parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/260167 | 4 | A recurrence is given by
$f[0]=2x$, $f[1]=3x^3-x^2+x+1$,
$$
f[n]=(x^{2^n}+1)f[n-1]+(x^{2^n}+1)(x^{2^n-1}+1)
$$
How does the PRODUCT of the nonzero coefficients of $f[n]$ scale with $n$?
| https://mathoverflow.net/users/9766 | Product of polynomial coefficients of a recurrence | It follows by induction from the recursion that for all $n\ge2$ the polynomial $f\_n$ writes as $f\_n=(x^{2^{n}}+1)(x^{2^{n-1}}+1)g\_n $ where $g\_n$ satisfies
$$\begin{cases} g\_2:=3x+2 \\ g\_n:=(x^{2^{n-2}}+1)g\_{n-1}+1,\quad & \mathrm{if }\; n>2\end{cases}$$
and has degree $2^{n-1}-1$. Since $f\_n=(x^{2^{n}}+1)(x^{2... | 4 | https://mathoverflow.net/users/6101 | 260183 | 117,317 |
https://mathoverflow.net/questions/260190 | 1 | It's well-known (and fairly easy to prove) that over any topological space $X$, if a sheaf $\mathcal{F}\in\mathfrak{Ab}(X)$ is injective, then it must also be flasque. However, I've never seen mentioned that this is an equivalence, so I suspect there must be some example of a sheaf $\mathcal{F}\in\mathfrak{Ab}(X)$ that... | https://mathoverflow.net/users/94022 | Is there such a thing as a non-injective flasque abelian sheaf? | Just take a non-injective abelian group and a point in your space, and form the skyscraper sheaf at that point with stalk the abelian group.
| 10 | https://mathoverflow.net/users/1384 | 260192 | 117,321 |
https://mathoverflow.net/questions/260196 | 43 | [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function) is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum\_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when the real part is greater than $1$.
>
>
> >
> > **My questio... | https://mathoverflow.net/users/51189 | Could the Riemann zeta function be a solution for a known differential equation? | When posed properly, a long-standing open problem, but in the form you ask:
*Robert A. Van Gorder*, MR 3276353 [**Does the Riemann zeta function satisfy a differential equation?**](http://dx.doi.org/10.1016/j.jnt.2014.08.013), *J. Number Theory* **147** (2015), 778--788.
| 45 | https://mathoverflow.net/users/11142 | 260198 | 117,325 |
https://mathoverflow.net/questions/260209 | 2 | Let $I$ be an $n\times n$ identity matrix, $B$ be an $n\times n$ matrix with all the elements tending to zero. If we can expression $det(I+B)= 1 + f(B) + o(f^2(B))$, as all the elements of $B$ go to zero, where $f(B)$ is the first order expansion term, what is the exact form of $f(B)$? I have been searching for the ans... | https://mathoverflow.net/users/103887 | Exact first order perturbation expansion of matrix determinant | The exact form is $f(B) = \mathrm{tr\ } B.$
| 2 | https://mathoverflow.net/users/11142 | 260210 | 117,330 |
https://mathoverflow.net/questions/260124 | 5 | We recently need the following result about product space in an applied analysis paper:
Let $(M,d)$ be a metric space, and $\bigotimes^{k+1}M = \overbrace{M \times M \times \dots \times M}^{(k+1)-{\rm times}}$ be equipped with the product topology.
Let $U \subset \bigotimes^{k+1}M$ be a neighborhood of the hyperdiago... | https://mathoverflow.net/users/102458 | Neighborhoods of the hyperdiagonal in a product space | The result does not hold for general topological spaces.
For a counterexample, equip $M = \{a, b, c\}$ with the topology whose open sets are $\emptyset$, $M$, $\{a, b\}$, $\{b\}$ and $\{b, c\}$.
The set $U= \{a,b\}^3\cup \{b,c\}^3\subseteq M^3$ is an open neighborhood of the ternary diagonal.
Now let $V\subseteq ... | 2 | https://mathoverflow.net/users/75735 | 260217 | 117,332 |
https://mathoverflow.net/questions/260214 | 1 | Let $n\geq 2$ be a positive integer and $k$ be a number between $1$ and $n$. Recently, I came across the following question about $\mathbb Z/n\mathbb Z$ and I wonder if it was studied before. I'd be thankful for any references or any suggestions on how to approach the problem.
Find the smallest integer $t$ (or an upp... | https://mathoverflow.net/users/22733 | Covering a finite ring with arithmetic progressions | The case of prime $n$.
We may suppose that $j\ne 0$. Consider $t$ distinct elements $j/i\_1,\dots,j/{i\_{t}}$. If all of them belong to the set $\{k+1,\dots,n-1\}$, we get $t\leqslant n-k-1$. Thus for $t=n-k$ we may always find $r$ such that $j/i\_r\in \{1,2,\dots,k\}$ as you need. For $t=n-k-1$ this is not always p... | 3 | https://mathoverflow.net/users/4312 | 260219 | 117,333 |
https://mathoverflow.net/questions/260220 | 2 | Let $x = (a,b) \in \mathbb{Q}^2$ and let $p(x,t) = t^2-at+b$. Does there exist an involution $\tau$ of $\mathbb{Q}^2$ such that for all $\tau(x) \neq x$, $x \in \mathbb{Q}^2$ one of the polynomials $p(x,t)$ or $p(\tau(x),t)$ is irreducible in $\mathbb{Q}[t]$. This is not a homework question and I don't know if this que... | https://mathoverflow.net/users/nan | A question about involutions and polynomials | Of course, for example $\tau(x)\equiv x$. Well, if you want fixed point free involution, the answer is still yes since there are countably many irreducible polynomials and countably many reducible polynomials, you may match them. Maybe, you need something more about $\tau$, say, require it to be linear?
| 2 | https://mathoverflow.net/users/4312 | 260221 | 117,334 |
https://mathoverflow.net/questions/260213 | 5 | Let $X$ be a measurable space, and let $M(X)$ be the vector space of finite signed measures on $X$. Are there natural conditions on a linear functional $f:M(X)\rightarrow\mathbb{R}$ that are equivalent to the existence of a measurable function $g:X\rightarrow\mathbb{R}$ such that $f(\mu)=\int gd\mu$ for all $\mu\in M(X... | https://mathoverflow.net/users/83073 | Conditions under which a linear functional on a space of measures must be integration of a function | If you work in terms of topologies on M, then the way to go is to use the theory of dualities from functional analysis. The basic theory of these things can be found in books called "Topological Vector Spaces", such as Schaefer's or Bourbaki's.
In this case, the relevant duality is
$$
\langle \mu, f \rangle = \int\_... | 3 | https://mathoverflow.net/users/61785 | 260228 | 117,336 |
https://mathoverflow.net/questions/260225 | 1 | Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is convex and nonempty. Does it follow that there is a continuous function $f: X \rightarrow Y$ s.t. the graph of $f$ is co... | https://mathoverflow.net/users/11146 | Every open convex-valued multimap has global sections? | Yes, and this follows directly from a selection theorem in
>
> Michael, Ernest. “Continuous Selections. I.” Annals of Mathematics,
> vol. 63, no. 2, 1956, pp. 361–382. Second Series,
> [www.jstor.org/stable/1969615](http://www.jstor.org/stable/1969615).
>
>
>
Theorem 3.1''' on page 368 shows that a lower hem... | 1 | https://mathoverflow.net/users/35357 | 260231 | 117,337 |
https://mathoverflow.net/questions/260156 | 15 | Suppose we are given some polynomial with integer coefficients, which we regard as carving out an affine variety $E$, for example:
$$ 3x^2y - 12 x^3y^5 + 27y^9 - 2 = 0 \tag{$\*$} $$
(We might consider a bunch of equations, we might work over projective space, but let's keep it simple for now).
We are interested i... | https://mathoverflow.net/users/401 | Modular forms from counting points on algebraic varieties over a finite field | The correct setting for this construction turns out to be *projective* varieties, so let me suppose we have a smooth variety $X$ inside $\mathbf{P}^N$, for some $N \ge 1$, defined by the vanishing of some homogenous polynomials $F\_1, \dots, F\_r$ in variables $x\_0, \dots, x\_N$, with the $F\_i$ having coefficients in... | 22 | https://mathoverflow.net/users/2481 | 260232 | 117,338 |
https://mathoverflow.net/questions/260050 | 10 | **Question:** Let $k$ be an algebraically closed field of characteristic zero.
Let $X=\mathrm{Spec}\!~R$ be a scheme of finite type over $k$. Suppose $I\subset R$ is
an ideal and $f\in I$.
Assume that $df = 0$ in $\Omega\_{R/k}\otimes\_R R/I$.
Does it follow that there is some positive integer $k$ such that $f^k ... | https://mathoverflow.net/users/40862 | Functions with vanishing Kahler differentials along a subscheme | I am writing up the comments above as an answer, partly because they involve a couple of fun lemmas about blowing up. I tried to find a proof that does not use resolution of singularities, but I could not find one.
Let $X$ be a scheme, and let $\mathcal{J}$ be a quasi-coherent sheaf of ideals. Then the blowing up, $\... | 2 | https://mathoverflow.net/users/13265 | 260233 | 117,339 |
https://mathoverflow.net/questions/260244 | 3 | Let $G=\operatorname{GL}\_n(\Bbb C)$ act polynomially on some finite-dimensional complex vector space $V$. This means that the action is given by a morphism $\rho\colon G\to\operatorname{GL}(V)$ of algebraic groups which extends to a morphism $\Bbb C^{n\times n}\to\operatorname{GL}(V)$.
Let $v\in V$ be a point with f... | https://mathoverflow.net/users/9947 | Does convergence in orbit imply convergence in group for finite stabilizer? | Yes, there is always a subsequence which converges:
Since $H$ is finite there exists an $\epsilon>0$ such that the sets $h\cdot B(1,\epsilon)$ are disjoint, where $B(1,\epsilon)=\{x|d(1,x)\leq\epsilon\}$.
Consider the continuous map $f:G\to V$ which sends $g$ to $g\cdot v$.
This is a polynomial map which is also l... | 3 | https://mathoverflow.net/users/41644 | 260246 | 117,342 |
https://mathoverflow.net/questions/260218 | 1 | I'm having trouble understanding some parts of the paper "[Provably computable functions and the fast growing hierarchy](https://epub.ub.uni-muenchen.de/3843/1/3843.pdf)" by Buchholz and Wainer (1987).
On page 183 they say that their system has axioms corresponding to the "defining equations of each elementary functi... | https://mathoverflow.net/users/27742 | Elementary functions in a formalized PA | I think your first and last questions more or less answer each other: the elementary functions can be defined using quantifiers, so adding them as symbols makes the quantifier-free formulas more expressive (and, by extension, $\Sigma\_1$ and so on). This makes their final result stronger: it includes formulas which use... | 2 | https://mathoverflow.net/users/8991 | 260247 | 117,343 |
https://mathoverflow.net/questions/260017 | 6 | Given a multi-variable function $F$, denote the number of monomials by $N(F)$. For example, $N(x(x+y))=N(x^2+xy)=2$ and
$$
N(x(x+y)(x+y+z))=N(x^3+2x^2y+x^2z+xy^2+xyz)=5.
$$
Define the functions $f\_n=\prod\_{i=1}^n(y\_1+\cdots+y\_i)$ and $g\_n=\prod\_{i=1}^n(y\_1+\cdots+y\_i+x\_{i+1})$ and
$$
h\_n=\prod\_{i=1}^n(y\... | https://mathoverflow.net/users/66131 | Expanding into monomials | Let $\mathrm{CC}\_n$ denotes the set of Catalan codes of length $n$, i.e.
$$\mathrm{CC}\_n = \left\{ (c\_0, \dots, c\_{n-1})\in \mathbb{Z}\_{\geq 0}^n\ :\ c\_0+\dots+c\_i\leq i\ \text{for all}\ i=0,1,\dots,n-1 \right\}.$$
For $c\in \mathrm{CC}\_n$, denote $s(c) = n-c\_0-\dots-c\_{n-1}$.
Then
$$N(h\_n) = \sum\_{k=0}^n... | 3 | https://mathoverflow.net/users/7076 | 260253 | 117,345 |
https://mathoverflow.net/questions/260184 | 1 | Let $(A\_{1},A\_{2}, \ldots,A\_{k})$ be $k$ matrices in $M\_{n}(\mathbb{R})$.
Is there an algebraic formula, as a generalization of "Determinant" for $k=1$, to compute the joint spectrum of $(A\_{1},A\_{2}, \ldots,A\_{k})$?
The joint spectrum is the union of righ and left joint spectrum. The later is defined as fol... | https://mathoverflow.net/users/36688 | How to compute the joint spectrum? | Let us write $T=M\_n(\mathbb{R})$. We have a polynomial map $f:\mathbb{R}^k\times T^k\to Hom\_{\mathbb{R}}(T^k,T)$ which sends $((\lambda\_i),(A\_i))$ to the linear map $$(B\_i)\mapsto \sum\_i B\_i(A\_i-\lambda\_i).$$
It holds that $f((\lambda\_i),(A\_i))$ is not surjective if and only if $(\lambda\_i)$ is in the left ... | 2 | https://mathoverflow.net/users/41644 | 260254 | 117,346 |
https://mathoverflow.net/questions/257912 | 4 | I am reading the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences", the link is here:<https://pub.uni-bielefeld.de/publication/1780235>.
In the paper, $A$ is an artin algebra, and $A$-mod the category of (finitely generated) $A$-modules. Let $\Theta=\{\Theta(1), \dots, \T... | https://mathoverflow.net/users/83554 | Questions in the paper "The category of good modules over a quasi-hereditary algebra has almost split sequences" | This is probably just a mistake in the preprint. If you look at the corresponding paragraph in the published version
*Claus Michael Ringel*, MR 1128706 [**The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences**](http://dx.doi.org/10.1007/BF02571521), *Math. Z.* **20... | 1 | https://mathoverflow.net/users/18756 | 260256 | 117,348 |
https://mathoverflow.net/questions/260257 | 0 | Let $a,q$ be a positive integers. I am trying to evaluate the following sum:
$\sum\_{\substack{1<a<q \\(a,q)>1 \\ (a+1,q)>1}}1$. Is there a formula that exists to calculate such sums?
Here is an example:
Let $q=15$.
We have the following multiple of divisors:
1,2,3,5,6,9,10,12,15.
But since the two pairs (5,6) a... | https://mathoverflow.net/users/100898 | Sum of $\sum_{\substack{1<a<q \\(a,q)>1 \\ (a+1,q)>1}}1$ | At first, we count the number of residues $a$ for which both $a$ and $a+1$ are coprime with $n$. Let $q=\prod p\_i^{k\_i}$ be a factorization of $q$. For any $p\_i$, there exist $p\_i-2$ admissible remainders modulo $p\_i$ (forgotten remainders are 0 and $-1$), thus $(p\_i-2)p\_i^{k\_i-1}$ admissible remainders modulo ... | 6 | https://mathoverflow.net/users/4312 | 260260 | 117,350 |
https://mathoverflow.net/questions/260207 | 6 | In the definition of Minimal model of projective variety, some authors use of discrepancy, and some others omit this condition. I am wondering to know the advantage of discrepancy In the definition of Minimal model.
| https://mathoverflow.net/users/103797 | Advantage of discrepancy | Take $(X,B)$ and $(Y,B\_Y)$, be the log canonical pairs and assume we have birational map $φ:X⟶Y$, we need to take $d(E,X,B)≤d(E,Y,B\_Y)$ for any prime divisor $E$ on $X$. In fact discrepancy is measure of singularities, and this condition $d(E,X,B)≤d(E,Y,B\_Y)$ saying that the singularities of $Y$ is at least as good ... | 9 | https://mathoverflow.net/users/nan | 260265 | 117,352 |
https://mathoverflow.net/questions/260146 | 2 | Let $s=α+iβ$ be a complex number. Consider the Dirichlet series of the form $$f(s)=∑\_{n=1}^{∞}(a\_{n})/n^{s}$$
where $(a\_{n})\_{n≥1}$ is a real sequence.
We consider the class of Dirichlet series satisfying the following conditions:
(1) The real sequence $(a\_{n})\_{n≥1}$ is not identically zero.
(2) The funct... | https://mathoverflow.net/users/74668 | Does there exist a known Dirichlet series verifying all these conditions and have non trivial zeros off the critical line | What's wrong with the most obvious choice for a Dirichlet series that converges for $\text{Re}(s)>0$?
$$
f(s)=(1-2^{1-s})\zeta(s)=\sum\_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}
$$
converges for real $s>0$ by [Dirichlet's test](https://en.wikipedia.org/wiki/Dirichlet's_test)
and in the half plane by standard facts about t... | 4 | https://mathoverflow.net/users/6756 | 260274 | 117,355 |
https://mathoverflow.net/questions/9558 | 39 | **Update (January 17)**: The problem has now been solved by Daniel Ahlberg and Christopher Hoffman. (Thanks to Matt Kahle for informing us.)
---
Consider a square planar grid. (The vertices are pair of points in the plane with integer coordinates and two vertices are adjacent if they agree in one coordinate and d... | https://mathoverflow.net/users/1532 | The shortest path in first passage percolation | It seems that this problem was recently solved by Ahlberg and Hoffman.
<https://arxiv.org/abs/1609.02447>
| 7 | https://mathoverflow.net/users/4558 | 260278 | 117,357 |
https://mathoverflow.net/questions/260284 | 3 | Let $n\ge 2r$ be positive integers.
Is there a closed form for following finite summation involving in q-binomial coefficients
$$\sum\_{s=0}^r(-1)^sq^{\frac{s(s+1)}{2}}{n-2r+s\brack n-2r}\_q{n\brack r-s}\_q\,\,\,\,\,\, ?$$
I found this while studying q-Fibonacci/ Lucas polynomials.
What is the general approach ... | https://mathoverflow.net/users/54507 | What is the value of this sum involving q-binomials? | Doron Zeilberger has written a Maple code for checking and proving [ordinary binomial identities](http://www.math.rutgers.edu/~zeilberg/tokhniot/EKHAD) and their $q$-analogues. What you need in the present case is the package called [qEKHAD](http://www.math.rutgers.edu/~zeilberg/tokhniot/qEKHAD).
I just tested your s... | 6 | https://mathoverflow.net/users/66131 | 260288 | 117,359 |
https://mathoverflow.net/questions/260295 | 3 | The question is motivated by a simple example: the area of a ring is $\pi(R^2-r^2)$, where $R$ and $r$ are the radii of the outer and inner circles respectively. Let $C$ be the 'middle circle' with radius $(R+r)/2$. The area can also be written as $\int\_C (R-r) d\mathscr{H}$, where $\mathscr{H}$ is the 1-dim Hausdorff... | https://mathoverflow.net/users/103887 | volume of region between two manifolds | If $C\_1$ and $C\_2$ are the boundary components of an $\epsilon$-neighborhood of $C$, then the volume of the region is $2\epsilon$ times the volume of $C$. This is a special case of the Weyl tube formula, see e.g. the book "Tubes" by Alfred Gray.
| 3 | https://mathoverflow.net/users/98590 | 260300 | 117,361 |
https://mathoverflow.net/questions/260096 | 3 | The definition of $M\_n^\sharp$ in [OIMT10] is the unique sound, $(\omega,\omega\_1,\omega\_1+1)$-iterable mouse which is not $n$-small, but all of whose proper initial segments are $n$-small.
What is the general definition of $M\_n^\sharp(X)$ for an arbitrary set $X$? For reals $x$ it seems to be identical to the de... | https://mathoverflow.net/users/38602 | Definition of $M_n^\sharp(X)$ for arbitrary set $X$ | See following article by Foreman-Magidor-Schindler:
>
> [The consistency strength of successive cardinals with the tree property](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/div-classtitlethe-consistency-strength-of-successive-cardinals-with-the-tree-propertydiv/C7A69BE2768837CC7C8B35A... | 3 | https://mathoverflow.net/users/11115 | 260301 | 117,362 |
https://mathoverflow.net/questions/260298 | 13 | Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function holomorphic in the right half plane, bounded in the right half plane, that interpolates $\lambda^{z} \Big{|}\_{\mathbb{N}}$. It i... | https://mathoverflow.net/users/nan | Is the exponential function the sole solution to these equations? | The answer is yes. Let me address the last stated form of the question.
Functions $F$ with $F^{(n)}\geq 0$ are called totally monotone, and S. Bernstein
proved that all such functions are Laplace transforms of positive measures,
that is
$$F(x)=\int e^{xt}d\mu(t).$$
Your functional equation $F(x+1)=eF(x)$ then implies
... | 18 | https://mathoverflow.net/users/25510 | 260304 | 117,363 |
https://mathoverflow.net/questions/260275 | 2 | Let $\mathcal{L}=\{0,S,+,\cdot,=,<,X,R,S\}$ be the language of arithmetic with three additional predicate symbols $X(v)$, $R(v,u)$ and $S(v,u)$.
Let $\phi(x),\psi(x,y)$ and $\eta(x,y)$ be formulas in $\mathcal{L}$. Define the formula $\Theta\_e(\phi,\psi,\eta)$ in $\mathcal{L}$ for some $e\in \mathbb{N}$ which says:
... | https://mathoverflow.net/users/83598 | Bounded Arithmetic and Counting | Q1: Yes. The paper you linked to in the question actually proves the theorem for every pair of natural numbers $p,q$ such that $p$ has a prime factor that does not divide $q$ (in other words, $p$ does not divide any power of $q$).
Q2: No. Even at the best of times, you'd only get the conclusion that the classes are d... | 3 | https://mathoverflow.net/users/12705 | 260309 | 117,366 |
https://mathoverflow.net/questions/260170 | 3 | Let $A$ be a weak amenable Banach algebra and $I$ be a closed (two-sided) ideal of $A$. In general $\frac{A}{I}$ is not weakly amenable. Is there an example of this type of weak amenable Banach algebra?(Can you give me an example of a weak amenable Banach algebra with a non weak amenable quotient?)
| https://mathoverflow.net/users/27066 | Quotient of weak amenable Banach algebras | Let $E$ be a Banach space, let $E\hat{\otimes}E^\*$ be the projective tensor product of $E$ with it's dual $E^\*$. It can be considered as a Banach algebra and $E\hat{\otimes}E^\*$ is weakly amenable for every Banach space $E$ (see "Derivations iterated duals of Banach algebras" by Dales, Ghahramani and Grønbaek ($\sta... | 6 | https://mathoverflow.net/users/75338 | 260310 | 117,367 |
https://mathoverflow.net/questions/260316 | 6 | We have $n$ independent identically distributed random variables $X\_1$, $X\_2$, ..., $X\_N$, $X\_i=j$ with probability $1/k$ for $j=1, 2, ... k$. Let $Y\_j$ be a number of random variables $X\_i$, which are equal to $j$, i.e. $Y\_j=|i:X\_i=j|$.
I'm interested in mathematical expectation of $\min(Y\_1, Y\_2, ..., Y\_k)... | https://mathoverflow.net/users/97131 | Mathematical expectation of minimum of k random variables with fixed sum n | Your formula is correct.
Let $Y^{(n)}\_j:=\#\{i\leq n: X\_i=j\}$. Then, $Y^{(n)}=(Y\_1^{(n)},\dots,Y\_k^{(n)})$ can be viewed as a sum of $k$-dimensional i. i. d. random variables, with terms uniform among $(1,0,\dots,0)$, $(0,1,\dots,0)$, .... Note that $$\mathbb{E}Y^{(n)}=\left(\frac{n}{k},\dots,\frac{n}{k}\right),... | 5 | https://mathoverflow.net/users/56624 | 260323 | 117,371 |
https://mathoverflow.net/questions/260324 | 2 | Let $(\mathcal X,\mathcal D)\to T$ be a surjective holomorphic fibre space of K\"ahler manifolds of pairs such that fibers $(X\_s,D\_s)$ admit Ricci flat metric in bounded geometric sense (conic, Poincare singularities),i.e $Ric(\omega\_s)=[D\_s]$
Then, is there any asymptotic formula for $Ric(\omega\_s)$ and $\omega... | https://mathoverflow.net/users/86428 | Asymptotic formula for Ricci flat metric | On each fiber $(X\_s,D\_s)$ , if you kindly assume $D\_s$ is snc with conic singularities such that $K\_{X\_s}+D\_s$ is trivial, then $Ric(\omega\_s)=[D\_s]$ where $[D\_s]$ is the current of integration of divisor $D\_s$. Hence you need to know the asymptotic formula of current of integration $[D\_s]$. See page 150 , 3... | 4 | https://mathoverflow.net/users/nan | 260331 | 117,374 |
https://mathoverflow.net/questions/260337 | 1 | Let $S\_n$ be the symmetric group over $\{1,2,\ldots,n\}$. How to return elements of length $m$ in $S\_n$ using Sage? I try to find such function in Sage but didn't find one. Thank you very much.
Edit: $S\_n$ is the Coxeter group of type $A$ generated by $s\_1=(12), \ldots, s\_{n-1}=(n-1,n)$. The length of an element... | https://mathoverflow.net/users/11877 | How to return elements of a given length in a symmetric group using Sage? | You can enumerate words of a given length in any Coxeter group using the function `elements_of_length`. Here is an example of using this for a Coxeter group I studied recently:
```
sage: CM = CoxeterMatrix([[1,2,-1,-1,2],[2,1,2,-1,-1],[-1,2,1,2,-1],[-1,-1,2,1,2],[2,-1,-1,2,1]])
sage: G = CoxeterGroup(CM, base_ring=Z... | 2 | https://mathoverflow.net/users/5495 | 260339 | 117,377 |
https://mathoverflow.net/questions/260335 | 4 | It is well known that under the Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ representable functors are self-dual, i.e. fixed by the unit and counit of the adjunction: $$\text{Spec}({\cal O}(\hom(-,x)))\cong \hom(-,x)\qquad\qquad {\cal O}(\text{Spec}(\hom(y,-)))\c... | https://mathoverflow.net/users/7952 | Characterizing Isbell self-dual objects | A quick answer for now, which I might add more to later. Recall that for every adjunction $F \dashv G: C \to D$ there is the notion of "fixed point" of the adjunction which has two faces: either it is an object $c$ of $C$ for which the counit $\epsilon\_c: FGc \to c$ is an isomorphism, or it is an object $d$ of $D$ for... | 5 | https://mathoverflow.net/users/2926 | 260342 | 117,379 |
https://mathoverflow.net/questions/260341 | 12 | I wonder what would be a good/early reference for the fact:
>
> *rational points on the unit sphere (centered at the origin) are dense.*
>
>
>
Stereographic projection (from a rational point in the sphere) provides a bijection between rational points on the sphere and rational points in euclidean space, where ... | https://mathoverflow.net/users/39495 | reference request: rational points on the unit sphere | The earliest reference is surely Diophantus' *Arithmetica*. His "method of adequality" can be used to construct rational points on quadrics that approximate real points arbitrarily well (that is, starting from the existence of a rational point).
This is not of course how Diophantus phrases it, but that is what it co... | 16 | https://mathoverflow.net/users/17907 | 260347 | 117,382 |
https://mathoverflow.net/questions/260329 | 15 | Given topological spaces $X$ and $C$ we call $C$ a **coordinate space** for $X$ to mean that every open set $U \subset X$ is of the form $f^{-1}(V)$ for some open $V \subset C$ and continuous $f \colon X \to C$.
More generally if $\mathscr X$ is a class of spaces we call $C$ a **coordinate space** for $\mathscr X$ t... | https://mathoverflow.net/users/58082 | Does the class of Hausdorff spaces have a shared "Coordinate space"? | First of all, let me point out that $[0,1]$ is not a coordinate space for the class of completely regular spaces.
The definition of completely regular spaces says that for any closed $C \subset X$ and any point $p \notin C$, there is a continuous $[0,1]$-valued function (say $f$) mapping $p$ to $1$ and mapping every ... | 15 | https://mathoverflow.net/users/70618 | 260349 | 117,384 |
https://mathoverflow.net/questions/260103 | 4 | While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.
Let $X$ be a r.v., let $F\_{X}(s)$ be its p.g.f., and let
$$B\_{d}(s)=\sum\_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right... | https://mathoverflow.net/users/37757 | Total progeny of a Galton-Watson branching process - standard textbook question | The answer above is fine, nevertheless I make some hopefully useful supplementary remarks (the first two essentially reformulating Did's answer)
(1) It is well known (see e.g. Feller I, 3rd ed., p.299) that the generating function $r(s)$ of the total progeny in a Galton-Watson process (started with one individual)
w... | 1 | https://mathoverflow.net/users/48831 | 260352 | 117,386 |
https://mathoverflow.net/questions/260357 | 0 | In "HOLOMORPHIC FIBER BUNDLES WITH HYPERBOLIC FIBER 1" by H. L. ROYDEN, it was proved that a holomorphic fiber bundle with simply connected base
and Kobayashi hyperbolic fiber is trivial. Now given a complex Lie group $G$ and a holomorphic $G$-principle fiber bundle $E$ over a simply connected complex manifold, may one... | https://mathoverflow.net/users/102212 | Kobayashi hyperbolic complex Lie groups | A complex Lie group is never Kobayashi hyperbolic, as the exponential map applied to any complex line in the Lie algebra gives a complex line in the group.
| 4 | https://mathoverflow.net/users/13268 | 260359 | 117,388 |
https://mathoverflow.net/questions/260364 | 3 | Reference: H. Blaine Lawson, Spin Geometry, Page 72
**Problem:** Here Remark 10.5 states an internal symmetry in the KR-theory that for any compact space $X$ there are isomorphisms $$KR(X\times S^{0,p}) \cong KR^{-2p}(X\times S^{0,p}) ~~~~~~~~~~~~~(1)$$
for $p=1,2,4$. If $\mathbb R^{r,s} =\mathbb R^r \oplus \mathbb R... | https://mathoverflow.net/users/69190 | Recovering K-theory and KO-theory from KR-theory and Bott Periodicity Theory | Unless I am mistaken, $KU(X) = KR(X \times S^{0,1})$ for $X$ an ordinary space. To see why this is reasonable, unpack $X\times S^{0,1} = X\sqcup X$ with the obvious involution and check that Real vector bundles on $X \sqcup X$ are the same as ordinary (complex) vector bundles on (the first copy of) $X$. Indeed, given a... | 7 | https://mathoverflow.net/users/78 | 260369 | 117,393 |
https://mathoverflow.net/questions/260374 | 1 | Let $q=\prod P\_i^{k\_i}.$
I have learned [here](https://mathoverflow.net/questions/260257/sum-of-sum-substack1aq-a-q1-a1-q11) that $\# \{ 1 \leq a \leq q-1| (a,q)=1, (a+1,q)=1 \}= F(q)=\prod (p\_i-2)pi^{k\_i-1}
= q\prod (1-2/p\_i)$
Here was the idea behind it : For any $p\_i$, there exists $p\_i-2$ admissible remind... | https://mathoverflow.net/users/100898 | What is the general formula for $\# \{ 1 \leq a \leq q-2,| (a,q)=1, (a+2,q)=1 \}$ | When $q=\prod\_ip\_i^{k\_i}$ is odd,
$$\# \{ 1 \leq a \leq q-2\, \vert \, (a,q)=1, (a+2,q)=1 \}
=-1+q\prod\_i\left(1-\frac2{p\_i}\right),$$
because the only number you miss (compared to your former problem) is $q-1$ which satisfies $(q-1,q)=1$ and $(q-1+2,q)=1$.
If $q$ is even, then
$$\# \{ 1 \leq a \leq q-2\, \vert ... | 1 | https://mathoverflow.net/users/66131 | 260382 | 117,397 |
https://mathoverflow.net/questions/260322 | 5 | $\def\L{\mathfrak{L}}\def\Prof{\mathsf{Prof}}$ Recall that Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ allows us to define the functor
$$
\L : \Prof(A,B) \to \Prof(B,A)
$$
($\cal V$ is a cosmos in which $A,B$ are enriched categories) sending $K : A^° \times B \to... | https://mathoverflow.net/users/7952 | The $\mathfrak L$ functor on $\textsf{Prof}$ | This is the right Kan extension of $\hom\_A: A \nrightarrow A$ along $K: A \nrightarrow B$ in the bicategory of profunctors. Which is to say that for every profunctor (aka bimodule) $L: B \nrightarrow A$ there is a natural bijection between morphisms $LK \to \hom\_A$ and morphisms $L \to \mathcal{L}(K)$. Since $\hom\_A... | 3 | https://mathoverflow.net/users/2926 | 260384 | 117,398 |
https://mathoverflow.net/questions/260381 | 8 | I have run some [computations](https://www.wolframalpha.com/input/?i=%5Cfrac%7B(3*5*7*11*19*23)%7D%7B(3%2B5%2B7%2B11%2B19%2B23+%7D+is+rational+number) for some finitely primes to know the nature of the ratio below (the product of the first few primes over the sum of them), specifically if it is an integer for finitely ... | https://mathoverflow.net/users/51189 | How do i show that $\displaystyle\frac{\prod_{k=1}^np_k}{\sum_{k=1}^{n}p_k}$ is an integer for finitely many $n$? | I did experiments, and it looks like the density of those $n$ for which the ratio is an integer approaches a limit, and the limit is approximately $0.2187.$ Of this writing I went up to $1000000$ - (with *Mathematica*); the convergence seems reasonably rapid, however. Note that the density is **not** the same as the de... | 7 | https://mathoverflow.net/users/11142 | 260387 | 117,399 |
https://mathoverflow.net/questions/260392 | 8 | Define the sequence given by the finite sum
$$a\_n:=\sum\_{k=2}^{n+1}\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}.$$
>
> **Questions.**
>
>
> (1) Is $0<a\_n<1$?
>
>
> (2) Does the limit $\lim\_{n\rightarrow\infty}a\_n$
> exist? If yes, what is its value?
>
>
>
| https://mathoverflow.net/users/66131 | does this sum have a limit? | For each specific $k$ we have $$\binom{2k}k\binom{n+1}k\frac{k-1}{2^k\binom{4n}k}\to \binom{2k}k\frac{k-1}{8^k},$$
that suggests that the sum tends to $$\sum\_{k=2}^\infty \binom{2k}k\frac{k-1}{8^k}=1-\frac{1}{\sqrt 2},$$
as $$\sum\_{k= 0}^\infty \binom{2k}k x^k =(1-4x)^{-1/2};\,\\\sum\_{k= 0}^\infty k\binom{2k}k x^k =... | 5 | https://mathoverflow.net/users/4312 | 260399 | 117,403 |
https://mathoverflow.net/questions/260397 | 3 | Let $K$ be a positive integer and $C$ be any $K \times K$ non-singular matrix. For positive real numbers $q\_1, \dots, q\_K$, define
$$\Sigma(q\_1, \dots, q\_K) = CC' + diag(\frac{1}{q\_1}, \dots, \frac{1}{q\_K})$$
and
$$V(q\_1, \dots, q\_K) = I\_k - C'\Sigma^{-1}C.$$
**Question:** Are each of the diagonal entries... | https://mathoverflow.net/users/90626 | Log-convexity of conditional variances | It seems that a stronger claim holds. Let $D$ can be any positive definite matrix (not just a diagonal).
The $i$-th entry of the matrix in the OP is
\begin{equation\*}
v\_{ii}(D) = 1 - e\_i^TC'(CC' + D^{-1})^{-1}Ce\_i.
\end{equation\*}
**Key idea.** Notice that we can write $v\_{ii}(D)$ as the ratio
\begin{equatio... | 2 | https://mathoverflow.net/users/8430 | 260402 | 117,404 |
https://mathoverflow.net/questions/260160 | 5 | Consider the following definitions :
$C\subset M$ is convex if any $p,\ q\in C$ *all minimizing
geodesic* between $p$ and $q$ are in $C$
$C$ is totally convex if for $p,\ q\in C$, every geodesic between $p$ and $q$ are in
$C$
$C$ is locally convex if for $p\in C$ there is open set $U$ around $p$ s.t. $U\cap C... | https://mathoverflow.net/users/36572 | Totally convex, convex and locally convex sets | Let $M$ be the cylinder $\{x^2+y^2=1,\ |z| < 2\}$ plus hemispherical caps on both ends.
Let $C$ be the set $\{x < \frac{1}{2},\ x^2+y^2=1,\ |z| < 1\}$. $C$ is locally convex.
Then $C\_1$ is the cylinder $\{x^2+y^2=1,\ |z| < 1\}$, with all shortest geodesics between points in $C$.
And $C\_2$, with all geodesics be... | 3 | https://mathoverflow.net/users/nan | 260409 | 117,407 |
https://mathoverflow.net/questions/260411 | 6 | What information about a (square) matrix we earn if the inner product of its columns are known?
| https://mathoverflow.net/users/40723 | Inner product of columns of a matrix | You have determined $A^\*A$, or, alternatively, you know $A$ up to pre-multiplication by a unitary matrix $U$. So you know the $R$ factor of its QR factorization, and the factors $\Sigma$ and $V$ of its SVD. In particular, among other things, you know singular values and right singular vectors, the sign of its determin... | 16 | https://mathoverflow.net/users/1898 | 260412 | 117,408 |
https://mathoverflow.net/questions/260417 | 3 | Let kQ be the path algebra of a Dynkin quiver over a finite field with $q=p^n$ elements.
Let $f(Q,p^n)$ denote the exponent of the unit group of kQ. Is there an explicit formula for $f(Q,q)$?
| https://mathoverflow.net/users/61949 | Exponent of unit group of Dynkin quiver algebras | If $Q$ is an acyclic quiver, then the exponent of the unit group of $kQ$ is $(q-1)p^m$ where $m$ is chosen smallest so that $p^m$ is greater than the length of the longest path in $Q$.
Here is a proof. A unit of $kQ$ is of the form
$$u=\sum\_{v\in Q\_0}a\_v\epsilon\_v+r$$
where $Q\_0$ is the vertex set, $\epsilon\_... | 3 | https://mathoverflow.net/users/15934 | 260419 | 117,409 |
https://mathoverflow.net/questions/260407 | 3 | Let $F$ be a (totally real) number field, and $E$ a (totally imaginary) quadratic extension of $F$. We consider $U$ a unitary group (with respect to a given hermitian form over $E$). The question is:
>
> When is $U(F) \backslash U(\mathbf{A}\_F)$ compact?
>
>
>
I found some interesting reflexions about that in... | https://mathoverflow.net/users/43737 | Compactness of the automorphic quotient | There are actually several different conditions all of which are equivalent, and one can make various formulations of these conditions depending on whether one sticks to the reductive case and/or global fields of characteristic 0 vs. general global fields. So it may be confusing to sort out exactly what is equivalent t... | 9 | https://mathoverflow.net/users/81332 | 260426 | 117,410 |
https://mathoverflow.net/questions/260405 | 1 | Assume $A$ is a complex $\*$-algebra which is also a Baer\*-ring.
Q. Can we concluded that there exists a Hilbert space $H$ such that $A$ is embedded in $B(H)$ as a Baer\*-ring? What about when $A$ is finite dimensional?
| https://mathoverflow.net/users/84390 | A Baer *-ring which is not embedded into $B(H)$ | No. Begin with a faithful non-atomic probability measure on a space, $X,\mu$ (e.g., $[0,1]$ with Lebesgue measure). Then $L^{\infty}(X,\mu)$ is of course a vN algebra. Form $M(X,\mu)$, the algebra of measurable functions modulo zero ae. As is well-known and easy to verify, $M \equiv M(X,\mu)$ is the classical ring of q... | 4 | https://mathoverflow.net/users/42278 | 260430 | 117,412 |
https://mathoverflow.net/questions/260273 | 5 | Belyi's theorem says that if $X$ is a curve defined over a number field $K$ then $X\_{\overline{K}}$ admits a map to $\mathbb{P}^1\_{\overline{K}}$ which at most 3 branch points. Must there also exist a map to $\mathbb{P}^1\_K$ defined over $K$?
This may be a slightly lazy question, but I figure it's probably got a q... | https://mathoverflow.net/users/88840 | Does there exist a curve defined over a number field $K$ such that every map to $\mathbb{P}^1_K$ is ramified over at least 4 points? | To record nfdc23's answer:
The following is true:
If $X$ is a curve over a number field $K$, then it admits a map to $\mathbb{P}^1\_K$ defined over $K$ branched over at most 3 points.
This follows from Theorem 2.5 in Mochizuki's 2004 paper [here](http://www.kurims.kyoto-u.ac.jp/~motizuki/Noncritical%20Belyi%20Map... | 2 | https://mathoverflow.net/users/88840 | 260439 | 117,413 |
https://mathoverflow.net/questions/260423 | 5 | Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be an analytic function such that its restriction to any arbitrary analytic curve $\gamma$ passing through the origin $0\in \mathbb{R}^n$ attains a local minimum in $0$ and let say that $f(0)=0$.
Is $0$ a point of local minimum for $f$? If the answer is positive, does the sam... | https://mathoverflow.net/users/104001 | Does minimum of an analytic map restricted to analytic curves implies minimum? | As to the generalization to *infinite dimensional Hilbert spaces*, the lack of compactness makes the property fail for not even too delicate reasons. Consider the cubic functional $$f(x):=\sum\_{n\ge1} {1\over n}x\_n^2 - \sum\_{n\ge1} {x\_n^3}\ ,$$ on the Hilbert space $\ell\_2$. For any $C^2$ curve $\gamma:(-1,1)\to \... | 5 | https://mathoverflow.net/users/6101 | 260440 | 117,414 |
https://mathoverflow.net/questions/260450 | 5 | Denote the e.g.f. for the number of (unordered) rooted labeled trees on $n$ nodes by
$$\Phi(x)=\sum\_{n\geq1}\frac{n^{n-1}}{n!}x^n.$$
And, the related series $\Psi(x)=\sum\_{n\geq1}\frac{n^n}{n!}x^n$. Designate the operator $D:=x\frac{d}{dx}$. Some properties:
(1) $\Psi=D\Phi$;
(2) both $\Phi$ and $\Psi$ have radiu... | https://mathoverflow.net/users/66131 | an algebra generated by some known series | This has been considered by Dimitri Zvonkine, see his article "An algebra of power series arising in the
intersection theory of moduli spaces of curves
and in the enumeration of ramified coverings
of the sphere" (<https://arxiv.org/pdf/math/0403092.pdf>).
| 5 | https://mathoverflow.net/users/10881 | 260451 | 117,418 |
https://mathoverflow.net/questions/122072 | 14 | Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}\_p$).
Then there is a sheaf $\mathscr{K}\_m^M$ on $X$ (in the Zariski topology), for each $m \ge 1$, which comes from sheafif... | https://mathoverflow.net/users/2481 | Motivic cohomology and cohomology of Milnor K-theory sheaf | The previous answer contained a major error/misconception, and I apologize for the dealy in correcting it. The answer to the question is "yes" locally in the Zariski topology but "no" globally.
**Comparison of Milnor K-cohomology and motivic cohomology via edge maps:**
Motivic cohomology has Zariski descent and henc... | 10 | https://mathoverflow.net/users/50846 | 260452 | 117,419 |
https://mathoverflow.net/questions/260420 | 3 | General Statement
-----------------
Suppose we have a sequence of identically distributed but dependent random variables $(X\_n)\_{n\in \mathbb{N}}$ which take values on $\{0,\dots,m\}$ for some $m \in \mathbb{N}$ (suppose for all $n$, $X\_n \sim X$). Assume further that the $(X\_n)\_n$ are such that the correlation ... | https://mathoverflow.net/users/83088 | Show that $\mbox{Var}(\sum_{k=0}^{\infty} \delta\{L_{t-k} > k\}) \leq \mbox{Var}(L)$ | $\newcommand{\E}{\mathbb{E}}$
You practically answered your own question.
Denote by $A\_k$ the event "$X\_0 > k$". Then $X\_0=\sum\_{k=0}^{m-1} \delta(A\_k)$.
Denote by $B\_k$ the event "$X\_k > k$". Then $Y\_0=\sum\_{k=0}^{m-1} \delta(B\_k)$.
$A\_k$ and $B\_k$ have the same probabilities, hence $\E[X\_0]=\E[Y\_0... | 3 | https://mathoverflow.net/users/1061 | 260456 | 117,420 |
https://mathoverflow.net/questions/260454 | 5 | Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.
What is the appropriate quotient metric on the homogeneous space and what are the geodesics? Clearly, for example, $K=U(n-1)$ gives $\mathbb{C}P^{n-1}$ and the Fubini-Study geodesics which can be expr... | https://mathoverflow.net/users/41654 | Geodesics on Homogeneous Spaces of $SU(n)$ | I believe the answers to your and related questions are covered in Ch.X of
[1] Foundations of Differential Geometry, vol. II, Kobayashi, Nomizu
More specifically, let $\mathfrak g:=Lie(SU(n))$ and $\mathfrak k:=Lie(K)$ be Lie algebras of the corresponding groups. Since $SU(n)$ is a compact simple Lie group there i... | 5 | https://mathoverflow.net/users/40950 | 260459 | 117,422 |
https://mathoverflow.net/questions/260460 | 5 | I'm trying to find a simple proof that a monoid under Day convolution is equivalent to a lax monoidal functor (see [nCatLab](https://ncatlab.org/nlab/show/Day+convolution)). For simplicity, consider functors from $C$ to $Set$. For $F$ to be a monoid there has to be a natural transformation:
$ \mu : F \otimes F \to F ... | https://mathoverflow.net/users/34546 | From a coend over a pair to a coend over the tensor product | This can be seen via a sequence of isomorphisms involving ends and co-ends, using the presentation of natural transformations via ends:
$$
\begin{array}{cl}
& \int\_{xy} Fx \times Fy \to F (x \otimes y) \\
\cong& \int\_{xy} Fx \times F y \to (\int\_z C(x \otimes y,z) \to Fz) & (1) \\
\cong& \int\_{xyz} Fx \times Fy ... | 8 | https://mathoverflow.net/users/104012 | 260462 | 117,424 |
https://mathoverflow.net/questions/260469 | 7 | What is the simplest way to prove that each finite game is also determined? I know that a game is said to be determined if one of the players has a winning strategy. I was hoping to prove by contradiction that assuming in a finite game Player 1 does not have a winning strategy it needs to be the case that Player 2 has ... | https://mathoverflow.net/users/103143 | Determined, finite games | There are numerous proofs of what I call the fundamental theorem of finite
games.
**Theorem.** (Fundamental theorem of finite games)
In any finite two-player game of perfect information, one of the players has a winning strategy.
**Proof 1.** Back-propagation through the game tree. Label the
nodes with the player ... | 14 | https://mathoverflow.net/users/1946 | 260471 | 117,426 |
https://mathoverflow.net/questions/260470 | -3 | Some of my computations [here](https://www.wolframalpha.com/input/?i=zeta(s)(zeta(1-s)),+s%3D0.5%2B500i) showed to me that the imaginary part of $\displaystyle\ \zeta(s)\zeta(1-s)=0$ for $\operatorname{Re}(s)=\frac{1}{2}$, really i w'd like to know if there is any paper showed this result that is true for all s with a ... | https://mathoverflow.net/users/51189 | Is the imaginary part of $\displaystyle\ \zeta(s)\zeta(1-s)=0$ for $\operatorname{Re}(s)=\frac{1}{2}$ | $\overline{\zeta(s)\zeta(1-s)} = \zeta(\overline{s})\zeta(\overline{1-s}) = \zeta(1-s)\zeta(s)$
($1-s=\overline{s}$ for $Re(s)=1/2$)
So this is indeed a purely real number. $\zeta(\overline{s}) = \overline{\zeta(s)}$ holds because $\overline{\zeta(\overline{s})}-\zeta(s)$ is analytic and zero for real $s>1$, hence ... | 10 | https://mathoverflow.net/users/43383 | 260472 | 117,427 |
https://mathoverflow.net/questions/260385 | 0 | Let $M$ be a $n \times n$ matrix over the finite field of two elements that satisfies the following property$\colon$ the total number of 1's in each row coincides with one in each column. In other words, there is a number $N$ such that
1. The number of non-zero elements in each row of $M$ is $N$.
2. The number of non... | https://mathoverflow.net/users/103978 | Find the minimum distance of some bad binary code | Unfortunately, the nice property that every distinct pair of rows yields another row by mod 2 vector addition has been withdrawn. However, it may be possible to regard M as having maximal subsets of rows where this property holds, and still say something useful about M.
The comment above regarding Hadamard matrices g... | 0 | https://mathoverflow.net/users/3402 | 260484 | 117,433 |
https://mathoverflow.net/questions/260432 | 8 | For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^\*$ via $x \mapsto \langle x, -\rangle$.
It is often stated in the literature that this does not work in full generality for a Hilbert module over a $C^\*$ algebra. For example, attempts to define the adjoint of a m... | https://mathoverflow.net/users/86526 | Counterexample to Riesz representation for Hilbert modules | Take $A= \mathcal{C}([0,1])$ and $H$ the ideal of $A$ of functions that vanish at $0$.
$H$ is a Hilbert $A$ module (as any ideal, with the natural multiplication of $A$ and the scalar product $(x,y)=x^\*y$ of $xy^\*$ depending on if you are talking of right or left modules) the inclusion of $H$ into $A$ is a continuo... | 5 | https://mathoverflow.net/users/22131 | 260496 | 117,437 |
https://mathoverflow.net/questions/260506 | 2 | in DaPrato/Zabczyk's book "Second Order Partial Differential Equations in Hilbert Spaces", there is a useful proposition (Prop. 1.2.8) about a particular calculation of a Gaussian integral in Hilbert spaces: Take a symmetric operator $M$ in a Hilbert space, a centred Gaussian measure $N\_Q$ on $H$ with covariance opera... | https://mathoverflow.net/users/88505 | complex version of Gaussian integral | $$\int\_H \exp\left\{\frac{1}{2}\langle My,y\rangle + \langle b\_1 + ib\_2, y\rangle\right\}N\_Q(dy)=$$
$$= \frac{\exp\left\{\frac{1}{2}\langle b\_1+ib\_2,Q^{1/2}(1-Q^{1/2}MQ^{1/2})^{-1}Q^{1/2}(b\_1+ib\_2)\rangle\right\}}{\sqrt{\det (1-Q^\frac{1}{2}MQ^\frac{1}{2})}}$$
the inner product is defined here without complex... | 3 | https://mathoverflow.net/users/11260 | 260508 | 117,442 |
https://mathoverflow.net/questions/260478 | 3 | In Bill Casselman's notes on root systems (<http://www.math.ubc.ca/~cass/courses/tata-07a/Roots.pdf>), I am confused about the proof of the result $s\_{\alpha}(\beta)^{\vee} = s\_{\alpha^{\vee}}(\beta^{\vee})$ for any roots $\alpha,\beta$ (Corollary 3.10).
In these notes a root system is a quadruple $(V,\Sigma, V^{\... | https://mathoverflow.net/users/38145 | In a non-reduced root system, $s_{\alpha}(\beta)^{\vee} = s_{\alpha^{\vee}}(\beta^{\vee})$ | Proof of uniqueness in Method 1: it suffices to show that $\langle \beta, l - \alpha^{\vee} \rangle = 0$ for all roots $\beta$. Write
$$(\beta + \alpha\mathbb{Z}) \cap \Sigma = \{ \beta + n\_i \alpha, i = 1, ... , r \}$$
this is nonempty.
Then
$$0 = \sum\limits\_{i=1}^r \langle \beta + n\_i \alpha, l \rangle =... | 2 | https://mathoverflow.net/users/38145 | 260521 | 117,446 |
https://mathoverflow.net/questions/260519 | 2 | Let $S$ and $C$ denote the Fourier sine transform and the Fourier cosine transform, respectively, i.e.,
\begin{align\*}
S f(k) &= \sqrt{\frac2\pi} \, \int\_0^\infty f(x) \sin(kx)\,dx, \\
C f(k) &= \sqrt{\frac2\pi} \, \int\_0^\infty f(x) \cos(kx)\,dx.
\end{align\*}
Recall that $S$ and $C$ are unitary selfadjoint opera... | https://mathoverflow.net/users/69194 | Fourier sine and cosine transforms and Laguerre polynomials | Here is a derivation starting from the Fourier transforms given in [Orthogonal polynomials on the unit circle associated with the Laguerre polynomials.](http://www.ams.org/journals/proc/2001-129-03/S0002-9939-00-05821-4/S0002-9939-00-05821-4.pdf) The Fourier cosine and sine transforms of $\phi\_m$ are equal to the real... | 4 | https://mathoverflow.net/users/11260 | 260527 | 117,448 |
https://mathoverflow.net/questions/260536 | 1 | I'm really having a hard time finding a book that covers distributional equations such as $xT = \delta(x)$ or distributional differential equations. Any suggestions?
| https://mathoverflow.net/users/84693 | Distributional equations: reference request | This is the problem of division of distributions. The general existence theorem is the Malgrange-Ehrenpreis Theorem. The first two volumes of Hörmander's "The Analysis of Linear Partial Differential Operators" is a canonical reference. See also [this book](http://link.springer.com/book/10.1007/978-3-319-20140-5) by Ort... | 3 | https://mathoverflow.net/users/7410 | 260537 | 117,449 |
https://mathoverflow.net/questions/260529 | 1 | Let $\mathscr{A}$ be an abelian category, and $S\subset\mathscr{A}$ some set of objects in $\mathscr{A}$ (i.e. $S$ is not a proper class), so we can fairly easily define the category $\mathscr{C}\_S$ to be the smallest full subcategory of $\mathscr{A}$ containing $S$ such that $\mathscr{C}\_S$ is closed under subobject... | https://mathoverflow.net/users/94022 | Is the minimal Serre subcategory containing a set also a set? | Not in general, even if by "small" you mean "essentially small" (i.e., there is a small set of isomorphism classes of objects). First of all, it is possible for a single object to have a proper class of subobjects (if this does not happen, the category $\mathscr{A}$ is called *well-powered*). I don't know an explicit e... | 4 | https://mathoverflow.net/users/75 | 260539 | 117,451 |
https://mathoverflow.net/questions/260534 | 4 | Let $K$ be a field and $\alpha\_i$ ($i=1,\dots,n$) be Galois conjugates. Let $L=K(\alpha\_1,\dots,\alpha\_n)$ and $G=Gal(L/K)$. We embed $G$ in $S\_n$ by its action on $\alpha\_i$. Let $H$ be another subgroup of $S\_n$, acting on the variables $X\_i$ ($i=1,\dots,n$). Let $R=K[X\_1,\dots,X\_n]^H$ be the ring of invarian... | https://mathoverflow.net/users/37103 | Galois group and invariant polynomials | **Lemma 1:** Let $H$ be a subgroup of $\mathfrak{S}\_n$, and $K$ a field. If $(\xi\_1,\ldots,\xi\_n)$ and $(\xi'\_1,\ldots,\xi'\_n)$ are two $n$-tuples of elements of $K$, then t.f.a.e.:
1. there exists $\sigma\in H$ such that $\xi'\_i = \xi\_{\sigma(i)}$ for all $i$,
2. for every $f \in K[X\_1,\ldots,X\_n]^H$ we hav... | 6 | https://mathoverflow.net/users/17064 | 260543 | 117,454 |
https://mathoverflow.net/questions/260479 | 3 | **Background**
Given a category $C$, one can consider the category $Fam(C)$ of set-indexed familiies of objects in $C$. Formally, the objects are pairs $(X,F)$ in which $X$ is a set and $F:X\rightarrow C$ is a functor and a map $(X,F)\rightarrow (Y,G)$ is pair $(f,g)$ where $f:X\rightarrow Y$ is a function and $g:F\R... | https://mathoverflow.net/users/102957 | The "Family" functor for infinity-categories | It's apparent that $Fam$ is functorial, right?
So you start with an $\infty$-locus $C$ and some localization $f:C\to C'$ thereof. You want to show that $Fam(C')$ is an $\infty$-topos; in particular, maybe it's a localization of $Fam(C)$ in some natural way (indeed, implicit in your question is the idea that $Fam(f)$ ... | 1 | https://mathoverflow.net/users/28033 | 260551 | 117,457 |
https://mathoverflow.net/questions/260550 | 1 | [In mathematics](https://en.wikipedia.org/wiki/Inverse_function), an inverse function is a function that "reverses" another function: if the function $f$ applied to an input $x$ gives a result of $y$, then applying its inverse function $g$ to $y$ gives the result $x$, and vice versa. i.e., $f(x) = y$ if and only if $g(... | https://mathoverflow.net/users/51189 | When does $f^{-1}=\frac{1}{f}$ with $f$ a function mapping $\mathbb{R}^{*}$ to $\mathbb{R}$? | You can't have $f^{-1}(x) = 1/f(x)$ with both sides defined for all $x \in \mathbb R$. Namely, if $t = f^{-1}(0)$ then $f^{-1}(t) = 1/f(t) = 1/0$ is undefined.
EDIT:
With the correction that $f$ maps $\mathbb R^\* = \mathbb R \backslash \{0\}$ to itself, here is one class of solutions. Take any $f\_0$ that maps $(0,1... | 10 | https://mathoverflow.net/users/13650 | 260553 | 117,459 |
https://mathoverflow.net/questions/260549 | 3 | Let $f(x)=erfi(a+x)$ and $g(x)=e^{cx}$ with
\begin{align\*}
f^{(n)}(x)=\frac{2}{\sqrt{\pi}}e^{(a+x)^2}\sum\_{m=0}^{n-1}\sum\_{j=0}^{m}\frac{\binom{m}{j}(-1)^j(a+x)^{2m-n+1}}{m!}\\
\prod\_{p=1}^{n-1}(2m-2j-p+1).
\end{align\*}
and
\begin{align\*}
g^{(n)}(x)=c^n e^{cx}
\end{align\*}
I want to find an expression for $ (f(x... | https://mathoverflow.net/users/103013 | nth derivative of error function | To avoid "extended discussion", here is what I said.
\begin{align} (f(x).g(x))^{(n)} &=\sum\_{k=0}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)\\
&=f(x)g^{(n)}+\sum\_{k=1}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x) \\
&=\frac2{\sqrt{\pi}}c^ne^{cx}\int\_0^{a+x}e^{t^2}dt
+\sum\_{k=1}^{n}\binom{n}{k}f^{(k)}(x)g^{(n-k)}(x).
\end{align... | 3 | https://mathoverflow.net/users/66131 | 260555 | 117,460 |
https://mathoverflow.net/questions/260566 | 0 | Suppose $\ell\_1=E\oplus F$ where $E$ and $F$ are linear subspaces with $E\cap F=\{0\}$, and $||x|| = ||x\_1||+||x\_2||$ for all $x\in \ell\_1$ with $x=x\_1+x\_2$ with $x\_1\in E$ and $x\_2\in F$. Must $E$ and $F$ be norm closed?
| https://mathoverflow.net/users/20300 | $L$-decompositions of $\ell_1$ | I think that "yes", because the projections to these subspaces are continuous, and the range of a continuous projection in a Banach space is closed.
| 4 | https://mathoverflow.net/users/85406 | 260568 | 117,461 |
https://mathoverflow.net/questions/260535 | 1 | I would like to know if there is any known result on dispersive estimates for Schrodinger operator with magnetic potential in one dimension.
There is a lot of literature for three dimensional magnetic operators, but i was not able to find anything in dimension one.
| https://mathoverflow.net/users/54552 | Dispersive estimates for one dimensional magnetic Schrodinger operators | In 1D, a change of variables is sufficient to transform a first order perturbation (i.e., a magnetic potential) into a 0th order perturbation (i.e. an electric potential). Thus you can apply the results concerning the electric case; there are several. Actually this trick works even for the coefficient of the second ord... | 2 | https://mathoverflow.net/users/7294 | 260572 | 117,463 |
https://mathoverflow.net/questions/228258 | 8 | Recall the Hales-Jewett Theorem:
>
> **HJT**: Given a finite alphabet $A$ and some $r \in \mathbb{N}$, there is some $H \in \mathbb{N}$ such that whenever $A^H$, the set of all length-$H$ words from $A$, is $r$-colored, there is a monochromatic combinatorial line.
>
>
>
[A *combinatorial line* is defined as fo... | https://mathoverflow.net/users/70618 | The Hales-Jewett Theorem for an infinite alphabet | I returned to this problem recently, and was finally able to answer my own question (though not in the way I'd hoped):
>
> **Theorem:** The principle HJT($\kappa$) is false for every infinite cardinal $\kappa$.
>
>
>
As a consolation prize, we can at least get:
>
> **Theorem:** Suppose $A$ is countable and... | 2 | https://mathoverflow.net/users/70618 | 260599 | 117,471 |
https://mathoverflow.net/questions/260563 | 5 | I know Law has a tensor product, is closed with respect to that tensor product, and it has coproducts. Does it have products?
My best guess at the cartesian product of Lawvere theories is the "intersection" of the theories:
say $Th\_1$ has a sort $X,$ function symbols $f\_i\colon X^{n\_i} \to X$ and a set of equation... | https://mathoverflow.net/users/756 | Does the category of Lawvere theories have products? | According to
Fajtlowicz, S.
Birkhoff's theorem in the category of non-indexed algebras.
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 1969 273-275.
a product of algebras and varieties was introduced by W. Narkiewicz. The terminology ``nonindexed product'' was used. It is stated in the above paper tha... | 4 | https://mathoverflow.net/users/75735 | 260600 | 117,472 |
https://mathoverflow.net/questions/260556 | 8 |
>
> What is the current state of knowledge of the group homology of $SL\_2(k[t,t^{-1}])$?
>
>
>
I am mostly interested in the case $k$ is algebraically closed of characteristic zero. The most recent work I am aware of is these two papers of Knudson from 1996-7:
<http://www.ams.org/mathscinet-getitem?mr=1375567... | https://mathoverflow.net/users/35353 | State of the art knowledge about homology of $SL_2(k[t,t^{-1}])$ | There are two papers that could be interesting to you. First, there is a paper of Kevin Hutchinson:
* K. Hutchinson. On the low-dimensional homology of ${\rm SL}\_2(k[t,t^{-1}])$.
J. Algebra 425 (2015), 324–366.
He uses the amalgamation sequence from Knudson's paper and computes the boundary maps to show that secon... | 8 | https://mathoverflow.net/users/50846 | 260632 | 117,481 |
https://mathoverflow.net/questions/259834 | 26 | I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:
1. Some multilinear algebra including the Kernel Theorem and Volterra composition,
2. Some Fourier analysis including the Bochner-Schwartz Theorem,
3. An introduction to wavelets with... | https://mathoverflow.net/users/7410 | Nice applications for Schwartz distributions | Unsurprisingly, the topics that occur to me have various connections to number theory (and related harmonic analysis) (and unclear to me what might have already been done in your course...):
EDIT: inserted some links... EDIT-EDIT: one more...
* Genuinely distributional proof of Poisson summation: <http://www.math.u... | 27 | https://mathoverflow.net/users/15629 | 260634 | 117,482 |
https://mathoverflow.net/questions/260554 | 5 | I'm trying to write a program (with javascript or python) that samples a random lozenge tiling of a hexagon with Propp - Wilson's coupling from the past algorithm. I'm quite clear of the framework of the algorithm, but I don't know which is the most efficient way to encode a tiling, or whether different encoding method... | https://mathoverflow.net/users/70248 | implementing Propp-Wilson's Coupling From the Past on Lozenge Tilings of a Hexagon | You should represent it as a plane partition. But you shouldn't represent a plane partition as a 3D array of 0/1 for absent/present. Most people think of a partition as a set of natural numbers, not a Ferrers diagram. The Ferrers diagram has more symmetry, but usually the set (or decreasing sequence) of numbers is the ... | 3 | https://mathoverflow.net/users/4639 | 260636 | 117,483 |
https://mathoverflow.net/questions/260633 | 0 | Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\subseteq y$, and
4) $\forall x,y\in P,$ $x\cap y \neq \varnothing$.
For each $x\in P$, let $1\_x\in \mathbb R\_+^N$ be... | https://mathoverflow.net/users/104107 | Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another? | **Counterexample.**
$N=\{0,1,2,3,4,5,6\}$
$P=\{u,v,w,y,z\}$
$u=\{0,1,3,5\},\ v=\{0,2,4,6\},\ x=\{0,1,2\},\ y=\{0,3,4\},\ z=\{0,5,6\}$
$\alpha=\frac12(1\_u+1\_v),\ \beta=\frac13(1\_x+1\_y+1\_z)$
| 0 | https://mathoverflow.net/users/43266 | 260637 | 117,484 |
https://mathoverflow.net/questions/260638 | 5 | I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. I am not quite sure whether this is research-level, but let me state some context first:
An old result of functional analysis tells us that a symmetric (in the sense that $(Ax,y)=(x,Ay)$, for a... | https://mathoverflow.net/users/85967 | On the domains and extensions of unbounded operators | Yes, you've got it right. Given an unbounded self-adjoint operator $A$ with domain $D(A) \subset H$, using Zorn's lemma you can produce an everywhere defined operator $A'$ on $H$ which extends $A$. (In fact you can produce many such operators; the extension is highly non-unique.) By Hellinger–Toeplitz, $A'$ cannot be s... | 7 | https://mathoverflow.net/users/4832 | 260640 | 117,486 |
https://mathoverflow.net/questions/260619 | 17 | Suppose one has a smooth non-degenerate curve $\gamma: [0,1] \to {\bf R}^n$ into Euclidean space (thus $\gamma'$ never vanishes), with the property that the velocity $\gamma'(t)$ and acceleration $\gamma''(t)$ are always parallel, thus
$$ \mathrm{dim}\ \mathrm{span}( \gamma'(t), \gamma''(t) ) \leq 1$$
for all $t \in [0... | https://mathoverflow.net/users/766 | A variant of the Monge-Cayley-Salmon theorem? | Setting aside the assumption that $\phi$ be a polynomial mapping for the moment (however, see below for a construction of a large family of polynomial solutions), if one makes the 'nondegeneracy' assumptions
1. $\mathrm{dim}\ \mathrm{span}\bigl( \phi\_s(s,t), \phi\_t(s,t)\bigr) =2 $,
2. $\mathrm{dim}\ \mathrm{span}\b... | 14 | https://mathoverflow.net/users/13972 | 260641 | 117,487 |
https://mathoverflow.net/questions/260624 | 4 | Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$.
Is the following estimate true, and, if so, how can it be proved?
\[\left|\sum\_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^{o(1)},\]
where $f(x)=x^{o(1)}$ means $\lim\_{x \rightarrow \infty}\frac{\log f(x)}{\log x}=0... | https://mathoverflow.net/users/98416 | An Exponential Sum Restricted to Primes | No, this is definitely not true. First of all, there's no way we should expect better than square-root cancellation in an exponential sum without an incredible amount of structure, which the primes do not possess. Second, it's known to be false for certain rational numbers, and conditionally known to be false for all r... | 6 | https://mathoverflow.net/users/5091 | 260656 | 117,494 |
https://mathoverflow.net/questions/260518 | -1 | Let $A$, $B$ and $C$ be three random variables and $p\_{A,B,C}=p\_Ap\_Bp\_{C|A,B}$ and $q\_{A,B,C}=p\_Aq\_{B|A}p\_{C|A,B}$ be two distributions on them. Then, we can conclude that?
\begin{align\*}
\lvert I\_p(A,B;C)-I\_q(A,B;C)\rvert\leq\log\lvert\mathcal{B}\rvert.
\end{align\*}
where $\mathcal{B}$ is the alphabet set ... | https://mathoverflow.net/users/68835 | Bounding difference of two mutual information with different distributions on random variables | The set-up is the following.
>
> We have random variables $A,B,C$ and $A', B', C'$. We know $A$ and $A'$ have the same distribution, and for all $(a,b)$, the random variable $C | (A,B)=(a,b)$ has the same distribution as $C' | (A', B') = (a,b)$.
>
>
>
Under the above hypothesis, the claim you want is in fact f... | 1 | https://mathoverflow.net/users/22512 | 260688 | 117,506 |
https://mathoverflow.net/questions/260702 | 3 | Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a\_\min,a\_\max]$, with $a\_\min$, $a\_\max>0$.
>
> **Question:** Are there any upper bounds on the condition number of the eigenvector matrix associated to $A$ as $n$ increases?
>
>
>
The motivation of this question comes from the $n\tim... | https://mathoverflow.net/users/62673 | Upper bounds on the condition number of the eigenvector matrix | **No, there are no such bounds.**
Let $a>0$ be a fixed real and $\varepsilon\ll a$. Take the non-diagonalizable matrix $A=\begin{bmatrix}3a & 0 & 0\\ 0 & 0 & \varepsilon \\ 0 & 0 & 0 \end{bmatrix}$. Let $V$ be a symmetric, orthogonal matrix that maps $\begin{bmatrix}\sqrt {3}\\0\\0\end{bmatrix}$ to $\begin{bmatrix}1\... | 4 | https://mathoverflow.net/users/1898 | 260706 | 117,512 |
https://mathoverflow.net/questions/258112 | 1 | If i take $v\in H^1(\Omega)$ where
$$
H^1(\Omega)=\{u\in L^2(\Omega), \frac{\partial u}{\partial x\_i}\in L^2(\Omega), i=1,\ldots,N\}
$$
$\Omega$ is bounded open set from $\mathbb{R}^N$
What is the sufficient condition for a real function $f:\mathbb{R}\rightarrow \mathbb{R}$ to have that $f\circ v\in H^1(\Omega)$ ... | https://mathoverflow.net/users/49045 | An acting condition for a superposition operator from $H^1(\Omega)$ to $H^1(\Omega)$ | The following is stated in the paper
*Moshe Marcus and Victor J. Mizel*, MR 531975 [**Complete characterization of functions which act, via superposition, on Sobolev spaces**](http://dx.doi.org/10.2307/1998689), *Trans. Amer. Math. Soc.* **251** (1979), 187--218.
as Theorem 1 (note that $T\_f$ is the superposition ... | 2 | https://mathoverflow.net/users/11512 | 260709 | 117,515 |
https://mathoverflow.net/questions/156332 | 13 | *All structures are countable with countable signature.*
Given a structure $\mathcal{A}$, the *age* of $\mathcal{A}$, $Age(\mathcal{A})$, is the set of structures isomorphic to finitely-generated substructures of $\mathcal{A}$ (see <https://en.wikipedia.org/wiki/Age_(model_theory)>). (This isn't really a set, but we ... | https://mathoverflow.net/users/8133 | "Fraïssé limits" without amalgamation | This question has been around for a long time, and the OP himself has known the answer for almost as long! But just in the interest of putting some references out there, I'll give an answer:
Let $\mathbb{K}$ be a class of finite structures. Then $Gen(\mathbb{K})$ contains a unique isomorphism class $M\_{\mathbb{K}}$ ... | 9 | https://mathoverflow.net/users/2126 | 260713 | 117,517 |
https://mathoverflow.net/questions/260696 | 1 | I have checked in [Andrew Odlyzko](http://www.dtc.umn.edu/~odlyZko/zeta_tables/index.html), Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I would like to know if there is an example of zeros as $s=0.5+at... | https://mathoverflow.net/users/51189 | Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer? | As explained by Peter Humphries in the comments, the famous conjecture here is that the imaginary part of the non-trivial zeros of $\zeta(s)$ is transcendental (or equivalently, linearly independent over $\mathbb{Q}$).
I'm not sure of the origins of the conjecture, but it was certainly around before the Rubinstein-Sa... | 5 | https://mathoverflow.net/users/43108 | 260718 | 117,520 |
https://mathoverflow.net/questions/260607 | 14 | Given four random lines in $\mathbb{R}P^3$, how many lines intersect all of those lines?
In the recent paper [Probabilistic Schubert Calculus](http://arxiv.org/abs/1612.06893v1), Peter Bürgisser and Antonio Lerario
discuss this question and much more general versions of it.
In **Proposition 6.7**, they determine the ... | https://mathoverflow.net/users/39495 | Expected number of lines meeting four given lines or "what is 1.72..." | The integrand is periodic modulo $\pi$ in each variable, so it suffices to integrate each variable over $[0, \pi]$ and replace the constant factor by $2^{-7}$.
If we were to apply a change of variables (e.g. set $x = \cos(s\_1)$ and similarly for the other five variables), we would have an integral of a piecewise-alg... | 11 | https://mathoverflow.net/users/39521 | 260719 | 117,521 |
https://mathoverflow.net/questions/260677 | 2 | Let $A$ be a finite dimensional algebra . Let $P\_{\inf}$ be the full subcategory of modules having finite projective dimension and $P\_r$ the subcategory of modules having projective dimension bounded by $r$. Does $P\_{\inf}$ being contravariantly finite imply that $P\_r$ is contravariantly finite for all $r$?
| https://mathoverflow.net/users/61949 | Contravariant finiteness of subcategories | Not in general, although there may be some classes of algebras where it holds. For a simple counterexample $A$ consider the path algebra (over an infinite field) of the quiver $$1 \stackrel{a, b}\Rightarrow 2 \stackrel{c}{\rightarrow} 3$$ modulo the relation $ca=0$. This algebra has global dimension 2, so in particular... | 3 | https://mathoverflow.net/users/11791 | 260721 | 117,522 |
https://mathoverflow.net/questions/260731 | 1 | [A while ago](https://mathoverflow.net/questions/138870/vopenkas-principle-for-non-first-order-logics) I asked about versions of Vopenka's principle for logics other than first order. Unfortunately, there doesn't seem to be much there; most logics yield the same notion, while "big" logics like $\mathcal{L}\_{\infty\ome... | https://mathoverflow.net/users/8133 | Theories with the infinitary Vopenka property | The answer to the question is yes, assuming $VP$ holds and thus, in the terminology [from your earlier question](https://mathoverflow.net/questions/138870/vopenkas-principle-for-non-first-order-logics), that $VP(\mathcal{L}\_{\omega\_1 \omega})$ holds. Namely let $T$ be any unstable theory with countably many countable... | 2 | https://mathoverflow.net/users/26705 | 260733 | 117,525 |
https://mathoverflow.net/questions/260635 | 13 | [nCatLab](https://ncatlab.org/nlab/show/Day+convolution) states that a monoid in the Day-convolution monoidal category is equivalent to a lax monoidal functor. In the Haskell community this result is used to explain the construction of the applicative functor. [Rivas and Jaskelioff](https://arxiv.org/abs/1406.4823) sho... | https://mathoverflow.net/users/34546 | Monoid under Day convolution and lax monoidal functor: Is strength necessary? | You are correct.
The most straightforward structure that you have on $[C, Set]$ is that of a multicategory (<https://ncatlab.org/nlab/show/multicategory>). Given functors $F\_1, F\_2, ..., F\_n$ (n can be 0) and $F$ the set of multimorphisms $(F\_1, F\_2, ..., F\_n) \rightarrow F$ is defined as the set
$$Nat(F\_1(X\... | 2 | https://mathoverflow.net/users/39004 | 260734 | 117,526 |
https://mathoverflow.net/questions/260736 | 4 | A space $X$ is called weakly-Lindelöf if every open cover $\mathcal{U}$ has a countable subcover $\mathcal{U'} \subseteq \mathcal{U}$ such that $\cup \mathcal{U}'$ is dense in $X$.
This class seems to be of interest in functional analysis (spaces of continuous functions). Every Lindelöf space or space with a dense Li... | https://mathoverflow.net/users/2060 | more examples of non-weakly Lindelöf spaces | It is not hard to show that a weakly-Lindelof metric space is separable (basically the same proof used for Lindelof spaces works). Hence $l^\infty$ or your favorite non-separable metric space is an example of a non-weakly-Lindelof.
Another example is $\omega\_1$ with the order topology (which is also first-countable ... | 4 | https://mathoverflow.net/users/17836 | 260738 | 117,528 |
https://mathoverflow.net/questions/260739 | 0 | We consider the infinite product:
$$\frac{1}{c}\prod\_{n \geqslant 1} \frac{c^n}{c^n+1} = \frac{1}{c}\frac{c}{c+1}\frac{c^2}{c^2+1}\frac{c^3}{c^3+1} \cdots$$
For which real value of $c > 1$ has the product its maixmum?
Background of Question: Expected number of equilibria in multi-person games on trees of uniform... | https://mathoverflow.net/users/104154 | Maximise specific infinite product | Optimal $c$ is the unique positive root of $\sum k/(1+c^k)=1$, numerically about 2.43. I doubt that it is expressable via known constants and elementary functions.
| 3 | https://mathoverflow.net/users/4312 | 260742 | 117,529 |
https://mathoverflow.net/questions/260754 | 0 | Let $\mathcal{P}$ denote the class of positive semidefinite symmetric bilinear forms $A : \mathbb{R}^{n} \times \mathbb{R}^n \to \mathbb{R}$. Let
$$
X(\mathcal{P}) := \{(x,y) \in \mathbb{R}^n \times \mathbb{R}^n \colon \, \forall A \in \mathcal{P}, \,\, A(x,y) \ge 0 \}.
$$
Clearly $\Delta := \{ (x, x)\in \mathbb{R}^n\... | https://mathoverflow.net/users/nan | Nonnegative locus of the class of positive semidefinite symmetric bilinear forms | Maybe I am misunderstanding the problem because this solution looks simple.
Essentially, you are asking if for every pair $x,y$ of linearly independent vectors there is a positive definite matrix $A$ such that $x^\*Ay<0$.
Let $x,y$ be two fixed linearly independent vectors. Take an invertible matrix $V$ whose firs... | 1 | https://mathoverflow.net/users/1898 | 260759 | 117,533 |
https://mathoverflow.net/questions/260617 | 4 | Let
$\mathcal{H}=L^2(\mathbb{R}^3)$,
$H\_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian)
and
$H=H\_0+V(\vec{x})$
(where $V(\vec{x})$ is the operator of multiplication by a bounded continuous function $V(\vec{x})$; $H$ is a perturbation of $H\_0$ by a bounded operator).
Co... | https://mathoverflow.net/users/47256 | Scattering of relativistic particle by long-range potential | It seems that long-range scattering for the operator $\sqrt{-\Delta+M^2}$ has not been dealt with in the literature. On the technical side, I would start with Chapter XXX of Hörmander [*The Analysis of Linear Partial Differential Operators IV*](http://www.springer.com/de/book/9783642001178), where long-range scattering... | 2 | https://mathoverflow.net/users/69194 | 260763 | 117,535 |
https://mathoverflow.net/questions/260672 | 7 | *I'm asking this question in the most model-ambiguous way I can since this is the kind of answer i'm looking for.*
There are various explicit constructions of the Whitehead and Postnikov towers. I'm trying to understand what exactly characterizes these construction.
>
> **Postnikov tower:** A Postnikov tower of (... | https://mathoverflow.net/users/22810 | Simple characterization of Postnikov & Whitehead towers? | Indeed, the properties you stated characterize Postnikov and Whitehead towers. A nice conceptual way of justifying this is by using $k$-connected / $k-$truncated factorization systems.
To fix terminology, a $k$-connected map is one whose all homotopy fibers are $k$-connected and a $k$-trunceted map is one whose all h... | 5 | https://mathoverflow.net/users/12547 | 260764 | 117,536 |
https://mathoverflow.net/questions/260722 | 1 | It is well known that if $T$ is a nonsingular transformation of a standard probability space $(X,\mu)$ and there exists an ergodic measure preserving transformation $S$ of $(X,\mu)$ such that $T$ commutes with $S$, then $T$ is measure preserving.
I would like to know of there is an approximate version of this fact. M... | https://mathoverflow.net/users/30721 | Nonsingular transformation commuting with approximately measure preserving transformation | No. I don't think there can be any result of that type. Let $X=\{0,1\}^{\mathbb N}$ and let $S$ be the dyadic odometer (I like to write elements of $X$ so they are infinite on the left; in this case, the map $S$ is just add 1 with carry to the left -- the way it should be!). Equip $X$ with the measure that is the produ... | 0 | https://mathoverflow.net/users/11054 | 260773 | 117,539 |
https://mathoverflow.net/questions/260762 | 4 | I didn't get the argument of Example 7.3.11, page 123, from the representation theory book of Peter Webb, available also online at:
<http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf>
In this example it is shown that the trivial module $\mathbb Z$ does not have a projective cover over $ZG$-mod where $G$ ... | https://mathoverflow.net/users/103474 | Non-existence of projective covers | The mod-2 reduction is a functor from $\mathbb{Z}G$-modules to $\mathbb{F}\_2G$-modules that preserves direct sums , because for instance it can be realized as tensoring with $\mathbb{F}\_2$. Also if $M$ has $\mathbb{Z}$-rank $r$ then its mod-2 reduction has dimension at least $r$ as an $\mathbb{F}\_2$-vector space. In... | 6 | https://mathoverflow.net/users/21848 | 260775 | 117,540 |
https://mathoverflow.net/questions/260595 | 2 | In the [nlab entry on uniform spaces](https://ncatlab.org/nlab/show/uniform+space#examples_2) they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped with the uniform structure generated by:
$$
\bigg\{\big\{ (f,g... | https://mathoverflow.net/users/30366 | Cartesian powers of uniform spaces | E.g. $C(\mathbb{R} ,[0,1])$ in this function space uniformity is metrisable in this uniformity, using the $\sup$-metric $d(f,g) = \sup \{|f(x) - g(x)|: x \in \mathbb{R}\}$. While the product $\mathbb{R}^{[0,1]}$ is very non-metrisable (not even first countable) in the product topology. These are quite different beasts.... | 3 | https://mathoverflow.net/users/2060 | 260779 | 117,542 |
https://mathoverflow.net/questions/260771 | 5 | (I apologize in advance for these simple questions, I am a beginner trying to go through Scholze's paper Perfectoid Spaces).
Let $(R, R^+)$ be an affinoid $k$-algebra as defined in Scholze's paper Perfectoid Spaces, Definition 2.6. In particular, there exists a subring $R\_0 \subset R$ such that $aR\_0$, $a \in k^\ti... | https://mathoverflow.net/users/80739 | simple questions on topological rings arising in the context of Perfectoid Spaces | Firstly, I don't think you should start to learn about adic spaces by thinking about perfectoid spaces. The sensible examples of adic spaces for a beginner to think about are sane Noetherian things like the closed polydisc etc. It's quite hard/messy to do any explicit basic calculations (of the sort one would like to d... | 15 | https://mathoverflow.net/users/1384 | 260782 | 117,544 |
https://mathoverflow.net/questions/260778 | 3 | That is, is there a problem that all problems in the polynomial hierarchy can be reduced to in polynomial time, but which some PSPACE problem cannot be reduced to in polynomial time? Clearly if the polynomial hierarchy collapses, then there is such a problem if and only if $PH\neq PSPACE$. Is it known whether there is ... | https://mathoverflow.net/users/83073 | Should we expect there to be a problem that is PH-hard but not PSPACE-hard? | Alexander Shen suggests the following language that is PH-hard,
but does not seem to be PSPACE-hard:
STRANGE TQBF $f \in L \Leftrightarrow \forall y\_1 \exists y\_2 \exists y\_3 \forall y\_4 \forall y\_5 \forall y\_6 \forall y\_7 \ldots f(y\_1, y\_2, \ldots, y\_n) =1$.
(So there are $O(\log n)$ changes of quanti... | 6 | https://mathoverflow.net/users/31356 | 260785 | 117,545 |
https://mathoverflow.net/questions/260792 | 1 | EDIT:
The observation described here was due to a bug in the code used to generate the data (as commented by Lucia), which renders this question irrelevant.
The answer, however, is worth reading.
The following relation is quite easy to derive
$$(\sum\_{i=1}^n \frac{1}{i^R})^2 = \sum\_{i=1}^n \frac{\sigma\_1(i)}{i^R}+... | https://mathoverflow.net/users/104170 | Question on the zeta and sigma functions | My original response was wrong in an embarrasing way, so let me fix it. As Lucia remarked, the OP's observation was due to a programming error, and the general picture is as follows. We have
$$\epsilon\_R(n)=\left(\sum\_{m=1}^n \frac{1}{m^R}\right)^2-\sum\_{m=1}^n \frac{d(m)}{m^R}=\sum\_{m=n+1}^{n^2}\frac{c(m)}{m^R},$$... | 7 | https://mathoverflow.net/users/11919 | 260805 | 117,555 |
https://mathoverflow.net/questions/260545 | 10 | They say that all mathematics problems eventually reduce to linear algebra or combinatorics. I have reduced mine to proving a solution exists for the following set of inequalities but have no idea how to proceed.
**Question:** For $n \geq 2$, fix once and for all, a permutation $\tau \in S\_n, \tau \neq (1,n)(2,n-1)\... | https://mathoverflow.net/users/2720 | Existence of solution to these inequalities | For a given $\tau\in S\_n$ and for any $j\in[n]$ define the numbers
$$a\_j:=\chi\_{\Delta}(j)-\chi\_{\Delta}(j-1)-\chi\_{\Delta }(n-j)+\chi\_{\Delta}(n-j+1),$$
where $\chi\_\Delta:\mathbb{Z}\to\{0,1\}$ denotes the characteristic function of the set $\Delta:=\Delta(\tau)\subset\mathbb{Z}$. Also, with $c:=5n-a\_n$, defin... | 8 | https://mathoverflow.net/users/6101 | 260818 | 117,560 |
https://mathoverflow.net/questions/260820 | 4 | Divide the 1-dimensional sphere $S^1$ into arcs.
Label each vertex between two arcs as either 1 or 2.
Count the number of "full arcs" - arcs with both 1 and 2 at their endpoints.
Divide these full arcs to two kinds: "positive" (1-2 going clockwise) and "negative" (2-1).
Then, it is clear that the number of positive ... | https://mathoverflow.net/users/34461 | Labeling a triangulated sphere | The labeling gives a piecewise linear map from $S^n$ to a simplex $\Delta$ with $n+1$ vertices, where label $k$ means the point is sent to vertex $v\_k$, and we extend linearly on simplices. The map must have degree $0$ because $\Delta$ is contractible. The degree can be computed from the preimage of any point in the i... | 9 | https://mathoverflow.net/users/2954 | 260824 | 117,561 |
https://mathoverflow.net/questions/260393 | 7 | Consider a pair of principal bundles $P \to M$ and $P' \to M'$ with groups $G$ and $G'$, respectively. A morphism from $P$ to $P'$ is a pair $(\Phi, \phi)$ where $\phi: G \to G'$ is a Lie group homomorphism and $\Phi: P \to P'$ is a fibre bundle map which is equivariant with respect to $\phi$ in the sense that $\Phi(u.... | https://mathoverflow.net/users/56938 | Morphisms of principal bundles with different structure groups and associated bundles | **New answer:** Here is a more abstract solution: [tensor product of modules](https://mathoverflow.net/questions/103141/what-is-the-functor-tensor-product) and balanced products are both (enriched) [coends](https://ncatlab.org/nlab/show/end), defined in exactly the same way. So if you pretend all of the topological gro... | 5 | https://mathoverflow.net/users/56938 | 260825 | 117,562 |
https://mathoverflow.net/questions/260654 | 3 | In the paper, [Topologically Defined Classes of Commutative Rings](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=0ahUKEwj47P6Zp9_RAhXFFiwKHabcAB8QFgghMAE&url=http%3A%2F%2Fwww.mat.uniroma3.it%2Fusers%2Ffontana%2Fdocuments%2FFontana-AnnMatPA-80.pdf&usg=AFQjCNGIKDglVjFV737cTRGhxjXD1UDq8... | https://mathoverflow.net/users/47763 | Localization of the pullback diagram | First, you expect that the pullback (as a special kind of limit) should be the target of your map. Thus you expect a map:
$$(S\_A \times\_{S\_C}S\_B)^{-1} D\longrightarrow S\_A^{-1}A \times\_{S\_C^{-1}C}S\_B^{-1}B$$
Denote $S=S\_A \times\_{S\_C}S\_B$. In this setting you have $S=v'^{-1}(S\_A)\cap u'^{-1}(S\_B)$. So tha... | 1 | https://mathoverflow.net/users/16678 | 260836 | 117,564 |
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