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https://mathoverflow.net/questions/275197 | 5 | Let $W=\bigcup\_{n=1}^\infty S\_n$ be the union of all symmetric groups $S\_n$. For an element $w\in W$, denote by $\mathfrak{S}\_w$ the Schubert polynomial associated to $w$, and by $\partial\_w$ the divided difference operator associated to $w$.
**Question:** Let $u,v\in W$. Why is
$$
\partial\_u(\mathfrak{S}\_u\m... | https://mathoverflow.net/users/66288 | Positivity of coefficients of a polynomial derived from Schubert polynomials | I found the relevant literature for this question:
<https://arxiv.org/abs/0705.4546>
<https://arxiv.org/abs/1409.7056>
The keyword is "skew divided difference operator". In the first paper, Conjecture 1 states explicitely what I was trying to solve. An affirmative answer to this conjecture would immediately imply... | 2 | https://mathoverflow.net/users/66288 | 275883 | 122,872 |
https://mathoverflow.net/questions/275876 | 16 | If I got it right, there is a theorem due to M. Artin that on a projective complex manifold any point has a Zariski open neighbourhood which is a $K(\pi, 1)$ space.
I have two questions about it.
1. Is there a proof of this result written in English? (I am really bad at French.)
2. Is it true that a point has an *ar... | https://mathoverflow.net/users/9833 | A theorem of M. Artin | **Quick answer.**
I. See M. Olsson *On Faltings’ method of almost etale extensions*, chapter 5. He discusses there a version of this fact over a dvr, but I think you can easily extract what you want, if you *really* don't want to look at SGA.
II. Yes. One possible precise statement is: a smooth scheme $X$ over a fi... | 24 | https://mathoverflow.net/users/3847 | 275885 | 122,874 |
https://mathoverflow.net/questions/275886 | 1 | If ${\cal U}$ is any ultrafilter on $\omega$, the pair $(\omega,{\cal U}\cup \{\emptyset\})$ is a connected topological space. Is there a non-principal ultrafilter ${\cal U}$ on $\omega$ such that we have $(\omega,{\cal U}\cup \{\emptyset\})\cong (\omega,{\cal U}\cup \{\emptyset\})^2$?
| https://mathoverflow.net/users/8628 | Products of spaces with an underlying free ultrafilter as topology | No. Note that in the space $X = (\omega, \mathcal{U}\cup\{\emptyset\})$, every set is either open or closed. However in the space $X^2$, neither $\overline{\Delta} = \{(m,n) : n\ge m\}$ nor $\underline{\Delta} = \{(m,n) : n < m\}$ can be open, since neither of these sets contains a rectangle of the form $A\times B$ wit... | 4 | https://mathoverflow.net/users/11233 | 275889 | 122,875 |
https://mathoverflow.net/questions/275892 | 4 | Given a (strict) braided monoidal category $(\mathcal{C},\otimes,I)$ with braiding $b$ and a Hopf algebra $H$ in $\mathcal{C}$. There is a category Rep($H$) of modules over $H$ in $\mathcal{C}$. Do these have a canonical monoidal structure? My candidate for an action on $X\otimes Y$ for $H$-modules $X$ and $Y$ is
\begi... | https://mathoverflow.net/users/111720 | Do the modules over a Hopf algebra in a braided monoidal category form a monoidal category? | Yes, this is true in a braided monoidal category and your candidate for the action is correct.
A reference is Proposition 2.5 in
**Majid, Shahn.** Algebras and Hopf algebras in braided categories. *Advances in Hopf algebras* (Chicago, IL, 1992), 55--105, Lecture Notes in Pure and Appl. Math., **158**, Dekker, New ... | 4 | https://mathoverflow.net/users/33854 | 275893 | 122,876 |
https://mathoverflow.net/questions/275869 | 1 | What is a complete description for the configuration of zero locus of the algebraic curve $C$ defined by $$yP(x,y)-xQ(x,y)=0$$
where $P,Q \in \mathbb{R}[x,y]$ are arbitrary polynomials of degree $2$.
What is the (sharp) maximum number of connected components of $\mathbb{R}^2 \setminus C$?
This question is motivat... | https://mathoverflow.net/users/36688 | The configuration of the zero locus of certain polynomial | Your curve is a planar cubic, and such were classified completely recently by one Isaac Newton. An even more recent version can be found in this 1928 Annals of Math article:
```
Canonical Forms of Plane Cubic Curves Under Euclidean Transformations
R. S. Burington and H. K. Holt
Annals of Mathematics
Second Series, V... | 1 | https://mathoverflow.net/users/11142 | 275901 | 122,877 |
https://mathoverflow.net/questions/275903 | 2 | Let $K$ be a field of characteristic zero, $\bar{K}$ its algebraic closure and $X$ a smooth, projective $K$-scheme. We know the Galois descent theory for quasi-coherent sheaves defined on $X\_L$ for a finite extension $L$ of $K$. This gives a cocycle condition on a quasi-coherent sheaf on $X\_L$ to descend to $X$ (see ... | https://mathoverflow.net/users/45397 | Galois descent for absolute Galois group | You need a continuity condition on the cocycles, otherwise it is probably false. The coherent sheaf automatically has a model over a finite Galois extension $L$ of $K$ contained in the fixed algebraic closure, and the continuity condition tells you that you can choose $L$ so that the cocycle factors through $Gal(L/K)$.... | 3 | https://mathoverflow.net/users/111447 | 275920 | 122,883 |
https://mathoverflow.net/questions/273866 | 8 | For a subset $X$ of $S\_n$ (the symmetric group of degree $n$) define $C(X)$ to be the union of all conjugates of elements of $X.$
**Question 1** What is that called?
**Question 2** Over all transitive subgroups $H$ of $S\_n,$ which one has the *smallest* $C(H)?$
| https://mathoverflow.net/users/11142 | transitive subgroups of $S_n$ | Since nothing else came up, I'm posting this as an answer.
For $n$ a power of a prime $p$, the answer is any group of exponent $p$, acting regularly.
Indeed, if $n$ is a power of $p$, then a Sylow $p$-subgroup $P$ of $H$ is transitive, and any element of order $p$ in the center of $P$ will be fixed-point-free, so $... | 4 | https://mathoverflow.net/users/22377 | 275923 | 122,886 |
https://mathoverflow.net/questions/275928 | 6 | Let $P$ denote the set of primes in $\mathbb{N}$. For $k\in \mathbb{N}, k\geq 2$ set $$M\_k = \big\{p\in P: \{kp-1, kp+1\}\cap P \neq \emptyset\big\}.$$
Is there $k\in \mathbb{N}, k\geq 2$ such that $M\_k$ is infinite?
| https://mathoverflow.net/users/8628 | Primes that are "almost" multiples of each other | You're asking whether, for some $k \ge 2$, there are infinitely many primes $p$ such that either $kp-1$ or $kp+1$ is prime. That would mean there are infinitely many primes such that $kp-1$ is prime or infinitely many primes such that $kp+1$ is prime. Of course, $k$ had better be even. If $k$ is even, Dickson's conject... | 12 | https://mathoverflow.net/users/13650 | 275931 | 122,890 |
https://mathoverflow.net/questions/275899 | 5 | Let $\xi\_1,\xi\_2,\ldots$ be i.i.d. positive random variables with infinite mean $\mathbb E[\xi\_i]=\infty$, and consider the random walk
$$T\_n=\sum\_{i=1}^n\xi\_i,\qquad n\in\mathbb N.$$
Here's an example that illustrates why I'm interested in such questions.
>
> **Example.** If $(S\_n)\_{n\in\mathbb N}$ is a si... | https://mathoverflow.net/users/50406 | Local limit theorems for positive random walks | Yes, there is a general result, see Chapter 9 of Gnedenko-Kolmogorov book.
The theorem says that if $\xi\_i$ are i.i.d with values in $\mathbb{Z}$ such that $$\text{gcd}\{s-s':\mathbb{P}(\xi\_1=s)>0,\;\mathbb{P}(\xi\_1=s')>0\}=1,$$ and $b\_n^{-1}(T\_n-a\_n)$ converges in distribution to a stable law, then also the l... | 6 | https://mathoverflow.net/users/56624 | 275953 | 122,898 |
https://mathoverflow.net/questions/275880 | 4 | I would like to find a closed form for the following series involving the Bessel function $J\_k(z)$:
$$
\sum\_{k=0}^{+\infty}\frac{(\mu)\_{k}}{k!(\lambda)\_{k}}t^k\left(\frac{z}{2}\right)^{k}J\_{k+\nu}(z),
$$
where $(a)\_{k}$ is the Pochhammer symbol. Actually, I am trying to compute a probability density function ... | https://mathoverflow.net/users/78781 | Infinite summation formula of Bessel functions | Concerning the finite sum, I am not sure you can get it (maybe for particular values of $ \nu $). Here is a possible "closed" expression for your sum. Start by writing $ \lambda := \mu + \alpha $ with $ \alpha > 0 $ and
$$ \frac{ (\mu)\_k }{ (\lambda)\_k } = \frac{ (\mu)\_k }{ (\mu + \alpha)\_k } = \mathbb{E}(\beta\_{\... | 1 | https://mathoverflow.net/users/109373 | 275962 | 122,900 |
https://mathoverflow.net/questions/275972 | 0 | Does $\int\_{0}^{T} \frac{1}{T}\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{\left(t-t\_{0}\right)^{2}}{2\sigma^{2}}}e^{i\omega t}dt$ have a closed-form solution?
| https://mathoverflow.net/users/97057 | Integral involving a gaussian function | If an error function is allowed:
$$\frac{e^{\frac{1}{2} i \omega \left(i \sigma ^2 \omega +2 t\_0 \right)} \left(\text{erf}\left(\frac{i \sigma ^2 \omega + t\_0 }{\sqrt{2}
\sigma }\right)-\text{erf}\left(\frac{i \sigma ^2 \omega +t\_0 -\text{T}}{\sqrt{2} \sigma }\right)\right)}{2 \text{T}}$$
For
$$\int\_0^T \fr... | 1 | https://mathoverflow.net/users/89654 | 275973 | 122,905 |
https://mathoverflow.net/questions/275957 | 8 | Consider the discrete case:
Shannon's entropy is $H(x)=-\sum\limits\_i^n p(x\_i) log\space p(x\_i)$.
Probability based on prefix-free Kolmogorov complexity is $R(x\_i)=2^{-K(x\_i)}$ where $K(x\_i)$ is the prefix-free Kolmogorov complexity of $x\_i$.
What is the relation between $R(x\_i)$ and $p(x\_i)$? Are they e... | https://mathoverflow.net/users/14024 | What is the precise relation between Kolmogorov complexity and Shannon's entropy? | What follows, which is all standard stuff (and should be in, say, Li and Vitanyi's monograph, or Downey and Hirschfeldt perhaps), might be the sort of relationship you vaguely recall seeing.
Your $R(x):= 2^{-K(x)}$ is not itself a probability measure, but it is a lower semi-computable semi-measure; and it is *optimal... | 9 | https://mathoverflow.net/users/4137 | 275976 | 122,907 |
https://mathoverflow.net/questions/275949 | 0 | The following concept arose when studying some properties of a real world computer network.
Let $G=(V,E)$ be a finite, simple, undirected graph. If $v\in V$ we set $N(v)=\{y\in V:\{v,y\}\in E\}$. Let $Z\neq \emptyset$ be a set. A map $c:V\to Z$ is said to be an "almost coloring" of $G$ if for all $v\in V$ the color o... | https://mathoverflow.net/users/8628 | "Almost" vertex coloring | No for $G=K\_{2,1}$ (two edges with common vertex) and $H=K\_{1,1}$ (an edge). Both chromatic numbers are equal to 1, and almost chromatic numbers are 2 and 1 respectively.
| 2 | https://mathoverflow.net/users/4312 | 275977 | 122,908 |
https://mathoverflow.net/questions/275865 | 28 | Let $P$ be a real polynomial of exact degree $2n$ ($n \geq 1$) whose zeros are real numbers and such that
\begin{equation\*}
P(j) \geq 0
\quad \text{for any} \quad
j \in \mathbb{Z}.
\end{equation\*}
Does there exist non-negative real numbers $\alpha\_0,\alpha\_1,\ldots,\alpha\_n,$ with at least one of the $\alph... | https://mathoverflow.net/users/112434 | Polynomials non-negative on the integers | Occasionally I wish somebody could give me a good whack on the head to keep my brains running and the older I get, the more frequently I need it. The problem is actually trivial.
I will prefer to think that $P$ is non-negative on some disjoint with integers arithmetic progression $\Lambda$ with step $1$ . Then we nee... | 29 | https://mathoverflow.net/users/1131 | 275991 | 122,912 |
https://mathoverflow.net/questions/275580 | 2 | This question is basically a rerun of [this one](https://math.stackexchange.com/q/2265236/4280), but I am curious about its resolution and couldn't really find info online on it, and no answers were given at the other site.
The rational sequence topology on the reals (an example to be found in Steen and Seebach's "C... | https://mathoverflow.net/users/2060 | Covering properties of the rational sequence topology | Summarizing partial answers I posted earlier as comments:
>
> Since $X$ is locally countable, $l(X) \ge |X| = \mathfrak{c}$. If $X$ were metacompact then we should have $l(X) \le d(X) = \aleph\_0$. This settles
> questions 1 and 2.
>
>
> That $X$ is countably metacompact can be shown by observing that
>
>
> * ... | 1 | https://mathoverflow.net/users/10075 | 276993 | 122,914 |
https://mathoverflow.net/questions/276997 | 1 | Let $R$ be a ring with more than 1 element, and let $A$ be a non-empty set. We call a map $c:R\to A$ an *ideal coloring* if for every nonempty ideal $I$ with $I\neq\{0\}$ the restriction $c|\_I$ is not constant (that is, the elements of every nontrivial ideal receive at least 2 "colors").
Is there a ring that can be ... | https://mathoverflow.net/users/8628 | "Coloring" the ideals of a ring | Every ring can be ideal-colored with $2$ colors. Let $0$ be colored by $0$, and every other element by $1$.
| 1 | https://mathoverflow.net/users/110389 | 277000 | 122,917 |
https://mathoverflow.net/questions/277001 | 11 | Suppose we are given $A \subseteq \mathbb{N}$ with $\lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0$. For $k\in \mathbb{N}, k\geq 2$ we set $$M\_A(k) = \{a\in A: ka \in A\}.$$
Does there exist $k\in \mathbb{N}, k\geq 2$ such that $M\_A(k)$ is infinite?
| https://mathoverflow.net/users/8628 | Multiples in sets of positive upper density | Not necessarily: you can in fact have $M\_k(A)=\varnothing$ for all integer $k\ge 2$. This was shown by Besicovitch ("[On the density of certain sequences of integers](https://link.springer.com/content/pdf/10.1007%2FBF01448032.pdf)", *Math. Ann.* **110** (1935), no. 1, 336–341) who has constructed a set (of positive in... | 15 | https://mathoverflow.net/users/9924 | 277003 | 122,918 |
https://mathoverflow.net/questions/275965 | 2 | Let $G$ be a connected reductive group over $\mathbb{C}$. Let $F=\mathbb{C}((t))$ and $\bar{F}$ its algebraic closure. Let $c$ be a Cartan subalgebra of $g$ and $N$ its normalizer. It is written in *Fixed point varieties on affine flag manifolds* by Kazhdan and Lusztig that Cartan subalgebras of $G(F)$ are classified b... | https://mathoverflow.net/users/58689 | classifying non-split Cartan subalgebras | The field $\mathbb C((t))$ is $C\_1$ by a theorem of Lang (1952). This implies that $H^1(Gal(\bar F/F,G(\bar F))$ is trivial. Hence, for any 1-cocycle $n\_\gamma$ of $N(\bar F)$ there is $g\in G(\bar F)$ with $n\_\gamma=g^{-1}{}^\gamma g$. Then $\tilde c:=(\mathrm{Ad} g)(c)$ is a Cartan subalgebra which is defined over... | 5 | https://mathoverflow.net/users/89948 | 277006 | 122,920 |
https://mathoverflow.net/questions/277014 | 3 | Let ${\cal E}$ denote the collection of open sets of $\mathbb{R}$ with respect to the Euclidean topology. It is well known that $|{\cal E}| = 2^{\aleph\_0}$. Is there an injective map $f:{\cal E}\setminus \{\emptyset\} \to \mathbb{R}$ such that $f(U)\in U$ for all $U \in {\cal E}\setminus \{\emptyset\}$?
| https://mathoverflow.net/users/8628 | Picking a real for every non-empty open set in $\mathbb{R}$ | Choose a well-ordering of $\cal{E}\backslash\{\emptyset\}$ with order type the initial ordinal of size $2^{\aleph\_0}$. Consider injective functions $f$ from initial segments of $\cal{E}\backslash\{\emptyset\}$ to $\mathbb{R}$ satisfying $f(U)\in U$. The set of such functions is partially ordered by extension, and Zorn... | 6 | https://mathoverflow.net/users/5263 | 277016 | 122,922 |
https://mathoverflow.net/questions/275940 | 2 | I have a question about Feller processes.
In this paper ["On the doubly Feller property of resolvent"](https://projecteuclid.org/download/pdfview_1/euclid.kjm/1492194850) the transition semigroup $(P\_t)\_{t \ge 0}$ of a Hunt process (on a metric space $E$) is said to have the Feller property if the following two con... | https://mathoverflow.net/users/68463 | On necessity of Feller property | The Feller property is *sufficient* (bot not necessary) for the existence of a Hunt process associated with $(P\_t)$. As you observe, condition (2) in the Feller property is (in isolation) also a necessary consequence of the Hunt property.
| 1 | https://mathoverflow.net/users/42851 | 277018 | 122,923 |
https://mathoverflow.net/questions/277015 | 9 | Which are the configrations $P\subset \mathbb{R}^2$ of points, such that the following property holds:
>
> **Property M** (for **M**onochromatic): Every two-coloring of $\mathbb{R}^2$ contains a monochromatic copy of $P$.
>
>
>
(By "copy" we allow Euclidean motions and also scaling by a positive factor. By "m... | https://mathoverflow.net/users/39495 | Monochromatic point sets in two-colored plane | I think the best starting point are surveys by R. L. Graham; for example:
Euclidean Ramsey Theory, Handbook of discrete and computational geometry, 153–166,
CRC Press Ser. Discrete Math. Appl., CRC, Boca Raton, FL, 1997.
(<http://dl.acm.org/citation.cfm?id=285879>)
Recent trends in Euclidean Ramsey theory, Discret... | 8 | https://mathoverflow.net/users/24076 | 277037 | 122,932 |
https://mathoverflow.net/questions/277030 | 8 | Non-trivial zeros play an important (main) role in the distribution of prime numbers.
Are there theorems in which trivial zeros play an important (main) role?
| https://mathoverflow.net/users/nan | Are trivial zeros of the zeta function important? | Since you tagged this 'reference request', I'm going to answer with a theorem in my own paper *"Euler, the symmetric group, and the Riemann zeta function."*
Let $\pi$ be a permutation in the symmetric group $S\_n$. An *ascent* is an occurrence of $\pi(j)<\pi(j+1)$ for $1\le j\le n-1$. For
example, the permutation $(... | 4 | https://mathoverflow.net/users/6756 | 277038 | 122,933 |
https://mathoverflow.net/questions/277029 | 2 | The definition of first selection principle is well known: $S\_1(A,B)$.
Let $A$ and $B$ be families of sets. The symbol $S\_1(A, B)$ denotes the statement:
for each sequence $(A\_n : n \in \omega)$ of elements of $A$ there is a sequence
$(x\_n : n\in \omega)$ such that each $x\_n$ is an element of $A\_n$, and $\{x\_n ... | https://mathoverflow.net/users/112417 | Definition of $S_1(A,B)$ | I understand your question to mean "must we chose an element $x\_n$ for all $n$?". (This is in accordance with the comments.)
**Answer.** Officially, the answer is "Yes", you must chose an element for each $n$. However, as pointed out in the comments, often you can prove this makes no difference if you are allowed to... | 2 | https://mathoverflow.net/users/2415 | 277042 | 122,935 |
https://mathoverflow.net/questions/275682 | 3 | Given integers $a,b$, we say a polynomial $f(x) \in \mathbb{Z}[x]$ is an $(a,b)$-filter, if $f(x)$ splits completely into linear factors modulo an odd prime $p$ only if $p=a \pmod b$. For example $x^2+1$ is a $(1,4)$-filter and $(x^2+1)(x^2+2)$ is a $(1,8)$-filter.
**Problem**: Determine all pairs $(a,b)$ for which ... | https://mathoverflow.net/users/81443 | Polynomials splitting into linear factors modulo certain primes | **Claim.** An $(a,b)$-filter exists if and only if $a\equiv 1\pmod{b}$.
**Proof.** Assume that $f(x)\in\mathbb{Z}[x]$ is an $(a,b)$-filter. Let $K$ be a number field containing all the roots of $f(x)$ and all the $b$-th roots of unity. Let $p$ be an odd unramified prime that splits completely in $K$ (there are infini... | 5 | https://mathoverflow.net/users/11919 | 277051 | 122,940 |
https://mathoverflow.net/questions/277039 | 3 | Let $C$ be a family of convex sets in $\mathbb{R}^d$ and assume further that $C$ is closed under translation: for all $A\in C$ and $x\in\mathbb{R}^d$, we have $A+x\in C$.
Let $P:\mathbb{R}^d\to\mathbb{R}^k$ be a linear projection and define $C'$ to be the image of $C$ (elementwise) under $P$. Question: is it ever the... | https://mathoverflow.net/users/12518 | VC dimension under projection | Yes, of course.
*Claim 1:* the VC dimension of translations of a fixed triangle on the plane is at most $3$.
*Proof:* Take any $4$ points consider the $3$ points lying farthest in the directions of the outer unit normals to the triangle sides. If you cover them all, you have to cover the fourth point as well, so no... | 3 | https://mathoverflow.net/users/1131 | 277054 | 122,942 |
https://mathoverflow.net/questions/277048 | 3 | We know that planar graphs have $O(1)$ degree.
We know balanced (each color has same number of vertices) complete bipartite graphs have genus $O(n^2)$.
>
> 1. If maximum and average degree are $O(n^\alpha)$ where $\alpha\in[0,1]$ then is genus also $O(n^\alpha)$?
> 2. If maximum degree is $O(n^\alpha)$ where $\al... | https://mathoverflow.net/users/10035 | Is bipartite graph genus bound by $O(\mbox{max deg})$? | The answer is no. [Hossein Namazi, Pekka Pankka, Juan Souto](https://arxiv.org/abs/1208.2130) showed that [expander graphs](https://en.wikipedia.org/wiki/Expander_graph) have genus that is linear in the number of vertices. You can construct bipartite, bounded degree expanders.
| 5 | https://mathoverflow.net/users/1061 | 277058 | 122,944 |
https://mathoverflow.net/questions/277057 | 8 | Let $f:[0,1]^n \rightarrow [0,1]^n$ be a continuous mapping. Brouwer's fixed point theorem says that $f$ has a fixed point, i.e., some $x$ such that $f(x) = x$.
Suppose we have a continuous family, i.e., a continuous function $f:[0,1]^n \times [0,1] \rightarrow [0,1]^n$. Then for each $r \in [0,1]$ we have that there... | https://mathoverflow.net/users/61129 | Continuity of mapping sending a function to its (brouwer) fixed point | The answer is yes. Fixed points of $f$ are zeros of the continuous $g(x)=f(x)-x$.
So one has to prove the following: If $g:[0,1]^n\times[0,1]\to[0,1]^n$ is continuos,
and for each $r\in[0,1]$, $g(.,r)$ has exactly one zero $x\_r$ then $r\to x\_r$ is
continuous. Consider the set
$$E=\{ (x,r):g(x,r)=0\}\subset [0,1]^{n+1... | 7 | https://mathoverflow.net/users/25510 | 277063 | 122,946 |
https://mathoverflow.net/questions/277027 | 4 | I am reading through Chapter III of Bourbaki, *Lie Groups and Lie Algebras*, and many proofs cite the Bourbaki volume *Differentiable and Analytic Manifolds*. I can't find this book anywhere. Does it actually exist?
| https://mathoverflow.net/users/38145 | Existence proof of Bourbaki, Differentiable and Analytic Manifolds | This book exists of course, it is even translated into Russian (as all Bourbaki books are), and it is easy to find in Russian. I have it. However this book is only a
"Fascicule de résultats", which means that it is a sort of resume (definitions and statements without proofs) of a much larger book which does not exist (... | 6 | https://mathoverflow.net/users/25510 | 277065 | 122,948 |
https://mathoverflow.net/questions/277052 | 23 | The elementary divisor theorem was originally proved by a calculation on integer matrices, using elementary (invertible) row and column operations to put the matrix into Smith normal form. That is the matrix is zero off the diagonal, and on the diagonal each entry divides the one below it.
This calculation immediate... | https://mathoverflow.net/users/38783 | Can one prove the elementary divisor theorem for PIDs by elementary matrix operations? | The answer is **no**: it is not possible, in general, to reduce a matrix over a principal ideal domain (PID) to a diagonal (or trigonal) matrix by means of elementary row and column operations. (This topic has been discussed in [this MO post](https://mathoverflow.net/questions/251103/principal-ideal-ring-does-there-exi... | 13 | https://mathoverflow.net/users/84349 | 277068 | 122,950 |
https://mathoverflow.net/questions/277069 | 87 | *Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid mistakes please treat them as such and try to focus on the main issue raised if at all possible.*
As the title suggest... | https://mathoverflow.net/users/22810 | What is homology anyway? | Let's take coefficients in a field $k$, for simplicity.
On 2): the singular cohomology of a topological space $X$ is the dual of its singular homology, almost by definition. But if $X$ is a space for which singular cohomology is not the same as sheaf cohomology, then the sheaf cohomology of $X$ need not have a predua... | 61 | https://mathoverflow.net/users/7721 | 277071 | 122,951 |
https://mathoverflow.net/questions/277070 | 6 | Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are finite dimensional.
First I want the example of this.
Second, is there any sources about the proof of this statemen... | https://mathoverflow.net/users/112538 | Tensor product space with projective norm is incomplete | The projective tensor product $\ell\_1\widehat{\otimes}X$ is naturally isometrically isomorphic to the $\ell\_1$-sum of countably many copies of $X$. The uncompleted tensor product $\ell\_1 \odot X$ is then the linear span of elements of the form $(\xi\_n x)$, where $(\xi\_n)$ is in $\ell\_1$ under this identification,... | 6 | https://mathoverflow.net/users/15129 | 277072 | 122,952 |
https://mathoverflow.net/questions/277087 | 12 | There is a well known problem of LeBrun-Salamon:
are there any non-symmetric compact quaternionic-Kahler
manifolds of positive scalar (and Ricci) curvature?
It is hard and still unsolved:
[Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?](https://mathoverflow.net/questions/78793/qu... | https://mathoverflow.net/users/3377 | Compact quaternionic Kahler manifolds of negative curvature: examples | Any (Riemannian) symmetric space admits a cocompact lattice. This is due to A. Borel, [Compact Clifford-Klein forms of symmetric spaces,](http://www.sciencedirect.com/science/article/pii/0040938363900260) Topology 2, 1963, pp.111-122. The quaternionic hyperbolic space is symmetric and quaternionic-Kahler.
| 8 | https://mathoverflow.net/users/1573 | 277107 | 122,960 |
https://mathoverflow.net/questions/277113 | 16 | It can easily be shown that if a complex polynomial $P$ leaves invariant $\mathbb{Z}$ ($P(\mathbb{Z}) \subseteq \mathbb{Z}$) then it must be a linear combination (with integer coefficients) of Hilbert polynomials $H\_k$, i.e. polynomials of the form : $$ H\_k(X) : = \frac{X(X-1)\cdots (X-k+1)}{k!} $$
Now, what happen... | https://mathoverflow.net/users/112561 | Polynomials leaving invariant the Gaussian integers | Your question is related to the study of *(generalized) numerical polynomials*: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \subseteq R\}$ is a subdomain of $K[x]$, whose elements are called the *numerical polynomials over $R$* (in one variable $x... | 13 | https://mathoverflow.net/users/16537 | 277116 | 122,964 |
https://mathoverflow.net/questions/215899 | 10 | I must confess I hardly know anything about probability theory. Still, I'm interested in the following: Much like point-free topology, where one basically replaces topological spaces by their locales of open sets, I figured there is a way to do something similar with $\sigma$-algebras and with probability spaces.
> ... | https://mathoverflow.net/users/78650 | A Point-free probability theory? | Point-free probability theory, is treated in Kappos's book: Probability algebras and Stochastic spaces. Academic Press, 1969.
| 6 | https://mathoverflow.net/users/112567 | 277122 | 122,966 |
https://mathoverflow.net/questions/277011 | 1 | Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective module $M$.
1. Is $A$ selfinjective?
2. Is $A$ local?
| https://mathoverflow.net/users/61949 | Algebra with all modules non-rigid | No, the path algebra of an $A\_2$-quiver provides a counterexample. Let $M$ be a (finite dimensional) module. Then $M=S(1)^{n\_1}\oplus S(2)^{n\_2}\oplus P(1)^{n\_3}$ where $S(1)=I(1)$ is injective non-projective and $S(2)=P(2)$ is projective non-injective. That $M$ is non-projective is thus equivalent to $n\_1\neq 0$,... | 2 | https://mathoverflow.net/users/15887 | 277127 | 122,970 |
https://mathoverflow.net/questions/277096 | 3 | I am interested in finding the extreme points of the following set of distributions
\begin{align}
\mathcal{P}= \left\{F: \int\_{\mathbb{R}} |x|^k dF(x)=c \right\}
\end{align}
where $k,c>0$.
I know that [this paper](https://www.jstor.org/stable/pdf/3689944.pdf?refreqid=excelsior%3A6c1f2a909f95192f8b30edf1e31ad3de) b... | https://mathoverflow.net/users/69661 | Extreme points of set of probability measures $\mathcal{P}= \{F: \int_{\mathbb{R}} |x|^k dF(x)=c \}$ | What the paper asserts in this case is $ex\mathcal P=\{F:F=(1-t)\delta\_{x}+t\delta\_y, t\in[0,1], (1-t)|x|^k+t|y|^k=c, x+y\neq0\}$. This includes singletons $\delta\_x$ with $|x|^k=c$.
What was pointed out is not the existence of three-point masses as extreme points, but that the proof (that only convex combinations... | 1 | https://mathoverflow.net/users/75422 | 277129 | 122,972 |
https://mathoverflow.net/questions/277130 | 0 | Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective indecomposable module $M$, with the condition that at least one such module exists.
1. Is $A$ selfinjective?
2. Is $A$ local?
(answer is no,see the answer by Jeremy Rickard)
Two other questions:
... | https://mathoverflow.net/users/61949 | Algebra with all modules non-rigid 2 | Let $A=kQ/\text{rad}(kQ)^2$, where $Q$ has two vertices, a loop at vertex $1$ and an arrow from vertex $1$ to vertex $2$. I think it has five indecomposable modules, all of which are either projective or injective apart from the simple $S\_1$ at vertex $1$, and $\text{Ext}^1\_A(S\_1,S\_1)\neq0$.
| 2 | https://mathoverflow.net/users/22989 | 277136 | 122,973 |
https://mathoverflow.net/questions/277103 | 1 | Let $f:\mathbb{C}\to\mathbb{C}$ be meromorphic or even entire. Let $z:\mathbb{C}\to\mathbb{C}$ be such that it holds
$$z(t+1) = f(z(t)).$$
If $f$ is entire and we choose carefully $z$ can be constructed to be also entire. This is done usually by choosing an unstable direction of a fixed point and then propagate analyti... | https://mathoverflow.net/users/18812 | Finding the "orthogonal" map of a given 1d map | If $f$ is rational, then $g$ is also rational. It is a very rare, exceptional
situation when such two functional equations are satisfied. All such cases have been explicitly described by J. Ritt in Permutable rational functions, Trans. Amer. Math. Soc. 25 (1923), no. 3, 399–448.
If $f$ is entire transcendental, the s... | 3 | https://mathoverflow.net/users/25510 | 277137 | 122,974 |
https://mathoverflow.net/questions/277121 | 2 | This is an update of an [older question](https://mathoverflow.net/questions/276997/coloring-the-ideals-of-a-ring), suggested in [this comment](https://mathoverflow.net/questions/276997/coloring-the-ideals-of-a-ring?noredirect=1#comment682204_276997) by Zach Teitler.
Let $R$ be a ring with more than 1 element, and let... | https://mathoverflow.net/users/8628 | Ideal colorings of rings | There are such rings. I will give a nonunital example
first, then explain how to modify it to a unital example.
Given a colored ring, I call an ideal *monochromatic*
if all nonzero elements have the same color.
I call a coloring of a ring *valid* if no ideal
of more than $2$-elements is monochromatic.
**First exam... | 2 | https://mathoverflow.net/users/75735 | 277147 | 122,982 |
https://mathoverflow.net/questions/277149 | 3 | I have a sequence of Radon measures (on some set $X$, compact subspace of $\mathbb{R}^d$ so nothing too fancy), say $\mu\_n$, which are actually $L^1(X)$ functions. In the limit I want to prove that I obtain a measure supported on a finite set $\{x\_1, \ldots, x\_N\}$.
One way to prove this seems to be taking a (con... | https://mathoverflow.net/users/112577 | Discrete support for Radon measures | Let $\mu$ be your limiting measure and $F$ be your finite set. Let $B$ be any closed ball disjoint from $F$, and let $f\_B$ be a positive continuous function which equals 1 on $B$ and is supported in $F^c$. (For instance, you can take $f\_B(x) = \max(1 - n d(x,B), 0)$ for sufficiently large $n$.) By assumption, $\int f... | 1 | https://mathoverflow.net/users/4832 | 277164 | 122,986 |
https://mathoverflow.net/questions/277145 | 2 | Let $1\leq p<\infty$. Suppose that an operator $T:X\rightarrow Y$ has a factorization $T=RS$, where $S:X\rightarrow l\_{p}$, $R:l\_{p}\rightarrow Y$ are compact operators.
Question: Let $\epsilon>0$. Are there operators $B:X\rightarrow l\_{p}$, $A:l\_{p}\rightarrow Y$ and $\lambda=(\lambda\_{j})\_{j}\in c\_{0}$ such... | https://mathoverflow.net/users/41619 | A question on p-factorable operators | Yes. If $T$ has finite rank this follows from the fact that every finite dimensional subspace of $\ell\_p$ is contained in a finite dimensional superspace of $\ell\_p$ that is $1+\epsilon$-isomorphic to the $L\_p$ space of its dimension.
For the general case of compact $T$, write $T= \sum\_{n=0}^\infty T\_n$ with ea... | 2 | https://mathoverflow.net/users/2554 | 277172 | 122,990 |
https://mathoverflow.net/questions/277173 | 2 | Let $X$ be a topological space, $A$ a sheaf of (unital and associative but not necessarily commutative) rings on $X$. Suppose $M$ is a simple quasicoherent $A$-module and $U$ an open subset of $X$. Is $M|\_U$ a simple $A|\_U$-module?
| https://mathoverflow.net/users/36720 | Is the restriction of a simple sheaf of modules simple? | Yes. Restriction is an exact functor split by its right adjoint (pushforward). If $M|\_{U}$ has a proper non-zero quotient $N$, then $i\_\*N$ receives a non-zero and thus injective map from $M$. However, this means that $M|\_U$ must map injectively to $N$, which is a contradiction.
| 3 | https://mathoverflow.net/users/66 | 277176 | 122,992 |
https://mathoverflow.net/questions/277174 | 6 | Perhaps I sould ask this question on a physics forum, but I am curious about answers coming from mathematicians.
Calabi-Yau manifolds are examples of Ricci-flat Kaehler manifolds. As we know, in the semi-riemannian case, Ricci flat metrics describes solutions for Einstein field equations on vacuum. So my question is:... | https://mathoverflow.net/users/94097 | Is there any relativistic interpretation on considering Kaehler-Einstein metrics and Calabi-Yau manifolds? | First, a purely mathematical remark: it is not so easy to construct Riemannian Ricci-flat metrics on compact manifolds. Ricci flat Kähler (= Calabi-Yau) metrics give a large class of examples and are "easy" to obtain: by Yau's theorem, it is enough to check a complex geometric condition (vanishing first Chern class) on... | 5 | https://mathoverflow.net/users/25309 | 277181 | 122,994 |
https://mathoverflow.net/questions/256754 | 3 | Say we are working in $\mathbb{F}^{2n}\_2$ where vectors can be written as pairs $(a,b)$ with $a,b \in \mathbb{F}^{n}\_2$. Given a list of basis vectors for an $n-1$ dimensional subspace $S \subset \mathbb{F}^{2n}\_2$, I want a polynomial time algorithm that can test membership of an input vector $(a,b)$.
This versio... | https://mathoverflow.net/users/99673 | Binary subspace membership testing with signed vectors | Took me a while but I figured it out. Say I already know that either $(0,a,b) \in S$ or $(1,a,b) \in S$ and I want to find out which for some $(a,b)$.
Define a regular inner product $(a,b) \cdot (a',b') = a^Ta' + b^T b'$. I can compute the null space $\text{Null}(S)$ of a space $S$ with respect to this inner product ... | 0 | https://mathoverflow.net/users/99673 | 277185 | 122,997 |
https://mathoverflow.net/questions/277166 | 5 | I am reading the paper <http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf> fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\mathcal{K}$ is a reflective subcategory of a LFP category $\mathcal{L}$ closed under filtered colimits and such that the ... | https://mathoverflow.net/users/24563 | About small $\omega$-orthogonality classes and Gabriel-Ulmer duality | I understand where is the mistake. $r$ does preserve finite presentability (the proof is straightforward and it is due to the fact that it is a left adjoint of a functor preserving filtered colimits). But there is no reason for $A'=iX$ to be finitely presentable.
| 3 | https://mathoverflow.net/users/24563 | 277204 | 123,002 |
https://mathoverflow.net/questions/273930 | 1 | I have the following triangular system
\begin{equation}
\begin{pmatrix}
1 & & & & \\
\mu\_1 & 2 & & & \\
\mu\_2 & \mu\_1 & 3 & \\
\vdots & \vdots &\ddots & \ddots & \\
\mu\_{n-1} & \mu\_{n-2} & \dots & \mu\_1 & n
\end{pmatrix}
\begin{pmatrix}
c\_{n-1} \\
c\_{n-2} \\
c\_{n-3} \\
\vdots \\
c\_0
\end{pmatrix... | https://mathoverflow.net/users/102057 | Solving Linear System with Noisy Input | You are right. You need the forward error to answer your question. The term $\| x - y \|$ in the inequality from your book corresponds to $\| c - \hat{c} \|$ in your example. Hence, to bound $\| x - y \|$ you first multiply the inequality by $\| x \|$ to get
\begin{equation}
\|x-y\| \leq
\frac{\epsilon}{1 - \epsilo... | 1 | https://mathoverflow.net/users/112614 | 277226 | 123,008 |
https://mathoverflow.net/questions/277219 | 3 | Let $P\_1,\dots,P\_4$ be four projective plane conics without a common point. A dimension count tells us that there is a 9-dimensional space of quadratic relations among the defining equations $p\_j=0$ for $P\_j.$ Indeed, we have a $24=4\times 6$-dimensional vector space of coefficients that is mapped to the 15-dimensi... | https://mathoverflow.net/users/11100 | relations between 4 plane conics | Maybe this is something like what you have in mind:
Let $V$ be a (complex) vector space of dimension $3$. It is easy to show that a generic subspace $P\subset S^2(V^\*)$ of dimension $4$ can be written as
$$
P = \mathrm{span}\{\ {x\_1}^2,\ {x\_2}^2,\ {x\_3}^2,\ x\_1x\_2{+}x\_2x\_3{+}x\_3x\_1\ \}
$$
for some basis $x... | 4 | https://mathoverflow.net/users/13972 | 277229 | 123,010 |
https://mathoverflow.net/questions/277212 | 7 | This is a question on literature about cohomology of arithmetic groups.
Let $M$ denote a quaternion algebra over $\mathbb Q$ and assume it is non-split over $\mathbb R$. Fix a maximal order $\Lambda$ in $M$ and for any ring $R$ let
$$
M(R)=\Lambda\otimes\_{\mathbb Z}R,\qquad G(R)=M(R)^\times/R^\times.
$$
Let $p$ be a p... | https://mathoverflow.net/users/nan | Cohomology of certain arithmetic groups | We can write $G(\mathbb Q\_p)/ G(\mathbb Z\_p) =GL\_2(\mathbb Q\_p)/GL\_2(\mathbb Z\_p)$ as the set of vertices of a tree (the Bruhat-Tits tree) of degree $p+1$.
The group $\Gamma\_p$ acts on this tree with finite stabilizers. Hence the cohomology of the quotient $\Gamma\_p \backslash G(\mathbb Q\_p)/ G(\mathbb Z\_p... | 6 | https://mathoverflow.net/users/18060 | 277232 | 123,012 |
https://mathoverflow.net/questions/277151 | 5 | It is well known that the only homogeneous surfaces in $\mathbb{R}^3$ are the spheres, cylinders or planes. My question is about other examples in dimension $4$. Such a surface should have "constant curvature" but it seems that there is no good scalar invariant such as mean curvature or Gauss curvature...except perhaps... | https://mathoverflow.net/users/9253 | homogeneous surface in $\mathbb{R}^4$ | I'm rearranging my answer a little bit because I realized that I overlooked an apparent possibility (that turns out not to occur), and I didn't want my answer to be misleading:
If the surface in Euclidean $\mathbb{R}^4$ has positive Gauss curvature and is homogeneous, it will be complete and hence compact. Hence the ... | 8 | https://mathoverflow.net/users/13972 | 277236 | 123,014 |
https://mathoverflow.net/questions/277187 | 6 | Does anyone have any ideas on howto verify $$\sum\_{n,m=0}^\infty \frac{\Gamma(n+m+3x)}{\Gamma(n+1+x)\Gamma(m+1+x)}\cdot \frac{1}{3^{n+m+3x-1}} = \Gamma(x)$$ for $x>0$?
I posted this question also on [math.stackexchange](https://math.stackexchange.com/questions/2364283/double-series-equals-gamma-function).
This is... | https://mathoverflow.net/users/101850 | Double Series involving Gamma function | This problem can be reduced at least formally to a compact double integral, which might be easier to solve.
Starting with the integral representation for the Gamma function, we write the double sum as an integral of the square of the confluent hypergeometric function ${}\_1F\_1$, then apply analogue of Euler's transf... | 4 | https://mathoverflow.net/users/82588 | 277237 | 123,015 |
https://mathoverflow.net/questions/275946 | 2 | Given the algebra $A=K[x]/(x^n)$ for some field $K$ and natural number $n \geq 2$ with enveloping algebra $A^e=A \otimes\_K A$.
It is easy to see that the 1. Hochschild cohomology of $A$ is nonzero since $Ext\_{A^e}^{1}(A,A)=\underline{Hom\_{A^e}}(A,\Omega^{2}(A))=\underline{Hom\_{A^e}}(A,A) \neq 0)$, where the 2. equa... | https://mathoverflow.net/users/61949 | First Hochschild cohomology of $A=K[x]/(x^n)$ | Consider $M=A\oplus A$ as a left $A$-module. Denote an $A$-basis by $e\_1,e\_2$. In order to define the action of $A$ from the right, we just need to give an $A$-linear map $M\to M$ whose $n$-th power is zero. So define $e\_1x = xe\_1$ and $e\_2x = xe\_2 + x^{n-1}e\_1$. This module is not isomorphic with the direct sum... | 2 | https://mathoverflow.net/users/41644 | 277238 | 123,016 |
https://mathoverflow.net/questions/277213 | 0 | This is a special case of a [question](https://mathoverflow.net/questions/275489/nowhere-dense-covering-number-of-a-connected-t-2-space) that has not been answered so far.
If $(X,\tau)$ is a connected $T\_2$ space with more than 1 point, we define its *nowhere dense covering number* $\nu(X)$ by the smallest cardinali... | https://mathoverflow.net/users/8628 | Connected $T_2$-spaces with nowhere dense covering number $3$ | Since the union of two nowhere dense sets is nowhere dense, the number $\nu(X)$ is always infinite. For meager spaces it is countable. But it cannot be equal to 3.
| 2 | https://mathoverflow.net/users/61536 | 277242 | 123,017 |
https://mathoverflow.net/questions/268561 | 1 | Let $L$ be the left shift operator on $\ell^2(\mathbb{Z})$ with trace $\tau(T) := \langle T \delta\_0, \delta\_0 \rangle$.
How can I show that the Brown measure of $L$ is the uniform measure on the unit circle?
The Brown measure is defined as follows: For each $z \in \mathbb{C}$ define $\nu\_{z}$ to be the spectra... | https://mathoverflow.net/users/64507 | Brown measure of left shift operator | Since the two-sided shift is a normal operator this is not really a question about the Brown measure, but about the spectral measure, so it suffices to compute the $\*$-moments $\tau(L^nL^{\*m})$. For an introduction to Brown measure you might also see Chapter 11 of the book
<https://www.math.uni-sb.de/ag/speicher/publ... | 4 | https://mathoverflow.net/users/112626 | 277252 | 123,021 |
https://mathoverflow.net/questions/277253 | 55 | The (very nice) final problem of [IMO 2017](http://imo-official.org/year_info.aspx?year=2017) asked contestants to show:
>
> If $S$ is a finite set of lattice points $(x,y)$ with $\gcd(x,y)=1$, then there is a nonconstant homogeneous polyonmial $f \in \mathbb Z[x,y]$ such that $f(x,y) = 1$ for all $(x,y) \in S$.
> ... | https://mathoverflow.net/users/70654 | IMO 2017/6 via arithmetic geometry | The set $S$ gives rise to a subscheme (which let's also denote by $S$) of $\mathbb{P}^1\_{\mathbb{Z}},$ because relatively a prime pair $(x,y)$ corresponds to a section of $\mathbb{P}^1\_{\mathbb{Z}}\rightarrow\operatorname{Spec}\mathbb{Z}$, and we take the union of these divisors in $\mathbb{P}^1\_\mathbb{Z}$.
Now, ... | 33 | https://mathoverflow.net/users/51424 | 277254 | 123,022 |
https://mathoverflow.net/questions/277223 | 9 | This is closely related to this question: [Eigenvalues of a matrix with binomial entries](https://mathoverflow.net/q/275911/16615).
We consider the matrix:
$$M\_{ij} = 4^{-j}\binom{2j}{i}$$
where it is understood that the binomial coefficient $\binom{m}{k}$ is zero if $k<0$ or $k>m$. The indices $i,j$ traverse a ... | https://mathoverflow.net/users/16615 | Inverse of a matrix with binomial entries | Let's refer everything to square matrices indexed from $0$ to $h$, that I will denote as
$$
{\bf M}\_{\,h} = \left\| {\;f(n,m)\;} \right\|\_{\,h}
$$
with $n$ being the row index and $m$ the column index.
I will then denote by
$$
\left( {f(n) \circ {\bf I}\_{\,h} } \right)
$$
the diagonal matrix whose entries are equ... | 6 | https://mathoverflow.net/users/89279 | 277269 | 123,025 |
https://mathoverflow.net/questions/277274 | 10 | Let $\pi:\mathcal X\to B$ be a family of Kaehler manifolds then if we take $B'\subset B$ be the set of parameters such that $X\_b$ admit Kaehler-Einstein metric(with zero, negative, or positive Ricci curvature) , then $B'$ is Zariski open subset of $B$ always?
| https://mathoverflow.net/users/111160 | Zariski open subset on family of Kaehler-Einstein manifolds | The answer depends on diameter bounds of the fibers $(X\_b,\omega\_b)$ and the $L^\infty$ norm of the Ricci curvature. (You can derive the answer by using a nice paper of J. Cheeger-A.Naber. See the nice survay paper of [Donaldson](http://www.intlpress.com/site/pub/pages/journals/items/sdg/content/vols/0019/0001/a005/)... | 10 | https://mathoverflow.net/users/nan | 277277 | 123,026 |
https://mathoverflow.net/questions/277259 | 3 | Let $(X\_1,\ldots,X\_n)$ be a collection of random variables. For $\alpha\ge1$, let us say that these are $\alpha$-weakly dependent if for all $1\le k\le n$
and all $1\le i\_1<\ldots< i\_k$,
we have
$$
\alpha^{-k}\le
\frac{ P(X\_{i\_1},\ldots,X\_{i\_k})}
{\prod\_{j=1}^k P(X\_{i\_j})}
\le
\alpha^k
.
$$
Obviously, the $n... | https://mathoverflow.net/users/12518 | A notion of weak dependence | This reminds me of the notion of $(\epsilon, k)$-wise independence for random bit vectors. That is, given a set of $n$ random binary bits $X\_i \sim \text{Bernoulli}\left(\frac{1}{2}\right)$, they are said to be $(\epsilon, k)$-wise independent if for any $S \subset [n], |S| = k,$ we have that $\left| \text{Pr}\left(\c... | 1 | https://mathoverflow.net/users/112658 | 277283 | 123,029 |
https://mathoverflow.net/questions/277228 | 12 | The Dirac belt trick produces a nice 3-dimensional geometric object with symmetry group $Spin(3) = SU(2)$: a 2-sphere with a properly embedded framed ray (usually presented by using orientations to reduce the framing of the ray to a single normal vector field, then integrating this to give a "belt" of finite width), wi... | https://mathoverflow.net/users/112607 | Is there a higher dimensional analogue of the Dirac belt trick? | If I understand the question correctly, the same "belt" construction works in higher dimensions. The belt should have a full framing of its normal bundle ($n−1$ normal vectors; using the orientation of $\mathbb R^n$, you can get by with $n−2$ normal vectors; adding the orientation tangent to the belt gives $n$ vectors ... | 6 | https://mathoverflow.net/users/284 | 277290 | 123,033 |
https://mathoverflow.net/questions/277155 | 5 | Where I could find relationships between Legendre and Chebyshev polynomials?
For example I found with maple
$$ P\_n(\cos\theta)=\sum\_{k=0}^n(-1)^{n+k}\frac{2-\delta\_{k0}}{4^n}
\binom{n-k}{\frac{n-k}{2}}\binom{n+k}{\frac{n+k}{2}}\cos(k\theta)$$
The sum runs over $n+k$ even, and $\delta\_{k0}=1$ if and only if $k=0$. ... | https://mathoverflow.net/users/94200 | Relation between Legendre and Chebyshev polynomials | Both the Legendre and Chebyshev polynomials are particular cases of Jacobi polynomials $P\_n^{(\alpha,\beta)}(x)$. A general connection formula of the type $$P\_n^{(\gamma,\delta)}(x)=\sum\_{k=0}^nc\_{n,k}^{\gamma,\delta;\alpha,\beta}P\_k^{(\alpha,\beta)}(x)$$ can be found on page 256 of the book [Mourad E.H. Ismail, *... | 2 | https://mathoverflow.net/users/4953 | 277296 | 123,035 |
https://mathoverflow.net/questions/277292 | 6 | I am looking for references on the geometry of knot groups. For instance, I am interested in the following question:
>
> When is a knot group relatively hyperbolic?
>
>
>
Hyperbolic knots are known to have a group which is relatively hyperbolic, but is the reciprocal true? Is the answer easier when restricted ... | https://mathoverflow.net/users/43559 | Relatively hyperbolic knot groups | The following theorem, which can be deduced from a [combination theorem](https://arxiv.org/abs/math/0203258) of Dahmani, is stated as Theorem 7.2.2 in our book *[3-manifold groups](http://www.ems-ph.org/books/book.php?proj_nr=195)* .
>
> **Theorem:** Let $N$ be a compact, orientable, irreducible 3-manifold with emp... | 9 | https://mathoverflow.net/users/1463 | 277297 | 123,036 |
https://mathoverflow.net/questions/277270 | 8 | I am reading Macdonlad's book on "symmetric functions and Hall polynomials" and I have difficulty figuring out an identity which involves hook-lengths. I would like to ask for a hint.
Let $\lambda=(\lambda\_1,\dots, \lambda\_k)$ be a partition and define $\mu\_i=\lambda\_i+k-i$, for $1\leq i\leq k$. The hook-length ... | https://mathoverflow.net/users/91357 | An identity involving hook-lengths | The first identity says that
$$\{h(x) : x \in \lambda\} \cup \{\mu\_i - \mu\_j : 1 \leq i < j \leq k\} = \bigcup\_{i=1}^k \{j : 1 \leq j \leq \mu\_i\}$$
as multisets. The second identity then follows because the multisets of exponents are the same on the LHS and RHS.
| 5 | https://mathoverflow.net/users/51668 | 277312 | 123,041 |
https://mathoverflow.net/questions/277330 | 3 | Given three eigenvectors and three eigenvalues, how would you go about finding BOTH non-symmetric matrix A and symmetric matrix B?
EDIT 7/27:
Sorry for not being specific enough~ In the problem I am given three linearly independent 4x1 eigenvectors u1, u2, and u3 and their respective eigenvectors. I found out that ev... | https://mathoverflow.net/users/112687 | Finding non-symmetric matrix given real eigenvalues and eigenvectors | The question is really unclear. In the following, I assume that the matrices you're looking for are 3x3 matrice, otherwise the answer is trivial.
If your given eigenvectors are linearly independent, then your matrix is completely determined by those three vectors. It is symmetric if and only if they are orthogonal.
... | 2 | https://mathoverflow.net/users/111917 | 277332 | 123,046 |
https://mathoverflow.net/questions/277343 | 9 | One of the nice features of the first admissible ordinal after $\omega$, i.e. $\omega\_1^{CK}$, is that it is the collection of ordinals whose order type is that of a computable well-ordering on $\omega$.
Is a similar thing true for all other admissible sets? Specifically suppose $\alpha$ is an admissible ordinal, $... | https://mathoverflow.net/users/8106 | Order type of $\alpha$-computable well-orderings | [Surprisingly, no there isn't!](http://www.sciencedirect.com/science/article/pii/0003484379900251) See also [this paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.101.3332&rep=rep1&type=pdf).
Roughly, if $\alpha$ is sufficiently stable, then the next admissible is much greater than the supremum of the ... | 8 | https://mathoverflow.net/users/8133 | 277345 | 123,050 |
https://mathoverflow.net/questions/277295 | 6 | Is $H^4(PSL(2,\mathbb{Z}),\mathbb{Z})$ known? I ask this in response to the recent calculation of the same cohomology group for $\mathrm{Co}\_0$ and $\mathrm{Co}\_1$.
| https://mathoverflow.net/users/4177 | Fourth cohomology of the modular group | From Jeremy Rickard's comments, the group cohomology (with coefficients a module with trivial action) of a free product of (discrete) groups is sent to direct sum (eg Proposition 1.3.16.3 in C. Löh, *[Group Cohomology & Bounded Cohomology](https://wwwmath.uni-muenster.de/u/clara.loeh/topologie3_ws0910/prelim.pdf)* (pdf... | 6 | https://mathoverflow.net/users/4177 | 277347 | 123,051 |
https://mathoverflow.net/questions/277382 | 5 | It is not completely clear how [Bridges, Richman and Youchuan](http://www.sciencedirect.com/science/article/pii/S0019357797891318) treated sets in their paper. Example is in the following lemma (Lemma 7 on p. 7):
>
> Let $U$ and $V$ be (**inhabited** to mean $\exists u \in U, \exists v \in V$) sets of a Banach spac... | https://mathoverflow.net/users/42302 | Sets in constructive mathematics | **The lemma has an existence hypothesis; it is this existence hypothesis that allows the proof to “choose” a point when necessary.**
$\newcommand{\x}{\vec{x}}$Specifically, the lemma assumes that $U \cup V$ is dense. That says (by definition) that for every suitable $x$ and $\varepsilon$, there exists some point in $... | 12 | https://mathoverflow.net/users/2273 | 277386 | 123,063 |
https://mathoverflow.net/questions/277337 | 1 | Let $\mathrm{G}$ be a semi-simple algebraic group over $\mathbb{C}$ and $V$ be a finite dimensional representation of $\mathrm{G}$. Let $x \in V$ be a non zero vector such that the variety $\mathrm{G}.[x] \subset \mathbb{P}(V)$ is closed.
Is there an effective criterion to decide when the projective dual of $X$ is th... | https://mathoverflow.net/users/37214 | dense orbit projective dual homogeneous space | This is certainly a very rare phenomenon. First of all, since $Gx$ generates an irreducible submodule of $V$ you may restrict your attention the case that $V$ is irreducible.
Next, the dual variety of $X$ is almost always of codimension one. The few exceptions were classified in Knop-Menzel (1987).
Now assume we ar... | 3 | https://mathoverflow.net/users/89948 | 277393 | 123,066 |
https://mathoverflow.net/questions/277387 | 5 | Can the Hirzebruch Surface $F\_2:=\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(2))$ be obtained by some GIT quotient of $\mathbb{P}^4$ (or $\mathbb{C}^4$)?
| https://mathoverflow.net/users/nan | Hirzebruch Surface F2 | Depending on the interpretation of your question, the answer is *Yes*.
In fact the Hirzebruch surface $\mathbb{F}\_n =\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(n))$ is the quotient of $X = \mathbb{A}^2 \setminus \{0\} \times \mathbb{A}^2 \setminus \{0\}$ with respect to the group action
$$ \mathbb{G}\_m^2 \times X \to... | 6 | https://mathoverflow.net/users/5101 | 277395 | 123,067 |
https://mathoverflow.net/questions/277380 | 1 | Many results in probability theory/random matrix theory/etc require probability distributions with finite fourth moments; what is the measure of such probability distributions (in the space of probability measures)? While the question "what RVs has finite fourth moment" is answered [here](https://mathoverflow.net/quest... | https://mathoverflow.net/users/97686 | Measure of bounded fourth (and below) moment distributions? | I'll answer the topological version. Let's work in one dimension for simplicity, so we consider the space $\newcommand{\PR}{\mathcal{P}(\mathbb{R})}\PR$ of probability measures on $\mathbb{R}^1$.
Most of the commonly encountered metrics on $\PR$ induce the same topology: the [weak topology](https://en.wikipedia.org/w... | 4 | https://mathoverflow.net/users/4832 | 277399 | 123,069 |
https://mathoverflow.net/questions/277405 | 9 | Let $R$ be a subring of $\mathbf{No}$, the set of surreal number. We try to construct $\tilde{R}$, the Cauchy completion of $R$, just like the ordinary Cauchy completion for metric space.
In the following we only consider the sequences in $R$ indexed by (i.e. with length equal) $\mathrm{Cf}(R)$, the cofinality of $R$... | https://mathoverflow.net/users/74664 | Surreal number: trying to construct complete ordered fields | In *Fields of surreal numbers and exponentiation* (Fund. Math. 167 (2001), pp. 173-188, doi:[10.4064/fm167-2-3](https://doi.org/10.4064/fm167-2-3)), Lou van den Dries and I show that $O\_\alpha$ is an ordered field if and only if $\alpha$ is an epsilon number (see Corollary 4.9). Moreover, for epsilon $\alpha$, $O\_\al... | 12 | https://mathoverflow.net/users/18939 | 277415 | 123,077 |
https://mathoverflow.net/questions/210428 | 1 | In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform.
In *free* (noncommutative) probability theory, sums of random variables correspond to *free* convolutions of distribution functions, which plays ... | https://mathoverflow.net/users/29961 | Relating the R-transform in free probability to noncommutative group representations | The $R$-transform is related to harmonic analysis around free products of groups.
Actually, the computational machinery for the $R$-transform was also found independently from Voiculescu and about the same time, by Woess, Cartwright and Soardi, and McLaughlin, in a more restricted setting of random walks on free produc... | 4 | https://mathoverflow.net/users/112626 | 277422 | 123,080 |
https://mathoverflow.net/questions/277406 | 4 | Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type:
$$
F(\lambda,u) = \lambda u - G(u) = 0,
$$
where $G \in C^1(X,X)$ is such that $G(0)=0$ and $G'(0)$ is compact (where ' is a Fréchet derivative).
If $\lambda\_0$ is an eigenvalue for $G'(... | https://mathoverflow.net/users/90127 | Crandall & Rabinowitz Theorem, bifurcation curves | Here is a simple example in $R^2$: $G(u)=(-u\_1+u\_2^3,u\_2-u\_1^3)$.
| 4 | https://mathoverflow.net/users/12120 | 277423 | 123,081 |
https://mathoverflow.net/questions/277426 | 10 | How does one show that for a given packing body $B$ (a finite set of integers) there is a periodic packing of the integers by disjoint translates of $B$ that achieves as its density the supremum of the set of densities achieved by all periodic packings?
I know (via a compactness argument) that the supremum is achieve... | https://mathoverflow.net/users/3621 | Maximal packings of the integers | Once you have it for some packing (and the standard mumbo-jumbo about achieving (as true density) the supremum of upper densities over all packings, periodic or not), the (upper) density of certain not too long intervals $A---A$ ($A$ here is some not too short packing pattern) is positive (because every sufficiently lo... | 8 | https://mathoverflow.net/users/1131 | 277431 | 123,083 |
https://mathoverflow.net/questions/277208 | 7 | Let $K$ be a multiplicatively written semigroup (either commutative or not) and $H$ a subsemigroup of $K$. We say that $H$ is *divisor-closed* (in $K$) if $x \in H$ for all $x, y \in K$ such that $x \mid\_K y$ (i.e., $y = uxv$ for some $u, v \in K$) and $y \in H$.
Accordingly, we say that a semigroup $S$ is *annular... | https://mathoverflow.net/users/16537 | For which abelian groups $G$ does the monoid of zero-sum sequences over $G$ embed into a ring as a divisor-closed subsemigroup? | Figured it out (sorry for answering my own question). I'll prove the following:
>
> **Lemma.** Let $H$ be a linearly orderable monoid and $R$ a domain whose group of units is trivial. Then $H$ embeds as a divisor-closed submonoid into the multiplicative monoid of the monoid ring $R[H]$.
>
>
>
This will show t... | 4 | https://mathoverflow.net/users/16537 | 277434 | 123,086 |
https://mathoverflow.net/questions/277416 | 8 | Let X be a variety. Suppose $\mathcal{E\_1}$ and $\mathcal{E\_2}$ are two vector bundles on X. Is there an example such that $\mathbb{P}(\mathcal{E\_1})$ and $\mathbb{P}(\mathcal{E\_2})$ are isomorphic as varieties but not as $\mathbb{P}^n$-bundles over X?
| https://mathoverflow.net/users/nan | Projective bundle | Here is one method of constructing many such examples.
>
> **Claim.** Let $(X,H)$ be a projective variety, let $\phi \colon X \stackrel\sim\to X$ be an automorphism, and let $\mathscr L$ be a line bundle on $X$ with $P\_H(\mathscr L,n) \neq P\_H(\mathcal O\_X,n)$ such that $\phi^\* \mathscr L \not \cong \mathscr L$... | 8 | https://mathoverflow.net/users/82179 | 277437 | 123,088 |
https://mathoverflow.net/questions/277450 | 8 | Are there two smooth independent vector fields $X,Y$ on $S^3$ with $[X,Y]=Y$?
| https://mathoverflow.net/users/36688 | Independent vector fields $X,Y$ on $S^3$ with $[X,Y]=Y$ | No. If such a pair existed, they would be tangent to a codimension $1$ foliation of $S^3$. Such a foliation must have a Reeb component, in particular, a compact leaf, $T\subset S^3$, which would be a torus (since its tangent bundle would be trivial). If $\xi$ and $\eta$ were the $1$-forms on $T$ dual to the basis $X$ a... | 21 | https://mathoverflow.net/users/13972 | 277452 | 123,094 |
https://mathoverflow.net/questions/275720 | 4 | Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}\_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}\_{\geq 2}$.
Then $V$ is uniquely determined by a pair of smooth characters $\alpha,\beta$ of $\mathbb{Q}\_p^{\times}$. Breuil and Berger gives in [their paper... | https://mathoverflow.net/users/69289 | Smooth intertwining operators | As Breuil say in the paper you cite that these are *algebraic* Intertwining and so do not bother much about convergence.
| 1 | https://mathoverflow.net/users/69289 | 277461 | 123,099 |
https://mathoverflow.net/questions/277298 | 3 | Sweedler's 4-dimensional Hopf algebra admits a one-parameter family of triangular structures given by
\begin{equation}
R\_{\lambda}:=1\otimes1+1\otimes g+g\otimes 1-g\otimes g-\frac{\lambda}{2}(x\otimes x-gx\otimes x+x\otimes gx+gx\otimes gx), \lambda\in \mathbb{C}.
\end{equation}
Is there a value for $\lambda$ such t... | https://mathoverflow.net/users/111720 | Is Sweedler's Hopf algebra factorizable? | The answer is no. The easiest to see this is the following: Consider the element $$X:=(R\_{\lambda})\_{21}R\_{\lambda}\in H\otimes H.$$ The Hopf algebra is factorizable if and only if $X$ has maximal length in the tensor product. But if this would have been the case, $X$ would have also had a maximal length in the quot... | 4 | https://mathoverflow.net/users/41644 | 277469 | 123,103 |
https://mathoverflow.net/questions/277002 | 7 | Can you recommend some reference books that use software like **MATLAB** and **Mathematica** to deal with the basic topics in
* *analysis of PDE* (the ones you can find in **Strauss**' book *Partial Differential Equations: An Introduction*)
and
* *numerical analysis of PDE*?
| https://mathoverflow.net/users/nan | Books and resources on PDEs that use Mathematica and Matlab | **4 reference books for the study of PDE with MATLAB:**
>
> Coleman, Matthew P. An introduction to partial differential equations with MATLAB. Second edition. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL, 2013.
>
>
>
Thorough treatment of PDEs and their applic... | 2 | https://mathoverflow.net/users/89429 | 277475 | 123,105 |
https://mathoverflow.net/questions/277326 | 0 | Let $X$ be a projective variety over an algebraically closed field of characteristic zero. Let $\eta$ be a generic point of $X$ and $x$ be a closed point. By <http://stacks.math.columbia.edu/tag/054F> there exists a discrete valuation ring $R$ and a morphism $\mbox{Spec}(R) \to X$ such that the fraction field of $R$ ma... | https://mathoverflow.net/users/45397 | Can the specialization map be flat | I am just writing my comment as an answer. Since you refer to $X$ as a variety, I assume that $X$ is an integral scheme. I will also assume that $x$ does not equal $\eta$, i.e., I will assume that $X$ is not a singleton. In that case, there exists a flat morphism $\text{Spec}(R)\to X$ from the spectrum of a DVR to $X$ ... | 2 | https://mathoverflow.net/users/13265 | 277477 | 123,106 |
https://mathoverflow.net/questions/277494 | 8 | I asked [Turing degree of a turing machine with access to an (arbitrary) nonstandard integer](https://mathoverflow.net/questions/275639/turing-degree-of-a-turing-machine-with-access-to-an-arbitrary-nonstandard-inte/), not thinking about the possiblity that this could depend on the model used. The question was not formu... | https://mathoverflow.net/users/65915 | What is the Turing degree associated with an ultrafilter $U$? | Great question! This is something that Uri Andrews, Mingzhong Cai, David Diamondstone, and I looked at in a recent (still unpublished) paper.
First of all, let's note that there's an important undefined notion: the Turing degree of an ultrafilter. Only *sets of natural numbers* have Turing degrees, and an ultrafilter... | 11 | https://mathoverflow.net/users/8133 | 277496 | 123,110 |
https://mathoverflow.net/questions/277493 | -1 | For positive integers $m, d\in \mathbb{N}$ consider the following statement:
>
> $\mathsf{S}(m,d):$ There is $N\in\mathbb{N}$ such that the complete graph $K\_m$ on $m$ vertices is a minor of any finite simple undirected graph $G=(V,E)$ with diameter $d$ and $|V|\ge N$.
>
>
>
Does $\mathsf{S}(m,d)$ hold for an... | https://mathoverflow.net/users/8628 | Complete minors in graphs of bounded diameter | No, because there exists trees with small diameter.
For example, the star on $N$ vertices has diameter $2$ and does not even contain a $K\_3$ minor.
| 3 | https://mathoverflow.net/users/39146 | 277501 | 123,111 |
https://mathoverflow.net/questions/277498 | 1 | I am currently interested in minimum weight regular d-spanners (i.e. d-factors) of complete graphs. When searching the internet for related articles, I came across [this one](http://www.cmm.uchile.cl/~rhoeksma/files/cornelissen_hoeksma_manthey_narayanaswamy_rahul_waanders-approximating_connected_graph_factors.pdf), whi... | https://mathoverflow.net/users/31310 | Tutte's Reduction of Minimum Weight d-Factors to Matching | Given a graph $G$ we construct $\hat{G}$ such that $G$ has a $d$-factor if and only if $\hat{G}$ has a perfect matching. For each vertex $x$ of $G$ we take $d$ vertices $x\_1, \dots, x\_d$ in $\hat{G}$. For each edge $e = \{x,y\}$ of $G$ we take two vertices $e\_x$ and $e\_y$ in $\hat{G}$. We then create for edges:
1... | 1 | https://mathoverflow.net/users/51668 | 277508 | 123,115 |
https://mathoverflow.net/questions/277487 | 11 | What is already known about rigid line arrangements? By line arrangement, I mean a unions of lines in $\mathbb{P}^2\_{\mathbb{C}}$ with fixed incidences. (Written in notation, I mean a collection of lines $\mathcal{L}$, a collection of points $\mathcal{P}$, and a set of incidences $\mathcal{I}\subset \mathcal{L}\times\... | https://mathoverflow.net/users/16356 | Rigid line arrangements | You have to be a bit more careful with your definition of arrangements and their rigidity. Once, you are, then "Mnev universality" provides an enormous supply of rigid arrangements in $P^2$ (even on scheme-theoretic level). See Theorem 1.3 in
my paper with John Millson ["On representation varieties of Artin groups, pr... | 1 | https://mathoverflow.net/users/21684 | 277514 | 123,116 |
https://mathoverflow.net/questions/277530 | 4 | Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$).
A (cancellative) monoid/semigroup $S$ is *right reversible* if for all $a,b \in S$, it follows that $Sa \cap Sb \neq\emptyset$. This is the Ore condition for semigroups, and allows one t... | https://mathoverflow.net/users/42278 | Right reversibility of submonoids of nilpotent groups | It follows from Lemma 2 in Grigorchuk's <https://page-one.live.cf.public.springer.com/pdf/preview/10.1007/BF01138837> that all subsemigroups of a group $G$ are right reversible iff it contains no free semigroup on 2 generators.
A free subsemigroup is not right reversible. Conversely suppose $S$ is not right reversib... | 4 | https://mathoverflow.net/users/15934 | 277531 | 123,122 |
https://mathoverflow.net/questions/237751 | 8 | Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an *extension* of $R$ if there is a subring $\tilde{R}$ in $S$ isomorphic to $R$.
Let $S$ be an extension of $R$, and $$\phi: R\to \tilde{R}\subset S$$
be a ring isomorphism.
We say that... | https://mathoverflow.net/users/85489 | Polynomial roots in the ring extension | I have found a very simple and demonstrative construction of the ring extension, which came from non-commutative generalization of Hamilton-Caley's Theorem.
Let $R$ be a ring and $f(x) = x^m-\sum\limits\_{j=0}^{m-1}f\_jx^j\in R[x]$ be a monic polynomial.
We identify ring $R$ with a subring $\tilde{R} = \{\mathrm{diag... | 4 | https://mathoverflow.net/users/85489 | 277541 | 123,126 |
https://mathoverflow.net/questions/277538 | 3 | Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}\_X$ be its algebra of differential operators.
Consider $\mathcal{C}=\mathcal{D}\_X$-$\text{mod}$, the stable $\infty$ category of $\mathcal{D}\_X$-modules (unbounded complexes of $\mathcal{D}\_X$-modules localized at quasi-isomorphisms). Assumi... | https://mathoverflow.net/users/22810 | Who are the compact generators in the derived category of $\mathcal{D}_X$-modules? | One place where this is discussed is
*Drinfeld, Vladimir; Gaitsgory, Dennis*, [**On some finiteness questions for algebraic stacks**](http://dx.doi.org/10.1007/s00039-012-0204-5), Geom. Funct. Anal. 23, No. 1, 149-294 (2013). [ZBL1272.14005](https://zbmath.org/?q=an:1272.14005).
| 4 | https://mathoverflow.net/users/6263 | 277549 | 123,128 |
https://mathoverflow.net/questions/275754 | 9 | Let $f:[a,b]\to\mathbb R$ be a Henstock-Kurzweil-integrable function (short: HK-integrable).
>
> Can $f$ always be written as a sum of a Lebesgue-integrable function and a function which has a classical primitive, i.e. are there $f\_1\in L^1([a,b])$ and an everywhere differentiable function $F:[a,b]\to\mathbb R$ s... | https://mathoverflow.net/users/66283 | Decomposition of Henstock-Kurzweil-integrable functions | The answer is negative. Consider the space $\Delta'=\Delta'([a,b])$ of functions, which are classical derivatives of functions $[a,b]\to\mathbb R$. Assume the claim was true. Then any HK-integrable $f:[a,b]\to\mathbb R$ had a representation $f=f\_1+f\_2$, where $f\_1\in L^1([a,b])$ and $f\_2\in\Delta'$. Let $g$ be a mu... | 4 | https://mathoverflow.net/users/66283 | 277554 | 123,130 |
https://mathoverflow.net/questions/277573 | 3 | I'm trying to count the number of binary strings of length $n$ with the properties described below. Say we break the string into substrings (starting from left to right) of consecutive $0$'s or $1$'s. We must have:
1. The first substring can be of any length (from $1$ to $n$).
2. The last (ending at $n$) substring ca... | https://mathoverflow.net/users/29228 | Number of binary strings with 'at least two consecutives' constraints | We start considering words with no consecutive equal characters at all. These words are called Smirnov words or Carlitz words. (See example III.24 *Smirnov words* from *[Analytic Combinatorics](http://algo.inria.fr/flajolet/Publications/books.html)* by Philippe Flajolet and Robert Sedgewick for more information.)
A ... | 10 | https://mathoverflow.net/users/51505 | 277578 | 123,133 |
https://mathoverflow.net/questions/277585 | 2 | Let $A \subseteq A[w] \subseteq C$ be three Noetherian integral domains, over a field $k$ of characteristic zero, with $A \subseteq C$ an algebraic ring extension (in particular, $w$ algebraic over $A$).
Further assume that: **(1)** $A$ and $C$ are unique factorization domains (UFD's).
**(2)** $A \subseteq C$ is ro... | https://mathoverflow.net/users/72288 | Sandwich theorem for UFD's | That is not true. Let $A$ be a polynomial ring in two variables, $k[s,t]$. Let $C$ be the $A$-subalgebra of the fraction field generated by the fraction $s/t$, i.e., $C=k[(s/t),t]$. Both $A$ and $C$ are unique factorization domains (roughly by Gauss's Lemma). Since $A$ is integrally closed in its fraction field, the ex... | 7 | https://mathoverflow.net/users/13265 | 277591 | 123,135 |
https://mathoverflow.net/questions/277589 | 8 | Let $M$ be a smooth manifold, and $N$ is a submanifold. My question is simply that, does there exist a metric $g$ on $M$ so that $N$ is totally geodesic?
In general the answer might be 'no'. But it might be possible that, when we assumed additional conditions on $M$ or $N$, the answer became 'yes'. This question is k... | https://mathoverflow.net/users/69190 | Find a metric so that a given submanifold is totally geodesic | Suppose $N$ is closed. Let $p:U\to N$ be a tubular neighborhood of $N$ in $M$. So $p:U\to N$ is a vector bundle. Now let $g\_N$ be a Riemannian metric on $N$ and $g\_U$ be a fiber metric on $U$, and let $\nabla$ be a linear connection on the vector bundle $p:U\to N$. Split $TU= V\oplus H$ into the vertical and horizont... | 5 | https://mathoverflow.net/users/26935 | 277596 | 123,137 |
https://mathoverflow.net/questions/277610 | 4 | I know that $A$ and $A^t$ have the same characteristic polynomial. But I'm looking for some picture of why they should have the same set of eigenvalues.
Maybe slightly more concrete question is whether or not you can say something about bases of $A^t$ given an eigenbasis of $A$. Does knowing one give you a computati... | https://mathoverflow.net/users/92401 | geometric intuition for A and A-transpose having the same eigenvalues | Suppose $v\_j$ is an eigenbasis of $A$ with eigenvalues $\lambda\_j$, so that $A v\_j = \lambda\_j v\_j$. Then for all dual vectors $f$ we have
$$\langle f, A v\_j \rangle = \lambda\_j \langle f, v\_j \rangle$$
where $\langle -, - \rangle$ denotes the dual pairing. If $f\_i$ denotes the dual basis to $v\_j$, so tha... | 2 | https://mathoverflow.net/users/290 | 277612 | 123,145 |
https://mathoverflow.net/questions/277550 | 1 | Let $X\rightarrow T$ be a fibre bundles with smooth projective fibre $F$ and $X$ and $T$ are also smooth. Let $D$ is relative effective Weil divisor. Suppose $W\_1 $ and $W\_2$ are relative subvarieties which are isomorphic(by a relative map say $\phi$). Let $L:=\mathcal{O}(D)$. Let $L\_t|\_{W\_{1,t}}\cong L\_t|\_{W\_{... | https://mathoverflow.net/users/nan | Relative divisors | I am posting my comment above as an answer. Let $T$ be a smooth, projective curve of genus $g\geq 1$. Let $F$ be $\mathbb{P}^3$. Let $X$ be the product $\mathbb{P}^3\times T$ with its projection. Let $L$ be $\text{pr}\_{\mathbb{P}^3}^\*\mathcal{O}(1)$. Let $w:W\to T$ be a finite, étale morphism of degree $d>1$ such tha... | 1 | https://mathoverflow.net/users/13265 | 277614 | 123,146 |
https://mathoverflow.net/questions/277613 | 6 | Let $E(z,s):=\pi^{-s}\Gamma (s) \sum\_{(m,n)=1}\frac{y^s}{|mz+n|^{2s}}$ be the real-analytic Eisenstein series.
It satisfies the functional equation $E(z,s)=E(z,1-s)$ with two poles at $s=0,1$.
The method I know to prove this is to calculate the Fourier coefficients individually. They are either divisor functions o... | https://mathoverflow.net/users/2666 | Alternative way to prove the functional equation for Eisenstein series? | As usual, the functional equation on the Dirichlet series side comes from a theta function on the modular side.
---
Using the poisson summation formula we show $$\vartheta\_z(x) = \sum\_{(c,d) \in \mathbb{Z}^2} \exp(-\pi x \frac{|cz+d|^2}{|\Im(z)|}) = x^{-1} \vartheta\_z(1/x)$$
Then let $$E\_z(s) = \sum\_{\gamm... | 6 | https://mathoverflow.net/users/84768 | 277616 | 123,147 |
https://mathoverflow.net/questions/277615 | 12 | I understand that the top Stiefel Whitney class is an obstruction for the tangent bundle of a manifold to have a trivial line sub-bundle. I am looking for a counterexample when removing the word "trivial", i.e: A compact manifold $M$ of dimention n such that $w\_n(TM)\neq0$ (or equivalently $\chi(M)$ is odd) and there ... | https://mathoverflow.net/users/111049 | A Compact Manifold with odd Euler characteristic whose tangent bundle admits a field of lines | I believe there is no example satisfying all your constraints. If I recall (my memory is a little foggy on this) the result likely goes back to Hopf, and one of his variations on the Poincare-Hopf index theorem. This question might be addressed in the Milnor and Stasheff text. Here is one way to argue the point.
Say... | 16 | https://mathoverflow.net/users/1465 | 277620 | 123,149 |
https://mathoverflow.net/questions/277417 | 0 | Let $(X,d)$ be a metric space.
An (intern) horofunction is a function of the form $h\_y:x\mapsto d(x,y)-d(x\_0,y)$, where $x\_0$ is a fixed point.
Now, the map $y\mapsto h\_y$ is one-to-one and continuous (with respect to pointwise convergence topology) so that there is an embedding $X\hookrightarrow \mathrm{Lip}\_{x\_... | https://mathoverflow.net/users/111917 | First introduction of horofunctions | For those who are interested in this, I found a reference in Gromov's book Metric structures for Riemannian and Non-Riemannian spaces (Section 3.11.$\frac{2}{3}\_+$).
Although horofunctions are often attributed to Gromov, they were introduced by Kuratowski in 1935 in the paper "Quelques problèmes concernant les espac... | 0 | https://mathoverflow.net/users/111917 | 277625 | 123,150 |
https://mathoverflow.net/questions/277119 | 6 | In the hope of completing the rich tapestry of complemented (or not) topological vector subspaces, I would like to know (maybe it is immediate for specialists)
whether the space of analytic functions is complemented within the space of infinitely differentiable ones. I begin with the one-variable case
... and make ... | https://mathoverflow.net/users/25256 | Is this closed subspace of Fréchet space complemented | $H(\Omega) $ is not complemented in $C^\infty (\Omega)$ e.g. for the unit disc in $\mathbb C $. This follows from the structure theory of Frechet spaces: The space of smooth functions is isomorphic to $s^\mathbb N$ and has a certain property (DN$\_{loc}$) of Vogt. If $H (\Omega) $ were complemented it would also have t... | 6 | https://mathoverflow.net/users/21051 | 277626 | 123,151 |
https://mathoverflow.net/questions/277646 | 3 | I am currently confused with the moment of non-homogeneous compound Poisson process and a Brownian Motion. I know that generally Poisson Process and Brownian Motion are independent if they are adapted to the same filtration. But what if the intensity of the Poisson Process and the Brownian Motion are correlated?
For ... | https://mathoverflow.net/users/112804 | Poisson process with stochastic intensity correlated with a Brownian Motion | This is only a partial answer, as the computations can become quite involved.
We have
$$ \mathbb{E}( X\_t J\_t ) = \mathbb{E}\left( X\_t Q \int\_{ \mathbb{R}\_+ \times [0, t] } d\mu \right) = \mu\_Q \mathbb{E}\left( X\_t \int\_{ \mathbb{R}\_+ \times [0, t] } d\mu \right). $$
by independence of $Q$ with the other pro... | 2 | https://mathoverflow.net/users/109373 | 277657 | 123,156 |
https://mathoverflow.net/questions/277655 | 24 | I recently stumbled upon the following identity, valid for any real numbers $\alpha\_1,\dots,\alpha\_n$ and $\lambda\_{n1} \leq \dots \leq \lambda\_{nn}$:
$$ \mathrm{det}( e^{\alpha\_i \lambda\_{nj}} )\_{1 \leq i,j \leq n} = V(\alpha) \int\_{GT\_\lambda} \exp( \sum\_{i=1}^n \sum\_{j=1}^i \lambda\_{ij} (\alpha\_{n+1-i... | https://mathoverflow.net/users/766 | Reference for exponential Vandermonde determinant identity | This looks like a special case of a formula by Samson Shatashvili related to the HCIZ integral as mentioned in Ryan's answer. Compare, in particular the two ways of computing $\langle 1\rangle$ given by Equations 3.2 and 3.4 in ["Correlation Functions in The Itzykson-Zuber Model"](https://arxiv.org/abs/hep-th/9209083) ... | 14 | https://mathoverflow.net/users/7410 | 277666 | 123,158 |
https://mathoverflow.net/questions/277665 | 2 | Given a 1+1 dimensional wave equation ($c$ constant) plus a small ($\left|k\right|\ll 1$, $k$ imaginary (?)) third order derivative in $x$ term,
$$
f\_{tt}=c^2\ f\_{xx}+k\ f\_{xxx}
$$
is there a sensible answer to what is the propagation speed?
OK, IF there is such an answer, it seems it would be dependent on freque... | https://mathoverflow.net/users/29625 | propagation speed for a modified wave equation | One can certainly prove something along these lines; here's a straightforward perturbative statement: Let's impose the initial conditions $f(0,x)=u$, $f\_t(x,0)=0$ (for simplicity), with (as you propose) $\widehat{u}$ compactly supported, by $|\xi|\le C$. Let's also take $u\in\mathcal S$, $\|u\|\_2=1$. Finally, let me ... | 1 | https://mathoverflow.net/users/48839 | 277667 | 123,159 |
https://mathoverflow.net/questions/277629 | 4 | By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,\*)$ such that both $\cdot,\*$ are associative, and the following property holds (see [this](http://jas.shahroodut.ac.ir/article_616.html) )
$$
x\*(y\cdot z)=x\*y\*z\;\; ; \;\; \forall x,y,z\in S?
$$
The right and two-sided cases and group-... | https://mathoverflow.net/users/40520 | Regarding a new algebraic structure | One example is that $\cdot$ is ordinary multiplication of integers and $x\*y=(x\cdot y) \bmod b.$ One could as well let $\cdot$ be ordinary addition. That is essentially your example except that you are not requiring $b$ to be a positive integer.
More generally, Let $\sim$ be an equivalence relation on $S$ and $\cdot... | 5 | https://mathoverflow.net/users/8008 | 277674 | 123,163 |
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