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https://mathoverflow.net/questions/208323 | 10 | As a non expert in the theory of topological operads, I find it pretty hard, to understand what algebras for little balls/cubes/something operads are.
For all the other famous operads I know (like Lie, Com, Ass ect.) associated algebras are meanwhile given in terms of generators and relations, which makes them unders... | https://mathoverflow.net/users/21965 | What are algebras for the little n-balls/n-cubes/n-something operads exactly? | An $E\_n$ algebra (an algebra over the little $n$-cubes operad, etc.) is intuitively an object with $n$ compatible monoid structures. All of the subtlety in this theory lies in making "compatible" precise; in particular it is not a property but a structure.
Here are some examples.
* In $\text{Set}$, an $E\_1$ alge... | 13 | https://mathoverflow.net/users/290 | 208351 | 99,481 |
https://mathoverflow.net/questions/208341 | 64 | Recently I discovered the differential identity
$$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$
valid for any odd natural number $k$; for instance $\frac{d^6}{dx^6} (1+x^2)^{5/2} = \frac{225}{(1+x^2)^{7/2}}$. This identity was surprising at first, since u... | https://mathoverflow.net/users/766 | Is this differential identity known? | It's the Rodrigues formula for [Gegenbauer polynomials](https://en.wikipedia.org/wiki/Gegenbauer_polynomials) $C\_n^{(\alpha)}(x)$, in the special case $\alpha=-n/2$ when the polynomial is just unity.
The general formula reads
$$C\_n^{(\alpha)}(x)=\frac{(-2)^n}{n!}\frac{\Gamma(n+\alpha)\Gamma(n+2\alpha)}{\Gamma(\al... | 85 | https://mathoverflow.net/users/11260 | 208362 | 99,485 |
https://mathoverflow.net/questions/203076 | 2 | I am looking for a previous reference to the following very simple geometric lemma, which I use in my paper [[arXiv:1503.03462]](http://arxiv.org/abs/1503.03462):
Let $P$ be the parabola $y=x^2$. Let $a,b,c,d$ be four points on $P$ sorted from left to right, and let $z$ be the point of intersection of the segments $a... | https://mathoverflow.net/users/27742 | Reference to parabola lemma | Not only power of point, but many things about circles may be defined for parabolas with vertical aces. For example, we may define the angle between lines $y=ax+b$ and $y=cx+d$ as $a-c$. Then parabola passing two points $A$, $B$ is a locus of points $P$ for which angle between lines $PA,PB$ is fixed. More advanced exam... | 2 | https://mathoverflow.net/users/4312 | 208364 | 99,487 |
https://mathoverflow.net/questions/208303 | 2 | For given values of $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m$, how can I find the value of:
$$
\max\_{x \in [0,1]^n} \|Ax+b \|\_1
$$
Or is this problem NP-hard?
| https://mathoverflow.net/users/74516 | How can I find the maximum value of this function? | Note that since the objective is convex,
there are optimal solutions that are extreme points of the feasible region, i.e. we can assume all $x\_i \in \{0,1\}$.
We can encode an Ising hamiltonian in this problem, with spins $\sigma\_i = (-1, 1)$ corresponding to $x\_i = (0, 1)$.
Thus a term
$$J \sigma\_1 \sigma\... | 3 | https://mathoverflow.net/users/13650 | 208370 | 99,490 |
https://mathoverflow.net/questions/135593 | 16 | Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int\_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the following inequality may be true.
$$f(x):= (x^2+1)N + xn-(xN+n)^2 > N^2$$
where the dependency of $n$ and $N$ on $x$ are absor... | https://mathoverflow.net/users/32660 | A normal distribution inequality | Here is a complete solution. The idea is to kill the entries of $N$ in two steps, by applying two appropriately constructed first-order differential operators, which will result in a simple elementary expression:
Let $b:=f-N^2$. As noted by cardinal, $b$ is an even function. So, it is enough to show that $b>0$ on $[0... | 6 | https://mathoverflow.net/users/36721 | 208373 | 99,491 |
https://mathoverflow.net/questions/208374 | 10 | This is problem 15.3 in Arnie Miller's [problem list](http://www.math.wisc.edu/~miller/res/problems.pdf):
(Juhasz) Suppose there exists $\langle A\_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega\_1$ and for each $\alpha \in L$, $A\_{\alpha}$ is an unbounded subset of $\alpha$,... | https://mathoverflow.net/users/2689 | Is $\clubsuit_{\omega_1}$ enough to get Suslin tree? | The answer is negative apparently. It is consistent relative to ZFC that all Aronszajn trees are
special and that the club principle holds:
<http://home.mathematik.uni-freiburg.de/mildenberger/postings/paperspdf/988_2014_10_15no.pdf>
| 10 | https://mathoverflow.net/users/57583 | 208383 | 99,493 |
https://mathoverflow.net/questions/208372 | 4 | this post is concerned with functionals defined in measures. Consider the following functional
$$\mathcal{W}[\mu]=-\int\_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$
were we define $-\log\vert0\vert=+\infty$ and $\mu$ is a finite, non negative Radon measure in $\mathbb{R}$. It is clear that if $\mu$ does g... | https://mathoverflow.net/users/62252 | On the domain of functionals in measure with singular kernels | It is well known (to those who know it well) that the Hausdorff dimension of a set is closely related to its capacities. More precisely, if we define the capacitary dimension of a set $A\subset\mathbb R^n$ as the infimum of those $s>0$ for which
$$
I\_s(\mu) = \int \!\!\!\int |x-y|^{-s}\, d\mu(x)d\mu(y)=\infty
$$
for a... | 8 | https://mathoverflow.net/users/48839 | 208387 | 99,494 |
https://mathoverflow.net/questions/208368 | 4 | Let $D$ be the digraph on $2^d$ vertices with $d2^d$ edges that we obtain by directing each edge of the $d$-dimensional hypercube in both directions.
>
> Can we partition the edges of $D$ into $d$ directed Hamiltonian cycles?
>
>
>
An alternative formulation would be to ask whether the hypercube has a so-calle... | https://mathoverflow.net/users/955 | Can we find 3 disjoint directed Hamiltonian cycles in the cube? | It looks like the titled question (d=3) is not directly answered: I will hint at how to
show the answer is no.
At each vertex, there are two ways that the decomposition can go. I like
to call them black or white, or clockwise and counterclockwise. If we look
at a cube face and pick all of one type or the other, we in... | 3 | https://mathoverflow.net/users/35626 | 208395 | 99,499 |
https://mathoverflow.net/questions/50497 | 16 | This is a follow up to this MO question: [Free symmetric monoidal $(\infty,n)$-categories with duals](https://mathoverflow.net/questions/49732/free-symmetric-monoidal-infty-n-categories-with-duals)
Freed-Hopkins-Lurie-Teleman define a classical field theory as a symmetric monoidal functor $I$ from $n$-cobordism to th... | https://mathoverflow.net/users/8320 | Fully dualizable objects in classical field theories | All of the objects in these iterated span categories are fully dualizable. See [this paper](http://arxiv.org/abs/1409.0837) by Rune Haugseng.
| 5 | https://mathoverflow.net/users/290 | 208404 | 99,505 |
https://mathoverflow.net/questions/208386 | 5 | Let $K$ be a number field and $A/K$ an abelian variety over it.
>
> Can it be that $K(A[n])$ does not contain a primitive $n$-th rooth of unity?
>
>
> If the answer is yes is it always possible to bound the bad $n$ uniformly in the degree of the number field and the dimension of $A$?
>
>
>
Both references... | https://mathoverflow.net/users/74548 | n-th root of unity in n-th division field of abelian variety? | Actually, this is an exercise in Serre's Lectures on Mordell--Weil Theorem:
$K(A[n])$ always contains $\mu\_n$ if $char(K)$ does not divide $n$ and $A$ is an abelian variety of positive dimension over $K$. (You don't need to assume that $K$ is a number field)
Here is a solution. First, it suffices to check the cas... | 16 | https://mathoverflow.net/users/9658 | 208405 | 99,506 |
https://mathoverflow.net/questions/208400 | 3 | We define the *lower density* of a set $A\subseteq \mathbb{N}$ by
$$
\operatorname{ld}(A) \ := \ \liminf\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.
$$
For $A,B\subseteq \mathbb{N}$, we set
$$
A + B \ := \ \{a+b: a\in A, b\in B\},
$$
and similarly
$$
A \cdot B \ := \ \{a\cdot b: a\in A, b\in B\}.
$$
Further let ${... | https://mathoverflow.net/users/8628 | Suprema of lower density of sums and products of sets with lower density 0 | Anthony Quas has already answered the first part.
The second supremum is $1$ as well -- just let $A$ be the set of
positive integers all of whose prime divisors are $2$ or
$\equiv 1 (\!\!\mod 4)$, and let $B$ be the set of positive integers
all of whose prime divisors are $\equiv 3 (\!\!\mod 4)$.
Then $A \cdot B = \mat... | 4 | https://mathoverflow.net/users/28104 | 208420 | 99,509 |
https://mathoverflow.net/questions/191365 | 8 | Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in $L^1(\mathbb{R}^n)$ such that the distributional gradient if a vector valued finite measure)?
| https://mathoverflow.net/users/56191 | Dual or pre-dual of BV | Depending on the locally convex topology you take on $BV\_c(\mathbb{R}^n)$ (BV functions with essentially compact support in $\mathbb{R}^n$), its dual is given by so called charges (or strong charges) as defined in "De Pauw, Pfeffer - Distributions for which div v = F has a continuous solution - Comm. Pure Appl. Math.,... | 4 | https://mathoverflow.net/users/63927 | 208432 | 99,512 |
https://mathoverflow.net/questions/208431 | 2 | Let $R$ be a ring. Recall that a module $M$ is called finitely presented if there is an exact sequence
$R^n \to R^m \to M \to 0$.
with $n,m \in \mathbb{N}$.
A well known result states that any module $M$ is a filtered direct limit of finitely presented modules. A bit more generally, we can call a module $M$ 2-finit... | https://mathoverflow.net/users/74573 | Direct limit and finite presentation of modules | Assume that a f.p. module $M$ is filtered direct limit of modules $(M\_i)$.
Recall that this $M$ f.p. means that $\mathrm{Hom}(M,--)$ commutes with inductive limits. So the identity of $M$ can be lifted to $M\to M\_i$ for $i$ large enough (say $i\ge i\_0$), let $j\_i$ be the lift $M\to M\_i$, and let $N\_i$ be the ke... | 6 | https://mathoverflow.net/users/14094 | 208437 | 99,515 |
https://mathoverflow.net/questions/208418 | 6 | Suppose $k$ is a field of characteristic zero, and $R$ is a $k$-algebra. The $R$-module of Kähler differentials $\Omega\_{R/k}$ of $R$ over $k$ with generators $\{dr\}\_{r \in R}$ is the module subject to the following relations:
1. $dr = 0$ if $r \in k$.
2. $d(r+s) = dr + ds$.
3. $d(rs) = r\,ds + s\,dr$.
We have a... | https://mathoverflow.net/users/nan | Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations? | You want something that plays the role of cotangent bundle in nice cases, and be defined in general.
It may not be clear, a priori, that the universal derivation $d:R\to \Omega\_{R/k}$ will give a reasonable object but it does. If you do a few examples, you will see that it behaves quite well.
If $R=k[x\_1,\ldots, x\_n... | 6 | https://mathoverflow.net/users/4144 | 208438 | 99,516 |
https://mathoverflow.net/questions/208429 | 8 | Let $k$ be a commutative ring. Feel free to assume it's a field.
Let $X$ be a set. This question is only interesting when $X$ is infinite.
Write $k^X$ for the $k$-algebra of functions $X \to k$, with the algebra operations defined pointwise.
>
> What are the $k$-algebra homomorphisms $k^X \to k$?
>
>
>
Tr... | https://mathoverflow.net/users/586 | For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$? | As explained in the comments, if $k$ is a field, then $k$-algebra homomorphisms $k^X\to k$ are in bijection with $|k|^+$-complete ultrafilters on $X$ (that is, ultrafilters closed under $|k|$-fold intersections, or equivalently, ultrafilters such that whenever you partition $X$ into $|k|$ pieces one of the pieces must ... | 4 | https://mathoverflow.net/users/75 | 208458 | 99,523 |
https://mathoverflow.net/questions/208153 | 1 | I'm reading a proof of del Busto (<http://arxiv.org/pdf/alg-geom/9410018v1.pdf>) and am confused at one step. I'm sure its quite easy, but the key trick hasn't come to me yet.
The context here is that $X$ is a nonsingular (complex) projective surface and A an ample divisor. For $\kappa\_1,\ldots,\kappa\_r$ positive i... | https://mathoverflow.net/users/13288 | A key step in Del Busto's effective Matsusaka theorem | It seems to me that $[\frac 1nF\_n]$ is exc. and there is a short exact sequence inducing $$W(k)→V(k+1)→H^1((−∑k\_iE\_i)|\_{\sum \eta \_iE\_i})$$ which has positive degree on a bunch of (non-reduced $\mathbb P^1$'s and so vanishes. If each $\eta \_i\in \{0,1\}$, this is clear. If not proceed by induction via the short ... | 1 | https://mathoverflow.net/users/19369 | 208473 | 99,525 |
https://mathoverflow.net/questions/208407 | 1 | Let $(H \subset G)$ be an inclusion of finite groups.
Let $\{ g\_i \ \vert \ i \in I=[1, \dots ,n] \}$ a subset of $G$ of double coset representatives, i.e. $$G = \coprod\_{i \in I} Hg\_iH$$
On the algebra $\mathbb{C}G$ we consider the elements $$e\_i = \frac{1}{\vert H \vert} \sum\_{g \in Hg\_iH} g$$
They generat... | https://mathoverflow.net/users/34538 | Is a prime index inclusion of finite groups, separating? | I think the answer for prime index inclusions is Yes. I see the question from the following point of view: For each double coset $D= HgH$, you form the element
$$ e\_D := \frac{1}{\lvert H \rvert} \sum\_{x\in D} x \in \mathbb{Q} G . $$
The product of two such elements is an $\mathbb{N}$-linear combination of such elem... | 1 | https://mathoverflow.net/users/10266 | 208474 | 99,526 |
https://mathoverflow.net/questions/208475 | -3 | Without the use of calculator how to calculate $\log\_{10} ~2$, $\log\_{10} ~3$?
| https://mathoverflow.net/users/74593 | Without the use of a calculator, how to calculate the logarithm of 2 and 3 in base 10 | See the note
N. David Mermin: Logarithms!, *Am. J. Phys.* **46** (1978), 101-105
(<http://dx.doi.org/10.1119/1.11104>).
| 1 | https://mathoverflow.net/users/14830 | 208477 | 99,527 |
https://mathoverflow.net/questions/208476 | 1 | Let $R$ be an excellent, Henselian, discrete valuation ring with algebraically closed residue field and $\hat{R}$ be the completion of $R$. If I understand correctly, the residue field of $\hat{R}$ is the same as that of $R$. Is there any classification of subrings $R'$ of $\hat{R}$ containing $R$ such that the induced... | https://mathoverflow.net/users/58203 | Completion of discrete valuation ring | The transcendence degree of $\hat{R}$ over $R$ can be large. If $R$ is the Henselization of $k[t]\_{(t)}$ then the transcendence degree is max(${2^{\aleph\_0}}$,$|R|$), and there are many ways to construct transcendental power series. So $R'$ is rather arbitrary.
| 3 | https://mathoverflow.net/users/59248 | 208481 | 99,529 |
https://mathoverflow.net/questions/208493 | 0 | The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method.
Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not?
Solution: As part of my regular involvement on this problem, I have found the following set of integer solutions by... | https://mathoverflow.net/users/74601 | Special type Diophantine equations with integer solutions | If one could prove the explicit $abc$-conjecture, i.e., with an explicit constant $C$, then one would have explicit bounds on the heights of the rational points in Falting's theorem. If this is not available, one can try other methods. For example, consider the Fermat quartics $x^4+y^4=c$. For $c=17$ this is Serre's cu... | 4 | https://mathoverflow.net/users/32332 | 208503 | 99,539 |
https://mathoverflow.net/questions/208506 | 1 | Is there a finite non-ablelian group $G$ generated by $x$ and $y$ such that $1+x+y$ is a zero divisor in the rational group algbera $\mathbb{Q}[G]$ and also $x^2$, $y^2$ and $(x^{-1} y)^2$ are all non-trivial?
| https://mathoverflow.net/users/19075 | Zero divisors of the form $1+x+y$ in the rational group algebra | If $G$ has a cyclic quotient $G/N$ of order $3$ in which $x$ and $y$ map to the two non-identity elements, then $1+x+y$ acts as zero on the non-trivial linear complex representations of $G/N$. So $1+x+y$ does not have full rank in the regular representation of $G$, and is therefore a zero divisor.
For example, take $... | 7 | https://mathoverflow.net/users/22989 | 208513 | 99,544 |
https://mathoverflow.net/questions/208525 | 14 | Let $S = \{D\_1, D\_2, D\_3, \ldots \}$ be the set of all prime discriminants
(or positive prime discriminants) of quadratic number fields. For such a
discriminant let $\chi\_j(n) = (\frac{D\_j}n)$ be the associated Dirichlet
character and
$$ L(1,\chi\_j) = \sum\_{n \ge 1} \frac1n \Big(\frac{D\_j}{n}\Big) $$
the va... | https://mathoverflow.net/users/3503 | Geometric Mean of $L(1,\chi)$ for quadratic Dirichlet characters | The distribution of values of $L(1,\chi\_d)$ as $d$ varies over fundamental discriminants has been extensively studied. For example, see this paper of [Granville and Soundararajan](http://arxiv.org/abs/math/0206031) which gives uniform such results (and discusses other references and history). The main result there sho... | 15 | https://mathoverflow.net/users/38624 | 208528 | 99,546 |
https://mathoverflow.net/questions/208464 | 4 | Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.
If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the group $G$ ?
Any reference and observation would be appriciated. I also wonder whether is there any such nonsolvable grou... | https://mathoverflow.net/users/47344 | The groups with symmetric subgroups lattice | You should have a look at Roland Schmidt's book *Subgroup lattices of groups*, especially Chapter 8, "Dualities of subgroup lattices". In this chapter, groups $G$ are studied such that there is another group $\overline{G}$ (a "dual"), such that the subgroup lattice of $G$ is isomorphic to the reverse of the subgroup la... | 5 | https://mathoverflow.net/users/10266 | 208530 | 99,547 |
https://mathoverflow.net/questions/208532 | 1 | I know very little about how computers work, so please excuse my ignorance!
I think of the [Glasgow Haskell Compiler](https://www.haskell.org/ghc/) as a program that eats up extensional type theory and spits out a program which can be run on my computer.
>
> **Question 1:** Would it be possible to write a com... | https://mathoverflow.net/users/4002 | Functional programing and intensional type theory | GHC doesn't eat extensional *dependent* type theory. The type theory of GHC has some fancy bells and whistles that allow you to fake some things you can do with dependent types, but it isn't truly dependently typed. There are, however, "compilers" that eat true (extensional) dependent type theory, such as [Agda](http:/... | 7 | https://mathoverflow.net/users/49 | 208533 | 99,549 |
https://mathoverflow.net/questions/208439 | 7 | Let $(M,g)$ be a Riemannian manifold. Recall that the $k^{th}$-Pontryagin class is a topological invariant which, by classical Chern-Weil theory, can be represented using the so-called Pontryagin forms. These forms are obtained by appropriate constant linear combinations and wedge multiplications of the closed $4p$-for... | https://mathoverflow.net/users/74576 | Pontryagin Forms and Special Holonomy | Here are a few remarks to show that your Questions 1 and 2 are almost equivalent. (What the answers are is another matter, and I expect that to be somewhat difficult, as I'll explain below.) Of course, Question 1 is a special case of Question 2 in the OP's situation because, if a closed form is a scalar multiple of $\o... | 4 | https://mathoverflow.net/users/13972 | 208537 | 99,551 |
https://mathoverflow.net/questions/208485 | 14 | Two years ago, I made a conjecture [on stackexchange](https://math.stackexchange.com/questions/334448/1-a21-b21-c2-8abc-a-b-c-in-mathbbq-has-infinitely-many-sol/336803#336803):
Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$
I have found som... | https://mathoverflow.net/users/38620 | Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$ | Call (the projective completion of) your surface $S$. It admits three double covers of $\mathbb P^2$, namely $$\pi\_1(a,b,c)=(a,b), \quad\pi\_2(a,b,c)=(a,c),\quad\pi\_3(a,b,c)=(b,c).$$ Each double cover induces an involution, so we get
three involutions $$\sigma\_1,\sigma\_2,\sigma\_3:S\to S.$$ These involutations don'... | 18 | https://mathoverflow.net/users/11926 | 208543 | 99,554 |
https://mathoverflow.net/questions/208496 | 4 | For $n\in\mathbb{N}$ let $S\_n$ denote the set of permutations on the set $\{1,\ldots,n\}$. Set $$E\_n = \big\{\{\pi\_1, \pi\_2\}: \pi\_1,\pi\_2\in S\_n \land \exists k\_1 < k\_2 <\ldots <k\_r\leq n: \pi\_2=(k\_1 \cdots k\_r)\circ \pi\_1\big\}.$$
(In other words, $\pi\_2$ can be generated from $\pi\_1$ with a **monoton... | https://mathoverflow.net/users/8628 | Chromatic number of graph defined on the set of permutations | In case it helps here is a Sage function that should generate these graphs (if I got the definition right)
```
def g(n):
gens = []
A = SymmetricGroup(n)
for el in A:
c = el.cycles()
if len(c) == 1 and sorted(c[0]) == list(c[0]):
gens += [el]
return Graph(A.cayle... | 6 | https://mathoverflow.net/users/1737 | 208546 | 99,555 |
https://mathoverflow.net/questions/208561 | 1 | I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the diophantine equation $y^2-3x^2=1$? Any help would be appreciated.
| https://mathoverflow.net/users/74634 | How to solve a quadratic diophantine equation | The fundamental solution is $2+\sqrt{3}.$ The general solution is $x\_n+\sqrt{3}y\_n=(2+\sqrt{3})^n,$ that is $$x\_n=\frac{(2+\sqrt{3})^n+(2-\sqrt{3})^n}{2}.$$ This is sequence A001075 at <https://oeis.org>. Writing it as a recurrence sequence we have $x\_0 = 1, x\_1 = 2, x\_n = 4x\_{n-1} - x\_{n-2}$. It follows that $... | 1 | https://mathoverflow.net/users/74606 | 208564 | 99,562 |
https://mathoverflow.net/questions/208568 | 19 | Suppose one has an $n \times n$ orthogonal matrix $M$:
$$
\left(
\begin{array}{ccc}
0.239326 & 0.846726 &
0.475161 \\
0.768893 & 0.13356 &
-0.625272 \\
0.592897 & -0.514992 &
0.619077 \\
\end{array}
\right)
$$
Because it is orthogonal, $M^T M = I$.
Suppose one entry of $M$ is erased, say $M(2,2)$:
$$
\left(
\begi... | https://mathoverflow.net/users/6094 | How much redundancy resides in an $n \times n$ orthogonal matrix? | $O(n)$ is a manifold of dimension $n(n-1)/2$, so "generically" one might hope to recover up to $n(n+1)/2$ entries from the other $n(n-1)/2$. This won't quite work, however. Given any set $A$ of rows and any set $B$ of columns, generically we need at least one entry to survive that is either in both rows $A$ and columns... | 11 | https://mathoverflow.net/users/13650 | 208570 | 99,565 |
https://mathoverflow.net/questions/208562 | 5 | Suppose I am given a symmetric matrix $G\_{ij}$ with $G\_{ii} = 2$. Can I always find an invertible integer matrix $S$ such that $(S^T G S)\_{ii}=2$ and $(S^T G S)\_{ij} \leq 0$ for $i \neq j$? Is there a practical algorithm to do so?
If you'd like a particular challenge, I'd like to know the answer for
$$G = \begin{... | https://mathoverflow.net/users/297 | From a (not positive definite) Gram matrix to a (Kac-Moody) Cartan matrix | There is an algorithm, based on a 1907 article of Hurwitz I mention sometimes, based in turn on the tree of Markov numbers.
We begin with a ternary quadratic form $\langle 1,1,1,r,s,t \rangle.$ The (Lehman) discriminant of this is
$$ 4 + rst - r^2 - s^2 - t^2. $$ We would like to know whether we can find replacemen... | 3 | https://mathoverflow.net/users/3324 | 208573 | 99,568 |
https://mathoverflow.net/questions/208574 | 8 | When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$ ($n$ being an integer) , i .e., when does $(-1+\sqrt[3]{2})^n$ not have a non-zero term in $\sqrt[3]{4}$. As you might have noticed, I'm interested in solving the diophantine equation $x^3-2y^3=1$ using this specific method. Is there any way I can use Skolem's p... | https://mathoverflow.net/users/74634 | When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$? | The only solutions of $x^3-2y^3=1$ are $(x,y)=(1,0)$ and $(x,y)=(-1,-1)$.
I don't know whether there's a nice Skolem-style proof, but here
this happens to be unnecessary because $(1,0)$ and $(-1,-1)$ are the
only *rational* solutions and this can be proved by a Fermat-style descent:
the Weierstrass form is $Y^2 = X^3 ... | 23 | https://mathoverflow.net/users/14830 | 208575 | 99,569 |
https://mathoverflow.net/questions/208586 | 1 | Two vector bundles $E$ and $F$, are said two be S-equivalent if they have isomorphic gradients. My question is: Is it possible to caracterise this properity using transition functions?
Thanks
| https://mathoverflow.net/users/66528 | s-equivalence and transition function | Yes. Suppose for simplicity that $E$ and $F$ are extensions of two stable bundles with the same slope, say of ranks $r$ and $s$. Then you can choose their transition matrices of the form $\pmatrix{ (g\_{ij}) & (a\_{ij})\\ 0 & (h\_{ij})}$, with $(g\_{ij}) \in GL\_r(\mathcal{O})$ and $(h\_{ij}) \in GL\_s(\mathcal{O})$, a... | 2 | https://mathoverflow.net/users/40297 | 208595 | 99,574 |
https://mathoverflow.net/questions/204433 | 14 | I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation $a$ (and bounds on its degree and height). A fairly standard idea for how to do this is to pick a large constant $N$ and a... | https://mathoverflow.net/users/3121 | Computing minimal polynomials using LLL | The theory of "layered lattices" is an attempt to deal
with these issues. The 2011 thesis of Lenstra's student
Erwin Dassen is devoted to this subject:
<https://openaccess.leidenuniv.nl/handle/1887/18264>
| 6 | https://mathoverflow.net/users/18645 | 208597 | 99,575 |
https://mathoverflow.net/questions/208578 | 2 | I have an array of numbers lets call it $p$ , where $p[r]={k+r-1\choose k-1}$
I want to find the sum of all the elements of $p$ taken $n$ at a time .
$0\le r\le k$
For instance, for $k=3$ ,$n=2$ , i need
$$p[0]^2+p[1]^2+p[2]^2+p[3]^2+p[0]\cdot p[1]+p[0]\cdot p[2]+p[0]\cdot p[3]+p[1]\cdot p[2]+p[1]\cdot p[3]+\\+... | https://mathoverflow.net/users/74641 | Sum Of n numbers taken $k$ at a time, where numbers are of form $r\choose k$ | I'm not sure why this received a downvote. You are asking for the
coefficient of $x^n$ in $F(x)=1/\prod\_{r=0}^k (1-{k+r-1\choose
k-1}x)$. Since the denominator of $F(x)$ has distinct explicit zeros,
one can explicitly find the partial fraction expansion of $F(x)$ and
then expand each term. For instance, when $k=3$ we ... | 7 | https://mathoverflow.net/users/2807 | 208600 | 99,577 |
https://mathoverflow.net/questions/208599 | 5 | Given a perfect field $F$ of prime characteristic the ring of Witt Vectors $W(F)$ is a discrete valuation ring. For example, $W(\mathbb{F}\_p)$ is the ring of $p$-adic integers. Is it possible to embed an arbitrary unramified discrete valuation ring of mixed characteristic into a Witt Vector ring of a perfect field?
| https://mathoverflow.net/users/23240 | discrete valuation ring and ring of witt vectors | I think this is possible, if I understand your question correctly. If $R$ is a discrete valuation ring of mixed characteristic $(0,p)$ with residue field $k$ and maximal ideal $pR$, then $R$ is a Cohen ring for $k$. Cohen rings are unique up to (generally non-unique) isomorphism, and there is a construction of the Cohe... | 6 | https://mathoverflow.net/users/4351 | 208603 | 99,578 |
https://mathoverflow.net/questions/208606 | 6 | I saw it conjectured at <http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0309.0317.ocr.pdf> that "discrete subgroups with property $(T)$ may have modest subgroup growth." (Page 5, directly above the heading for section 4.)
According to the author, this conjecture was supported by some examples.
I have three ... | https://mathoverflow.net/users/74653 | Do discrete groups with property $(T)$ have "modest" subgroup growth? | This is now outdated: in 1994 there were very few known sources of Property T groups: lattices on the one hand (for which congruence subgroup property was true or unknown, and hyperbolic groups with no information on their finite index subgroups). This has dramatically changed. Restricting to cases for which we have in... | 13 | https://mathoverflow.net/users/14094 | 208608 | 99,580 |
https://mathoverflow.net/questions/208604 | 17 | Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P\_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$.
>
> Is it true that $\sum\_{p \in \mathscr P\_P} \frac1 p$ diverges?
>
>
>
The case of polynomials with degree $2$ can be studied by ... | https://mathoverflow.net/users/38851 | About the prime divisors of values of polynomials | We will assume that $f$ is irreducible over $\mathbb{Q}$. Let $\gamma$ be a root of $f$, and let $K = \mathbb{Q}(\gamma)$. By Chebotarv's density theorem, the set $P\_f$ of primes $p$ with the property that $f(x) \equiv 0 \pmod{p}$ has at least one root in $\mathbb{F}\_p$ has positive relative density in the set of rat... | 11 | https://mathoverflow.net/users/10898 | 208609 | 99,581 |
https://mathoverflow.net/questions/208621 | 0 | According to the prime number theorem there are about $n/\ln(n)$ primes less than $n$. This value is a limit but it could fluctuate. My question is, is there a known bound on this fluctuation? i.e. are there functions $f$ and $g$ such that $\forall n>100, f(n) < \mathrm{numPrimes}(n) \lt g(n)$? How tight is this bound?... | https://mathoverflow.net/users/33725 | Min number of primes up to n | Let $\pi(x)$ denote the prime counting function. Pierre Dusart has [shown](http://arxiv.org/abs/1002.0442) that
$$\pi(x) \geq \frac{x}{\log(x)-1} \qquad \qquad \mbox{ for }x\geq 5393$$ and $$\pi(x) \leq \frac{x}{\log(x)-1.1}\qquad \qquad \mbox{ for } x\geq 60184.$$
Many other results of a similar flavor are also show... | 6 | https://mathoverflow.net/users/nan | 208622 | 99,587 |
https://mathoverflow.net/questions/208627 | 2 | Suppose we have positive-definite matrices $A$, $B$, if $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $ \alpha A \geq B $? If it has, then what is it?
| https://mathoverflow.net/users/70621 | If $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $\alpha A \geq B $? | It seems the following.
Let $A$ and $B$ be $n\times n$ matrices and $S\subset\Bbb R^n$ be the unit sphere. Then positive definiteness of the matrix $A$ is equivalent to
$$m(A)=\min\{(Ax,x):x\in S\}>0.$$
Similarly, we have
$$m(A-B)=\min\{(A-B)x,x):x\in S\}>0.$$
Put
$$M(B)=\max\{(Bx,x):x\in S\}>0.$$
Let ... | 5 | https://mathoverflow.net/users/43954 | 208636 | 99,591 |
https://mathoverflow.net/questions/208640 | 0 | Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking into this subject but I couldn't find this subject in numerical analysis. I was interested in this for approximating expe... | https://mathoverflow.net/users/74661 | Finding the distribution of a random variable numerically with sample data? | IANAS, but I'll try an answer.
The problem is known as [density estimation](https://en.wikipedia.org/wiki/Density_estimation). A first option (which you probably will not find satisfactory) is an average of Kronecker deltas centered at the sampled points. Or you can replace the deltas with Gaussians or other shapes (... | 1 | https://mathoverflow.net/users/1898 | 208642 | 99,593 |
https://mathoverflow.net/questions/208375 | 11 | The injections (coprojections) of a coproduct in a category are very often monomorphisms. For instance, this happens in any [extensive category](http://ncatlab.org/nlab/show/extensive+category) (essentially by definition) and also in any category with zero morphisms (since in that case they are *split* monos). However,... | https://mathoverflow.net/users/49 | A cosmos where coproduct injections are not monic | Let me recall that injectivity of coproduct's injections follows from the distributivity of products over coproducts (rather than full extensivity of coproducts). Since every cartesian closed category is (obviously) distributive we cannot find counterexamples in cartesian closed categories.
However, the proof of inje... | 7 | https://mathoverflow.net/users/13480 | 208643 | 99,594 |
https://mathoverflow.net/questions/208645 | 23 | In group theory the number of Sylow $p$-subgroups of a finite group $G$, is of the form $kp+1$.
So it is interesting to discuss about the divisors of this form. As I checked it seems that for an odd prime $p$, there is not any divisor $a$ of $p^4+1$, where $1<a<p^4+1$ and $a=kp+1$, for some $k>0$.
Could you help me a... | https://mathoverflow.net/users/31045 | divisors of $p^4+1$ of the form $kp+1$ | There's none indeed.
**Lemma**: if $1<m<n$ are coprime integers then $mn+1$ does not divide $n^4+1$.
First observe that for any $m,n$, of $mn+1$ divides $n^4+1$, then it divides $n^4m^4+m^4=((nm)^4-1)+1+m^4$, and since $mn+1$ clearly divides $((nm)^4-1)$, we deduce that it also divides $1+m^4$.
To prove the lemma... | 37 | https://mathoverflow.net/users/14094 | 208652 | 99,598 |
https://mathoverflow.net/questions/208023 | 2 | Let $(X,g)$ be a $m$-dimensional complex, hermitian, spin manifold and let us denote by $S\_{\mathbb{C}}$ its complex spinor bundle. Then:
$S\_{\mathbb{C}}\simeq \Lambda\_{\mathbb{C}}(X)$
Let $\nabla$ be any metric connection on $T\_{\mathbb{C}}X$ that lifts to a connection $\nabla^{S\_{\mathbb{C}}}$ on $S\_{\mathb... | https://mathoverflow.net/users/66688 | Constant spinors from constant forms | If you also assume that $\nabla$ preserves the complex structure, it is true that $\nabla^{S\_{\mathbb C}}\eta=0$ if and only if $\nabla\Omega=0$. The assumptions mean that the unitary frame bundle of $X$ admits an extension with structure group the appropriate spin group, and that the principal connection inducing $\n... | 4 | https://mathoverflow.net/users/64141 | 208654 | 99,599 |
https://mathoverflow.net/questions/208663 | 6 | Is the group $G$ with the presentation $\langle x,y \;|\; x^7=1, y^2 x y=x^4\rangle$ solvable? infinite?
I have computed by GAP the following fators of the derived series of $G$:
$G/G'\cong C\_3 \times C\_7$, $G'/G'' \cong C\_2 \times C\_2 \times C\_2 \times C\_7$, and $G''/G^{(3)}$ and $G^{(3)}/G^{(4)}$ are elementary... | https://mathoverflow.net/users/19075 | A finitely presented group with two simple relations | The group $G$ is not solvable, since its quotient
$$
\tilde{G} := \langle x, y \ | \ x^7 = 1, y^2xy = x^4, y^{15} = 1\rangle
$$
is a group of order $423360$ such that $\tilde{G}'' \cong {\rm PSL}(3,4)$.
| 7 | https://mathoverflow.net/users/28104 | 208665 | 99,604 |
https://mathoverflow.net/questions/208662 | 2 | Now, today, my friend tell me this problem was posted by American Mathematical Monthly (Vol. 111, No. 2 Feb., 2004), p. 165 by Wu wei Chao ,and It is said that this problem is unsolved, until now. Find the diophantine-equation rational points
$$2\sqrt{\dfrac{q^2-1}{p^2-1}}=\sqrt{q}(\sqrt{p+1}-\sqrt{p-1})$$
or
$$\dfra... | https://mathoverflow.net/users/38620 | Find all rational solutions of this diophantine-equation? | The number of rational solutions to your equation is finite. In short: your equation defines a genus $3$ curve, as follows from a straightforward computation and an application of Riemann--Hurwitz; finally, by Faltings' theorem, the number of rational points on a curve of genus $>1$ is finite.
One shows this as follo... | 13 | https://mathoverflow.net/users/17907 | 208666 | 99,605 |
https://mathoverflow.net/questions/208529 | 5 | Following is the wonderful Euler's partition identity:
$$\prod\_{i=1}^\infty (1 - x^i) = 1 + \sum\_{k=1}^\infty (-1)^k \left (x^{(3k^2-k)/2} + x^{(3k^2+k)/2} \right )$$
I'm wondering if there is similar expansion for infinite product
$$\prod\_{i=1}^\infty (1 - x^{2i-1})$$
We know that the inverse of that is the gen... | https://mathoverflow.net/users/43438 | Euler-like identity for partition function | If $F(x)$ is a $D$-finite power series then there are finitely many points $w\_1, \ldots, w\_n \in \mathbb{C}$ such that $F(x)$ extends to a meromorphic function defined on any simply-connected domain in $\mathbb{C}$ not containing any of the $w\_i$. (This is a restatement of (a) on page 185 of Stanley, *[Differentiabl... | 3 | https://mathoverflow.net/users/7709 | 208671 | 99,607 |
https://mathoverflow.net/questions/177823 | 7 | Let $G$ and $H$ be finite groups and $\gcd(|G|,|H|)=1$.
Suppose the lattice of all subgroups of $G\times H$ is self-dual. Is the lattice of all subgroups of $G$ self-dual?
| https://mathoverflow.net/users/47958 | Self-duality of the subgroup lattice of $G\times H$ | The answer is Yes. This is seen as follows:
First, let us say that a group $G$ has a *dual* $\overline{G}$ (which is another group), if there is an order-reversing bijection $\delta\colon L(G) \to L(\overline{G})$ from $L(G)$, the subgroup lattice of $G$, onto the subgroup lattice $L(\overline{G})$ of $\overline{G}$... | 2 | https://mathoverflow.net/users/10266 | 208683 | 99,614 |
https://mathoverflow.net/questions/208625 | 9 | Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H\_2(M; \mathbb Z/2\mathbb Z)$. Unfortunately the paper is known to be flawed, as it would imply (as explained in the paper) that there... | https://mathoverflow.net/users/1198 | Homology class of a Lagrangian Klein bottle | The homology class of a Lagrangian Klein bottle is non-zero in any *ruled* symplectic four-manifold, e.g. in $S^2\times S^2$ with a product symplectic form. This was first proved by Shevchishin (Izvestia Math., 2009, 73:4, 797-859) and then a shorter proof was given by Nemirovski (Izvestia Math., 2009, 73:4, 689-698). ... | 7 | https://mathoverflow.net/users/74682 | 208687 | 99,616 |
https://mathoverflow.net/questions/208680 | 13 | A positive integer is *multiplicatively even (odd)* if, when decomposed into primes, the sum of the exponents is even (odd).
A subset of the integers is *thick* if it contains arbitrarily long intervals $\{n,n+1,\cdots,n+m\}$.
Is the set of multiplicatively even numbers thick?
It is easy to see that the set of mult... | https://mathoverflow.net/users/18698 | Is the set of multiplicatively even numbers thick? | You are asking about sign patterns of the [Liouville function](http://en.wikipedia.org/wiki/Liouville_function). While conjecturally all finite sign patterns appear infinitely often, this has been surprisingly hard to establish. One of the best results thus far is [by Hildebrand](http://www.ams.org/mathscinet-getitem?m... | 17 | https://mathoverflow.net/users/766 | 208689 | 99,617 |
https://mathoverflow.net/questions/120468 | 11 | Here is a simple question regarding the standard Lyndon basis for the free Lie Algebra. Suppose I take two lyndon words $m$ and $n$ and their standard bracketings $B(m)$ and $B(n)$ as elements in the free Lie algebra. Suppose further that $m < n$, so that $mn$ is a Lyndon word.
My question is when we express the brac... | https://mathoverflow.net/users/11392 | Bracket of lyndon words? | Yes it is of this form because any Lie polynomial
"begins" by a Lyndon word, in particular, the least
monomial of $B(l)$ is $l$. Then
$$
B(m)=m+\sum\_{m<u\atop |u|=|m|} \alpha^{(u)}u;\ B(n)=n+\sum\_{m<v\atop |v|=|n|} \alpha^{(v)}v\qquad \mathbf{(LB)}
$$
then $[B(m),B(n)]$ has only words of length $|m|+|n|$ and it... | 5 | https://mathoverflow.net/users/25256 | 208716 | 99,629 |
https://mathoverflow.net/questions/208713 | 2 | Let $G$ be finite abelian group and $K$ a field such that $char(K)$ does not divide the order $r$ of $G$. For each divisor $d$ of $r$ let $\omega\_d$ be a primitive $d$-root of unity and $a\_d:=\frac{\mid \{ a\in G \mid o(a)=d \} \mid}{dim\_K(K(\omega\_d))}$. By a theorem of Perlis and Walker (see e.g. [Perlis/Walker](... | https://mathoverflow.net/users/57804 | Irreducible characters of finite abelian groups | This is all fairly standard, but here goes. If $M$ is an irreducible $KG$-module, then (since $G$ is Abelian) we obtain a homomorphism $\theta: G \to {\rm End}\_{KG}(M)^{\times}$, so $\theta(G)$ is a finite Abelian subgroup of the group of units of a division algebra (using Schur's Lemma). Hence ${\rm Im} \theta$ is cy... | 2 | https://mathoverflow.net/users/14450 | 208725 | 99,632 |
https://mathoverflow.net/questions/208712 | 6 | It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on $l^2(\Gamma)$, is a ${\rm II}\_1$ factor iff $\Gamma$ is an [ICC group](http://en.wikipedia.org/wiki/Infinite_conjugacy_class_pr... | https://mathoverflow.net/users/34538 | The (Hecke) double coset von Neumann algebra | $L(\Gamma,G)$ is the algebra of endomorphisms of the representation $l^2( \Gamma/G)$ (with $\Gamma$ acting by left multiplication). This answer your second optional question. Also, by classical results on $W^\*$-categories, the category of normal representations of $L(\Gamma,G)$ will be equivalent to the category of un... | 5 | https://mathoverflow.net/users/22131 | 208748 | 99,639 |
https://mathoverflow.net/questions/208728 | 1 | Let $A, B, C$ be known matrices, and let $X$ be an unknown matrix.
Given that $C = AXB \Leftrightarrow \text{vec}(C) = K \text{vec}(X)$, where $\text{vec}(\cdot)$ denotes the vectorization of a matrix, $K = (B^{T} \otimes A)$ and $\otimes$ is the Kronecker product, it is easy to infer the least squares solution for $... | https://mathoverflow.net/users/74697 | arg min_X ||A X B - C||^2, with X diagonal | Yes, it is possible. Notice that $AXB$ is a linear function in each $x\_i$, so you can write it as $Mx$, for some $M\in\mathbb{C}^{n^2\times n}$. With a little work one can find out that the $i$th column of $M$ is given by the Kronecker product of the $i$th column of $A$ and the $i$th row of $B$, so one can write this ... | 1 | https://mathoverflow.net/users/1898 | 208751 | 99,642 |
https://mathoverflow.net/questions/208731 | 5 | Let $k$ be an algebraically closed field of characteristic 0, let $G$ be any group and $N\unlhd G$ a normal subgroup. Let $U$ be a finite-dimensional and irreducible $kG$-module, such that $U$ is also an irreducible $kN$-module. Moreover, let $V$ be a finite-dimensional irreducible $k(G/N)$-module (so it is also a $kG$... | https://mathoverflow.net/users/8338 | Irreducibility of the tensor product of two finite-dimensional irreducible group representations | $$ \mathrm{End}\_{kG}(U \otimes\_k V) = \left(\mathrm{End}\_k(U \otimes\_k V)^N\right)^{G/N} = \mathrm{End}\_k(V)^{G/N} = k. $$
The second equality holds because $\mathrm{End}\_{kN}(U) = k$ (thanks to the assumption that $k$ is algebraically closed).
EDIT: Using Chevalley's theorem is probably overkill. Let $k$ be an... | 12 | https://mathoverflow.net/users/74720 | 208759 | 99,646 |
https://mathoverflow.net/questions/208738 | 3 | What is sum of degrees of the irreducible complex characters of the alternating groups?
The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra of the group algebra $\mathbb{C} A\_n$ for all $n\in \mathbb{N}$.
The corresponding sum for the symmetric group is kn... | https://mathoverflow.net/users/57804 | Sum of irreducible complex character degrees for alternating groups | For small $n$, you can compute the sums of irreducible complex character degrees of ${\rm A}\_n$ with [GAP](http://www.gap-system.org):
```
gap> sums := List([3..15],n->Sum(List(CharacterDegrees(AlternatingGroup(n)),Product)));
[ 3, 6, 16, 46, 126, 448, 1366, 5356, 18568, 76296, 297012, 1264264, 5412928 ]
```
Howe... | 3 | https://mathoverflow.net/users/28104 | 208762 | 99,647 |
https://mathoverflow.net/questions/208742 | 3 | Let $A$ be a ring, and let $M$ be a right $A$-module. Then $M$ is flat if and only if for each left $A$-module $N$ we have that $Tor^1\_A(M,N) = 0$. Becasuse $Tor$ commutes with filtered direct limits, and since every left module is such a limit of finitely presented $A$-modules, $M$ is flat if and only if $Tor^1\_A(M,... | https://mathoverflow.net/users/74706 | Can you test flatness on $FP_3$-modules? | Answering my own question in comments:
Let $k$ be a field, and $A=k\oplus V$, where $V$ is an infinite-dimensional square-zero ideal. Then I think $A$ has no non-projective $FP\_3$-modules (using your terminology), as the kernel of any map between finitely generated free modules, that's not a surjection onto a direct... | 5 | https://mathoverflow.net/users/22989 | 208768 | 99,649 |
https://mathoverflow.net/questions/208723 | 0 | I want to quickly solve the following linear least-squares problem
$$\min\_{x \in \mathbb{R}^n} \left\| A x - b \right\|\_2^2$$
with a special sparse structure where each row in $A$ has only up to $4$ consecutive non-zero entries. This makes its normal matrix
$$C = A^T A$$
a positive definite $7$-*band* matrix ... | https://mathoverflow.net/users/74696 | Solving sparse linear least squares or a positive definite 5-band matrix system fast | The LU factorization of $C$ along with forward and backward substitution works well in this case. The factorization can still be done completely in-place. So, there is no need to touch or create other off-band elements.
| 0 | https://mathoverflow.net/users/74696 | 208769 | 99,650 |
https://mathoverflow.net/questions/208782 | 1 | I am working on doing explicit computations finding class groups of quaternions over $\mathbb{Q}$, and the following question (with $n=3$) was a clear choking point:
Given a point $P = [x\_0, \dots, x\_n] \in \mathbb{P}^{n}(\mathbb{F}\_p)$, is there a method beyond iterating through multiples $(kx\_0, \dots, kx\_n)$ of... | https://mathoverflow.net/users/69868 | Finding the "minimum norm" of points in projective space above a prime field | The lifts to ${\bf Z}^{n+1}$ of a point in projective $n$-space mod $p$
constitute $L - L\_0$ where $L\_0 = p{\bf Z}^{n+1}$ and $L \supset L\_0$
is a sublattice of index $p^n$ in ${\bf Z}^{n+1}$. At least when
(as in your case) $n$ is a fixed small number and $p$ is large,
your question is thus tantamount to the Shorte... | 4 | https://mathoverflow.net/users/14830 | 208786 | 99,652 |
https://mathoverflow.net/questions/208785 | 0 | Is there somone help me to show that if this problem have positive Answer :
Problem :Can every non-discrete topological group G be algebraically gen-
erated by a nowhere dense subset ?
Thank you for any help .
| https://mathoverflow.net/users/51189 | Can every non-discrete topological group G be algebraically generated by a nowhere dense subset? | According to [this paper on open problems in topological algebra](http://www.researchgate.net/profile/Taras_Banakh/publication/221662575_Some_Open_Problems_in_Topological_Algebra/links/0fcfd50b52c10ec422000000.pdf), this is an open problem (it's listed as Problem 9 in particular).
>
> Problem 9 (Protasov). Can ever... | 2 | https://mathoverflow.net/users/41862 | 208790 | 99,654 |
https://mathoverflow.net/questions/208783 | 3 | This is exercise 7.7 from [Martin Hairer's Rough Path notes](http://www.hairer.org/notes/RoughPaths.pdf).
>
> Verify that $\mathbb{X}\_{s,t}=\int\_s^t X\_{s,r} \otimes dX\_r$ where the integral is to be interpreted in the sense of (4.22) (I'll define this later), taking $(Y,Y') $ to be $(X,I)$. In fact, check that... | https://mathoverflow.net/users/nan | Rough path theory- Verify that $\mathbb{X}_{s,t}=\int_s^t X_{s,r} \otimes dX_r$ | Actually, I found my mistake. Once we get to:
$$\sum\limits\_{[u,v]\in P} Y\_u \otimes Y\_{u,v}+Y'\_u\otimes Y'\_u\mathbb{X}\_{u,v}=\sum\limits\_{[u,v]\in P} X\_u \otimes X\_{u,v}+\mathbb{X}\_{u,v}$$
Without loss of generality let $X\_s=0$, so:
$$X\_{u} \otimes X\_{u,v}+\mathbb{X}\_{u,v}=X\_{s,u} \otimes X\_{u,v}... | 2 | https://mathoverflow.net/users/nan | 208794 | 99,655 |
https://mathoverflow.net/questions/208753 | 6 | If $X$ and $Y$ are topological spaces, let us write $X \preceq Y$ whenever $X$ embeds in $Y$.
Earlier today, I asked the question:
>
> *Is this a well-quasi-order on the completely metrizable spaces?*
>
>
>
This was short-sighted, as Tom Goodwillie has pointed out in the comments that the closed surfaces giv... | https://mathoverflow.net/users/70618 | Spaces that can't be embedded in the plane | It seems that the answer to the question is negative. However, it should be noted that I did not work out all details for the following counterexample.
Consider a graph $G$ that is not planar. Replace every vertex of $G$ by a segment and every edge of $G$ by a double-sided topologist's sine curve, obtaining a space $... | 3 | https://mathoverflow.net/users/15650 | 208798 | 99,656 |
https://mathoverflow.net/questions/204742 | 4 | I would like to know if there is a reference for the fact that the following diagram commutes:
$$
\begin{array}{ccccccccc}
0 & \to & H\_\*(X) \otimes H\_\*(Y) & \to & H\_\*(X\times Y) & \to & \mathrm{Tor}\_1^{\mathbb Z}(H\_\*(X),H\_\*(Y)) & \to & 0 \\
&& \downarrow && \downarrow && \downarrow & \\
0 & \to & H\_\*(Y) \o... | https://mathoverflow.net/users/22052 | Reference request: Flipping the factors in the Künneth formula | Larry Taylor just posted a preprint to the arXiv,
<http://arxiv.org/abs/1506.02063>
called *Splitting the Künneth formula*. Commutativity of your diagram is stated as Corollary 7.2, suggesting that the author is not aware of a published proof.
| 5 | https://mathoverflow.net/users/8103 | 208803 | 99,658 |
https://mathoverflow.net/questions/208804 | 0 | Is there a graph $G$ such that $\chi(G)$ is finite, but there is no [total coloring](http://en.wikipedia.org/wiki/Total_coloring) with finitely many colours?
| https://mathoverflow.net/users/nan | Graph with finite chromatic number but infinite total chromatic number | Yes there is: consider the complete bipartite graph $K\_{\omega,\omega}$, which has chromatic number 2, but every total coloring requires infinitely many colors.
| 2 | https://mathoverflow.net/users/8628 | 208805 | 99,659 |
https://mathoverflow.net/questions/208792 | 8 | Gabai's property R theorem is:
If the 0-surgery manifold of a knot $K$ is homeomorphic to $S^1\times S^2$, then $K$ is the unknot.
Recently, 3-manifold topology has been developed rapidly by Agol, Wise and many other mathematicians.
Is there an another simple proof for property R conjecture?
| https://mathoverflow.net/users/63984 | Simple proof for property R conjecture | Property R was reproved by Gordon and Luecke in the course of solving the knot complement problem - see [Corollary 3.2](http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0965210-7/home.html). They prove the stronger result (as did Gabai) that zero-frame surgery on a knot is [irreducible](http://en.wikipedia.o... | 16 | https://mathoverflow.net/users/1345 | 208809 | 99,660 |
https://mathoverflow.net/questions/208808 | 5 | I vaguely recall having read somewhere a definition similar to (but probably not exactly the same as) the following.
>
> **Definition (Blob)** Let $S\subset \mathbb{R}^n$ be a set, and $p \in S$. The *Blob* of $S$ at $p$ is defined to be the subset
> $$ \left\{v \in \mathbb{S}^{n-1} \middle| \exists (x\_n) \subse... | https://mathoverflow.net/users/3948 | Generalisation of "tangent space" to not-necessarily connected sets | Ah, I think I found the answer. (For future reference these notions apparently come up in *non-smooth analysis*.) The "Blob" I defined is closely (and rather trivially) related to the notion of the **contingent cone** or **Bouligand tangent cones** (the [French Wikipedia has a better article than the English on the sub... | 3 | https://mathoverflow.net/users/3948 | 208813 | 99,661 |
https://mathoverflow.net/questions/208829 | 4 | I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or differential geometry to numerical methods for PDEs.
A list of a few professors researching anything at all similar to ... | https://mathoverflow.net/users/74762 | Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? | M. Desburn at Caltech.
Anil Hirani at UIUC.
D. Arnold at Minnesota.
And a few more.
Quite a bit of the work in this area has originated from Jerry marsden or his collaborators.
Keywords "Discrete exterior calculus", "Discrete differential geometry",
"symplectic integrators" etc.
| 5 | https://mathoverflow.net/users/74763 | 208830 | 99,664 |
https://mathoverflow.net/questions/208825 | 5 | Suppose $C$ is a curve of genus $2$, does $C$ admit a surjective morphism onto some elliptic curve $E$?
| https://mathoverflow.net/users/nan | Can every genus $2$ curve be written as ramified cover of elliptic curve? | dhy's comment is correct, of course. You can also see that the answer is "no" by dimension counting. Fix a degree $d$. By a Riemann-Hurwitz computation, a degree $d$ cover of a genus $1$ curve by a genus $2$ curve is ramified over $2$ points (or doubly ramified over one point.) Given a positive integer $d$, a genus $1$... | 7 | https://mathoverflow.net/users/297 | 208836 | 99,666 |
https://mathoverflow.net/questions/208771 | 3 | Say we have a labeled, binary unrooted tree $T$, i.e. each node has either 1 or 3 neighbors.
Denote by $L(T)$ the set of leaves (degree-one nodes) of $T$.
For some $L \subseteq L(T)$, denote by $t(L)$ the smallest subtree of $T$ containing $L$.
That is, $t(L)$ is the minimal (in terms of nodes) connected induced s... | https://mathoverflow.net/users/56791 | Partitioning a binary tree into vertex-disjoint subtrees | **UPDATE.** My original answer addressed the case when $t(L)$ is defined as the minimal binary subtree (which may be rooted, i.e. have one vertex of degree 2). This is now posed as Case 1.
Case 2 down below addresses the case when $t(L)$ is the minimal arbitrary subtree (i.e., connected acyclic graph).
**Case 1. $... | 2 | https://mathoverflow.net/users/7076 | 208843 | 99,671 |
https://mathoverflow.net/questions/208827 | 3 | How do we show that
$$\prod\limits\_{k=1}^{n} \tau(k) = 2^{n (\log \log n + C) +
\phi(n)},$$
where $\tau(k)$ is the number of divisors of $k$, the constant $C$ is given by
$$C = \gamma + \sum\_{\nu = 2}^{\infty} \left\{ \log\_2 \left(1 + \frac{1}{\nu}\right) - \frac{1}{\nu} \right\} \left(\sum\limits\_{p \textrm{... | https://mathoverflow.net/users/62680 | Product of $\tau(k)$ | The proof relies on the following identity, which is straightforward to verify:
$$ \sum\_{k=1}^\infty\frac{\log\_2\tau(k)}{k^s}=\zeta(s)\sum\_p\sum\_{\nu=1}^\infty\frac{\log\_2\left(1+\frac{1}{\nu}\right)}{p^{\nu s}},\qquad\Re s>1.$$
Here $p$ runs through the prime numbers. This implies
$$ \sum\_{k=1}^\infty\frac{\log\... | 9 | https://mathoverflow.net/users/11919 | 208844 | 99,672 |
https://mathoverflow.net/questions/208845 | 4 | Consider the following fusion categorie $F(i)$ with integer parameter $i$. Simple objects are $1,a,A,B$ (where $a$ and $A$ are conjugates). Nontrivial fusion rules are $a\bigotimes{a}=A$ (and conjugate), $B\bigotimes{a}=B$ (and conjugate), $B\bigotimes{B}=1\bigoplus{a}\bigoplus{A}\bigoplus{i\*B}$ (i.e. multiplicity $i$... | https://mathoverflow.net/users/11504 | Fusion categories: If infinity were an integer | Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,A\}$, and is isomorphic to $\mathbb Z/3$.
The paper <http://arxiv.org/pdf/1401.1879v2.pdf> shows that there's only fin... | 7 | https://mathoverflow.net/users/5690 | 208851 | 99,673 |
https://mathoverflow.net/questions/208839 | 5 | Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces.
If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism whose geometric fibers are actual curves of genus at least two.)
Are there any other types of properties which one c... | https://mathoverflow.net/users/74766 | Algebraic spaces which are automatically schemes | For a scheme $S$, every proper finitely presented map $f:X \rightarrow S$ from an algebraic space $X$ admitting a line bundle $L$ that is ample on each geometric fiber (which of course forces such fibers to be projective schemes, essentially by the very definition of ampleness for algebraic spaces) is automatically a s... | 6 | https://mathoverflow.net/users/70739 | 208853 | 99,674 |
https://mathoverflow.net/questions/208820 | 14 | *Question*: Should we post on arXiv only papers in publishable shape (or very close)?
This question should be distinguished from the following:
[Should one post a paper on the arXiv if it is not intended to be published?](https://mathoverflow.net/q/201728/34538)
in the sense that a paper which has not a publis... | https://mathoverflow.net/users/34538 | Should we post on arXiv only papers in publishable shape (or very close)? | The documentation of arXiv has this to say (my emphasis):
>
> **Inappropriate format.** arXiv accepts **only** submissions in the form of an **article that would be refereeable by a conventional publication venue.** This **excludes** abstract-only submissions, submissions without references, book announcements or r... | 24 | https://mathoverflow.net/users/nan | 208856 | 99,675 |
https://mathoverflow.net/questions/208860 | 0 | Let $G=(V,E)$ be a graph and $\mathrm{Ind}(G)$ be the collection of its independent sets.
We call a graph automorphism $f:V \to V$ of $G$ *good* if it is non-trivial and $f(\mathrm{Ind}(G))=\mathrm{Ind}(G)$, i.e. $f$ maps every independent set of $G$ into another independent set of $G$.
(i) Is there a (elegant/min... | https://mathoverflow.net/users/13388 | Graph automorphisms that preserve independent sets | I thought that if $f$ is graph automorphism then it preserves both edges and non-edges. In that case, any automorphism would be good, and as @Turbo said, some graphs have trivial autormorphism groups only. See <https://en.wikipedia.org/wiki/Graph_automorphism>
| 4 | https://mathoverflow.net/users/74773 | 208865 | 99,680 |
https://mathoverflow.net/questions/208002 | 3 | The upper asymptotic density on $\mathbf Z$, viz. the function
$$
{\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup\_{n \to \infty} \frac{|X \cap [1,n]|}{n},
$$
has a ''symmetric variant'', which is given by the function
$$
{\sf d}\_{\rm ev}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup\_{n ... | https://mathoverflow.net/users/16537 | Who needs a symmetric upper asymptotic density on the integers? | Let $V$ be an affine variety defined over $\mathbb Z$, i.e., a variety defined by a (finite) set of polynomial equations in $\mathbb Z[X\_1,\ldots,X\_n]$. The set of integral points of $V$, denoted $V(\mathbb Z)$, is simply the set of integral solutions $\boldsymbol x =(x\_1,\ldots,x\_n)\in\mathbb Z^n$ to the set of eq... | 3 | https://mathoverflow.net/users/11926 | 208879 | 99,684 |
https://mathoverflow.net/questions/208874 | 5 | What is the best lower bound in terms of $k$ on the number of edges in a $3$-uniform hypergraph that is not $k$-colorable?
Thanks in advance.
| https://mathoverflow.net/users/68522 | Is it true that any $3$-uniform hypergraph that is not $k$-colorable must have $\Omega(k^3)$ edges? | The following paper of Alon shows that the quantity you're after, $m(k)$, the minimum number of edges of a $3$-uniform hypergraph which is not $k$-colourable, is indeed $\asymp k^3$.
More precisely, he shows that
$$ 2\left\lceil \frac{k}{3}\right\rceil \left\lfloor \frac{2k}{3}\right\rfloor^2 < m(k) \leq \binom{2k+... | 7 | https://mathoverflow.net/users/385 | 208881 | 99,685 |
https://mathoverflow.net/questions/208828 | 1 | Let $P\_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}\_{> 0}^n$, $\sum\_i p\_i = 1$ and define the following mapping:
$$
T(\mathbf{p}) = \mathbf{p} \lim\_{m \to \infty} P\_{\mathbf{p}}^m.
$$
Note that $T(\mathbf{p})$ can be seen as ... | https://mathoverflow.net/users/74761 | Row-stochasticity of the Jacobian matrix of a stationary distribution | There are a few issues here:
1. The limit $Q\_{\mathbf p}:=\lim\_m P\_{\mathbf p}^m$ may not exist in general.
2. It is unclear if the limit is differentiable in $\mathbf p$, even when it exists and $P\_{\mathbf p}$ is differentiable in $\mathbf p$.
3. It is unclear how you define the Jacobian (matrix?). For it be r... | 2 | https://mathoverflow.net/users/36721 | 208885 | 99,686 |
https://mathoverflow.net/questions/208886 | 11 | I am reading about 'Exponential Sums' in the book 'Analytic Number Theory' by Iwaniec and Kowalski. On page 199 they mention the bound:
$$|S\_f(N)|^2 \le N +2N^2q^{-1}+4(N+q)\log q \tag{1}$$
where, $\displaystyle S\_f(n) = \sum\limits\_{n=1}^{N} e^{2\pi i(\alpha n^2 +\beta n)}$ and $1 \le q \le 2N$ satisfies $\disp... | https://mathoverflow.net/users/62680 | Iwaniec-Kowalski Exponential Sum for Quadratic Function | Regarding your first question, observe that (2) is equivalent to
$$ |S\_f(N)|^2 \le 4N^2q^{-1}+4N\log q+q\log^2 q. $$
Hence, by (1), it suffices to show
$$ N +2N^2q^{-1}+4N\log q + 4q\log q \le 4N^2q^{-1}+4N\log q+q\log^2 q,$$
which in turn is equivalent to
$$ N+4q\log q \le 2N^2q^{-1}+q\log^2 q.$$
Here we have $N\leq ... | 12 | https://mathoverflow.net/users/11919 | 208888 | 99,687 |
https://mathoverflow.net/questions/208891 | 2 | Do there exist postive integers $a,b,c,x,y,p,q$ such $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are all Pythagorean triples? That is, does the system
$$\begin{cases}
a^2+b^2=c^2\\
x^2+y^2=a^2\\
p^2+q^2=b^2
\end{cases}$$
have a postive integer solution?
| https://mathoverflow.net/users/74676 | Is it possible that $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are Pythagorean triples simultaneously? | Yes, for example
$$ (a,b,c,x,y,p,q) = (145,10512,10513,143,24,7920,6912).$$
**P.S.** I found this example by looking for $(u,v)$ such that $u,v,u+v,u-v$ are all sums of two squares (cf. ARupinski's comment). I took $u=73$ and $v=72$.
| 8 | https://mathoverflow.net/users/11919 | 208892 | 99,688 |
https://mathoverflow.net/questions/208910 | 0 | I am trying to find an example of an algebra over a field of characteristic p (prime) which satisfies anti-symmetry and Jacobi identity but is not a lie algebra. i.e., [x,x] is not zero.
Can one provide a pattern or a general method to modify existing Lie algebras in order to save anti-symmetry and Jacobi but **not*... | https://mathoverflow.net/users/74797 | Lie Algebra, counterexample | You must have $p=2$ otherwise antisymmetry implies the alternate identity $[x,x]=0$.
The minimal example taylored on the pattern below I can provide is the (multiplicative) subalgebra (over the field $\mathbb{Z}/2\mathbb{Z}$) generated by
$$
E\_{12}+E\_{23}=
\begin{pmatrix}
0 & 1 & 0\\
0 & 0 & 1\\
0 & 0 & 0
\end{pma... | 6 | https://mathoverflow.net/users/25256 | 208912 | 99,696 |
https://mathoverflow.net/questions/208913 | 1 | Let $f\_1, f\_2,\ldots,f\_N$ be some affine polynomials. We consider the question if these polynomials have a common (affine) root. By homogenizing these polynomials, we can associate a projective resultant, in the literature, it is called Macaulay's or Dixon's resultant. But a problem arises: these $f\_i$ may have a c... | https://mathoverflow.net/users/11376 | Conditions for the consistency of a system of affine polynomials | The example of three linear equations in two variables is typical. The equations $ax+by+c=0$, $dx+ey+f=0$ and $gx+hy+i=0$ have a common root if
* $\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} =0$ AND one of the following two conditions is met
* EITHER at least one of $\det \begin{pmatrix} a ... | 5 | https://mathoverflow.net/users/297 | 208917 | 99,698 |
https://mathoverflow.net/questions/208915 | 1 | Let's call a nice matrix a square matrix of size $n$ with elements from $\{1,...,n\}$ such that every row and every column contains all the number $1,...,n$,
What is the number of nice matrices ?
a possible generalization is the following: the elements of the matrix are exactly $\{1,...,n^2\}$ and we ask that the s... | https://mathoverflow.net/users/66528 | Number of Nice Matrices | Ok, the posibillity of no formula existing fascinated me so I followed Wolfgang's advise (even if it was not meant for me) and found a formula on the internet. I quote:
>
> Theorem 3. Let $p(z)$ be any monic polynomial of degree $n$ and let $M\_n$ be the family of
> all $n × n$ matrices over $\{−1,+1\}$. Then
> ... | 4 | https://mathoverflow.net/users/41139 | 208920 | 99,700 |
https://mathoverflow.net/questions/208898 | -2 | Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C\_n(X)$ as in the book [The geometry of iterated loop spaces](http://www.google.com.sg/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CBwQFjAAahUKEwj7uO74wITGAhXlIqYKHZa-AJw&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D1... | https://mathoverflow.net/users/41075 | configuration space and iterated loop space | $C\_n(X)$ is not a monoid in any natural way; I never said it was. And the target $\Omega^n\Sigma^n X$ of $\alpha\_n$ has $n$ different loop products. The question is not meaningful as posed. Nevertheless the proof of Theorem 6.1 (the approximation theorem) answers it by describing $\alpha\_n$ as the fiber map of a map... | 5 | https://mathoverflow.net/users/14447 | 208922 | 99,701 |
https://mathoverflow.net/questions/208721 | 4 | Referring to [a previous question](https://mathoverflow.net/questions/199831/help-with-the-integral-int-0-infty-log-left1-fracs24-pi2-log/200106#200106), I am trying to do the following integral :
$$\phi(s)=i\int\_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log\sqrt{1+ix} \right )^{2}}{\pi ^{2}} \right ]-\log \left[... | https://mathoverflow.net/users/20782 | Help with a difficult integral | 
If asymptotics is enough, an accurate large-$s$ approximation is $\phi(s)\approx 2.8-\ln |s|$, see the plot above (the error is of order $1/s$, so almost invisible on the scale of the plot for the largest $s$); for small $s$ the function starts off quadratically,... | 3 | https://mathoverflow.net/users/11260 | 208928 | 99,702 |
https://mathoverflow.net/questions/208894 | 0 | I am trying to solve the following optimization problem for the vector $ y $, where $ A\_i $ are some given matrix (maybe low rank) and $ x\_i $ are unconstrained
$$ \min\_{y, x\_i} \sum\_{i=1}^J || y - A\_i x\_i ||\_2^2, \;\;\;\;\text{ subject to } ||y||\_2 = 1, \;\; || A\_i x\_i ||\_2 = 1 \;\; \forall i$$
I believe... | https://mathoverflow.net/users/29887 | Prove that the following two optimization problems are equivalent | The optimal values of the two optimization problems are not identical. Consider for example $J=3$, $A\_1=\begin{pmatrix} 1 \\0 \end{pmatrix}$ and $A\_2=A\_3=\begin{pmatrix} 0 \\1 \end{pmatrix}$. The optimal values for the first optimization problem are $y=\frac{1}{\sqrt{5}} \begin{pmatrix}\pm 1\\\pm 2\end{pmatrix}$. Th... | 2 | https://mathoverflow.net/users/35593 | 208929 | 99,703 |
https://mathoverflow.net/questions/207529 | 7 | Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ?
I don't think so but I couldn't find a reference.
| https://mathoverflow.net/users/17261 | Do non-normal states exist in the Solovay model? | The question is essentially already answered in the comments and answers, but I thought it might be valuable to combine these observations into something definitive.
Solovay's model is not unique. The model produced by Solovay's construction depends on the choice of ground model, and on the choice of generic filter. ... | 5 | https://mathoverflow.net/users/66833 | 208971 | 99,713 |
https://mathoverflow.net/questions/208966 | 6 | **EDIT:** Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov.
**Questions 1.** Can one define a measure $K$ on $S$ (thought to be an analogue of the Gauss curvature) satisfying the following properties:
(a) if the metric on $S$ is smo... | https://mathoverflow.net/users/16183 | Gauss-Bonnet formula for 2-dimensional Alexandrov spaces | The answer to question 1 is almost yes. This more or less follows from the work of A. D. Alexandrov on convex surfaces. See "The intrinsic geometry of convex surfaces" by Alexandrov, and also "Intrinsic geometry of surfaces" by A. D. Alexandrov and V. Zalgaller (which is written in a more general context).
Note: the ... | 5 | https://mathoverflow.net/users/8887 | 208978 | 99,715 |
https://mathoverflow.net/questions/208995 | 4 | I was wondering if there's any attempt to generalize the commutator for something general for more than two terms.
Here's what I was thinking of, for $[A,B]=AB-BA$, so for three terms:
$[A,B,C] = ABC+BCA+CAB-BAC-CBA-ACB$
where I have taken a plus sign for a cyclic permutation of $ABC$, and a minus sign for acycli... | https://mathoverflow.net/users/13904 | Generalizing the commutator and anti-commutator | This generalization is called a ternary commutator, and there are $n$-commutator generalizations, as explained [here:](http://arxiv.org/abs/1207.6312)
| 7 | https://mathoverflow.net/users/11260 | 208997 | 99,719 |
https://mathoverflow.net/questions/208991 | 1 | Let $C\subset \mathbb{R}^{n}$, $C'\subset\mathbb{R}^{m}$ be two convex sets with a non-empty interior. A function $F\: : \: C\to C'$ is said to be differentiable at $x\in C$ if there exists a linear map $dF\_{x}\: : \: \mathbb{R}^{n}\to\mathbb{R}^{m}$ such that
$$\quad \frac{|| F(y)-F(x)-dF\_{x}(x-y)||}{||y-x||}\to 0... | https://mathoverflow.net/users/41970 | Extension of a smooth function from a convex set | Some answers are in the paper
* MR1474553 (98i:46040)
Kriegl, A.(A-WIEN)
Remarks on germs in infinite dimensions. (English summary)
Acta Math. Univ. Comenian. (N.S.) 66 (1997), no. 1, 117–134.
pdf available via [EMIS](http://www.univie.ac.at/EMIS/elibm/journals/index.html)
From the review:
"Now what happens if... | 2 | https://mathoverflow.net/users/26935 | 209007 | 99,723 |
https://mathoverflow.net/questions/208994 | 3 | I need to choose $k$ pairs of numbers out of first $n$ natural numbers such that the elements in each pair are $l$ distance apart. For example, if $n = 10, k = 3$ and $l = 2$, $\{(1,3),(4,6),(7,9)\}$ is a valid choice. How many such valid choices are there? I worked out for $l = 1$ and it seems to be $\binom{n-k}{k}$.
... | https://mathoverflow.net/users/73880 | Choosing k pairs l distance apart from n numbers | Presumably $n > \ell$ to make this nontrivial.
Consider $V = \{1,\ldots, n\}$ as the vertices of a graph with edges $(i,i+\ell)$.
This has $\ell$ connected components corresponding to the congruence classes mod $\ell$, each of which is a path graph. If $\ell \mid n$, each component has $n/\ell$ vertices, otherwise ther... | 4 | https://mathoverflow.net/users/13650 | 209008 | 99,724 |
https://mathoverflow.net/questions/208802 | 2 | Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also
$$
f\_B:= \frac1{|B|}\int\_B f \, dx.
$$
Suppose $f \in L\_{\rm loc}^p(\Omega)$ for all $1<p<\infty$ and
$$\tag 1
\left|\{x \in B : |f(x)-f\_{2B}| > \lambda\}\right| \le c\_1 \exp\left(-c\_2\frac\lambda{\left[\fr... | https://mathoverflow.net/users/46298 | $BMO$-property via a John-Nirenberg type estimate? | In the simpler dyadic setting the answer to your question is no.
Consider independent Bernoulli ($\pm 1$) random variables, aka Rademacher functions, $X\_1,X\_2,\dots$. Take any function that is quadratic in them:
$$f := \sum\_{i < j} c\_{ij} X\_i X\_j,$$
$$\sum\_{i<j} |c\_{ij}|^2 < \infty$$
(it can include diagona... | 1 | https://mathoverflow.net/users/22758 | 209020 | 99,728 |
https://mathoverflow.net/questions/208985 | 3 | Let $D$ define the differential operator $-\frac{\partial^2}{\partial x^2}$ on $\mathbb R$.
Let $\xi\notin\mathbb R$ be a complex number.
Is it true that
$$
(D-\xi)C\_c^\infty({\mathbb R})
$$
is dense in $C\_c({\mathbb R})$, where the latter space has the usual locally convex inductive limit topology?
| https://mathoverflow.net/users/nan | Density of Laplace image | I introduce the following notation: let $D\_\xi = D-\xi$; let $\chi\colon \mathbb{R} \to \mathbb{R}$ be any smooth function such that $\chi(x) \equiv 0$, for $x<-1$, and $\chi(x) \equiv 1$, for $x>1$; also let $G\_\xi$ and $G^\pm\_\xi$ be the following integral operators
\begin{align}
G\_\xi[f](x) &= \int\_{\mathbb{R}... | 1 | https://mathoverflow.net/users/2622 | 209024 | 99,730 |
https://mathoverflow.net/questions/208983 | 5 | Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\delta>0$, $a \in R$ and $f \in C\_b([0,1])$ (continuous and bounded). (hence weak-\*-topology)
[Here](https://math.stack... | https://mathoverflow.net/users/74850 | Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space? | It is not sequential. Here is an explicit counterexample.
Notation: $\|\mu\|$ is the total variation norm of $\mu$, and $\bar{B}\_r \subset M([0,1])$ is the closed norm ball of radius $r$ centered at $0$. $\delta\_x$ is the Dirac measure which places a unit point mass at $x \in [0,1]$.
For each positive integer $n ... | 5 | https://mathoverflow.net/users/4832 | 209038 | 99,735 |
https://mathoverflow.net/questions/208140 | 1 | Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - \frac{1- G(x)}{G'(x)}$$ is monotonically non-decreasing.
Let $F\_1,F\_2 \colon [0,1] \to [0,1]$ be regular twice differentiab... | https://mathoverflow.net/users/38889 | Is regularity closed under products? | The answer is no. In the definition of the regularity, we assume that $G'>0$ on $(0,1)$ and the monotonicity in question is over $(0,1)$. Suppose for a moment that $G$ is any cumulative distribution function (cdf) on $\mathbb{R}$ such that $G=0$ on $(-\infty,0)$, $G=1$ on $[1,\infty)$, $G$ is twice differentiable on $(... | 2 | https://mathoverflow.net/users/36721 | 209046 | 99,736 |
https://mathoverflow.net/questions/209039 | 8 | The following problem is a general form of [another problem](https://mathoverflow.net/questions/209541/avoiding-countable-subgroups-of-a-group-homeomorphic-to-the-cantor-space) (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not appl... | https://mathoverflow.net/users/2415 | Avoiding countable subgroups of general uncountable groups | **Counterexample for Problem 1:** According to [this answer](https://math.stackexchange.com/a/1226896/111012) Saharon Shelah constructed a "[Jónsson group](https://en.wikipedia.org/wiki/J%C3%B3nsson_cardinal)" of order $\aleph\_1,$ i.e., an uncountable group $G$ in which every proper subgroup is countable. Choose an el... | 18 | https://mathoverflow.net/users/43266 | 209049 | 99,738 |
https://mathoverflow.net/questions/209045 | 5 | I just asked a question which is related to the one I'm about to ask, but I realized my question can be reduced to the following: let $G$ be a locally compact abelian group with Haar measure $\mu$, and $H$ a discrete subgroup of $G$. Then $G/H$ is locally compact with Haar measure $\bar{\mu}$.
I believe it should be... | https://mathoverflow.net/users/38145 | Why is it possible to normalize the Haar measure on the quotient? | This is possible. Suppose that $\mu(W)<\infty$ and that $\pi(W)$ is measurable.
Then $\mu(W)$ is the supremum of all compact sets $K$ contained in $W$, as $\mu$ is a Radon measure.
The same holds for $\pi(W)$.
For a compact set $K$ the measure $\mu(K)$ is the infimum of all integrals $\int\_Gf(x)d\mu(x)$ where $f\ge 0$... | 4 | https://mathoverflow.net/users/nan | 209053 | 99,740 |
https://mathoverflow.net/questions/209052 | 0 | Which integers $n>2$ have the following property?
There is a group $G$ such that
* $G^n \cong G$; and
* for all integers $k$ with $1<k<n$ we have $G^k\not \cong G$.
| https://mathoverflow.net/users/8628 | Groups with $G^n \cong G$ for some integer $n$ | This is possible with abelian groups for any $n$; see [this answer](https://mathoverflow.net/a/10194/75) to a very similar question.
| 5 | https://mathoverflow.net/users/75 | 209054 | 99,741 |
https://mathoverflow.net/questions/209010 | 10 | Let $C\_n$ be a sequence of rectifiable simple closed curves in $\mathbb{R}^2$ that converge to a rectifiable simple closed curve $D$ in the Hausdorff topology. It is easy to construct examples where
$$\limsup\_{n \mapsto \infty} \text{length}(C\_n) \neq \text{length}(D).$$
**Question 0**: I actually do not know an... | https://mathoverflow.net/users/74865 | Continuity of length and area | **The answer of question 1 is positive**. In fact, you can even replace the $\limsup$ by a $\liminf$: this is Golab's theorem that the $1$-dimensional Hausdorff measure is lower semi-continuous on the set of compact sets of the plane. I did not find a good reference to this theorem, but it is cited in [a paper by Rapha... | 5 | https://mathoverflow.net/users/4961 | 209057 | 99,742 |
https://mathoverflow.net/questions/209048 | 2 | Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that $P$ is elliptic on $M \setminus \{q\}$. Can we say that $(Pu, u) \cong \Vert u\Vert^2\_{H^1}$, or at least $(P(\varphi u... | https://mathoverflow.net/users/74880 | Elliptic regularity of second order pseudos | At least the answer to your second question is yes.
Let $\varphi\in C^\infty\_c(M\backslash\{q\})$ be given. Choose a function $\tau\in C^\infty\_c(M,[0,1])$ which is equal to $1$ in a neighbourhood of $q$ and which has a support disjoint from the support of $\varphi$. Put a Riemannian metric on $M$, so that we have ... | 1 | https://mathoverflow.net/users/58125 | 209059 | 99,743 |
https://mathoverflow.net/questions/209056 | 1 | It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is
What can be said about a finite group $G$ for which $\chi(x)=0$ for all irreducible character $\chi$ and elements $x\in G\setminus Z(\chi)$?
Here, $Z(\chi)=\{g\in G:|\chi... | https://mathoverflow.net/users/40723 | Groups with many vanishing elements | These appear to be the groups that I was asking about in [this older MO question](https://mathoverflow.net/questions/56380/characters-on-a-finite-group-with-extremal-behaviour) -- see in particular the answers by Isaacs and Ladisch.
| 3 | https://mathoverflow.net/users/763 | 209063 | 99,744 |
https://mathoverflow.net/questions/209069 | 2 | Let $S$ be the sub-algebra generated by the set $\{x\_1x\_2y\_3y\_4, x\_1x\_3y\_2y\_4, x\_1x\_4y\_2y\_3, x\_2x\_3y\_1y\_4, x\_2x\_4y\_1y\_3, x\_3x\_4y\_1y\_2\}$ of homogeneous polynomials of degree $4$. I need to study the projective variety $Proj(S)$. The relation among the above set of polynomials are $f\_1f\_6=f\_2f... | https://mathoverflow.net/users/74889 | sub-variety of (P^1)^4 | You can think of this ring as the semigroup ring of the semigroup $S$ generated by
$$(1,0,0,1),(-1,0,0,1),(0,1,0,1),(0,-1,0,1),(0,0,1,1),(0,0,-1,1).$$
The above semigroup elements correspond to $f\_1,f\_6,f\_2,f\_5,f\_3,f\_4$ respectively.
I believe that after saturation at the origin the semigroup ring is the polyto... | 3 | https://mathoverflow.net/users/38468 | 209073 | 99,746 |
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