearly $\text{Proj}k[z]\simeq \text{Spec}k$ (projective $0$-space is a point) and thus $\text{Proj}R[z]\simeq \text{Spec}R\times\_{\text{Spec}k}\text{Proj}k[z]\simeq \text{Spec}R\times\_{\text{Spec}k}\text{Spec}k\simeq \text{Spec}R$. The grading on $R[z]$ when $R=k[x]$ is not the grading on $k[x,z]$ by degree, $x^iz^j... | 4 | https://mathoverflow.net/users/nan | 334257 | 142,877 |
https://mathoverflow.net/questions/334268 | 3 | Consider two Banach spaces $E,F$ and a net $T\_\alpha : E \to F$ of continuous operators.
I know that for each $x \in E$ the net $T\_\alpha (x)$ is convergent in $F$ and it is easy to show that the limit $L(x)$ is a linear function.
Can one deduce that $L$ is bounded?
I thought originally that this is an obvious ... | https://mathoverflow.net/users/11552 | Question about pointwise convergence of operators | No, this works for sequences but not general nets. For a counterexample, let $T: E \to \mathbb{R}$ (or $\mathbb{C}$) be an unbounded linear functional. For each finite dimensional subspace $E\_0$ of $E$, the restriction of $T$ to $E\_0$ is bounded so by the Hahn-Banach theorem it can be extended to a bounded linear fun... | 3 | https://mathoverflow.net/users/23141 | 334269 | 142,878 |
https://mathoverflow.net/questions/334267 | 0 | Let $G$ be a regular connected simple graph on $n$ vertices with chromatic number $\chi$ and maximum degree $\Delta$. Then, it is implied that $G$ is $\chi$-partite. Suppose, we remove one of the partite set of vertices. Then, what would be the maximum degree of the induced subgraph formed by the remaining vertices?
... | https://mathoverflow.net/users/100231 | Highly asymmetric regular graph | Consider a complete $k$-partite graph $G$ where all parts have size $n/k >> k$. Then $\Delta = n(k-1)/k$, and even after removing a part the degree would still be $n(k-2)/k$ $=\Delta - n/k$. [I am not sure precisely what you mean by 'highly symmetric but you surely could remove some of the edges so that there are no gr... | 2 | https://mathoverflow.net/users/122188 | 334277 | 142,882 |
https://mathoverflow.net/questions/334283 | 2 | I'm facing the following problem:
Suppose that we have a finite field $\mathbb{F}\_p$ and an elliptic curve $E$ defined over it. Suppose that for $m\in \mathbb{Z}$ not multiple of the characteristic of the base field. So we have an isomorphism
$$E[m]\longleftrightarrow (\mathbb{Z}/m\mathbb{Z})^2$$
Suppose we know that ... | https://mathoverflow.net/users/129611 | Expressing a torsion point of an elliptic curve as a combination of the generators | There's probably an even quicker way, but here's one idea. Compute the Weil pairings $<P,Q>$ and $<R,Q>=<P,Q>^a$ and $<P,R>=<P,Q>^b$. Then you just have to solve the DLP twice in $\mathbb F\_q^\*$ to find $a$ and $b$, and there are subexponential algorithms for DLP over finite fields. And the Weil pairing is polynomial... | 3 | https://mathoverflow.net/users/11926 | 334287 | 142,886 |
https://mathoverflow.net/questions/334279 | 1 | I am a complex geometer trying to parse Vojta's conjecture on rational points, and I have a very basic misunderstanding (I apologize if this is too easy for MO).
Let $X$ be a variety over a number field. The conjecture states that for all $P$ outside of some Zariski closed subset,
$\sum\_{v \in S} \lambda\_{D,v}(P)... | https://mathoverflow.net/users/142054 | Local heights in Vojta's conjecture | If your rational curve $C$ intersects $H$ in three or more points, then you in fact won't be able to find infinitely many points in $C(\mathbb Q)$ whose height is entirely (or even mostly) coming from the finitely many places in $S$. For example, if the intersection is 3 points, you'd more or less need lots of solution... | 4 | https://mathoverflow.net/users/11926 | 334288 | 142,887 |
https://mathoverflow.net/questions/334172 | 9 | Suppose that $k$ is a commutative ring and that $A$ is an Azumaya
$k$-algebra. Then there is a well-known morphism from $Aut\_{Alg\_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)$. One way to phrase it is as follows. Every automorphism $\phi$ determines a $k$-linear autoequivalence
$$
\phi^\*: ... | https://mathoverflow.net/users/360 | On a morphism from the Brauer group to the Picard group | It seems to me that the involution of $Q \otimes Q$ that exchanges $a \otimes b$ and $b\otimes a$ is inner, which means that the homomorphism you describe should always be trivial.
Here is the argument, I hope it is correct. Suppose that $Q$ is trivial, that is, there is a projective $k$-module $V$ such that $Q \sime... | 7 | https://mathoverflow.net/users/4790 | 334293 | 142,889 |
https://mathoverflow.net/questions/334297 | 15 | It is conjectured that if $K$ is a convex body in $n$-dimensional Euclidean space, then there exists a point in the interior of $K$ which is the point of concurrency of normals from $2n$ points on the boundary of $K$. This has been proved for $n=2$ and $3$ by E. Heil. For $n=4$, a proof appeared (under a smoothness ass... | https://mathoverflow.net/users/136573 | Concurrent normals conjecture | Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $K\subseteq\mathbb R^n$ with closure $\overline K$, and we consider the "distance to $p$" function $d\_p:\partial K\to\mathbb R$ for $p\in\overline K$. Let $V\subseteq\overline K$ be the set o... | 27 | https://mathoverflow.net/users/35353 | 334299 | 142,892 |
https://mathoverflow.net/questions/334184 | 2 | Let $f(x,y) = ax^2 + bxy - cy^2$ be an *indefinite*, irreducible, and primitive binary quadratic form. That is, we have $\gcd(a,b,c) = 1$ and $\Delta(f) = b^2 - 4ac > 0$ and not equal to a square integer.
Written in this way, it is clear that $f$ represents $a$ primitively, since $f(1,0) = a$. Suppose that $f$ also ... | https://mathoverflow.net/users/10898 | Representation of two related integers by the same binary quadratic form | This question can be answered by general theory, at least when $\Delta$ is a fundamental discriminant (so it's not a square times a smaller discriminant). Assuming this, the general procedure for finding the forms of discriminant $\Delta$ that represent a number $n>1$ primitively is the following. Let the prime factori... | 8 | https://mathoverflow.net/users/23571 | 334301 | 142,894 |
https://mathoverflow.net/questions/334040 | 0 | Let $X,Y$ be two Hausdorff spaces and $F:X\to 2^Y$ be a multi-valued mapping. We says that $F$ is *lower semicontinuous* at $x\_0\in X$ if for each $y\_0\in F(x\_0)$ and any neighborhood $U\in \mathcal N(y\_0)$, there exist a neighborhood $V$ of $x\_0$ such that
$$
F(x) \cap U\neq \emptyset \quad\text{for all}\quad x ... | https://mathoverflow.net/users/80191 | Lower semicontinuity of a multi-valued map $F:X\to 2^Y$ in term of net | You're probably right about the statement of the theorem.
Indeed, many textbooks (especially older ones) call nets "generali(s/z)ed sequences"
I think it should say:
>
> Given a net $x: \mathcal{I} \to X$ such that $x \to x\_0$ in $X$, there is a directed set $\mathcal{D}$ and a net $y:\mathcal{D} \to Y$ such that... | 1 | https://mathoverflow.net/users/2060 | 334306 | 142,896 |
https://mathoverflow.net/questions/334312 | 2 | Padoa's inequality is named after Alessandro Padoa (1868-1937):
Let $a$, $b$, $c$ be sidelengths of a given triangle $\triangle ABC$ then
$$(b+c-a)(c+a-b)(a+b-c) \le abc .$$
>
>
> >
> > **My question:** Is the following generalization of Padoa's inequality corect?
> >
> >
> >
>
>
>
Let $a\_i>0$ for $1... | https://mathoverflow.net/users/122662 | Is a generalization of Padoa inequality correct? | Since $a\_i=\frac{S-b\_i}{n-1}=\frac{\sum\_j b\_j -b\_i}{n-1}$ we can write
$$\prod\_{i=1}^n a\_i=\prod\_{i=1}^n \left(\frac{\sum\_{j\neq i}b\_i}{n-1}\right)\geq \prod\_{i=1}^n \left(\prod\_{j\neq i}b\_i\right)^{1/n-1}=\prod\_{i=1}^n b\_i$$
by the Arithmetic-Geometric mean inequality.
| 5 | https://mathoverflow.net/users/2384 | 334313 | 142,897 |
https://mathoverflow.net/questions/331894 | 9 | In the appendix of [this](https://arxiv.org/pdf/math/9801017.pdf) paper. It is proved that Bass' conjecture for $K\_n$ implies the rational Beilinson-Soulé conjecture for $K\_n$. Then at the end the author claims that the same method can be applied to prove that the Bass' conjecture implies the Parshin's conjecture but... | https://mathoverflow.net/users/127776 | Bass' conjecture implies the Parshin's conjecture | I think the claim in the original paper linked above is not correct. It is either totally wrong or it is meant to be something else (like etale motivic version of Bass conjecture) which again I'm not sure whether the same argument given for the Beilison-Soule works or not (I cannot see how it works.). As a reference yo... | 2 | https://mathoverflow.net/users/127776 | 334319 | 142,900 |
https://mathoverflow.net/questions/334248 | 7 | The function is given by
$f(X) = (AX^{-1}A^\top + B)^{-1}$ where $X$, $A$, and $B$ are $n \times n$ positive definite matrices.
I'm trying to find the Lipschitz constant such that $\| f(X)-f(Y) \| \leq L \|X-Y\|$ where $X \geq 0$ and $Y \geq 0$. Motivated by Lemma 3.1 in Nonlinear Systems(H. Khalil, 3rd Ed.), I trie... | https://mathoverflow.net/users/140940 | Lipschitz constant of a function of matrix | I assume $X\ge0$ means $u^\top X u\ge0$, and that $B$ *is definite positive* $$\inf\_{\|u\|=1} u^\top B\, u:=\beta>0.$$ I also assume matrix norms are the Euclidean operator norms.
Compute the differential by the chain rule, as suggested in comments by F.Poloni:
$$Df(X)H=(AX^{-1}A^{\top}+B)^{-1}AX^{-1}\cdot H\cdot X... | 3 | https://mathoverflow.net/users/6101 | 334323 | 142,901 |
https://mathoverflow.net/questions/306439 | 3 | **Update:** A year ago, but the first answer is not clear with me. I bounty this question again.
>
> **My question:** I am looking for a proof or counterexample to the following inequality:
>
>
>
If $n \in \mathbb{N}$, and $a\_1 \ge a\_2 \ge \cdots \ge a\_n \ge 0$ and $\alpha\_1 \ge \alpha\_2 \ge \cdots \ge \a... | https://mathoverflow.net/users/122662 | A rearrangement inequality for exponentiation function | The first inequality is not true for $n=2$, for example
as =
```
0.0912 0.2256
```
alphas =
```
0.8281 1.7010
```
LHS =
```
0.0471
```
RHS =
```
0.0574
```
found by the following Matlab/Octave program:
n=2;
for iter=1:1e4
as=cumsum(rand(1,n));
alphas=cumsum(rand(1,n));
LHS=sum(as.^alphas... | 2 | https://mathoverflow.net/users/100927 | 334332 | 142,903 |
https://mathoverflow.net/questions/333131 | 3 | My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.
The characterization of the closed Poincaré dual is given [here](https://i.stack.imgur.com/oSgf9.png) (the "(5.13)") in Section 5.5. This has $\int\_M \omega \wedge \e... | https://mathoverflow.net/users/138519 | Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $? | A definition is what one makes it to be. Sometimes there are choices and it is difficult to say whether one choice is more correct than another. There is virtue in consistency. In the discussion of Poincaré duality in DFAT, I put forms with no restrictions before forms with compact support. To be consistent with this, ... | 16 | https://mathoverflow.net/users/142097 | 334342 | 142,906 |
https://mathoverflow.net/questions/334311 | 11 | Russell O'Connor wrote in 2009 ([link](http://r6.ca/blog/20091101T231201Z.html)):
>
> PRA has consistency strength equivalent to the well-foundness of $\omega^\omega$, which can be stated again as the termination of some other program on all inputs. Presumably this equivalence is proved in a still weaker system.
> ... | https://mathoverflow.net/users/4177 | What system suffices to show the strength of PRA is $\omega^\omega$? | First, this is not quite the right question. The implication as such is going to be provable in a trivial base theory; the most important question is what “well-foundedness of $\omega^\omega$” means in the statement. Well-foundedness is not directly expressible in the language of first-order arithmetic, it has to be ap... | 10 | https://mathoverflow.net/users/12705 | 334344 | 142,908 |
https://mathoverflow.net/questions/334231 | 3 | I am trying to convince myself that a naïve definition of the Wall self intersection number should not work for odd-dimensional manifolds. Namely, let $X^{2n-1}$ be a smooth oriented closed manifold and let $f : S^n \to X$ be a map from a sphere. Also, let's assume that $X$ and $S^n$ are based and $f$ preserves base-po... | https://mathoverflow.net/users/99414 | Wall self-intersection invariant for odd-dimensional manifolds? | I doubt that what you are proposing as the receptacle for the obstruction is the correct abelian group. For one thing, you are not taking into account the involution on the canonical double cover of the double point manifold, i.e.,
$\{(x,y)\in S^n\times S^n\setminus \Delta\_{S^n} |f(x) = f(y)\}$ (where we have assume... | 3 | https://mathoverflow.net/users/8032 | 334346 | 142,909 |
https://mathoverflow.net/questions/334250 | 2 | Let $t>0$ and $\delta\in\big(0,\frac12\big)$ be fixed. For any $k\in\mathbb{N}$ let $I\_k,J\_k\in\mathbb{N}$ be finite subsets of natural numbers with cardinalities denoted as $|I\_k|,|J\_k|$, respectively. Now define numbers
$\hspace{70pt}C\_k:=\max\Big\{\max\limits\_{i\in I\_k}i^{2t}\sum\limits\_{j\in J\_k}j^{2t},\... | https://mathoverflow.net/users/75127 | Choosing finite subsets of natural numbers | As @Krzysztof noticed, there was a mistake in the original post, which was that I was looking at $2t+2\delta$ and perceived it (for some unknown reason) as $2t(1+\delta)$. So here is the correctedversion:
Let $I,J$ be the interval lengths and $i,j$ be the top numbers in the intervals.
Then the sum of integers in $I$ ... | 7 | https://mathoverflow.net/users/1131 | 334359 | 142,915 |
https://mathoverflow.net/questions/334370 | 0 | I would like to find all integer solutions of the following system:
$$a+b+c+ab+ac+bc=-2,$$
$$a,b,c\le a+b+c-1.$$
One solution is $2,2,-2$. Is it possible to describe all others?
| https://mathoverflow.net/users/13441 | All the integer solutions of a certain semi-algebraic system | Here is a rough idea, the rest should not be too hard to fill in.
First, the given condition implies $a+b,b+c,c+a\geqslant 1$. Namely, $a+b+c$ is positive. Moreover, there is at least one negative number among, and not two. Suppose therefore $c<0$, and let $c=-d$ (I abuse the notation a bit here). Then, observe that,... | 1 | https://mathoverflow.net/users/127150 | 334372 | 142,920 |
https://mathoverflow.net/questions/334366 | 8 | So, I want to read the proof of Mordell-Weil theorem and so, I picked up the book 'Arithmetic of Elliptic Curves' by J. Silverman and J.S. Milne's Elliptic Curves book. But after going through both the books as well as Anthony Knapp's Elliptic curve book....I noticed up one thing, that Silverman takes way longer than M... | https://mathoverflow.net/users/124242 | Can someone suggest a path to study Mordell-Weil theorem for someone studying on their own? | Milne's and Knapp's books are excellent. As for my AEC, it covers lots of stuff that's not needed if you just want to get to Mordell-Weil. So for AEC, here's a path to MW:
1. If you know some algebraic geometry, skip chapters I and II, refer back as needed.
2. Read Ch. III through Sec. 7
3. Reach Ch. IV through Sec.... | 18 | https://mathoverflow.net/users/11926 | 334374 | 142,921 |
https://mathoverflow.net/questions/334085 | 7 | Given a topological space $X$, we have the notion of the fundamental groupoid $\Pi\_1(X)$.
[Here](https://ncatlab.org/nlab/show/fundamental+groupoid), the fundamental groupoid $\Pi\_1(X)$ is made into a topological groupoid giving a topology on the morphism set.
One can then talk about $\Pi\_1(\Pi\_1(X))$. Is this... | https://mathoverflow.net/users/118688 | On fundamental groupoid of fundamental groupoid | The discussion at the linked nlab page points to a few different accounts of topologies on the fundamental groupoid, but I'm not feeling energetic enough to track them all down.
So let me just consider the simplest thing. Let $X$ be a topological space and consider the following topologically-enriched groupoid, which... | 10 | https://mathoverflow.net/users/2362 | 334375 | 142,922 |
https://mathoverflow.net/questions/334347 | 10 | Let $W$ be $B^4$ or $S^3 \times I$. Let $Y$ be a properly embedded surface in $W$. Let $f : W \to W$ be a diffeomorphism which is the identity near $\partial W$. Very little is known about $\pi\_0(\mbox{Diff($W$) rel $\partial W$})$ (see [page 4 of this survey by Hatcher](http://pi.math.cornell.edu/~hatcher/Papers/Diff... | https://mathoverflow.net/users/284 | Can an exotic diffeomorphism of the 4-ball change the isotopy class of an embedded surface? | Isn't this relatively obvious for $W=B^4$ (i.e. it only took me several hours to realize it was trivial)? Isotope $Y$ to $Y'$ by an isotopy $g\_t$ into a small collar neighborhood of $\partial W$ (which we can do by general position), then $f(Y')=Y'$ since you've assumed $f$ is the identity in a neighborhood of $\parti... | 11 | https://mathoverflow.net/users/1345 | 334386 | 142,927 |
https://mathoverflow.net/questions/334376 | 7 | I'm wondering if the following strengthening of the Besicovitch covering theorem holds: Suppose $A\subset\mathbb R^n$ is a bounded subset and suppose $x\mapsto r\_x$ is a function $A\to(0,\infty)$. Is it possible to choose constants $0<\lambda<1$ and $N\in\mathbb N$ (both depending only on the dimension $n$) so that th... | https://mathoverflow.net/users/110236 | Stronger version of Besicovitch covering theorem | I think the statement is false. Consider $n = 1$. Let $A = \{x\_i\}\_{i \in \mathbb{N}}$ where $x\_i = 2^{-i}$.
Define $r\_i$ as follows: if $k^2 \leq i < (k+1)^2$, let $r\_i = 2^{-i} - 2^{-(k+1)^2}$.
Notice that $x\_{k^2}$ only belongs to $B(x\_{k^2}, r\_{k^2})$ and no other ball. So any cover of $A$ must includ... | 4 | https://mathoverflow.net/users/3948 | 334399 | 142,934 |
https://mathoverflow.net/questions/334265 | 2 | Let $X$ be a Banach space and $S\subseteq X$ be a subspace such that the unit sphere of $S$ is weakly compact. If $Y^\*=X$ for some separable Banach space $Y,$ is it true that $S$ is separable?
| https://mathoverflow.net/users/136860 | Weaky compact subset of Banach space with separable predual | Assume that Samya Kumar Ray meant that the unit ball of $S$ is weakly compact. If $S$ is really a sphere, the fact that the answer is "Yes" follows from the argument outlined by Bill Johnson (and the additional assumptions are not needed).
The answer is "yes" anyway. Possibly a bit more straightforward argument than ... | 3 | https://mathoverflow.net/users/85406 | 334402 | 142,935 |
https://mathoverflow.net/questions/334398 | 2 | I'm self-working on two theorems on Lie transformation group from the book Kobayashi transformation group in differential geometry, one is the following
>
> **Theorem** Let $\mathfrak{S}$ the group of differentiable transformation of manifold $M$ and $\mathcal{S}$ be the set of all vector fields $X\in\mathfrak{X}(M... | https://mathoverflow.net/users/97758 | Lie transformation group and the transformation of smooth structure from normal connected subgroup | Since $\mathcal{G}^\*$ is an open subgroup of a topological group $\mathcal{G}$, it is a union of topological components of $\mathcal{G}$. The proof: if some point $g$ lies outside of $\mathcal{G}^\*$, then left translation by that point moves $\mathcal{G}^\*$ to an open set $g\mathcal{G}^\*$. If this set is not disjoi... | 1 | https://mathoverflow.net/users/13268 | 334403 | 142,936 |
https://mathoverflow.net/questions/334281 | 3 | I am looking for a function $f\,:\,\mathbb{R}^n\to\mathbb{R}$ which is separately convex and increasing but not convex.
That is to say, the function is convex and increasing in each coordinate while the others variables are fixed but (globally) the function is not convex on $\mathbb{R}^n$.
A example in $\mathbb{R}... | https://mathoverflow.net/users/142055 | Monotone function which is separately convex but not convex | Let $f(x)$ be as follow
$$
f(x)=\left\{
\begin{array}{ll}
x+1 \quad \text{when}\quad x\geq 0,\\
e^x\quad \text{otherwise}.\\
\end{array}
\right.
$$
Then, set $g(x,y)=f(x)^2+22f(x)f(y)+f(y)^2$. It is a simple matter to check that $g$ satisfy the wanted properties. Besides, $g$ is not convex on $\mathbb{R}^2$ since ${... | 2 | https://mathoverflow.net/users/142055 | 334423 | 142,937 |
https://mathoverflow.net/questions/334409 | 2 | Let $M$ be a von Neumann algebra. If $x$ is positive, then Lemma 2.1(3) of the paper "Order Isomorphisms of Operator Intervals in von Neumann Algebras" (Mori, Integral Equations and Operator Theory, 2019, <https://arxiv.org/abs/1811.01647>) states that if $p$ is the support projection of $x$, then the order interval $[... | https://mathoverflow.net/users/46472 | Order isomorphic order intervals | (It is always hard to know what a "simple argument" is. One Mathematician's "simple" can be another Mathematician's 3 page argument. However, I think the follow is fairly "simple").
We can suppose our von Neumann algebra $M$ acts non-degenerately on $H$.
I *think* it's not so hard to reduce (by cutting down by the su... | 2 | https://mathoverflow.net/users/406 | 334425 | 142,938 |
https://mathoverflow.net/questions/334448 | 6 | Let $\left(W\text{, }S\right)$ be a Coxeter system. For every $w\in W$ let us write $\left|w\right|$ for the length of $w$. Set $\lambda\left(e\right)=1$ where $e\in W$ denotes the neutral element of the group and define coefficients $\lambda\left(w\right)\in\mathbb{Z}$ recursively via \begin{eqnarray} \sum\_{v\in W \t... | https://mathoverflow.net/users/64444 | Vanishing of certain coefficients coming from Coxeter groups | The answer is yes. I claim there are at most $2^n$ elements of $W$ for which $\lambda(w) \neq 0$. Specifically, let $\vee$ be the join in [weak order](https://en.wikipedia.org/wiki/Bruhat_order#Definition), which is a semi-lattice. If $\{ s\_1, s\_2, \ldots, s\_j \} \subseteq S$ and $s\_1 \vee s\_2 \vee \cdots \vee s\_... | 8 | https://mathoverflow.net/users/297 | 334451 | 142,944 |
https://mathoverflow.net/questions/334440 | 5 | The Cramér–Wold theorem states that, if $X$ is a random variable living in $\mathbb{R^d}$, then the distributions of all one-dimensional projections of $X$ uniquely determine the distribution of $X$.
But can this be made algorithmic? I have an algorithm that approximately finds the distribution of any one-dimensional... | https://mathoverflow.net/users/39142 | Algorithm to sample from unknown probability distribution, given its projections? | This is too long for a comment. As noted above, this is [Radon inversion problem](https://en.wikipedia.org/wiki/Radon_transform#Relationship_with_the_Fourier_transform), which can be reduced to the Fourier inversion problem. As is noted in Wikipedia, the inverse problem is ill-posed, which roughly speaking means that n... | 5 | https://mathoverflow.net/users/56624 | 334460 | 142,949 |
https://mathoverflow.net/questions/334368 | 29 | Many students' first introduction to the difference between classes and sets is in category theory, where we learn that some categories (such as the category of all sets) are class-sized but not set-sized. After working with such structures, we discover that it is still worthwhile to sometimes treat them as if they wer... | https://mathoverflow.net/users/3199 | How dangerous are set-size assumptions? | This answer repeats some of the material in other answers but I think it is not entirely redundant.
As you seem to have realized, the assertion that a theory is consistent is a *much* weaker statement than the assertion that the theory does not prove false arithmetical statements (or is "arithmetically sound", to use... | 12 | https://mathoverflow.net/users/3106 | 334469 | 142,952 |
https://mathoverflow.net/questions/117910 | 15 | The customary formulation of the Axiom of Infinity within Zermelo-Fraenkel set theory asserts the existence of an inductive set: a set $ I$ with $\varnothing\in I$ such that $x\in I$ implies $x\cup\{x\}\in I$. Since the intersection of any nonempty set of inductive sets is itself inductive, an instance of the Axiom Sch... | https://mathoverflow.net/users/15819 | Can one exhibit an explicit Kuratowski infinite set without invoking Replacement? | The answer is no. Let ZC' be ZFC without replacement and infinity and with the assertion there is a Kuratowski infinite set. We will construct a model $M$ of ZC' such that only hereditarily finite elements of $M$ are fixed under all automorphisms of $M.$ The idea is generate a model from a $\mathbb{Z}^2$-array of objec... | 9 | https://mathoverflow.net/users/109573 | 334472 | 142,953 |
https://mathoverflow.net/questions/334473 | 8 | If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) *homomorphism* $G\to A$? This seems like it can't always be true, and also should be some kind of classical result, but I'm not... | https://mathoverflow.net/users/4177 | When does $BG \to BA$ loop to a homomorphism? | If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically equivalent to a homomorphism $\phi:G\to A$. This follows from the main result of
*Scheffer, Wladimiro*, [**Maps betwe... | 13 | https://mathoverflow.net/users/8103 | 334477 | 142,954 |
https://mathoverflow.net/questions/334485 | 6 | A friend designed a drinking game with a lucky wheel of 30 distinct icons. When playing, each one takes turn to spin the wheel, and write down the items until the first one who gets the item that has already appeared then he drinks. And he asked what the average (expectation) of turns is. The formula is not so difficul... | https://mathoverflow.net/users/90295 | An expansion from Ramanujan related to birthday problem | We have $$Q(n)=\sum\_{k=0}^{n-1} (1-1/n)(1-2/n)\ldots(1-k/n).$$
Write $1-x/n=e^{-x/n-x^2/(2n^2)-\ldots}$, then
$$Q(n)=\sum\_{k=0}^{n-1} \exp\left(-\frac{k(k+1)}{2n}-\frac{k(k+1)(2k+1)}{12n^2}-\ldots\right).$$
If $k\geqslant n^{1/2+0.00001}$, the corresponding summands are super-polynomially small and do not rely on the... | 7 | https://mathoverflow.net/users/4312 | 334489 | 142,960 |
https://mathoverflow.net/questions/334435 | 3 | Suppose $f$ is a continuous function from a unit cube $[0,1]^n$ to itself, then $f$ has at least a fixed point. Further suppose $f$ is smooth, $0$ is a regular value of $f(x)-x$, and the fixed points are possibly on the boundary. In reading some textbook, I find that the fixed points on the boundary can not be involved... | https://mathoverflow.net/users/141685 | Is there a fixed-point index theorem that treats the fixed points on the boundary? | If you consider a self-map $f:M \to M$ a compact manifold with boundary then the map can be deformed outside the boundary and there are no fixed points on the boundary.
If you consider the map (and deformations) preserving the boundary $(f,\partial f) :(M,\partial M) \to (M,\partial M)$ then you may use the fixed poi... | 1 | https://mathoverflow.net/users/46230 | 334492 | 142,962 |
https://mathoverflow.net/questions/285826 | 8 | Consider a category $\mathsf C$ admitting a quadruple adjunction as below.
$$(\Pi\_0 \dashv \text{disc} \dashv \Gamma \dashv \text{codisc}) :
\mathsf{C}
\stackrel{\stackrel{\longrightarrow}{\longleftarrow}}{\stackrel{{\longrightarrow}}{{\longleftarrow}}}
\mathsf{B}
\;$$
A good example is the category of locally ... | https://mathoverflow.net/users/69037 | Sufficient cohesion and conservativity of "underlying stuff" | Let ${\phi : \mathrm{disc} \rightarrow \mathrm{codisc}}$ be the canonical map.
The map ${\Gamma \phi }$ is an iso in the base so, if $\Gamma$ is conservative, $\phi$ is an iso.
In other words, if $\Gamma$ is conservative, the original string of adjoints is a quality type. So, if the original string is sufficiently ... | 2 | https://mathoverflow.net/users/121350 | 334496 | 142,963 |
https://mathoverflow.net/questions/302185 | 10 | This question is related to, but much more specific than, [this one.](https://mathoverflow.net/questions/36452/explicit-description-of-all-morphisms-between-symmetric-groups)
For $k \leq n$, let $a(k,n)$ denote the number of conjugacy classes of subgroups of the symmetric group $S\_n$ which are isomorphic to $S\_{n-k... | https://mathoverflow.net/users/33089 | Counting symmetric subgroups of symmetric groups | After a year I have realized that this is quite simple (given some well-known things about small index subgroups of symmetric groups).
**Claim:** Indeed, $A(k)=\lfloor k/2 \rfloor$ + 1 for all $k$.
Proof: Fix $k$. Suppose we have an embedding $S\_{n-k} \hookrightarrow S\_n$; this is the same as a faithful action of... | 1 | https://mathoverflow.net/users/33089 | 334501 | 142,965 |
https://mathoverflow.net/questions/334503 | 3 | Let $L/K$ be a geometric Galois extension of function fields over $\mathbb F\_q$. Let $\chi$ be a non-trivial irreducible character of $\text{Gal}(L/K)$. According to Michael Rosen's *Number Theory in Function Fields* p.131, Weil's proof of the Riemann Hypothesis for curves over finite fields shows that the Artin L-fun... | https://mathoverflow.net/users/133679 | Bounds on Artin conductors over function fields | It's not completely clear from your question if you want lower or upper bounds.
For lower bounds, the Odlyzko bounds come from analysis of the $L$-function. The lower bound we get by analyzing the $L$-function in the function field case is
$$ \deg\_K \mathfrak{f}(\chi) \geq (2 -2g\_K) \chi(1) $$ coming from the obvio... | 3 | https://mathoverflow.net/users/18060 | 334509 | 142,967 |
https://mathoverflow.net/questions/334510 | 11 | This is a very basic beginners question about Chern-Simons theory. The configurations that we sum over to get the partition function are given by a Lie-algebra valued 1-form $A$ on a topological 3-manifold, called the connection. More precisely, $A$ is a connection of a principal Lie-group bundle over the manifold.
W... | https://mathoverflow.net/users/115363 | Importance of the principal bundle in Chern-Simons theory | In quantum Chern-Simons theory with gauge group $G$ (compact Lie), a field on a 3-manifold $M$ is a principal $G$-bundle with a connection $A$. The partition function/path integral associated to $M$ is supposed to be the integral over the (generally infinite-dimensional) space ("stack") $\mathcal{F}$ of all principal $... | 11 | https://mathoverflow.net/users/102390 | 334517 | 142,969 |
https://mathoverflow.net/questions/334507 | 4 | What is the asymptotic growth of the sequence
$$a\_n:=\sum\_{k\geq 0} 3^k c\_{n,k},$$
as $n\rightarrow\infty$, where $c\_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s?
A composition of $n$ is a sum $n=c\_1+c\_2+\cdots+c\_p$, with all the $c\_i$ positive.
The first values of the seq... | https://mathoverflow.net/users/31441 | Count weighted integer compositions | When corrected, $a\_n$ is the [OEIS sequence A052528](https://oeis.org/A052528) whose first nine values are
$\,1,1,4,8,22,52,132,324,808,\dots\,$ and it has a linear recurrence.
The combinatorial recurrence from its combination definition leads immediately to
$$ a\_n = 2a\_{n-2} + \sum\_{k=0}^{n-1} a\_k $$
where the... | 7 | https://mathoverflow.net/users/113409 | 334527 | 142,971 |
https://mathoverflow.net/questions/334528 | 10 | **Motivation:**
This question is motivated by wondering to what extent "natural" theories are linearly ordered (or at least ordered in a [directed](https://en.wikipedia.org/wiki/Directed_set) manner) by their (first-order) arithmetic consequences, in analogy to the phenomenon that "natural" theories seem to be linear... | https://mathoverflow.net/users/2362 | Does ZF+AD have any unusual arithmetic consequences? | For (1) and (2), the key point here is **absoluteness**. For example, suppose $V$ is a model of ZFC + enough large cardinals. Then $L(\mathbb{R})^V$ satisfies ZF + DC + AD. But $L(\mathbb{R})^V$ and $V$ have the same natural numbers, hence satisfy the same arithmetic sentences. So if $(\*)$ is a "big" large cardinal ax... | 10 | https://mathoverflow.net/users/8133 | 334529 | 142,972 |
https://mathoverflow.net/questions/334512 | 7 | This question is related to a classification of rational maps in terms of properties of their Julia set.
Let $f= z^2 + c$, with $c\in \mathbb{C}$ be a quadratic polynomial such that its Julia set $J(f)$ is connected.
* Q1: If there exists a relatively open set of $J(f)$ that is (the support of) a smooth curve. Is... | https://mathoverflow.net/users/142190 | Smooth Julia set for quadratic polynomials | The answer to a) is yes, and this was proved by Fatou in 1919.
[Sur les équations fonctionnelles
Bulletin de la S. M. F., tome 48 (1920), p. 208-314](http://archive.numdam.org/article/BSMF_1920__48__208_1.pdf).
There are many generalizations of this fact. For one generalization, and further references you may look to
... | 7 | https://mathoverflow.net/users/25510 | 334530 | 142,973 |
https://mathoverflow.net/questions/334495 | 0 | Consider a power of cycle graph $C\_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}\_n$. Supposing I remove an independent set of vertices of the form $\{i,i+k+1,\ldots,\left\lfloor\frac{n}{k+1}\right\rfloor+i\}$ or a single verte... | https://mathoverflow.net/users/100231 | A vertex transitive graph has a near perfect/ matching missing an independent set of vertices | Yes, it is possible to find a perfect/near perfect matching in the case of powers of cycles when one non-singleton set of maximal independent vertices of the given form is removed. This is because, the cycle $$\{i-1;i+1;i+2\ldots;i+k;i+k+2;\ldots\ldots i-2(=(i-1)+(n-1)\bmod n);i-1\}$$ is Hamiltonian in the induced subg... | 1 | https://mathoverflow.net/users/100231 | 334544 | 142,979 |
https://mathoverflow.net/questions/334526 | 2 | I am getting stuck in a detail in a paper. It's about the axi symmetric Navier Stokes equations
$$u\_t - \nu\,\Delta u + u\cdot \nabla u + \nabla p=0$$
We consider in cylindrical coordinates $u=(u^r, u^\theta,u^z)$. And we have the following vorticity equation in cylindrical form.
$$\omega^r = \frac{1}{r}\frac{\partial... | https://mathoverflow.net/users/142197 | Cylindrical coordinates in axis symmetric flow | There's a typo in the last two expressions you write: $u^{\theta } /r$ should be $u^r /r$.
Then, it seems to check out (for any well-behaved $J$): In the first expression, just insert the expressions for the $\omega $ in terms of the $u$; in the second expression, note
$$
\nabla \times (u^{\theta } e\_{\theta } ) = e... | 1 | https://mathoverflow.net/users/134299 | 334546 | 142,980 |
https://mathoverflow.net/questions/334545 | 1 | A book embedding of a graph G consists of placing the vertices of G on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is a measure of the quality of a book embedding which is the minimum number of pages in which the graph G can be embedded.
If... | https://mathoverflow.net/users/42816 | Bookthickness of covering space | The graph of the icosahedron is a 2-fold cover of $K\_6$; this covering can be induced by the covering of the projective plane by the sphere. The graph of the icosahedron is planar and Hamiltonian, so its page number is at most $2$, and since it is not outerplanar (as pointed out by Bullet51), its page number is exactl... | 3 | https://mathoverflow.net/users/24076 | 334547 | 142,981 |
https://mathoverflow.net/questions/321152 | 1 | Apologies in advance for the naive and rather speculative question.
In [this blog post](https://golem.ph.utexas.edu/category/2017/07/the_geometric_mckay_correspond_1.html) by John Baez, he paints a (perhaps not original) picture of how one might expect that the minimal resolution of a Kleinian singularity $\mathbb{C}... | https://mathoverflow.net/users/65875 | Resolution of Kleinian Singularities using Hilbert schemes of points | Exceptional divisors are realized by Hecke correspondences, pairs $(I\_1, I\_2)$ of $G$ invariant ideals with $I\_1\subset I\_2$ such that $\mathbb C[x,y]/I\_1$ is the regular representation, and $\mathbb C[x,y]/I\_2$ is the regular minus an irreducible representation $\rho\_i$. Hecke correspondences were originally us... | 3 | https://mathoverflow.net/users/3837 | 334552 | 142,982 |
https://mathoverflow.net/questions/334483 | 4 | Let $G$ be a finite group, and $X$ be a free $G$-space. Moreover, assume that $X$ has a homotopy type of a CW-complex. Does $X$ have $G$-homotopy type of a $G$-CW complex also?
**Edit:** My main motivation for this question is as follows. Milnor has shown that if $Y$ has a homotopy type of a CW-complex and X is a com... | https://mathoverflow.net/users/45532 | Homotopy type of G-CW-structure | There is a paper by Stefan Waner from 1980, I think it's called "Equivariant Classifying Spaces", in which he proves an equivariant version of Milnor's theorem. It might do what you want.
| 4 | https://mathoverflow.net/users/8103 | 334569 | 142,985 |
https://mathoverflow.net/questions/334414 | 3 | I have a big problem to solve this system:
$\Delta f-hf^2=0$
$p|\nabla f|^2+hf^3=0$
where $h$ and $p$ are constants (with $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on a 3-manifold ($f:M \rightarrow \mathbb{R}$, where $M$ is a 3-manifold not compact), $\Delta f$ is Laplacian of $f$ a... | https://mathoverflow.net/users/111304 | PDE system problem to find the metric | There are plenty of solutions that are not flat. In fact, the general solution (up to diffeomorphism) depends on two functions of three variables. Here is how one can write them down:
First, since $p$ and $h$ are nonzero, we can write $f$ in the form $f = -(p/h)x$ where $x$ is now a function on $M$ that satisfies the... | 7 | https://mathoverflow.net/users/13972 | 334582 | 142,989 |
https://mathoverflow.net/questions/334541 | 1 | The standard frame for $S^3$ consists of $X\_i,X\_j,X\_k$ with $X\_i(a)=ia, X\_j(a)=ja, X\_k(a)=ka$ where $i,j,k$ are standard quaternion numbers, $a\in S^3$, and the multiplication is the quaternion multiplication.
This global frame on $S^3$ gives us a trivialization of $TS^3\simeq S^3\times \mathbb{R}^3$. So the Hopf... | https://mathoverflow.net/users/36688 | Periodic orbit for certain vector field on $S^3$ (à la Seifert conjecture) | I think the question is perhaps confusing, since the term Hopf vector field is usually reserved for your $X\_i$ vector field (which is tangent to the fibers of the standard Hopf fibration). As I understand, you are instead referring to the vector field which, at a point $p$ with $P(p) = (a,b,c) \in S^2 \subset \mathbb{... | 6 | https://mathoverflow.net/users/66405 | 334584 | 142,990 |
https://mathoverflow.net/questions/334593 | 6 | Consider a polynomial $f
\in \mathbb C[x\_1,\dots ,x\_n]$. An atypical value of $f$ is a complex number about which $f:\mathbb C^n\to \mathbb C$ is not a *topological* fiber bundle. Writing $\mathrm{Atyp}(f)$ for the atypical values of $f$ we thus have a fiber bundle $$\mathbb C^n\setminus f^{-1}(\mathrm{Atyp}(f))\to \... | https://mathoverflow.net/users/69037 | Definition of geometric monodromy | As you say, a Hurewicz fibration defines a map from $\pi\_1(B)$ to the automorphisms of the fiber in the homotopy category.
But a topological fiber bundle has more structure than a Hurewicz fibration. In fact, a topological fiber bundle defines a map from $\pi\_1(B)$ to the mapping class group of the fiber (i.e. the ... | 3 | https://mathoverflow.net/users/18060 | 334599 | 142,993 |
https://mathoverflow.net/questions/334586 | 1 | What can we say about a polyconvex set $\omega\subset\mathbb{R}^N$ knowing the values of its Minkowski valuations?
I know that Hadwiger characterization theorem states that there are only $N+1$ valuations that can express any continuous, rigid motion invariant valuation as a linear combination. But what does that say... | https://mathoverflow.net/users/61629 | Minkowski functionals (valuations) | I doubt that much can be said about the shape of a convex body given its intrinsic volumes (this is a more common name for the basic valuations). Intuitively, this is because the space of convex bodies is "infinite dimensional".
Assume that a body $K$ has a hyperplane section with symmetries (for example, this sectio... | 2 | https://mathoverflow.net/users/98590 | 334604 | 142,997 |
https://mathoverflow.net/questions/334563 | 2 | Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and an integer $m$ is there a polynomial in $n$ algorithm to find a $\{-1,0,+1\}$ matrix $T$ such that $$\mathsf{det}(L+T\odot M)=m$$ where $\odot$ refers to $ij$th entry of $T\odot M$ being $T\_{ij}M\_{ij}$?
>
> 1. Is there any canonical approach to this problem... | https://mathoverflow.net/users/10035 | Matrix completion problem with determinant condition? | I will prove it is NP-complete if $T$ is restricted to $\pm 1$.
Let $k\_1,\ldots,k\_n$ be an arbitrary list of integers.
Suppose the cofactors of $L$ along the top row are $c\_1,\ldots,c\_n$ and all not zero. Define $M$ to be $k\_1/c\_1,\ldots,k\_n/c\_n$ along the top row and 0 everywhere else. Define $m=0$.
Now ... | 2 | https://mathoverflow.net/users/9025 | 334612 | 142,998 |
https://mathoverflow.net/questions/334617 | 11 | Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum\_{n\ge0}a\_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$.
Indeed let $P(X)=\sum\_{i=0}^lP\_i(T)X^i$ be the minimal polynomial of $f$ over $K(T)$. By differentiating $P(f(T))$, one has
$$Q(f(T))+... | https://mathoverflow.net/users/33128 | Derivative of an algebraic power series in positive characteristic | Let $p=\operatorname{char}(K)$.
By minimality of $P$, if $R(f(T))$ was zero, then $R(X)=0$. Thus, $p$ divides the degree of each nonzero coefficient of $P(X)$, so $P$ is not separable.
But the extension $K((T))/K(T)$ is separable, see: [Why is $K\_{\upsilon}|K$ separable for a global field $K$?](https://mathoverfl... | 6 | https://mathoverflow.net/users/128502 | 334623 | 143,000 |
https://mathoverflow.net/questions/334616 | 7 | Let $H$ be a $4$-uniform hypergraph on $[1..n]$, i.e. $H$ is a collection of $4$-element subsets of $[1..n]$. The elements of $H$ are called edges. A hypergraph is regular if every element of $[1..n]$ are in the same number of edges.
An **independent set** $I$ of $H$ is a subset of $[1..n]$ such that $I$ does not con... | https://mathoverflow.net/users/125498 | Independence number of $4$-uniform regular hypergraph | In general no. Partition the vertices onto $n/k$ subsets (I call them classes) of size $k$, where $k$ grows as $n^{2/3}$. Take into your hypergraph all 4-edges with the vertices in the same class. It has about $(n/k)k^4=nk^3\sim n^3$ edges, but each independent set contains at most $3$ vertices from each class, thus $O... | 6 | https://mathoverflow.net/users/4312 | 334625 | 143,002 |
https://mathoverflow.net/questions/285988 | 5 | There exists a Quillen equivalence between $HRModSpectra$ (model category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $Ch$ (model category of unbounded chain complexes of $R$-modules).
I was wondering what the Quillen functors are that give... | https://mathoverflow.net/users/110510 | Stable Dold-Kan correspondence and symmetric group actions | I'm posting a CW answer so that this doesn't remain open and unanswered, even though the OP said in the comments he figured it out. I just posted an [answer to an analogous question](https://mathoverflow.net/a/334622/11540) and in doing so found this question. The relevant paper is "[Stable model categories are categor... | 2 | https://mathoverflow.net/users/11540 | 334628 | 143,003 |
https://mathoverflow.net/questions/334621 | 0 |
>
>
> >
> > Is the inequality as follows true?
> >
> >
> >
>
>
>
Let $k > 0$, $a\_i$ Is a complex number for $1\le i\le n$ and let $$S:=a\_1+a\_2+....+a\_n$$ Suppose that $$b\_i:=S-ka\_i \quad\text{ for} \quad 1\le i\le n.$$
Then
$$k(|a\_1|+|a\_2|+...+|a\_n|) \le |b\_1|+|b\_2|+...+|b\_n|+k|S|$$
Equalit... | https://mathoverflow.net/users/122662 | Is $k(|a_1|+|a_2|+...+|a_n|) \le |b_1|+|b_2|+...+|b_n|+k|S|$ right? | Now it is false: take $n=4$, $k=1$, $a\_1=a\_2=a\_3=1$, $a\_4=-2$. Then $S=1$, $b\_1=b\_2=b\_3=0$, $b\_4=3$ , RHS equals 4 and LHS equals 5.
| 2 | https://mathoverflow.net/users/4312 | 334632 | 143,005 |
https://mathoverflow.net/questions/334594 | 7 | I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly for $G(\Bbb C)$)
But I'm struggling to understand some basic stuff,
It seems to me that there can't be finite index sub... | https://mathoverflow.net/users/142244 | Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups | (Essentially copied from comments)
It's just Borel-Harish-Chandra + Borel's density theorem (+ definition of Chevalley group). More precisely, $G$ is a $\mathbf{Q}$-defined group without nontrivial rational characters (since it has no nontrivial character at all), so Borel-Harish-Chandra implies $G(\mathbf{Z})$ is a ... | 6 | https://mathoverflow.net/users/14094 | 334634 | 143,006 |
https://mathoverflow.net/questions/334610 | 2 | Let $A$ and $B$ be two square real positive (all entries are positive) matrices that differ only in the first row. Let $\lambda\_A$ and $\lambda\_B$ be the maximal real eigenvalues of $A$ and $B$, respectively. Let $\lambda\_\*$ be the maximal real eigenvalue of the matrix $(A+B)/2$.
It is easy to see that $\lambda\_\*... | https://mathoverflow.net/users/64609 | The maximal eigenvalue of average of positive matrices | Let $u$ be an eigenvector of $M = (A+B)/2$ for $\lambda\_\*$. By
Perron-Frobenius we can choose $u \ge 0$. Now if $e\_j$ is the
$j$'th standard unit vector, $e\_j^T A = e\_j^T B$ for $j > 1$.
Thus for $j > 1$, $e\_j^T A u = e\_j^T B u = \lambda\_\* u\_j$. On the
other hand, $e\_1^T M u = \lambda\_\* u\_1$ implies that... | 6 | https://mathoverflow.net/users/13650 | 334639 | 143,007 |
https://mathoverflow.net/questions/334554 | 1 | How does one show that the projections in the [CAR algebra][1] do not form a complete lattice?
Background info: my paper with Scholz [2] paper works in the (infinite) CAR algebra and tries to define algorithmic randomness for its states. The analogs of effectively open sets are certain increasing sequences of project... | https://mathoverflow.net/users/142220 | Projections in CAR (Canonical Anticommutation Relation) algebra | As expressed in my comment, the finite-dimensional CAR algebras *do* have a complete projection lattice. Here I outline the proof that the CAR algebra with countably many degrees of freedom (equivalently, the CAR algebra of a separable Hilbert space) does not have a complete lattice of projections. The short summary is... | 1 | https://mathoverflow.net/users/61785 | 334640 | 143,008 |
https://mathoverflow.net/questions/334638 | 1 | Let $X$ be a smooth projective curve, and $E$ an indecomposable vector bundle on $X$ with $\mathrm{deg} E<0$. Is it true that $H^0(X,E)=0$?
This is true if $E$ is a line bundle, which means it is also true whenever $X$ is $\mathbb{P}^1$, since all vector bundles split here.
It is also true by results of Atiyah if $... | https://mathoverflow.net/users/142263 | Do negative indecomposable bundles on curves have sections? | Let $X$ be a smooth projective curve of genus $2$. First, show that $\mathrm{Ext}^1(\mathcal{O}\_X(-p),\mathcal{O}\_X)=1$. Then, we get an exact sequence
$$
0\rightarrow \mathcal{O}\_X\rightarrow E\rightarrow \mathcal{O}\_X(-p)\rightarrow 0
$$
We have that $h^0(E)=1$, but $E$ has degree $-1$.
| 2 | https://mathoverflow.net/users/124840 | 334641 | 143,009 |
https://mathoverflow.net/questions/334178 | 10 | Let $C$ be a small category, and consider the class of diagrams $G:D\to C$, with $D$ a small category, that have colimits in $C$. This is a proper class even when $C$ is very small, e.g. whenever $D$ has a terminal object $t$, any functor $G:D\to C$ has a colimit $G(t)$, and there is a proper class of small categories ... | https://mathoverflow.net/users/49 | Sufficient sets of colimits in small categories | I think the answer to (2) is affirmative under Vopenka's Principle. That is,
**Claim:**
Let $C$ be a small category, and assume Vopenka's Principle. Then there exists a small set of limit cones $L$ in $C$ such that for any category $D$ and any functor $F: C \to D$, $F$ preserves limits if and only if $F$ preserves ... | 6 | https://mathoverflow.net/users/2362 | 334653 | 143,011 |
https://mathoverflow.net/questions/334681 | 6 | I was wondering whether it is consistent to have $\frak{c} = \aleph\_{\frak{c}}$ where $\frak{c} = 2^{\aleph\_0}$ is the cardinality of the reals (over ZFC). If so, what interesting consequences of this statement are known (besides ¬CH)? I was curious about this because in some sense $\frak{c}$ is the largest possible ... | https://mathoverflow.net/users/50073 | What if $\mathbb{R}$ is in bijection with the cardinals less than $\frak{c}$? | Yes. Start with a model of $\sf CH$, then take the least fixed point with uncountable cofinality. Call that $\kappa$. Now add $\kappa$ Cohen reals.
Since fixed points form a club of ordinals, you can iterate the fixed points enumeration. Repeat that $\omega\_1$ times, then take the least one of cofinality $\omega\_1$... | 8 | https://mathoverflow.net/users/7206 | 334684 | 143,022 |
https://mathoverflow.net/questions/334562 | 13 | Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set.
*Is the set
$$
\big\{ \log{|P|} \, : \, P \in \mathbb{Z}[X] \setminus \{0\} \big\} \subset L^1(\mathbb{T})
$$
of functi... | https://mathoverflow.net/users/26522 | Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle? | It is still discrete though not uniformly. Since $\log|P|=\frac 12\log|p|$ where $p=P\bar P$ is a real non-negative trigonometric polynomial with integer coefficients, it is enough to work with $p$ instead of $P$. We have
$$
|\log p-\log q|\ge \frac{|p-q|}{\max(p,q)}.
$$
Now consider the outer function $f$ with $|f|=\f... | 9 | https://mathoverflow.net/users/1131 | 334691 | 143,023 |
https://mathoverflow.net/questions/334620 | 1 | Let $ M$ be a $2n$-dimensional compact and connected manifold.
Suppose there is $\Omega\in\Omega^{1}(M,\mathbb{C}) $ a closed complex form whose real and imaginary parts represent linearly independent rational cohomology classes and such that
$$ \omega^{n-1} \wedge \Omega \wedge \bar{\Omega}\neq 0$$ (Here $\omega$ is a... | https://mathoverflow.net/users/140797 | Submersion to $ T^{2}$ | 1. Note first that $\Lambda\subset \mathbb C$ is a subgroup generated by a finite collection of numbers (periods) of the type $\alpha+i\beta$, where $\alpha,\, \beta\in \mathbb Q$. Such a subgroup is clearly discreet so it is either $\mathbb Z$ or $\mathbb Z^2$. The only possibility for such subgroup to be isomorphic t... | 1 | https://mathoverflow.net/users/943 | 334695 | 143,024 |
https://mathoverflow.net/questions/334060 | 6 | Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field $X$ which satisfy $$(\*) \qquad \Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$$
What can be said about the struc... | https://mathoverflow.net/users/36688 | On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold | The following formula is known among the experts but hard to find in the literature, so I figure I will document it here. Throughout $(M,g)$ denote an arbitrary *pseudo-Riemannian* manifold, and $\nabla$ its Levi-Civita connection.
>
> **Definition** Given a vector field $X$, its corresponding *0th order deformati... | 6 | https://mathoverflow.net/users/3948 | 334696 | 143,025 |
https://mathoverflow.net/questions/334694 | 5 | Mixed Hodge structure is introduced by Deligne and it's very useful for studying complex algebraic varieties. We know $\text{MHS}$, the category of mixed Hodge structures is an abelian category. Where can I find a good reference for the computation of the (higher) extension groups inside MHS (and it's variations)? Carl... | https://mathoverflow.net/users/102104 | (Higher) extensions of mixed Hodge structures | Beilinson, *Notes on absolute Hodge cohomology* is another reference.
To answer your last question, the cohomological dimension of the category of mixed Hodge structures is $1$, i.e. there is no $Ext^i$ for $i>1$. See 1.10 of Beilinson's paper.
| 6 | https://mathoverflow.net/users/4144 | 334701 | 143,027 |
https://mathoverflow.net/questions/334637 | 4 | Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$ (i.e., the restriction of $\psi$ to the archimedian part $\mathbb{C}^\times$ of the idele group of $K$ has the form $z... | https://mathoverflow.net/users/132420 | Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM | **Edit:** Following Will Sawin's comments, I realized that my first answer contained many mistakes. I try to emend my answer, writing $G\_\mathbb{Q}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $G\_K=\operatorname{Gal}(\overline{\mathbb{Q}}/K)$ throughout.
First of all, your $r$ is actually equal to $1$:... | 5 | https://mathoverflow.net/users/18238 | 334708 | 143,030 |
https://mathoverflow.net/questions/334697 | 6 | Let $(M, g)$ be a closed smooth Riemannian manifolds with Anosov geodesic flow, does it implies that $M$ is an aspherical manifold?
I am thinking it is not known yet?
| https://mathoverflow.net/users/47336 | Does Anosov geodesic flow imply asphericity? | W. Klingenberg in [Riemannian Manifolds With Geodesic Flow of Anosov Type, Annals of Mathematics, Vol. 99, No. 1, 1974, pp. 1-13] proved among other things that a closed Riemannian manifold with Anosov geodesic flow does not have conjugate points, and hence the exponential map at any point is a covering map. In particu... | 12 | https://mathoverflow.net/users/1573 | 334711 | 143,032 |
https://mathoverflow.net/questions/334651 | 3 | Let $A \in \mathbb{R}^{m \times n}$.
Is it true that $$\min \limits\_{Q \in \mathbb{R}^{n \times m}}|I - QA|\_{\infty} < \frac{1}{2}$$ is criteria for the uniqueness of the 1-sparse solution to
$\min \limits\_{x \text{ s.t.} Ax=y} |x|\_1$ for any $y$.
If yes, where can I read about this result. I am not 100% sur... | https://mathoverflow.net/users/101237 | Uniqueness of l1 minimization | Yes, this is true. Since there exists a 'decoder' $Q$ which gives a value on the right hand side (close to or equal to) $\frac{1}{2}$, the pair $(A,Q)$ are $(\ell\_\infty, \ell\_1)$-instance optimal (see [this talk](http://math.univ-lille1.fr/~cempi/activites_scientifiques/FR/conf/files/Lecon3B.pdf) of Foucart, pg 75, ... | 3 | https://mathoverflow.net/users/118731 | 334713 | 143,033 |
https://mathoverflow.net/questions/334647 | 4 | Let $X\_i$ be a sequence of iid random variables, $E [X] = 0$, $E [X^2] = 1$ and $E [|X|^k] < \infty$ for some $k \ge 3$.
Classical local CLT says that the density function $f\_n$ of $\frac1{\sqrt n}\sum\_1^n X\_i$ satisfies that
$$
f\_n(x) - \phi(x)\left(1 + \sum\_{j=1}^{k-2} n^{-\frac j2}P\_j(x)\right) = o\left(n^{... | https://mathoverflow.net/users/44590 | Local central limit theorem far from the center | The asymptotics of the ratio
$$r\_n(x,y):=\frac1{f\_n(\sqrt nx)}f\_{n-1}\left(\frac{nx + y}{\sqrt{n-1}}\right)
$$
will very much depend on the tail asymptotics of the density (say $f$) of $X$.
E.g., if $X$ is standard normal, then $r\_n(x,y)\sim\exp\{-x^2/2-xy\}$ as $n\to\infty$.
If e.g. $f(x)\sim x^{-p}$ for s... | 3 | https://mathoverflow.net/users/36721 | 334718 | 143,035 |
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