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https://mathoverflow.net/questions/362560 | 3 | Consider a number field $K$, and let $v\_1, \cdots v\_n$ ($n \in \mathbb N$) be some finite (i.e. non-archimedean) places of $K$. Is the following true?
For every $\alpha \in K^\times$ there exists $\beta \in \mathcal{O}\_K$ for which $\alpha\beta \in \mathcal{O}\_K$ and
$$|\beta|\_{v\_j} = \frac{1}{\max \{1, |\alpha... | https://mathoverflow.net/users/157984 | Existence of algebraic integer with absolute value equal to reciprocal of maximum of $1$ and absolute value of a given algebraic number | The answer is already in the comments, but here again for completeness: It indeed follows easily from strong approximation:
Let $S=\{v\_1,\dots,v\_n\}\cup\{v:|\alpha|\_v>1\}$. By the strong approximation theorem one can find $\beta\in\mathcal{O}\_K$ that is close to $1$ at those $v\in S$ with $|\alpha|\_v<1$ and clos... | 1 | https://mathoverflow.net/users/50351 | 362639 | 152,502 |
https://mathoverflow.net/questions/362616 | 3 | The usual proof of the existence of an absolute value of a functional on a C\*-algebra $A$ uses the polar decomposition of normal functionals on $A^{\*\*}$, which relies on the compactness of the unit ball of $A^{\*\*}$ in the weak\*-topology.
Is it possible to derive the existence of an absolute value of a bounded l... | https://mathoverflow.net/users/99234 | Alternative proof of existence of absolute value of a functional on a C*-algebra | This is maybe a "backwards" answer to what you might have been hoping for...
An affirmative answer to the 2nd question would give the 1st question as well. Indeed,
if there is a $\*$-representation $\pi:A\rightarrow B(H)$ and $\xi,\eta\in H$ with $\|\varphi\| = \|\xi\| \|\eta\|$ and $\varphi(a) = \langle \pi(a)\xi, ... | 1 | https://mathoverflow.net/users/406 | 362642 | 152,503 |
https://mathoverflow.net/questions/362663 | 0 | I want to get checked if my attempt is okay.
First off, let me shortly describe what Polya's urn is:
A certain urn initially contains a red and a blue ball. We now repeatedly do the following : we (uniformly at random) pick a ball from the urn, and then put it back in together with an additional ball of the same ... | https://mathoverflow.net/users/157350 | Martingale convergence theorem in Polya's urn | The answer is yes, the distribution of $Y\_n$ converges to the uniform distribution on $[0,1]$.
More generally, if we initially have $r$ red and $b$ blue balls in the urn, then the distribution of the proportion of the red balls in the urn converges to the beta distribution with parameters $r,b$; see e.g. [Section 4... | 2 | https://mathoverflow.net/users/36721 | 362666 | 152,513 |
https://mathoverflow.net/questions/362578 | 29 | I uniformly mark $n^2$ points in $[0,1]^2$. Then I want to draw $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point. Surely, for a given constant c it is not always possible.
But it seems that for $c=100$ when n tends to infinity, the probability that such... | https://mathoverflow.net/users/4298 | A variation of the law of large numbers for random points in a square | Given $n^2$ i.i.d. uniform points in $[0,1]^2$, the goal is to draw a configuration of $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point.
We show below that $c$ must satisfy $c=\Omega(n^{1/3})$ for this to be typically possible:
In fact, $\Theta(n^{4/... | 26 | https://mathoverflow.net/users/7691 | 362672 | 152,518 |
https://mathoverflow.net/questions/362656 | 3 | If $\mathfrak{gl}\_n(\mathbb{R})$ denotes the Lie algebra of real $(n \times n)$-matrices, then the $\mathbb{R}$-diagonalizable matrices generate $\mathfrak{gl}\_n(\mathbb{R})$ as a Lie algebra.
If $\mathfrak{g}<\mathfrak{gl}\_n(\mathbb{R})$ is an arbitrary Lie subalgebra, then it need not be generated by the $\mathb... | https://mathoverflow.net/users/134603 | Linear Lie algebra generated by $\mathbb{R}$-diagonalizable matrices | Yes: if $K$ is a field of characteristic zero and $\mathfrak{g}$ is semisimple with no $K$-anisotropic factor (when $K=\mathbf{R}$, $K$-anisotropic means compact) then it is generated (and even linearly spanned) by its $K$-diagonalizable elements. And more generally if $\mathfrak{g}$ is perfect (trivial abelianization)... | 3 | https://mathoverflow.net/users/14094 | 362684 | 152,522 |
https://mathoverflow.net/questions/362681 | 0 | Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable Banach subspaces $\{X\_i\}\_{i \in I}$ of $X$ with the property that:
* $D\cap X\_i$ is dense in $X\_i$,
* $\bigcup\_{i \in ... | https://mathoverflow.net/users/36886 | Breaking up dense subset in non-separable space | For every countable subset $M\subseteq D$ set $X\_M=\overline{\text{span}(M)}$ (with the norm of $X$). These spaces seem to satisfy your requirements. Am I missing something?
| 4 | https://mathoverflow.net/users/21051 | 362686 | 152,524 |
https://mathoverflow.net/questions/362657 | 3 | Consider the Kirchhoff equation, given by
$$u\_{tt}-\left(1+\int\_{\mathbb{R}} u\_x^2\;dx\right)u\_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}\_+$$
where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. How to find the conserved quantitie of this equation?
| https://mathoverflow.net/users/156344 | How to find the conserved quantities of the Kirchhoff equation? | Some of them:
* Multiply by $u\_x$ and integrate, you get
$$ \int u\_{tt} u\_x ~dx = 0 $$
so
$$ \partial\_t \int u\_{t} u\_x ~dx - \int u\_t u\_{tx} ~dx = 0 $$
the second term integrates to zero.
* Multiply by $u\_t$ and integrate by parts you get
$$ \int u\_{tt} u\_t + (1 + \int u\_x^2 ~dx) u\_{xt} u\_x + uu\_t - u... | 2 | https://mathoverflow.net/users/3948 | 362697 | 152,525 |
https://mathoverflow.net/questions/362679 | 9 | For roughly the past month, I have been studying denesting radicals. For example: the expression $\sqrt[3]{\sqrt[3]2-1}$ is a radical expression that contains another radical expression, so this radical is *nested*. Is there a way to express this with radicals that are not (or not as) nested? Writing it in this way is ... | https://mathoverflow.net/users/115623 | A constant bizarrely related to the Fibonacci Numbers | First of all we get $A=1-\sqrt[3]{\frac{1}{2}}+\sqrt[3]{\frac{1}{4}}=\frac{3\sqrt[3]{2}}{2(\sqrt[3]{2}+1)}$......(1)
And from here you can proof by induction...
$A^{F\_n}=A^{F\_{n-1}}.A^{F\_{n-2}}$
If $A^{F\_{n-1}}=(\frac{3}{2})^{F\_{n-2}}(a\_{n-1}+b\_{n-1}\sqrt[3]{2}+c\_{n-1}\sqrt[3]{4})$
and
$A^{F\_{n-2}}=... | 6 | https://mathoverflow.net/users/156029 | 362711 | 152,532 |
https://mathoverflow.net/questions/362507 | 1 | Let $A$ be a fixed $n$ by $n$ real symmetric positive definite matrix with eigenvalues $\lambda\_1 \ge \lambda\_2 \ge \ldots \ge \lambda\_n > 0$, and let $f(A):=\sum\_{i=1}^n\log\lambda\_i$, and let $X$ be a random $n$ by $k$ matrix with real iid copies distributed according to $N(0,\sigma^2/k)$.
The regime
---------... | https://mathoverflow.net/users/78539 | Upper bound for $\mathbb P(|f(A+XX^T)-f(A)| > \epsilon)$, where $A$ is a fixed pd matrix and $X$ has random iid entries | Below, I provide a "high-probability" **non-asymptotic** bound (see (+) below) based on [non-linear Berry-Esseen theory](https://arxiv.org/pdf/0906.0177v2.pdf) developed by Iosif Pinelis. I'd be grateful if someone would kindly check that I didn't screw up anything. Thanks in advance!
---
Main tool: non-linear Be... | 1 | https://mathoverflow.net/users/78539 | 362726 | 152,536 |
https://mathoverflow.net/questions/362716 | 1 | Let $E$ be a separable $\mathbb R$-Banach space, $\rho\_r$ be a metric on $E$ for $r\in(0,1]$ with $\rho\_r\le\rho\_s$ for all $0<r\le s\le1$, $\rho:=\rho\_1$, $$d\_{r,\:\delta,\:\beta}:=1\wedge\frac{\rho\_r}\delta+\beta\rho\;\;\;\text{for }(r,\delta,\beta)\in[0,1]\times(0,\infty)\times[0,\infty)$$ and $(\kappa\_t)\_{t... | https://mathoverflow.net/users/91890 | Extension of spectral gap inequality in Wasserstein distance | I can answer assuming some regularity on the Markov semigroup, which I would expect to be satisfied in most cases. Specifically, assume local (in time) Lipschitz continuity on your Markov semigroup, i.e.
$$\forall s\_0>0, \exists C>0, \forall s\in[0,s\_0], \forall \mu\_1,\mu\_2 : \mathrm{W}(\mu\_1\kappa\_s,\mu\_2\kapp... | 1 | https://mathoverflow.net/users/4961 | 362732 | 152,538 |
https://mathoverflow.net/questions/362743 | 4 | I am wondering that if one can show $ZFC \vdash CON(\ulcorner ZFC-P \urcorner)$. There is an argument in Set Theory, An Introduction to Independence Proofs by Kunen (page 145), but I am confused about the proof.
Let $\phi$ be the formula for the coding of $ZFC-P$ in natural numbers, and $X\_{ZFC-P}=\{n\in \omega :\ph... | https://mathoverflow.net/users/159397 | Formal proof of $ZFC \vdash CON(\ulcorner ZFC-P\urcorner)$ | You can directly show from $ZFC$ that $\forall n \in X\_{ZFC-P}\, \colon \, ( H(\omega\_1) \vDash n)$. To see this remind yourself that $ \vDash$ is expressible by a single formula $\psi$, so that $H(\omega\_1) \vDash \varphi(z\_1,...,z\_m)$ iff $\psi(H(\omega\_1), \ulcorner \varphi \urcorner, \vec{z},1)$. Now for $n \... | 6 | https://mathoverflow.net/users/134910 | 362755 | 152,544 |
https://mathoverflow.net/questions/362695 | 4 | I learned that the average size in any ideal of subsets of $[n]$ is at most $n/2$, but I think the downward closed family of the subsets of $[n]$ also satisfied. I want to know how to proof it or it is wrong. A downward closed family $\mathcal{F}$ means for any $A \in \mathcal{F}, B \subseteq A$, we have $B \in \mathca... | https://mathoverflow.net/users/153820 | The average size of downward closed family of the subsets of $[n]$ is at most $n/2$? | Here is a pedestrian answer. If $\def\cF{\mathcal F}\cF$ is a downward closed subset of $\mathcal P([n])$, we have
$$\frac1{|\cF|}\sum\_{A\in\cF}|A|=\sum\_{i\in[n]}\Pr\_{A\in\cF}[i\in A].$$
Now, for any $i\in[n]$,
$$\Pr\_{A\in\cF}[i\in A]\le\frac12,$$
because the mapping $A\mapsto A\smallsetminus\{i\}$ provides... | 6 | https://mathoverflow.net/users/12705 | 362756 | 152,545 |
https://mathoverflow.net/questions/362757 | 0 | Let $X(N,N)$ be Wishart matrix with rank(X)=K in order to estimate the expectation of the trace of the square root of X i.e $X^{1/2}$ I want to know if is possible to use the unordered Wishart distribution function to estimate this value?
\begin{align}
E[trace(\sqrt X )]=?
\end{align}
| https://mathoverflow.net/users/144355 | expectation of the trace of the square root of wishart matrix | Since the trace is invariant under unitary transformations, you can work in a basis where $X$ is diagonal, with nonzero elements $x\_n$, $n=1,2,\ldots K$ on the diagonal; denote by $P(x)$ their marginal distribution; then
$$\mathbb{E}[{\rm tr}\, \sqrt X]=\mathbb{E}\left[\sum\_{n=1}^K\sqrt{x\_n}\right]=K\int P(x)\sqrt{... | 2 | https://mathoverflow.net/users/11260 | 362783 | 152,551 |
https://mathoverflow.net/questions/362775 | 2 | Consider an unrestricted wreath product $G = \prod\_X A \rtimes B$, where $A$ is a group and $B$ is some subgroup of $\mathrm{Sym}(X)$. I am wondering what the circumstances are under which $\prod\_X A$ is contained in (the closure of) the derived group $[G,G]$.
* If $X$ is finite, then $\prod\_X A \nleq [G,G]$, beca... | https://mathoverflow.net/users/4053 | Commutators in an unrestricted infinite wreath product | Then for the density question, the only obstruction is the existence of infinite orbits.
**Proposition:** let $B$ be a group, let $A$ be a nontrivial abelian group and $X$ a $B$-set. Then the intersection of $D(A^X\rtimes B)$ with $A^X$ is dense in $A^X$ (for the product topology, $A$ being discrete, and $D$ meaning ... | 3 | https://mathoverflow.net/users/14094 | 362784 | 152,552 |
https://mathoverflow.net/questions/362094 | 3 | I have been reading Bhatt's [notes on perfectoid spaces](http://www-personal.umich.edu/~bhattb/teaching/mat679w17/lectures.pdf) and I have stumbled upon a fact whose proof I am unable to understand. Specifically, in Remark 4.2.8 Bhatt describes the functor $A\mapsto A\_{!!}$ from $R^a$-algebras (almost $R$-algebras) to... | https://mathoverflow.net/users/30186 | Faithful flatness of left adjoint to almostification of algebras | To get this off the unanswered list, here is an answer provided by Anonymous and abx in the comments.
The proof uses the following characterization of faithful flatness and its analogue for almost modules:
>
> An $A$-algebra $B$ is faithfully flat if and only if the structure map $A\to B$ is injective and the quo... | 0 | https://mathoverflow.net/users/30186 | 362788 | 152,554 |
https://mathoverflow.net/questions/362779 | 0 | The questions was asked by me on Math StackExchange, but no answer appears, so I ask for help again.
Let $(X, d)$ be a complete (Hausdorff, separable, local compact and other nice properties you want) metric space and $\mathcal{M}$ be the space of local finite **full supported** Borel probability measures on $X$ (wit... | https://mathoverflow.net/users/90512 | The properties of total variation metric | **Answer to Question 3:** Yes, there is such a sequence. E.g., for all $n=0,1,\dots$ and all Borel $B\subseteq[0,1]$, let
$$\mu\_n(B):=\frac12\int\_B(2+\sin2\pi nx)\,dx.$$
Then $\mu\_n$ converges to $\mu\_0$ weakly but not in total variation. (Also, $\mu\_n$ is full supported for each $n$.)
**Concerning Question 1... | 2 | https://mathoverflow.net/users/36721 | 362792 | 152,555 |
https://mathoverflow.net/questions/362774 | 2 | I am requesting your help today trying to solve a somewhat odd problem. Is there a way to find through some numerical algorithm such as Newton's method the stochastic matrix $\boldsymbol{P}$ having stationary distribution $\boldsymbol{\pi}$ (column vector, given as an input) and lowest diagonal (whose elements are clos... | https://mathoverflow.net/users/159426 | Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal | If you make the objective to minimize the sum of the diagonal entries (i.e. the trace), your problem becomes a linear programming problem, solvable with readily available software (I think even Excel). In many cases the optimal solution will have all diagonal entries $0$.
EDIT: It may help to think of the problem thi... | 3 | https://mathoverflow.net/users/13650 | 362809 | 152,560 |
https://mathoverflow.net/questions/362735 | 3 | If $G$ is a labeled graph, the multi-affine characteristic polynomial (which depends on labeling) is defined by
$\Phi\_G(x\_1,...,x\_n)=\det(I\_x-A)$, where $I\_x$ is the diagonal matrix $diag\{x\_1,...,x\_n\}$ and $A$ is the adjacency matrix.
Since we can also write $\Phi\_G=\det( \sum\_{j=1}^n x\_jI\_j-A)$, where... | https://mathoverflow.net/users/1894 | How to prove that for the real stable characteristic polynomial $P=\Phi_T$ of a tree $T$, $P_iP_j-PP_{ij}=(\Phi_{T-[v_i,v_j]})^2$? | Let $B$ denote the $n\times n$ matrix $I\_x-A$. Let $[n]=\{1,2,\ldots,n\}$.
For $I,J\subset [n]$ with the same number of elements, let $D\_{I,J}$ denote the determinant of the matrix resulting from $B$ when deleting the rows indexed by the elements of $I$ and the columns indexed by the elements of $J$.
Assuming $i\ne... | 1 | https://mathoverflow.net/users/7410 | 362813 | 152,563 |
https://mathoverflow.net/questions/362815 | 2 | Let $G=SO\_n$ and fix a borel subgroup $P\_0$ of $G$. Let $P=MN$ be a standard maximal parabolic subgroup $G$ and $\sigma$ a cuspidal representation of $M$
Consider the normalized parabolic induced representation $\text{Ind}\_P^G(\sigma|\cdot|^z)$ and for sufficiently large $z$, we can define Eisenstein series $E(z,\... | https://mathoverflow.net/users/29422 | Question on the residual representation | In the positive cone/half-plane this is essentially never the case, because in that region the map "take residue at $z\_o$" is an intertwining map with non-trivial kernel (and image is the smaller quotient repn generated by the residue) from the principal series generated by the Eisenstein series, to the repn generated... | 3 | https://mathoverflow.net/users/15629 | 362817 | 152,565 |
https://mathoverflow.net/questions/362582 | 13 | Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$,
\begin{align\*}
\exp \left(t \rho\right), \exp \left(t \rho^{-1}\right) \in L^1\_{loc}.
\end{align\*}
Then, a conjecture of De Giorgi asserts that th... | https://mathoverflow.net/users/69642 | A conjecture of De Giorgi on weighted Sobolev spaces | I did some diggings and some readings and found out that the conjecture has been solved here
<https://link.springer.com/article/10.1134/S1064562413060173>
and extended recently to a wider context in
<https://www.degruyter.com/view/journals/crll/2019/746/article-p39.xml>
| 6 | https://mathoverflow.net/users/69642 | 362818 | 152,566 |
https://mathoverflow.net/questions/362790 | 6 | Below are two different stories about power operations for $\mathbb{E}\_\infty$-ring spectra, and I am struggling to see how they relate. In the following we let $R$ be an $\mathbb{E}\_\infty$-ring spectrum.
1. $R$ admits natural maps $R^{\wedge n}\_{h\Sigma\_n} \to R$. If $X$ is now a space, we may apply $(\,\cdot\... | https://mathoverflow.net/users/151698 | Two definitions of power operations --- how do they relate? | The first observation is that $U$ is representable in $CAlg(R)$ by $F(S)$ (with $F$ as in your question): $$Map\_{CAlg(R)}(F(S),A) \simeq Map\_{SMod}(S,A) \simeq U(A).$$
By some appropriately flowery version of Yoneda's Lemma, it follows that
$$ Hom(U,U) \simeq Map\_{CAlg(R)}(F(S),F(S)) \simeq Map\_{SMod}(S,F(S)) \si... | 8 | https://mathoverflow.net/users/102519 | 362832 | 152,571 |
https://mathoverflow.net/questions/362600 | 8 | Let $\pi:X\to S$ be a separated flat morphism of finite type of Noetherian schemes. Does $\pi$ necessarily factor as an open immersion followed by a proper flat morphism? The analogue of this question with the word "flat" replaced by "smooth" has a negative answer (consider an elliptic curve over $\mathbb{Q}\_p$ that h... | https://mathoverflow.net/users/nan | Does a flat compactification always exist? | This already fails if $S$ is regular of dimension $3$ and $\pi$ is quasi-finite. Indeed, let $X$ be a normal affine variety over $\mathbf C$ of dimension $3$ with an isolated non-Cohen–Macaulay singularity (e.g. an affine cone over a smooth projective surface $Y$ with $H^1(Y,\mathcal O\_Y) \neq 0$). By Noether normalis... | 4 | https://mathoverflow.net/users/82179 | 362834 | 152,572 |
https://mathoverflow.net/questions/362772 | 13 | What is the motivation behind naming the category O appearing in the theory of Lie algebras? Does O stand for something?
Here is a question [Why the BGG category O?](https://mathoverflow.net/questions/64931/why-the-bgg-category-o) that further confuses me. It seems like there is a notion of when a category is O, is it... | https://mathoverflow.net/users/127069 | Why the name O for category O? | From [Humphreys: Representations of semisimple Lie algebras in the BGG category O], notes for Chapter 1:
>
> The letter chosen to label the category is the first letter of a
> Russian word meaning “basic”
>
>
>
which is основной.
| 11 | https://mathoverflow.net/users/15292 | 362835 | 152,573 |
https://mathoverflow.net/questions/362797 | 2 | A graph $G=(V,E)$ is [*arc-transitive*](https://en.wikipedia.org/wiki/Symmetric_graph) if its symmetry group acts transitively on ordered pairs of adjacent vertices.
In general, the complement of an arc-transitive graph is not arc-transitive.
But I have a hard time finding an example of such a graph if I assume $\ma... | https://mathoverflow.net/users/108884 | A diameter 2 arc-transitive graph whose complement is not arc-transitive? | Complements don't even have to be edge transitive. Perhaps the simplest example is the wreath graph $W\_5$ (which is obtained by applying the construction below to $C\_5$).
Call two vertices "twins", if they have the same neighbourhood in $G$. Since the twin-relation is preserved under automorphisms, it suffices to c... | 4 | https://mathoverflow.net/users/97426 | 362838 | 152,575 |
https://mathoverflow.net/questions/362826 | 2 | I have (hopefully) a rather basic question about smooth elliptic partial differential
equations.
Let $L$ be a linear elliptic differential operator with polynomial coefficients in $\mathbb{R}^n, n>1.$ Let $u\in L^{\infty}(\mathbb{R}^n)$ be such that it has compact support
and $L(u)$ is supported on a finite set (as a... | https://mathoverflow.net/users/157028 | A basic question about elliptic pde | The answer is **Yes** and is a consequence of Holmgren's uniqueness Theorem. See for instance [Theorem 1.1.4 in this text.](https://webusers.imj-prg.fr/~nicolas.lerner/m2carl.pdf). The ellipticity serves here to ensure that there does not exist a characteristic hypersurface. Applying HUT, you obtain that $u\equiv0$ ove... | 0 | https://mathoverflow.net/users/8799 | 362854 | 152,582 |
https://mathoverflow.net/questions/362851 | 2 | I am studying about ideals of spatial (minimal) tensor product of $C^{\ast}$-algebras but I did not find any book/paper in which all the results are given.
>
> What are some results or folklore which are well known about the ideals(primitive/prime/modular) of spatial tensor products of $C^{\ast}$-algebras.
>
>
> ... | https://mathoverflow.net/users/129638 | Results which are known about ideals of spatial tensor product | Well, one source I know which "considers ideals of tensor products of $C^\*$-algebras" is David McConnell's thesis: [$C\_0(X)$-structure in $C^\*$-algebras, multiplier algebras and tensor products](http://www.tara.tcd.ie/handle/2262/80336).
| 1 | https://mathoverflow.net/users/406 | 362863 | 152,587 |
https://mathoverflow.net/questions/362749 | 1 | Consider two Girsanov measures $\mu\_1$ and $\mu\_2$ corresponding to drifts $F\_1(t)$ and $F\_2(t)$ respectively. By this, I mean that we have that $B(t)\sim F\_1(t)+\tilde B(t)$ where $\tilde B(t)$ is a Brownian motion under $\mu\_1$. Similarly for $\mu\_2$.
For $\lambda \in [0,1]$ we can consider the probability m... | https://mathoverflow.net/users/158968 | What is the drift for a convex combination of Girsanov measures? | Just take drift $F\_1$ with probability $\lambda$ and drift $F\_2$ w.p $(1-\lambda)$.
If you want an explicit probabilistic description in terms of the drifts $F\_1,F\_2$, just enlarge the probability space to support an
independent Bernoulli $B$ of parameter $\lambda$ and set the drift
$F=BF\_1+(1−B)F\_2$.
| 1 | https://mathoverflow.net/users/35520 | 362880 | 152,590 |
https://mathoverflow.net/questions/362881 | 4 | Given $2n+1$ fixed points: $A\_1, A\_2,....,A\_{2n+1}$ and point $P$. Let $B\_1$ is the reflection of $P$ in $A\_1$, $B\_2$ is the reflection of $B\_1$ in $A\_2$,...., $B\_{2n+1}$ is the reflection of $B\_{2n}$ in $A\_{2n+1}$.
>
>
> >
> > **My question:** Could You show that midpoints of $PB\_{2n+1}$ is fixed po... | https://mathoverflow.net/users/122662 | From a point and continuing reflection in $2n+1$ points then midpoint of the end point and the first point is fixed | The reflection of a point $p$ in a point $u$ is given (if we identify points with vectors) by $2u-p$. So the composition of two such reflections, say about points $u$ and $v$, is given by the translation $p\mapsto2v-(2u-p)=p+2(v-u)$. By induction, the composition of reflections in $u\_1,v\_1,u\_2,v\_2,\dots u\_n,v\_n$ ... | 4 | https://mathoverflow.net/users/6794 | 362888 | 152,594 |
https://mathoverflow.net/questions/362869 | 8 | I have posted it on Mathstackexchange but nobody replied.
Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex structure on $M$.
My naive question is: when does $\gamma$ bound a pseudoholomorphic curve? More precisely,
>
> wh... | https://mathoverflow.net/users/99042 | Which curves are boundary of pseudoholomorphic curves? | The 'moment conditions' that Ben McKay mentions are simply this: A closed curve $C$ in $\mathbb{C}^n$ bounds a compact Riemann surface (which might be singular) if and only if the integral around $C$ of any global holomorphic $1$-form on $\mathbb{C}^n$ vanishes.
One direction is just Stokes' Theorem: If $\omega$ is a... | 13 | https://mathoverflow.net/users/13972 | 362903 | 152,599 |
https://mathoverflow.net/questions/362802 | 10 | Let $X$ and $Y$ be smooth projective complex surfaces. If a diffeomorphism from $X$ to $Y$ maps subvarieties to subvarieties does it have to be holomorphic or antiholomorphic? Can we at least verify this for K3 surfaces?
| https://mathoverflow.net/users/nan | A diffeomorphism of complex surfaces mapping subvarieties to subvarieties | I am posting this answer because the following linear algebra proposition is too long for a comment.
**Lemma.** Let $A$, respectively $B$, be an invertible $\mathbb{R}$-linear operator on $\mathbb{C}^2$ that is $\mathbb{C}$-linear, resp. $\mathbb{C}$-conjugate linear.
**(1)**. For every $1$-dimensional $\mathbb{C}... | 10 | https://mathoverflow.net/users/13265 | 362910 | 152,601 |
https://mathoverflow.net/questions/362885 | 1 | I am looking for a simple proof that a non-negative polynomial in n variables has always even degree. I have proved it but using Artin's Theorem ( Every non-negative polynomial is the sum of squares of rational functions), but I think that there are simpler proofs.
I know that this question could be off-topic, but I ... | https://mathoverflow.net/users/159512 | A simple proof that non negative polynomials have even degree | This can be proved via an elementary argument.
Univariate case
---------------
Let's first look at the univariate case. Suppose $f(x) = x^{2r+1} + \sum\_{0 \le j \le 2r}a\_jx^j \in \mathbb R[x]$ is a nonnegative polynomial of odd degree $d=2r+1$, assumed to be unitary w.l.o.g. Then, in the limit as $x \longrightarr... | 6 | https://mathoverflow.net/users/78539 | 362918 | 152,604 |
https://mathoverflow.net/questions/357401 | 1 | Let $(R, \mathfrak m,k)$ be a Noetherian local ring such that the residue field $k$ is infinite. Let $n=\mu(\mathfrak m)$. Then $n=\dim\_k(\mathfrak m/\mathfrak m^2)$ . By fixing $x\_1,...,x\_n \in \mathfrak m$ such that $\bar x\_1,...,\bar x\_n\in \mathfrak m/\mathfrak m^2$ gives a $k$-vector space basis, we can ident... | https://mathoverflow.net/users/135389 | On a condition on ideals viwed as a Zariski open condition on co-tangent space | Case 1 was treated carefully in Appendix A of [J. Watanabe's paper](https://projecteuclid.org/download/pdf_1/euclid.nmj/1118780704) "$m$-full ideals". Case 2 can be treated the same way, as sketched below.
Let $J$ be either $mI$ or $\overline{mI}$ and $A=R/J$. Since $J:x \supset I$, we have that $l(A/xA)= l(0:\_Ax) ... | 1 | https://mathoverflow.net/users/2083 | 362924 | 152,607 |
https://mathoverflow.net/questions/362906 | 2 | Let $\Theta(s)$ be a Dirichlet series , and let $\beta$ be its abscissa of convergence:
$$\Theta(s)=\sum\_{n=1}^{\infty}\frac{\theta(n)}{n^{s}}\;\;\;\;\;\;\Re(s)>\beta$$
And let $\left\{a\_{n}\right\}\_{n\in\mathbb{N}}$ be a sequence, whose ordinary generating function is $g(x)$:
$$g(x)=\sum\_{n=0}^{\infty}a\_{n}x^{n}\... | https://mathoverflow.net/users/20782 | Binomial transform of Dirichlet series | Mellin transform interpolation is related formally at least to Newton series interpolation. One example is its use to [interpolate the Bernoulli polynomials](https://www.google.com/amp/s/tcjpn.wordpress.com/2015/01/12/umbral-composition-of-the-bernoulli-polynomials-and-the-riemann-zeta-function/amp/) from their relatio... | 2 | https://mathoverflow.net/users/12178 | 362929 | 152,609 |
https://mathoverflow.net/questions/339042 | 8 | In de Rham's classical book "Variétés Différentiables"
*de Rham, Georges*, Variétés différentiables. Formes, courants, formes harmoniques. 3e éd. revue et augmentée, Publications de l’Institut de Mathématique de l’Université de Nancago III. Actualités scientifiques et industrielles 1222 b. Paris: Hermann. X, 198 p. ... | https://mathoverflow.net/users/82672 | Odd differential forms | A form of odd type is the same thing as a [pseudoform](https://ncatlab.org/nlab/show/differential+form#twisted), as mentioned by Mike Shulman. This is a form twisted by the pseudo-scalar bundle $\Psi$. The issue with the term "density" is that it has multiple meanings. It can refer to forms twisted by $\Psi$ or by $\bi... | 4 | https://mathoverflow.net/users/147463 | 362936 | 152,612 |
https://mathoverflow.net/questions/362938 | 1 | Define a random walk on the integers $\mathbb{Z}$ with step distribution $F$ and initial state is zero which is a sequence $S\_n$ of random variables and its increments are iid random variables $X\_i$ with common distribution $F$, that is,
$$S\_n=\sum\_{i=1}^n X\_i$$
>
> Can we find a distribution $F$ such that fo... | https://mathoverflow.net/users/168083 | Conditional distribution for the random walk on $\mathbb{Z}$ | Suppose the possible increments are $+3$, $+2$, $-1$, and $-3$. The specific probabilities don't matter, as long as they're positive.
Then conditional on $S\_2=1$, we have $S\_1\leq 2$ with probability $1$ (the first two steps must be $2$ and $-1$ in some order).
But conditional on $S\_2=0$, we have $S\_1=3$ with... | 2 | https://mathoverflow.net/users/5784 | 362945 | 152,615 |
https://mathoverflow.net/questions/362947 | 4 | My question is concerned with the involution in Banach \*-algebras.
**1- Should the involution be assumed continuous in every Banach \*-algebra?**
If the answer is negative,
**2- Does there exist any characterization of Banach \*-algebras whose involution is continuous (isometry)?!**
| https://mathoverflow.net/users/84390 | Continuity of the involution in Banach *-algebras | For 1), this is § 36 (pp. 190) of Bonsall & Duncan: The following are equivalent for a Banach \*-algebra $A$, and the set $\text{Sym}(A')$ of all continuous self-adjoint linear functionals on $A$
* The linear involution $f \mapsto f^\*$ is continuous on $A$
* $\text{Sym}(A')$ is separating on $A$
* $\text{Sym}(A)$ is... | 7 | https://mathoverflow.net/users/5734 | 362949 | 152,616 |
https://mathoverflow.net/questions/362941 | 3 | I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{1}{\sqrt{3}}x \}$. The regions $A$ and $B$ are the right-angled triangles with angles $\{\frac{\pi}{4},\frac{\pi}{2},\frac{\... | https://mathoverflow.net/users/148980 | Conformal mapping between two right-angled triangles | There are two expressions depending on what you prefer: hypergeometric functions or elliptic functions.
1. Let $f$ be the Schwarz-Christoffel map of the upper half-plane onto $A$,
and $g$ the Schwarz-Christoffel map onto $B$ (both sending $(0,1)$ to $(0,1)$). Then your map $A\to B$ is $g\circ f^{-1}$. Explicit formul... | 3 | https://mathoverflow.net/users/25510 | 362956 | 152,618 |
https://mathoverflow.net/questions/362955 | 3 | Let $f$ be the characteristic function of a real-valued random variable $X$. It is known that if $f$ has a $k$-th order derivative (for some even $k$) then $\mu$ has a finite $k$-th order moment. Is there any reference or discussion the case when $k$ is odd? For example, if we only know that $f$ has its first order der... | https://mathoverflow.net/users/143284 | Differentiability of characteristic functions and moments of the corresponding measure | A good reference regarding this type of results is the book by Eugène Lukacs "Characteristic Function". For example, chapter 2.3 "Characteristic functions and moments" provides results in this direction. Theorems 2.3.1, 2.3.2 and 2.3.3 might be what you are looking for.
| 5 | https://mathoverflow.net/users/69642 | 362958 | 152,620 |
https://mathoverflow.net/questions/362953 | -2 | Let $A$ be a $C^{\ast}$-algebra. We say $A$ is weakly commutative if $ab^\*c=cb^\*a$ for all $a,b,c \in A$ and define weak center of $A$ as $$Z\_w(A)= \{ v \in A : av^\*c=cv^\*a \;\forall a,c \in A \}.$$
>
> Are these notions of weak commutativity and weak center the same as the usual notions of commutativity and c... | https://mathoverflow.net/users/129638 | Weak center is same as center for $C^{\ast}$-Algebra? | Not the same. The center of $M\_2$ is $\mathbb{C}\cdot I\_2$, but its weak center is $\{0\}$. E.g. $I\_2$ is not in the weak center because not all $a,c \in M\_2$ commute.
| 2 | https://mathoverflow.net/users/23141 | 362964 | 152,621 |
https://mathoverflow.net/questions/362963 | 4 | By Hitchin fibration I mean the usual morphism from the coarse moduli space of semi-stable Higgs bundles to the Hitchin base (i.e. the direct sum of spaces of global sections of powers of the canonical bundle).
Is this morphism known to be proper? If so what is the standard reference?
| https://mathoverflow.net/users/4096 | Is the Hitchin fibration proper? | Yes, it is proper. A reference for this fact is Theorem 6.11 in
C. Simpson, "*Moduli of representations of the fundamental group of a smooth projective variety II*", Pub. Mat. IHÉS, Tome 80 (1994), p. 5-79
| 8 | https://mathoverflow.net/users/116075 | 362965 | 152,622 |
https://mathoverflow.net/questions/362962 | 1 | Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R\_+$, where $f$ is an even continuous function.
Denote
$$
F(f):=\int\_0^{\infty}\int\_{S^n}f(y)g\left(\frac{|xy|}{t}\right)dy\frac{dt}{t^{n+1}},
$$
where $x \in S^n, \, t>0$, and function $g$ is such that
$$
\int\_{0}^{\infty}s^jg(s)ds=... | https://mathoverflow.net/users/122182 | Fourier Transform of an even function | This is not a full answer, just an outline. It may require additional regularity assumptions on $f$ and $g$. $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\SO}{SO}$ For $x\in\bR^{n+1}\setminus 0$ we set $$\bar{x}:=\frac{1}{|x|}x.$$
I assume that $xy$ denotes the inner product. Note that
$$
F[f](x)=\int\_0^\infty... | 5 | https://mathoverflow.net/users/20302 | 362973 | 152,624 |
https://mathoverflow.net/questions/362879 | 1 | I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series.
>
> Let $R$ be an associative unital ring and $M$ a unitary left $R$-module. Define the ***socle*** of $M$, denoted by $\text{soc}(M)$, to be the sum of all simple... | https://mathoverflow.net/users/33026 | References about transfinite socle series | As I said in comments, there is a fair amount of literature to be found by Googling "infinite socle series".
More specifically, a module $M$ for which (in the notation of the question) $\overline{\text{soc}}(M)=M$ is called a "semi-artinian module", and a ring $R$ for which every module is semi-artinian (or equivalen... | 1 | https://mathoverflow.net/users/22989 | 362985 | 152,626 |
https://mathoverflow.net/questions/362928 | 6 | Let $X$ be an infinite dimensional Banach space. Let $\Lambda\_{0}$ be the set of all finite dimensional subspaces of $X$ directed by the inclusion $\subseteq$. For each $\alpha\in \Lambda\_{0}$, let $I\_{\alpha}:=\{\beta\in\Lambda\_{0}:\alpha\subseteq \beta\}$. Then $\{I\_{\alpha}:\alpha\in \Lambda\_{0}\}$ is a filter... | https://mathoverflow.net/users/41619 | The Calkin representation for Banach spaces | The answer to question 1 is yes.
Suppose that $T^\*B\_{X^\*}$ is not compact. Since it is norm closed, there is
$\epsilon >0$
and an infinite subset $S$ of $B\_{X^\*}$ so that
$\|T^{\*}x\_1^\*-T^{\*}x\_2^\*\| > \epsilon$
for all
$x\_1^\*\not= x\_2^\*$
in
$S$. Let $x^\*$ be any weak$^\*$ limit point of $S$. For... | 6 | https://mathoverflow.net/users/2554 | 362988 | 152,627 |
https://mathoverflow.net/questions/362978 | 2 | Unfortunately I can't read Russian, I was wondering if there is an English translation of this paper
“The central limit theorem for stationary Markov processes”, Dokl. Akad. Nauk SSSR, 239:4 (1978), 766–767
I apologize in advance if this is not the suitable place to make this question.
| https://mathoverflow.net/users/98969 | English translation of a Russian paper by Gordin and Lifšic | This journal was transllated into English as
[Soviet Mathematics. Doklady](https://www.worldcat.org/title/soviet-mathematics-doklady/oclc/312040373) Many US libraries subscribed it.
If you have access to a university library, and it does not have it, use ILL.
Here is the exact reference for the translation:
Gordin,... | 6 | https://mathoverflow.net/users/25510 | 362995 | 152,631 |
https://mathoverflow.net/questions/362952 | 9 | Is there a condensation (continuous bijective mapping) from $D^{\aleph\_0}$ onto a metrizable compact space ?
$D$ - discrete space of cardinality $\aleph\_1$.
CH implies it is a positive answer. In general, I don’t know the answer.
| https://mathoverflow.net/users/112417 | Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space? | If $|D| < \aleph\_\omega$, then there is a condensation from $D^\omega$ onto $\omega^\omega$ (the Baire space) if and only if there is a partition of $\omega^\omega$ into exactly $|D|$ Borel sets.
As far as I know, this theorem was first proved by me and Arnie Miller in
>
> "Partitions of $2^\omega$ and complet... | 11 | https://mathoverflow.net/users/70618 | 363004 | 152,634 |
https://mathoverflow.net/questions/362651 | 3 | I was reading this [post](https://mathoverflow.net/questions/336489/existence-of-topologically-mixing-discrete-dynamical-system-on-manifold) and wondered. Does there exist a topologically transitive (TT) map $f:\mathbb{R}^n\to\mathbb{R}^n$ when $n\geq 2$? I know that post asks for compactness and topological mixing but... | https://mathoverflow.net/users/36886 | Existence of topologically transitive map on Euclidean space | Here is an explicit answer.
First, define a continuous tent map $w$ on $[1,2]$ define $w$ with $w(1)=w(2)=0$, and $w(3/2)=4$. Then define $w:[0,+\infty)\to [0,+\infty)$ by pasting scaled versions of this tent infinitely on both sides so that the graph consists of a sequence of congruent triangles.
Then for every op... | 1 | https://mathoverflow.net/users/53155 | 363010 | 152,637 |
https://mathoverflow.net/questions/363014 | 5 | Let $X$ and $Y$ be locally convex spaces, and $\varphi: X\to Y$ a linear continuous mapping. Suppose first that $S$ is a compact set in $X$. Then $\varphi$, being considered as a mapping from $S$ to $\varphi(S)$,
$$
\varphi\Big|\_S:S\to \varphi(S)
$$
is *open* in the sense that for any open set $U$ in $S$ (with respect... | https://mathoverflow.net/users/18943 | Are linear continuous mappings open on totally bounded sets? | Bounded sets, e.g., unit balls in normed spaces, are always totally bounded for weak topologies. So you only have to find such a ball with two incompatible weak topologies, which is easy to do—say the ball of a suitable dual space with the weak and the weak $\ast$ topology.
| 7 | https://mathoverflow.net/users/131781 | 363019 | 152,639 |
https://mathoverflow.net/questions/362976 | 1 | So I've been reading through [On the number of monochromatic Schur triples](https://www.sciencedirect.com/science/article/pii/S0196885803000101) by Datrovsky on finding the minimal number of Schur triples. This means you're trying to 2-colour the set of the smallest n positive integers in such a way as to minimize the ... | https://mathoverflow.net/users/1000 | Schur triples question | Fourier analysis is a well-established tool in additive combinatorics,
and counting questions are routinely translated into equivalent questions about the Fourier transform.
This survey of Gowers might give you an idea of the breadth and scope: <https://arxiv.org/abs/1608.04127>.
| 1 | https://mathoverflow.net/users/20598 | 363020 | 152,640 |
https://mathoverflow.net/questions/363021 | -1 | I don't have so much experience in category theory so my question may be stupid and non-sense.
1. There is a classical adjunction in algebraic geometry between the $M\rightarrow M^{\sim}$ and the global section in the affine case .
2. We know that if we deal with quasi-coherent sheaves is an equivalences of categorie... | https://mathoverflow.net/users/nan | prove classic proposition in algebraic geometry by category theory | I am not sure the example you choose is the most striking application of category theory in algebraic geometry
However, there is a similar question where indeed category theory makes the heart of the phenomenon way clearer.
Let $ f : X \longrightarrow Y$ be a proper morphism of schemes. You get a functor $Rf\_\* : D... | 4 | https://mathoverflow.net/users/37214 | 363023 | 152,642 |
https://mathoverflow.net/questions/363028 | 2 | Let $M$ be a (connected) complex manifold, $L$ be a local system on $M$ and $\mathcal{L}$ the vector bundle associated to $L$. If $L$ is indecomposable, does it imply that $\mathcal{L}$ is also indecomposable?
| https://mathoverflow.net/users/32151 | Indecomposability of local systems and vector bundles | No, this does not hold in general. For example, there do exist irreducible flat connections on the trivial holomorphic bundle of rank two over a compact Riemann surface of genus $g\geq2.$
On the other hand, if you consider irreducible local systems with unitary monodromy over compact Riemann surfaces then the associate... | 6 | https://mathoverflow.net/users/4572 | 363029 | 152,643 |
https://mathoverflow.net/questions/363034 | 3 | In the following, we work with additive categories.
We say that a category is weakly idempotent complete if all the epimorphisms which admit a section have kernel. It's equivalent to the dual statement: all the monomorphisms which admit a retract have cokernel.
A stronger notion is the notion of idempotent complete... | https://mathoverflow.net/users/152969 | Idempotent complete category which is not abelian | The OP asked for responses as an answer. I have made this answer CW, and encourage everyone to make a big list. I start with the answer that comes up in my work, and then compile the answers from the comments. (As this answer is CW, feel free to reorganize, remove this intro, etc.)
1. Project modules over a ring $R$.... | 3 | https://mathoverflow.net/users/78 | 363047 | 152,647 |
https://mathoverflow.net/questions/363027 | 1 | I have a question that troubled me for a long time.
If I have two convex discrete function $f(·)$ and $g(·)$ such that $f(·) \ge g(·)$. (may be not necessary?)
Let $x\_1 = \text{argmin } f(·)$, and $x\_2 = \text{argmin } g(·)$. How to prove that $x\_1 \le x\_2 $?
Actually, I want to find the upper bound of the... | https://mathoverflow.net/users/159579 | How to compare the minimums of two discrete convex functions? | First a note: This is not a research level question. My answer is too long for a comment.
I think that by discrete function $h$ you mean that $h$ is defined on some $K := \{\ldots,k,k+1,k+2,\ldots\}$, possibly unbounded and that $\Delta h(x) = h(x+1)-h(x)$. Then $h$ is convex iff $\Delta h$ is not decreasing. To fix ... | 0 | https://mathoverflow.net/users/100904 | 363051 | 152,648 |
https://mathoverflow.net/questions/363053 | 6 | Some elements of $L$ become constructible only in levels higher than its rank level. So I ask:
Let $V$ be such that $V = L$.
For which ordinals $\alpha$ do we have $V\_\alpha = L\_\alpha$?
Indeed, we have this for $\omega$ and if $\alpha$ is the class of all ordinals. But do we have this for other types of ordinals... | https://mathoverflow.net/users/151198 | For which ordinals do we have $V_\alpha = L_\alpha$? | Yes; in fact the first $\alpha>\omega$ with $V\_\alpha=L\_\alpha$ has cofinality $\omega$. To obtain this $\alpha$, define $f:Ord\to Ord$ by $f(\xi)=$ the smallest $\eta$ such that $V\_\xi\subseteq L\_\eta$. Such an $\eta$ exists because of the assumption that $V=L$. Write $f^n$ for the $n$-fold iterate of $f$. Then $\... | 16 | https://mathoverflow.net/users/6794 | 363056 | 152,650 |
https://mathoverflow.net/questions/362944 | 0 |
>
> Consider a Brownian bridge $B: [0,1]\to \mathbb{R}$ with $B(0)=B(1)=0$. Let $M[0, 1/2]=\max\_{x\in[0,1/2]}B(x)$. How to prove that
> $$\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$$
>
>
>
Actually, here is a "no big max" argument that could be used in the proof of construction Airy line ens... | https://mathoverflow.net/users/168083 | How to prove that a Brownian bridge $\mathbb{P}(M[0, 1/2]\geq s)\leq 2\mathbb{P}(B(1/2)\geq s/2)?$ | Let $B\_t:=B(t)$. We have to show that
\begin{equation}
P(M\_{1/2}\ge s)\le2P(B\_{1/2}\ge s/2) \tag{1}
\end{equation}
for $s\ge0$, where $M\_T:=\max\_{0\le t\le T}B\_t$ for $T\in(0,1)$.
We shall prove (1) by first obtaining an explicit expression for $P(M\_T\ge s)$.
Indeed, for $t\in[0,1]$, we can write
$$B\_t=W\_... | 1 | https://mathoverflow.net/users/36721 | 363068 | 152,653 |
https://mathoverflow.net/questions/363063 | 4 | Let $R$ be a ring (not necessary commutative) and let $P\_{\bullet}$ be a perfect $R$-bimodule (chain complex). I will denote the category of perfect right $R$-chain complexes by $\textbf{Perf}(R)$. The endofunctor $-\otimes\_{R}P\_{\bullet} :\textbf{Perf}(R)\rightarrow \textbf{Perf}(R)$ induces a map in algebraic $K$-... | https://mathoverflow.net/users/159597 | Induced map in K-theory by a "trivial" bimodule | No. Let $R=\mathbb{Z}\times\mathbb{Z}$, let $P$ and $Q$ be the projective modules $\mathbb{Z}\times0$ and $0\times\mathbb{Z}$, and let
$$P\_\bullet=\dots\longrightarrow0\longrightarrow P\otimes\_\mathbb{Z}P
\stackrel{0}{\longrightarrow}Q\otimes\_\mathbb{Z}P\longrightarrow0\longrightarrow\dots$$
| 8 | https://mathoverflow.net/users/22989 | 363069 | 152,654 |
https://mathoverflow.net/questions/363062 | 2 | Maximizing a hyperplane $\sum\_i a\_ix\_i$ where $a\_i\in\mathbb R$ and each $a\_i$ are fixed and non-negative and $x\_i$ are variables over a standard simplex $\sum\_i x\_i\leq 1$ with $0\leq x\_i$ always produces a vertex point on the simplex and maximization corresponds to $\max\_i a\_i$.
>
> 1. In infinite dime... | https://mathoverflow.net/users/136553 | Constructivity of two problems on a standard simplex? | You make an erroneous assumption in your question, as already in dimension 1 you need LLPO to know that the maximum is actually attained at some point.
We work constructively.
**Theorem:** *LLPO is equivalent to the statement that every affine map $[0,1] \to \mathbb{R}$ attains its maximum*
*Proof.* The general f... | 2 | https://mathoverflow.net/users/1176 | 363073 | 152,656 |
https://mathoverflow.net/questions/363055 | 5 | The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance:
* $S = T$ is the set of primes (leading to Goldbach's conjecture)
* $S$ is the set of squares and $T$ is the set of primes, leading to the... | https://mathoverflow.net/users/140356 | Goldbach conjecture and other problems in additive combinatorics | It seems what you are asking is "If we have a precise asymptotic for the number of elements of a set, can we solve binary additive problems involving that set?"
The answer in general seems to be `no'. Let's consider Goldbach's conjecture that every large integer $n$ is the sum of two primes. It is not hard to see fro... | 16 | https://mathoverflow.net/users/630 | 363078 | 152,658 |
https://mathoverflow.net/questions/363066 | 3 | Consider $n-$dimensional Euclidean ball centred at 0 with radius $\sqrt{n}$. We want to show that the uniform distribution $X$ in this ball is sub-gaussian and $||X||\_{\psi\_2}<C$ where $C$ is absolute constant.
Clarify: $X$ is subgaussian if $\langle X,x \rangle$ is subgaussian for any $x \in \mathbb{R}^n$ and $||... | https://mathoverflow.net/users/116621 | Uniform distribution in Euclidean ball is sub-gaussian | $\newcommand\Ga\Gamma$
For each unit vector $x$, the random variable (r.v.) $\langle X,x\rangle$ equals
$$V:=\sqrt n\,W\_nR$$
in distribution, where
$$W\_n:=\frac{Z\_1}{\sqrt{Z\_1^2+\dots+Z\_n^2}},$$
$Z\_1,\dots,Z\_n$ are iid $N(0,1)$ r.v.'s, and $R$ is a r.v. (independent of $V$ and) such that $0\le R\le1$.
So, it s... | 4 | https://mathoverflow.net/users/36721 | 363081 | 152,659 |
https://mathoverflow.net/questions/363083 | 14 | There is a positive density of odd numbers which are of the form $2^n+p$ (due to Romanoff), and a positive density which are not of this form (due to van der Corput and Erdos, see [this paper](https://doi.org/10.1090/mcom/3537) for a review and some results on the density). So, for some but not almost all odd numbers, ... | https://mathoverflow.net/users/24463 | Hamming distance to primes | See OEIS sequences [A067760](https://oeis.org/A067760) and [A076336](https://oeis.org/A076336). If $n$ is a dual Sierpiński number, there is no $k$ such that $n+2^k$ is prime. There is no prime with Hamming distance $1$ to the Sierpiński number $2131099$, and this may be the least positive integer with this property.
... | 24 | https://mathoverflow.net/users/13650 | 363089 | 152,664 |
https://mathoverflow.net/questions/363026 | 5 | *Also asked on MSE: [What is the best way to partition the $4$-subsets of $\{1,2,3,\dots,n\}$?](https://math.stackexchange.com/questions/3717109/).*
Consider the set $X = \{1,2,3,\dots,n\}$. Define the collection of all $4$-subsets of $X$ by $$\mathcal A=\{Y\subset X: Y\text{ contains exactly $4$ elements}.\}$$
I w... | https://mathoverflow.net/users/159578 | What is the best way to partition the $4$-subsets of $\{1,2,3,\dots,n\}$? | This problem can be reformulated in terms of graph coloring:
Let the graph $G=(V,E)$, $V=\mathcal A$, $(x,y)\in E \leftrightarrow x \cap y \geq 2$
Then a partition of $\mathcal{A}$ into groups $A\_1,A\_2,…,A\_m$ corresponds to a $m$-coloring of $G$.
The graph $G$ has degree no more than $6{n\choose 2}$, so $G$ ca... | 7 | https://mathoverflow.net/users/125498 | 363104 | 152,668 |
https://mathoverflow.net/questions/362989 | 0 | Let $H=(V, E)$ be a weighted hypergraph such that $V=A\cup B \cup C$, where $A,B,C$ are disjoint sets of size $n$, and $E=A\times B\times C$. In my particular case, $\forall e\in E$, $ wt(e)\in\{0,\pm 1, \pm 3\}$.
**I'm trying to figure out whether or not there is an efficient algorithm to compute the maximum cardina... | https://mathoverflow.net/users/159565 | Maximum-weight perfect matching in a 3-regular, complete, 3-partite hypergraph | Yup, NP-hard.
Consider the case where edges only have two weights. Then we have a hypergraph of light edges, and the problem is to pick a matching using as many light edges as possible. This is NP-hard, and in fact I believe it was one of Karp’s original 21 NP-complete problems.
<https://en.m.wikipedia.org/wiki/3-d... | 0 | https://mathoverflow.net/users/22512 | 363111 | 152,670 |
https://mathoverflow.net/questions/353990 | 4 | Let $p$ be an odd regular prime and $F$ be a $p$-rational number field containing $\mu\_p$. Equivalently, there is a unique prime $\mathfrak{p}$ above $p$ in $F$ and the $p$-class group is generated by $\mathfrak{p}$ [Gras' book on Class Field Theory IV.3.5]. Let $F\_\mathfrak{p}$ be the maximal pro-$p$ unramified outs... | https://mathoverflow.net/users/116598 | A question on p-rationality of number fields | I asked your question to my advisor Professor Movahhedi who introduced and studied the notion of $p$-rationality in his thesis.
Here I share his answer with you.
The two questions are somehow of different nature.
Denote by $F\_{S\_p}$ the maximal $p$-extension of the $p$-rational filed $F$ unramified outside the pr... | 5 | https://mathoverflow.net/users/152866 | 363113 | 152,672 |
https://mathoverflow.net/questions/363108 | 4 | I am reading Knapp's book "Lie groups beyond introduction". On page 369, he has described the following. Let $\mathfrak g$ be a real semisimple Lie algebra. Suppose $\theta\colon\mathfrak g\to \mathfrak g$ is a Cartan involution. Let $B$ be a nondegenerate symmetric bilinear form on $\mathfrak g$ which is $\theta $-inv... | https://mathoverflow.net/users/136860 | Generalization of Killing form | According to theorem 2.15, all Cartan subalgebras of a complex semi-simple Lie algebras are conjugated. I.e. there exists $\alpha \in \mathrm{Inn}(\mathfrak{g})$ such that $\mathfrak{h}\_1 = \alpha(\mathfrak{h}\_2).$ Since any invariant form $B$ is also invariant with respect to the group of inner automorphisms (sorry,... | 4 | https://mathoverflow.net/users/6818 | 363128 | 152,676 |
https://mathoverflow.net/questions/362265 | 17 | Let us use the standard notation for $q$-integers, $q$-binomials,
and the $q$-analog
$$
\operatorname{Cat}\_q(n) := \frac{1}{[n+1]\_q} \left[\matrix{2n \\ n}\right]\_q.
$$
I want to prove that for all integers $n\geq 0$, we have
\begin{equation}
\operatorname{Cat}\_q(n+2)
=
\sum\_{0\leq j,k \leq n}
q^{k(k+2) + j(n+2)}... | https://mathoverflow.net/users/1056 | Proof of certain $q$-identity for $q$-Catalan numbers | I managed to solve the problem,
in the last general conjecture, one can apply the $q$-Chu-Vandermonde theorem. After some simplification,
the resulting expression can be expressed as a ${}\_2\phi\_1$ q-hypergeometric series, where one again can apply the $q$-Chu-Vandermonde theorem.
Skipping lots of details the proof... | 5 | https://mathoverflow.net/users/1056 | 363159 | 152,689 |
https://mathoverflow.net/questions/362591 | 3 | Let $\pi:X\to \mathbb{P}^1\_{\mathbb{Z}}$ be a proper flat morphism with $X$ an integral scheme. Is $\pi\_\*\mathcal{O}\_X$ necessarily locally free?
| https://mathoverflow.net/users/nan | Proper and flat over $\mathbb{P}^1_{\mathbb{Z}}$ implies locally free | Like your [other question](https://mathoverflow.net/q/362600/82179), the answer to this one is related to miracle flatness:
**Theorem** (Miracle flatness). *Let $f \colon X \to Y$ be a finite dominant morphism of schemes with $Y$ regular. Then $f$ is flat if and only if $X$ is Cohen–Macaulay.*
See for example Tags ... | 4 | https://mathoverflow.net/users/82179 | 363164 | 152,694 |
https://mathoverflow.net/questions/363153 | 3 | **Edit:** This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and [migrated to its own post here](https://mathoverflow.net/questions/363294/bounding-the-probability-that-two-binomials-are-equal).
-... | https://mathoverflow.net/users/22512 | Inequality for difference of consecutive atom probabilities for binomial distribution | Concerning your first question:
Let $p\_k:=P(B\_{n,p}=k)$. We have to show that
\begin{equation\*}
p\_k-p\_{k+1}\ll\frac1{npq}, \tag{1}
\end{equation\*}
where $q:=1-p$ and $a\ll b$ means that $a\le Cb$ for some universal real constant $C>0$.
Clearly, without loss of generality (wlog)
\begin{equation\*}
1\ll npq.
\end... | 3 | https://mathoverflow.net/users/36721 | 363173 | 152,697 |
https://mathoverflow.net/questions/363148 | 6 | In their text *Foundations of Stable Homotopy Theory*, Barnes and Roitzheim define the suspension of a cofibrant object X of a pointed model category to be the pushout of the diagram $\*\leftarrow X\coprod X\to Cyl(X)$, where the second map is the structure map of the cylinder object. By contrast, there is a more manif... | https://mathoverflow.net/users/158123 | Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category? |
>
> if we don't assume properness, I don't even see why the first is homotopy-invariant!
>
>
>
The pushout of a diagram A←B→C in which all objects are cofibrant and one of the maps is a cofibration
is always its homotopy pushout in any model category,
see Proposition A.2.4.4 in Lurie's Higher Topos Theory.
Thi... | 9 | https://mathoverflow.net/users/402 | 363175 | 152,698 |
https://mathoverflow.net/questions/363181 | 2 | Given a set of vectors $S=\{v\_1, v\_2,...,v\_d\} \subset \mathbb{R}^{N}, \, N>d$, is there any algorithm to decide if there exist a vector with all coordinates strictly positive in the generating subspace $\langle S \rangle$?
I am aware of results like the Farkas Lemma (or variants as Gordan's Theorem, etc...).
Or ... | https://mathoverflow.net/users/98507 | Intersection of a vector subspace with a cone | Let
$$M = \begin{pmatrix} | & | & \cdots & | \\ v\_1 & v\_2 & \cdots & v\_d \\ | & | & \cdots & | \end{pmatrix}$$
and let $e\_1, \ldots, e\_N$ be the standard unit vectors. Then consider the linear programs indexed by $i = 1, \ldots, N$:
$$\begin{aligned}
&\text{maximize }\langle e\_i,Mx\rangle \\
&\text{subject to }... | 2 | https://mathoverflow.net/users/24463 | 363188 | 152,704 |
https://mathoverflow.net/questions/363172 | 5 | This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an *O-ring* if there exists an integer $n \ge 1$, a local field of zero characteristic (that is, a finite extension of $ \mathbb{Q}\_p$ for some prime $p$) with ring of integers $ \mat... | https://mathoverflow.net/users/3635 | Intrinsic characterisation of a class of rings | The following criterion came up when I was writing [this answer](https://mathoverflow.net/a/357315/82179) (but I did not end up using it there):
>
> **Lemma.** *Let $R$ be a commutative ring. Then $R$ is of the form $\mathcal O\_K/\mathfrak p^n$ for a finite extension $\mathbf Q\_p \subseteq K$ and $n \in \mathbf Z... | 6 | https://mathoverflow.net/users/82179 | 363193 | 152,707 |
https://mathoverflow.net/questions/356977 | 1 | *I was wondering whether the following category already has been used somewhere and whether it already has been named*.
Let us fix a field $k$ (or more generally a ring). An object is just a $k$-vector space and a morphism $f:U\rightarrow V$ consists of a choice of a sub-vectorspace $V\_f\subset V$ and a well-defined... | https://mathoverflow.net/users/3969 | category of non-welldefined linear maps | As Denis Nardin pointed out in the comments, the name *total correspondence* is a good choice.
| 1 | https://mathoverflow.net/users/3969 | 363201 | 152,711 |
https://mathoverflow.net/questions/363144 | 6 | Let $H$ be a closed subgroup of the compact Lie group $G$. Let $E$ be a continuous representation of $H$. In the book *"Representations of compact Lie groups"* by Bröcker and Dieck the induced representation of $E$ is defined as the vector space $iE$ of all continuous functions $f:G\to E$ satisfying $f(g\cdot h)=h^{-1}... | https://mathoverflow.net/users/36563 | Universal property of induced representation | You are writing a right adjoint to restriction so you have a natural $H$-module map
$$
iE\rightarrow E, \ (f(x):G\rightarrow E)) \mapsto f(1) .
$$
To cook up a map in the opposite direction, you need to use the fact that the category of $H$-modules is semisimple and choose a splitting map.
Now you use the fact the ca... | 2 | https://mathoverflow.net/users/5301 | 363208 | 152,715 |
https://mathoverflow.net/questions/363184 | 2 | Let $\mathcal{H}$ be a Hilbert space with orthonormal basis $\{\varphi\_{k}\}\_{k\in I}$. Take $\mathcal{H}^{\otimes n} := \overbrace{\mathcal{H}\otimes\cdots\otimes \mathcal{H}}^{\mbox{$n$ times}}$. An element of $\mathcal{H}^{\otimes n}$ can be expressed as:
$$\psi = \sum\_{\{k\_{1},...,k\_{n}\}\subset I}\alpha\_{k\_... | https://mathoverflow.net/users/152094 | Representation of an arbitrary element on a fermionic Fock Space | [This is not research level, so probably does not belong on MO, but I think the question is well-asked.]
If $\psi\in\wedge^n\mathcal{H}$ then by definition $\sigma^\*\psi = \epsilon\_\sigma \psi$ for each permutation $\sigma$ and so as $\epsilon\_\sigma \in \{\pm 1\}$ we have
$$ A\_n\psi = \frac{1}{n!} \sum\_\sigma \... | 1 | https://mathoverflow.net/users/406 | 363210 | 152,716 |
https://mathoverflow.net/questions/361902 | 0 | We know that the decision problem of classifying the graphs as class $1$ or class $2$ (with respect to edge coloring) is NP-complete. But, suppose we have to prove a graph to be in class $1$. Does it always imply that we have to find a polynomial time algorithm for the edge coloring of that graph? Like, is it possible ... | https://mathoverflow.net/users/100231 | Complexity of edge coloring of class 1 graphs | As written the question is ill-posed: class 1 or class 2 is a property of a particular graph, whereas polynomial time is a property of an algorithm applied to an infinite family of graphs.
Patrick Schnider points out in the comments that the first fix to the question doesn't work: if we have a polynomial time algorit... | 2 | https://mathoverflow.net/users/25485 | 363213 | 152,717 |
https://mathoverflow.net/questions/350370 | 12 | I consider $1,i,j,k,l,m,n,o$ the standard basis of the (complexified if you like) octonions ($\mathbb{O}$ for the octonions).
Let $a = x\_1.1 +\ldots + x\_8.o$, $b = x\_9.1+ \ldots + x\_{16}.o$ and $c = x\_{17}.1+ \ldots +x\_{24}.o$, where $x\_1, \ldots, x\_{24}$ are indeterminates over the base field (say $\mathbb{C}$... | https://mathoverflow.net/users/37214 | Characteristic polynomial of an $8 \times 8$ symmetric matrix with indeterminate entries related to octonionic multiplication | Let $a,b,c\in\mathbb{O}$ be octonions and consider the linear map $L:\mathbb{O}\to\mathbb{O}$ defined by
$$
L(x) = (b(cx))a = R\_aL\_bL\_c(x).
$$
One desires a formula for the characteristic polynomial of $S$, the *symmetric* part of $L$, i.e.,
$$
S(x) = \tfrac12\bigl(R\_aL\_bL\_c + {}^t(R\_aL\_bL\_c) \bigr).
$$
(I not... | 10 | https://mathoverflow.net/users/13972 | 363223 | 152,722 |
https://mathoverflow.net/questions/363115 | 3 | Suppose that $q$ is a prime power and $\xi, \eta\in \mathbb{F}\_q$ are nonzero. A computer calculation for $q<70$ suggests that the number $N$ of $4$-tuples
$(a,b,c,d)\in\mathbb{F}\_q^{4}$ satisfying $(ac-\xi bd)^2-(a^2-\xi b^2+1)(c^2-\xi d^2-\eta)=0$ is $q^3-q$.
**Question.** Is there some theory, or nice method, fo... | https://mathoverflow.net/users/23827 | Number of solutions of a degree 4 polynomial equation over a finite field | I will concentrate on the case of odd $q$ and $\xi,-\eta$ being squares, but the solution should be extendable to the remaining cases.
It is convenient to use the language of characters sums in order to compute $N$. To reduce your point counting problem to a character sum problem, the following observation is useful:... | 4 | https://mathoverflow.net/users/31469 | 363225 | 152,723 |
https://mathoverflow.net/questions/362975 | 3 | Let $F$ be an algebraically closed field of characteristic $p$ equipped with an absolute value $|\cdot|:F \rightarrow \mathbb{R}\_{\ge 0}$ with respect to which $F$ is complete.
>
> Define $|\cdot|\_{prod}$ on the ring $F\otimes \_{\mathbb F\_p} F$ in the following way. If $c\in F\otimes \_{\mathbb F\_p} F$, then
>... | https://mathoverflow.net/users/105386 | Norm on tensor product of fields | You can probably find this in most books on non-Archimedean functional analysis, see for instance Proposition 17.4 in Schneider's book.
The rough idea is to reduce to a tensor product of finite-dimensional spaces and then to norms associated to bases. You can now compute directly.
By the way, you probably want to a... | 2 | https://mathoverflow.net/users/4069 | 363227 | 152,724 |
https://mathoverflow.net/questions/363231 | 0 | I often encounter things of the form $x^\top M x$, where $M$ is symmetric positive (semi-)definite. Is there a term for that? I know related terms:
* We can say $M$ is a [bilinear form](https://en.wikipedia.org/wiki/Bilinear_form), $M(x,y) = x^\top M y$ with an induced norm, $|\!|x|\!|\_M := \sqrt{x^\top M x}$, so I ... | https://mathoverflow.net/users/159700 | Is there a name for $f(M, x) = x^\top M x$? | You can say that $x^\top M x$ is the quadratic form associated to $M$.
| 5 | https://mathoverflow.net/users/1106 | 363233 | 152,725 |
https://mathoverflow.net/questions/363228 | 1 | Let $G=\mathfrak{S}\_n$ be the symmetric group on $n$ elements. Via permuting the variables the polynomial ring $S=\mathbb{C}[x\_1,\ldots,x\_n]$ becomes an $G$-module. It is not hard to see that every irreducible representation of $G$ appears in $S$: For example the span of the orbit of the monomial $\prod\_{i=1}^nx\_i... | https://mathoverflow.net/users/36563 | Can every $\mathfrak{S}_n$-linear map be realized by a multiplication? | Yes. Take an injective map $A : U \to S$ and an injective map $B' : V \to S$.
Now compose $B'$ with the map that sends $x\_i$ to $x\_i^m$ for all $i$, where $m$ is greater than the degree of any polynomial in the image of $A$.
Then the composed map $A \otimes B : U \otimes V \to S$ will be injective because, as a m... | 1 | https://mathoverflow.net/users/18060 | 363235 | 152,726 |
https://mathoverflow.net/questions/363239 | 5 | One of the first things you learn in a programming 101 course is to write readable code, and to name your variables properly. This notion has seemingly never translated into mathematics. Everywhere you look, there are one letter constants, variables and functions, and an abundance of hard to remember symbols for operat... | https://mathoverflow.net/users/159703 | Why did mathematical notation stay so hard to read? | Mathematics uses very few variable names in any proof compared to the number of variable names occuring in typical programming languages. Variable names survive only for short passages, except for a small (less than a dozen) *global* variables. The names are subject to a host of conventions (for example, $\varepsilon$ ... | 5 | https://mathoverflow.net/users/13268 | 363241 | 152,729 |
https://mathoverflow.net/questions/363180 | 2 | In my work, I need to study the convergence of sequence defined by the non-linear recurrence relation
$$
u\_0,u\_1>0, \qquad \forall n\in \mathbb N, \; u\_{n+2}=a\ln(1+u\_n)+b\ln(1+u\_{n+1})
$$
with fixed $a,b> 0$.
For $a=b=1$ Wolfram gives the limit $-1-2 W\_{-1}\left(-\frac{1}{2\sqrt{e}}\right)$. (W is the Lambert ... | https://mathoverflow.net/users/126827 | Convergence for a non-linear second order difference equation | *I would suspect that non-classical arguments are needed to do so.*
All you need to know is that $t\mapsto \frac 1{1+t}$ is a decreasing function, so for $0<x\le x'$ we have $\frac{\log(1+x')}{\log(1+x)}\le \frac {x'}x$. This immediately implies that the mapping $T:(x,y)\mapsto (y,a\log(1+x)+b\log(1+y))$ is non-expan... | 5 | https://mathoverflow.net/users/1131 | 363242 | 152,730 |
https://mathoverflow.net/questions/363212 | 2 | Suppose we have a commutative square
$\require{AMScd}$
\begin{CD}
A @>{f}>> B\\
@V{h}VV @V{k}VV \\
C @>{g}>> D
\end{CD}
of quasi-categories, such that $f,g,h,k$ are cofibrations in the Joyal model structure (i.e. these are monomorphisms of simplicial sets). Suppose further that $A,B,C,D$ have all finite coproducts, and... | https://mathoverflow.net/users/89498 | Pushout of quasi-categories with finite coproducts | A pushout $B \coprod\_A C$ will almost never have all coproduct. the problem is that objects in $B \coprod\_A C$ are all either objects of $B$ or objects of $C$, so if $B \coprod\_A C$ has coproduct, it means that every time you take the coproduct of $b \in B$ with $c \in C$ it would have to be either in $B$ or in $C$.... | 3 | https://mathoverflow.net/users/22131 | 363245 | 152,731 |
https://mathoverflow.net/questions/363244 | 0 | I am currently reading *Boolean Function Complexity - Advances and Frontiers* by Stasys Jukna and on page 7 of the latest edition there is a paragraph titled **Boolean functions as set systems** with the following quote:
>
> By identifying subsets $S$ of $[n] = \{1, \cdots, n\}$ with their characteristic $0–1$ vect... | https://mathoverflow.net/users/159706 | Unknown notation in "Boolean function complexity" by Stasys Jukna | The notation $[n]$ is an $n$-element set of integers. Usually, it means $[n] = \{1,2,\ldots,n\}$, but sometimes it goes from 0 to $n-1$ (this will basically never matter, and just assume it’s the first one).
In general, $2^{S}$ could mean either the powerset of $S$ (i.e., all the subsets of $S$), or could mean the se... | 2 | https://mathoverflow.net/users/22512 | 363247 | 152,733 |
https://mathoverflow.net/questions/363257 | 4 | The classical definition of Hilbert modular cuspforms as given in say, Hida's "On $p$-adic Hecke algebras for $\mathrm{GL}\_2$ over totally real fields", defines them as holomorphic functions $f: \mathbb{H}^d \to \mathbb{C}$ satisfying the usual moebius transformation properties with respect to a chosen level subgroup ... | https://mathoverflow.net/users/143607 | anti-holomorphic Hilbert modular forms as global sections | Your second guess is spot on: these $J$-holomorphic forms are exactly $H^{k}$ of a sheaf on (a smooth compactification of) the Hilbert modular variety, where $k = |J^c|$. This is an instance of a much more general theory which is largely due to Harris. The canonical reference is
Harris, Michael. [Automorphic forms of... | 2 | https://mathoverflow.net/users/2481 | 363270 | 152,737 |
https://mathoverflow.net/questions/363232 | 1 | Disclaimer : I asked this question on MSE, I have no answer and I think it's better to ask it here.
Let $(A,\mathcal{W}, \mathcal{C})$ be a Waldhausen category with $A$ an additive category.
On one hand, we can define the ordinary limits $lim\_A$ of the underlying category $A$.
On other hand, we can define limits of ... | https://mathoverflow.net/users/152969 | Do limits in Waldhausen categories commute with ordinary limits? | Unless I'm misunderstanding your question, the answer is yes. Your second way of defining limits, via "the universal property of a diagram with some arrows in $\mathcal{F}$," is actually a special case of the normal definition of a limit, and limits commute. For your specific question of interest, the key observation i... | 1 | https://mathoverflow.net/users/11540 | 363271 | 152,738 |
https://mathoverflow.net/questions/363281 | 2 | Let $f: X\rightarrow S$ be a quasi-compact, quasi-separated, flat morphism of schemes, with $S$ locally Noetherian. Also, fix an integer $i\geq 1$.
**Question 1:** Does $H^i(X\_s,\mathcal{O}\_{X\_s})=0$, for all $s\in S$, imply that $R^if\_\ast\mathcal{O}\_X=0$?
The answer is *yes*, if $f$ is also assumed to be *pr... | https://mathoverflow.net/users/14349 | Does fibrewise vanishing of cohomology imply its vanishing (non-proper case)? | Here is one simple counterexample:
**Example.** Let $X = \mathbf A^2 \setminus 0$ and $S = \mathbf A^1$, where $f \colon X \to S$ is the first coordinate projection. The fibres are affine, so have no higher coherent cohomology.
But if $V \subseteq S$ is affine open containing $0$, then $f^{-1}(V)$ has an affine ope... | 10 | https://mathoverflow.net/users/82179 | 363282 | 152,739 |
https://mathoverflow.net/questions/363267 | 4 | For any cardinal $\alpha \in \omega\cup \{\omega\}$, let $[\omega]^\alpha$ denote the collection of subsets of $\omega$ having cardinality $\alpha$.
A *linear hypergraph* $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that whenever $e\neq e\_1\in E$ we have $|e\cap e\_1|\leq 1$.
A *color... | https://mathoverflow.net/users/8628 | Chromatic number of regular linear hypergraphs on $\omega$ | For $\alpha=\omega$ the answer is no. If $H=(\omega,E)$ is a linear hypergraph, then $E$ is countable; if $H=(\omega,E)$ is any hypergraph with $E\subseteq[\omega]^\omega$ and $E$ countable, then $\chi(H)\le2$.
For $3\le\alpha\lt\omega$ the answer is yes. It will be convenient to define the hypergraph on a countable ... | 4 | https://mathoverflow.net/users/43266 | 363283 | 152,740 |
https://mathoverflow.net/questions/363272 | 2 | The normalizer $N(\mathfrak{h})$ of the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{su}(3)$ is defined as
$$N=\left\{ x \in SU(3)\;|\; x^\dagger\mathfrak{h}x \in \mathfrak{h}\right\}$$
I would like to know explicitly what this subgroup looks like; for example, in terms of the exponentiation of generators of $\mathfr... | https://mathoverflow.net/users/159498 | Explicit Normalizer of SU(3) Cartan subalgebra | Reproduced from my [comment](https://mathoverflow.net/questions/363272/explicit-normalizer-of-su3-cartan-subalgebra#comment916426_363272); please let me know if it does not answer the question.
For your choice of $\mathfrak h$ (as the diagonal Cartan subalgebra in $\mathfrak{su}(3)$), we have that $C = \operatorname ... | 3 | https://mathoverflow.net/users/2383 | 363287 | 152,742 |
https://mathoverflow.net/questions/363275 | 0 | Given $v\_{ij} \in \{0,1\}$, $i \in \{1,2\}$, $j \in \{1,2,\ldots,n\}$. Let $X\_1, X\_2, \ldots, X\_n$ be random variables, $P[X\_i=1]=P[X\_i=0]=1/2$, $i \in \{1,\ldots, n\}$. By checking many examples, I think that the following is true: when $|v\_1|, |v\_2|$ are large enough,
\begin{align}
\frac{1}{2} \sum\_{x\_1=0}... | https://mathoverflow.net/users/11877 | Estimate an expression about probability about Bernoulli random variables | We shall assume that the $X\_i$'s are independent. The problem can be restated as follows: show that for some $h\in(0,1)$, all natural $n$, and all subsets $J$ and $K$ of the set $[n]:=\{1,\dots,n\}$
we have
\begin{equation\*}
S:=\sum\_{x,y}|P(X\_J=x,X\_K=y)-P(X\_J=x+1,X\_K=y+1)|\le2-h, \tag{1}
\end{equation\*}
where ... | 2 | https://mathoverflow.net/users/36721 | 363290 | 152,744 |
https://mathoverflow.net/questions/362414 | 0 | Let $\pi:X\to S$ be a proper morphism of Noetherian schemes. Is it possible that $\pi\_\*\mathcal{O}\_X\neq \mathcal{O}\_S$ but the natural map $\mathcal{O}\_s^\wedge\to (\pi\_\*\mathcal{O}\_X)\_s^\wedge$ is an isomorphism for all points $s\in S$? The completion is with respect to $\mathfrak{m}\_s$ on both sides.
| https://mathoverflow.net/users/nan | Completed stalks of the pushforward of the structure sheaf | This has little to do with morphisms, and follows immediately from the following commutative algebra lemma:
**Lemma.** *Let $R$ be a Noetherian ring, and $f \colon M \to N$ a morphism of finite $R$-modules. Then $f$ is an isomorphism if and only if $f^\wedge\_{\mathfrak m} \colon M^\wedge\_{\mathfrak m} \to N^\wedge\... | 2 | https://mathoverflow.net/users/82179 | 363304 | 152,747 |
https://mathoverflow.net/questions/363135 | 3 | Let us consider a sequence of continuous functions $g\_{q}:ℝ^2\to ℝ^2$. Let $(A\_{q})\_{q\geq 1}$ be a sequence of compact sets in $ℝ^2$. Assuming that each function $g\_{q}$ is topologically mixing in $A\_{q}$ for all $q\geq 1$, i.e., for every open subsets $U,V$ of $ℝ^2$ such that $U\cap A\_{q}$ and $V\cap A\_{q}$ ar... | https://mathoverflow.net/users/74668 | Diophantine equation that has an infinite number of positive integers solutions | It is just Pell's equation $m^2-Ny^2=1$, for $N=16q^6(q^2+1) $. Thus even for fixed $q$ it has infinitely many solutions, that was proved by Lagrange and you may find the proof in many textbooks.
| 3 | https://mathoverflow.net/users/4312 | 363305 | 152,748 |
https://mathoverflow.net/questions/363254 | 5 | This is probably a very easy question for experts in probability or measure theory.
I have a sequence of finite measures $\mu\_{n}$ on a non-compact metric space $X$ such that $\mu\_{n}$ converges to $\mu$ in the following sense:
$$ \int\_{X}fd\mu\_{n} \to \int\_{X}fd\mu \ \ \ \ \ \text{ for all f continuous with com... | https://mathoverflow.net/users/127739 | Tight sequence of measures | Since your space is Polish, $\mu$ is regular and there exists a compact set $C$ such that $\mu(X\setminus C)<\epsilon$ for each $\epsilon>0$. Since your space is locally compact, there is another compact set $C'$ such that $C$ is a subset of the interior of $C'$. The function given by $f\_n(x)=\max\{0,1-n\cdot d(C,x)\}... | 4 | https://mathoverflow.net/users/35357 | 363306 | 152,749 |
https://mathoverflow.net/questions/363101 | 2 | Intuition strongly suggests that there exist $\left\lfloor\frac{\binom{n}{k}}{\lfloor\frac{n}{k}\rfloor}\right\rfloor$ independent sets in the complement of a Kneser graph each having $\lfloor\frac{n}{k}\rfloor$ vertices in it. Is this true. If true, how to establish it?
A construction of such a set of cliques in the... | https://mathoverflow.net/users/100231 | Independent sets in complement of Kneser graphs | According to [p. 8], Baranyai's theorem [B] implies that the vertex set of the Kneser graph $K(n,k)$ can be partitioned into $\left\lceil\frac{\binom{n}{k}}{\left\lfloor\frac{n}{k}\right\rfloor}\right\rceil$ cliques of size $\left\lfloor\frac{n}{k}\right\rfloor$.
*References*
[B] Zs. Baranyai, *On the factorization... | 4 | https://mathoverflow.net/users/43954 | 363309 | 152,751 |
https://mathoverflow.net/questions/363286 | 6 | I am a PhD freshman working on topological graph theory and geometric group theory. I would like to learn some Bass-Serre theory. What do you think is the best introductory textbook in this topic? Thank you in advance for your recommendations.
| https://mathoverflow.net/users/159356 | Bass-Serre theory textbook | As mentioned by Andy Putman in the comments, the classical (and probably the best) references are Serre's book *Trees* and Scott and Wall's paper *Topological methods in group theory*.
Serre's approach is elementary and essentially self-contained, based on combinatorial arguments. Scott and Wall's approach is based o... | 7 | https://mathoverflow.net/users/122026 | 363311 | 152,752 |
https://mathoverflow.net/questions/362396 | 6 | For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$. A function $\mu:\omega^\omega\to \omega^X$ is called *monotone* if for any $f\le g$ in $\omega^\omega$ we have $\mu(f)\le\mu(g)$.
>
> **Questi... | https://mathoverflow.net/users/61536 | A monotone countably unbounded function from $\omega^\omega$ to $\omega^{\omega_1}$ | The answer to this problem is negative and follows from
>
> **Theorem.** For any monotone function $\mu:\omega^\omega\to\omega^{\omega\_1}$ there exists a countable infinite set $A\subset\omega\_1$ such that for every $f\in\omega^\omega$ the function $\mu(f){\restriction}A$ is bounded.
>
>
>
*Proof.* For every... | 1 | https://mathoverflow.net/users/61536 | 363329 | 152,757 |
https://mathoverflow.net/questions/363330 | 5 | The question is in the title:
>
> Who first claimed the existence / necessity of the empty set ? When did this happen ?
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>
>
Of course I know that the notation $\emptyset$ goes back to André Weil, and that the first axiom of ZF is that there exists an empty set, but I ask whether this is an older concept.
| https://mathoverflow.net/users/8799 | Historical origin of the empty set | "It can be justifiably argued that Boole had inventented the empty set" [in *The Mathematical Analysis of Logic* (1847)].
[The Empty Set, the Singleton, and the Ordered Pair](https://www.jstor.org/stable/3109881), Akihiro Kanamori (2003).
| 13 | https://mathoverflow.net/users/11260 | 363332 | 152,759 |
https://mathoverflow.net/questions/363327 | 1 | Given the expression
$T = \max(|\beta\_{\max}-1|,|\beta\_{\min}-1|)$
Can we relate $T$ to variance $\sigma^2$ of values $\beta\_i^2$, assuming we know $\sigma^2$, $\beta\_{\max}$ and $\beta\_{\min}$.
PS: What if values $\beta\_i^2$ are distributed with mean $\mu=1$.
| https://mathoverflow.net/users/75659 | Relating maximum/minimum value and variance of data points | Let $b\_i:=\beta\_i$. Then $1-T\le b\_i\le1+T$ for all $i$, and these inequalities provide the best bounds on the $b\_i$'s given the information we have. So,
$$A\le b\_i^2\le B \tag{1}$$
for all $i$, where
$$A:=\max[0,1-T]^2,\quad B:=(1+T)^2,$$
and inequalities (1) provide the best bounds on the $b\_i^2$'s given the... | 1 | https://mathoverflow.net/users/36721 | 363333 | 152,760 |
https://mathoverflow.net/questions/363336 | 11 | On p.177 of [Number Theory Revealed: A Masterclass](https://rads.stackoverflow.com/amzn/click/com/1470441586) by Andrew Granville, the author states that "One can ask for prime values of polynomials in two or more variables." (though he later mentions [Landau's $n^2+1$ conjecture](https://en.wikipedia.org/wiki/Landau%2... | https://mathoverflow.net/users/31084 | What is the significance of Friedlander-Iwaniec and related theorems? | I think questions about prime values of polynomials are considered inherently interesting.
All these questions are special cases of Bunyakovsky's conjecture, or, if you want, the Bateman-Horn conjecture. Certainly mathematics is not yet ripe for such problems.
You can also view them as special cases of the general ... | 14 | https://mathoverflow.net/users/18060 | 363339 | 152,762 |
https://mathoverflow.net/questions/363192 | 0 | If we are given a set of real linear inequalities then using elimination theory or just linear programming we can decide. If the program also has inequalities of form $2^x\leq g$ in addition to linear inequalities then is there a procedure when the conditions of exponentiation are base is constant and all involved vari... | https://mathoverflow.net/users/136553 | Linear programming with exponential inequalities and rational variables | Consider questions like whether $${\large{2^{2^{\sqrt{2^\phantom{o\!}}}}}}>\frac{19}{3}.$$
That inequality is true, but I think we don't know whether expressions like the left-hand side can be rational. So we don't know how accurately we need to compute towers of exponentials to see if inequalities hold.
In terms of ... | 0 | https://mathoverflow.net/users/nan | 363342 | 152,764 |
https://mathoverflow.net/questions/363346 | 6 | I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) algebra being a field $\mathbb{F}$ and operations being polynomials and field automorphisms) I am also interested in co... | https://mathoverflow.net/users/159768 | Algebraic geometry additionally equipped with field automorphism operation | You should look into [difference algebra](https://en.wikipedia.org/wiki/Difference_algebra), which is exactly this setting. (It's called difference algebra because the most classical version looked at the automorphism on polynomials that sends $f(x)$ to $f(x+1)$, which can be used to state the theory of difference equa... | 10 | https://mathoverflow.net/users/3711 | 363347 | 152,767 |
https://mathoverflow.net/questions/363285 | 2 | Let $f\in L^1(\mathbb{R})$ and continuous on $\mathbb{R}$ such that its Fourier transform $\hat f$ equals zero in a neighborhood of zero.
Let $F$ be function such that $\hat F$ exists and
$$\hat f(x) =x\hat F(x),\quad \forall x\in \mathbb{R}$$
Prove that $F\in L^1(\mathbb{R})$.
Any hints on how to prove that?
... | https://mathoverflow.net/users/126827 | Integrability of a function under a condition on its Fourier transform | We have $\widehat{f}=0$ near zero, so we can write $\widehat{F}=\widehat{g}\widehat{f}$ with a function $\widehat{g}\in C^{\infty}$, $\widehat{g}(x)=1/x$ for $|x|\ge a>0$.
Since $\widehat{g}'' = -(t^2 g)\,\widehat{}\in L^1$, we have $|g(t)|\lesssim 1/t^2$. Moreover, $g\in L^2\subseteq L^1\_{\textrm{loc}}$. So $g\in L... | 4 | https://mathoverflow.net/users/48839 | 363348 | 152,768 |
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