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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/185287 | 3 | I asked the question at MSE
<https://math.stackexchange.com/questions/982388/simple-finite-series-with-reciprocal-factorials>
but got no answer or comment (it is not a homework).
I'm trying to find the following sum:
$$
\sum\_{k=0}^n{1\over(n-k)!}{x^{k+2}\over k+2}.
$$
The most obvious way is to differentiate wrt to... | https://mathoverflow.net/users/60855 | Finite series with reciprocal factorials | According to Maple your sum is:
$$ \,{\frac {{x}^{2}{\_3F\_1(1,2,-n;\,3;\,-x)}}{2n!}} $$
```
1/2/n!*x^2*hypergeom([1, 2, -n],[3],-x)
```
| 2 | https://mathoverflow.net/users/12481 | 185289 | 92,119 |
https://mathoverflow.net/questions/185176 | 6 | Is there an infinite cardinal $\kappa$ for which the following statement (S) true?
(S) : There is a topology $\tau\_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq \kappa$ there is a binary relation $\sim$ on $\kappa$ such that $(X,\tau)\cong (\kappa,\tau\_\kappa)/\sim$.
(I'm trans... | https://mathoverflow.net/users/8628 | Universality with respect to quotients | The answer is no, by the argument given by Henno in his comment.
It true that there are exactly $2^{2^\kappa}$-many non-homeomorphic topologies on a set of size $\kappa \geq \aleph\_0$. See for example [this](https://math.stackexchange.com/a/65742) nice argument by Stefan Geschke. On the other hand it is obvious that t... | 4 | https://mathoverflow.net/users/17836 | 185290 | 92,120 |
https://mathoverflow.net/questions/185283 | 12 | Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such curves are also $K$-isogenous, I believe we can conclude that they are $K$-isomorphic, but I cannot find any reference or ... | https://mathoverflow.net/users/48499 | $j$-invariants of elliptic curves over finite fields | As Chris and Rene said, your second question is all about twists. See specifically *The Arithmetic of Elliptic Curves* (Springer), Proposition X.5.4, which says that if the characteristic of $K$ is not 2 or 3, then the twists of $E/K$ are in one-to-one correspondence with $K^\*/(K^\*)^n$, where
$$ n=\begin{cases} 2&\te... | 16 | https://mathoverflow.net/users/11926 | 185291 | 92,121 |
https://mathoverflow.net/questions/182631 | 4 | Let $A$ be a von Neumann algebra and $A\_\*$ its predual. One can define a topology on the set $vN(A)$of all von Neumann subalgebras of $A$ called the Effros-Marechal topology. It is characterized as the coarsest topology on $vN(A)$ such that for all $\varphi \in A\_\*$ the map
$ B \mapsto ||\varphi\_{|B}|| $
is contin... | https://mathoverflow.net/users/59115 | Characterization of hyperfinitness by the Effros-Marechal topology | After some research, I now think this statement is true for any von Neumann algebra $M$.
Indeed, U.Haagerup and C.Winslow give in their paper "The Effros-Marechal topology in the space of von Neumann algebras" various descriptions of this topology.
In particular, they define for a net $(N\_i)\_{i \in I}$ of von Neum... | 1 | https://mathoverflow.net/users/60863 | 185302 | 92,128 |
https://mathoverflow.net/questions/184788 | 9 | Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all possible $(2m)^k$ sums.
**Question.** Does there exists a number $k$ independent of $m$ such that the points in the l... | https://mathoverflow.net/users/24494 | Separating points in the plane II | Let me make the next step after Mirko Swirko.
In [quite a beautiful paper](http://cms.math.ca/10.4153/CJM-1965-072-1) A. Hajnal proves the following.
**Theorem.** Let $G$ be a $m$-saturated graph --- that is, $G$ does not contain a complete $(m+1)$-subgraph, but it does after adding any new edge. Then either $G$ ha... | 2 | https://mathoverflow.net/users/17581 | 185305 | 92,130 |
https://mathoverflow.net/questions/185299 | 0 | How should determine solutions to equations of this form?
$$e^{-f(x)} + b f(x) = ax$$
Here $f(x)>0$ is real valued. Also $a>0$, $b>0$.
| https://mathoverflow.net/users/60861 | How to solve $e^{f(x)} + a f(x) + bx = 0$ | $f(x) = W(-\exp(-ax/b)/b) + ax/b$
where W is one of the branches of the [Lambert W function](http://en.wikipedia.org/wiki/Lambert_W_function).
| 1 | https://mathoverflow.net/users/13650 | 185310 | 92,133 |
https://mathoverflow.net/questions/185210 | 8 | **Question:** Suppose $(M,J)$ is an almost complex manifold, and $X$ and $Y$ are two almost complex submanifolds (i.e. $J(T(X)) \subset T(X)$ and $J(T(Y)) \subset T(Y)$). Then must $X \cap Y$ also be an almost complex submanifold?
**Clarification:** If $X$ and $Y$ intersect transversally, then the answer is yes.
... | https://mathoverflow.net/users/nan | Intersections in almost complex manifolds | Let me answer the easy part now and then come back later for the (slightly) more delicate singular case: If $(M,J)$ is a real-analytic almost-complex manifold and $X,Y\subset M$ are almost-complex submanifolds of $M$ whose intersection $X\cap Y$ is a submanifold, then $X\cap Y$ is an almost-complex manifold.
To see t... | 11 | https://mathoverflow.net/users/13972 | 185311 | 92,134 |
https://mathoverflow.net/questions/185272 | 6 | Does somebody have a reference (or an argument why it should be true) for the following statement?
“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 \to S^3$ which takes $K$ to $K'$, then there is also an ambient isotopy $\eta : (S^3,K) \to (S^3,K')$.”
It seems to me ... | https://mathoverflow.net/users/60638 | Reference for a fact (?) on homeomorphic knot complements | The result they use is Moise's theorem:
Moise, Edwin E. (1952), Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics. Second Series 56: 96–114,
This states that a 3-manifold has a unique PL structure, and a unique smooth structure.
Moise's theorem was considere... | 13 | https://mathoverflow.net/users/1465 | 185319 | 92,138 |
https://mathoverflow.net/questions/185327 | 6 | Say that a *polynomial recurrence relation* (my terminology)
for $f\_i$ is:
* $k$ initial conditions setting $f\_1,\ldots,f\_k$ to integers ($\in \mathbb{Z})$.
* A recurrence equation of the form $f\_i =$ a polynomial in
$f\_{i-1},\ldots,f\_{i-k}$.
*Example 1*: ($k=2$): $\;f\_1=1,\; f\_2=1$, and $f\_i=f\_{i-1}+f\_{... | https://mathoverflow.net/users/6094 | Polynomial recurrence relation covering the integers (and then Gaussian integers) | For the first question, let $f\_1=0$, $f\_2=1$, $f\_3=-1$, and $f\_i=-f\_{i-1}+f\_{i-2}+f\_{i-3}$.
| 12 | https://mathoverflow.net/users/2807 | 185329 | 92,141 |
https://mathoverflow.net/questions/185314 | 4 | **The Question.** Suppose $0 < \alpha < \beta$ are fixed, and $a\_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below
\begin{equation\*}
\int\_0^{T} \Big| \sum\_{\alpha T < n < \beta T} a\_n n^{-it} \Big|^2 dt?
\end{equation\*}
**What I know.** The mean value theorem for Dirichlet polynom... | https://mathoverflow.net/users/2627 | A lower bound on the $L^2$ norm of a Dirichlet polynomial | There may be no such bound if $\alpha$ and $\beta$ are suitably large. For example, using the first derivative test (see e.g. Titchmarsh Chapter 4)
one can see that if $T\le t\le 2T$ then
$$
\sum\_{10T\le n \le 20T} n^{it} = O(1).
$$
Therefore
$$
\int\_T^{2T} \Big| \sum\_{10T \le n\le 20 T} n^{it} \Big|^2 dt \ll ... | 2 | https://mathoverflow.net/users/38624 | 185342 | 92,144 |
https://mathoverflow.net/questions/185348 | 4 | Suppose $\kappa$ is an almost huge cardinal. Using the characterization of Theorem 24.11 in Kanamori's book, one can show that if $j : V \to M$ is an embedding derived from an almost-huge tower of measures, then $M^{<j(\kappa)} \subseteq M$, but $M^{j(\kappa)} \nsubseteq M$. Also, there is some embedding with target $j... | https://mathoverflow.net/users/11145 | almost huge embeddings and stationary correctness | I think the answer is no. If $\kappa$ is huge, and we derive an almost-huge tower from the huge embedding, then the embedding computed from the tower has stationary correctness.
Suppose $h : V \to M$ is a huge embedding and $h(\kappa) = \lambda$. Then $M$ computes stationary subsets of $\lambda$ correctly by the clos... | 3 | https://mathoverflow.net/users/11145 | 185358 | 92,148 |
https://mathoverflow.net/questions/185338 | 16 | Godel proved the consistency of the axiom of choice with the axioms of $\sf ZF$ by showing that given any model of $\sf ZF$, there is a definable class which satisfies $\sf ZFC$.
The proof uses a lot of the power of $\sf ZF$, in particular it uses transfinite recursion which is equivalent to the axiom schema of repla... | https://mathoverflow.net/users/7206 | Is there an $L$ like inner model for $\sf Z$? |
>
> This answer is based on Adrian Mathias' majestic paper [The Strength of MacLane Set Theory](https://www.dpmms.cam.ac.uk/%7Eardm/maclane.pdf) (published in the Annals of Pure and Applied Logic, 2001).
>
>
>
Let $\sf{M}$ be the system of set theory whose axioms consist of Extensionality, Null Set, Pairing, Uni... | 21 | https://mathoverflow.net/users/9269 | 185362 | 92,149 |
https://mathoverflow.net/questions/185337 | 17 | When the Stirling numbers of the second kind were introduced by James Stirling in 1730, it was not combinatorially; rather, the numbers ${n \brace k}$ were defined via the polynomial identity
$$
x^n = \sum\_{k=1}^n {n \brace k} x(x-1)\ldots (x-(k-1)). ~~(\star)
$$
Most modern treatments of the Stirling numbers introduc... | https://mathoverflow.net/users/21690 | Who first noticed that Stirling numbers of the second kind count partitions? | [Niels Nielsen,](http://en.wikipedia.org/wiki/Niels_Nielsen_(mathematician)) who coined the name "Stirling number of the second kind", gives the partition counting interpretation in his 1904 book [Handbuch der Theorie der Gammafunktion](https://archive.org/details/handbuchgamma00nielrich), page 70.
| 19 | https://mathoverflow.net/users/11260 | 185371 | 92,153 |
https://mathoverflow.net/questions/185369 | 1 | Assume we have $n$ points $p\_0\ldots p\_{n-1}$ which form a discrete metric space $V$ with metric $d$. Can we define a function $f:V\rightarrow \mathbb{R}$ with $f(p\_0) = 0$, $f(p\_1) = d(p\_0,p\_1)$ and $d(p\_i,p\_j) \geq |f(p\_i) - f(p\_j)|$ for all $i,j$?
If the $p\_i$ are points in a vector space with inner pro... | https://mathoverflow.net/users/3816 | Generalised "projection" of a metric space | Take $f(x)=d(x,p\_0)/2-d(x,p\_1)/2+d(p\_0,p\_1)/2$.
| 2 | https://mathoverflow.net/users/4312 | 185372 | 92,154 |
https://mathoverflow.net/questions/185331 | 4 | A $2k\times 2k$ circulant matrix $\ C$ takes the form
\begin{align}
C= \begin{bmatrix} c\_0 & c\_{2k-1} & \dots & c\_{2} & c\_{1} \\
c\_{1} & c\_0 & c\_{2k-1} & & c\_{2} \\
\vdots & c\_{1}& c\_0 & \ddots & \vdots \\
c\_{2k-2} & & \ddots & \ddots & c\_{2k-1} \\
c\_{2k-1} & c\_{2k-2} & \dots & c\_{1} & c\_0 \\ \e... | https://mathoverflow.net/users/58802 | the eigenvalues of a generalized circulant matrix | $C\_1$ isn't quite a "generalized $2k \times 2k$ circulant",
because it does not include $C$ as a special case. It is, however,
block-circulant: let $A\_0,\ldots,A\_{k-1}$ be the $2\times 2$ matrices
$$
A\_j = \left[\begin{array}{cc}
c\_{2j} & -c\_{2j-1} \\ c\_{2j+1} & -c\_{2j}
\end{array}\right]
\quad(j=0,1,\ldots,k-... | 8 | https://mathoverflow.net/users/14830 | 185377 | 92,155 |
https://mathoverflow.net/questions/185343 | 4 | Let
$f(x) = \begin{cases}\ln\frac{x}{e^x-1}, \quad x > 0; \\ 0, \quad\qquad x=0; \\ \ln\frac{x}{e^x-1}, \quad x < 0. \end{cases}$
Power series in 0:
$f(x) = \sum\_{n=1}^{\infty} a\_n x^n = -\frac{x}{2} - \frac{x^2}{24} +
\frac{x^4}{2880} + \ldots$
I am interested in estimates (asymptotic) for the sum of the coeffi... | https://mathoverflow.net/users/60880 | Estimate of the sum Taylor's coefficients | If $|z|\le 5/4$ then one may check that
$$
\Big| \frac{e^{z}-1}{z} - 1 \Big| \le \sum\_{k=2}^{\infty} \frac{|z|^{k-1}}{k!} < 1,
$$
so that $\log (z/(e^z-1))$ (being the logarithm of a non-zero holomorphic function) is a holomorphic function in the region $|z|\le 5/4$ (indeed in a slightly bigger region). Therefore b... | 4 | https://mathoverflow.net/users/38624 | 185380 | 92,156 |
https://mathoverflow.net/questions/185003 | 3 | Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection
$$
A
= \left\{
\left[ \frac{i}{n},\frac{i + 1}{n} \right] \times
\left[ \frac{j}{n},\frac{j + 1}{n} \right] ~ \Bigg| ~
i,j \in \{ 0,\ldots,n - 1 \}
\right\}
$$
of $ n^{2} $ smaller squares ... | https://mathoverflow.net/users/50614 | A problem on chains of squares — can one find an easy combinatorial proof? | I used to think that Sperner's proof is simple enough, but if you don't like it, maybe it's because what you're really looking for is an algorithm.
I can show that there is no fast algorithm for your problem, at least not something faster than for Sperner's lemma.
This is because your question, if defined appropriately... | 2 | https://mathoverflow.net/users/955 | 185383 | 92,157 |
https://mathoverflow.net/questions/185395 | 11 | This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch.
EDIT:
This is an edited version. Before I asked about roots without qualification.
| https://mathoverflow.net/users/22051 | How many isolated roots can a polynomial in $z$ and $\overline{z}$ have? | $2n$ is incorrect. The correct upper estimate is $n^2$ (if the number is finite).
Indeed, let $P(z,\overline{z})$ be a polynomial of degree $n$. Writing $z=x+iy$ and $\overline{z}=x-iy$ we obtain one complex equation of the form $P^\*(x,y)=0$,
but one complex equation is equivalent to two real equations, each of degree... | 21 | https://mathoverflow.net/users/25510 | 185405 | 92,164 |
https://mathoverflow.net/questions/185407 | 6 | Let $f : \mathbb{C} \to \mathbb{C}$ be a polynomial map of degree $q > 1$. Consider $E\_n \subset \mathbb{C}$ the set of periodic points with period (dividing) $n$; generally, $|E\_n| = q^n$. Since all the peridic points are limited to a bounded subset of $\mathbb{C}$, we know that there are always two among them withi... | https://mathoverflow.net/users/26522 | Clustering of periodic points for a polynomial iteration of $\mathbb{C}$ | Consider $f(x) = x^2 - 2$. This has the property that $f(2 \cos (x)) = 2 \cos(2x)$, so that $f\_n(2 \cos (x)) = 2 \cos(2^n x)$ (where $f\_n$ is $f$ iterated $n$ times). Thus $E\_n = \{2 \cos (x): \cos(x) = \cos(2^n x)\}$, and the two
greatest members of $E\_n$ are both greater than $2 \cos(\pi 2^{1-n})\approx 2 - \pi^2... | 9 | https://mathoverflow.net/users/13650 | 185408 | 92,165 |
https://mathoverflow.net/questions/185373 | 0 | Find a rational function $R(x)$ such that:
$1)$ For $i\in\{1,\dots,g\}$, $x\_{i}=x\_{i-1}+g$ with $x\_0=0$.
$2)$ For $i\in\{0,\dots,g-1\}$ $R(x\_i)=R(x\_i+1)=\dots=R(x\_i+g-1)=i+1$.
$3)$ $R(x\_g)=g+1$.
So $R(x)$ is defined at $g^2+1$ points.
These points and the value taken by these points have an arithmeti... | https://mathoverflow.net/users/10035 | Minimum degree rational function interpolation | The answer is no. I assume that the degree of a rational function is the maximum of degrees of its numerator and denominator (in the irreducible form).
Consider any interval $(x\_i+j,x\_i+j+1)$ with $0\leq i\leq g-1$ and $0\leq j\leq g-2$. If the denominator of $R(x)$ has no roots on this interval, then $R'(x)$ shoul... | 4 | https://mathoverflow.net/users/17581 | 185414 | 92,167 |
https://mathoverflow.net/questions/185423 | 6 | Can one construct a harmonic function $f$ defined in the unit disk with the condition $f(0)≥1$ such that area of $\{z∈D:f(z)>0\}$ is small enough, i. e. for every $\epsilon>0$ does there exist a function harmonic in the unit disk with the condition $f(0) \ge 1$ s.t. the Lebesgue planar measure of $\{z∈D:f(z)>0\}$ is le... | https://mathoverflow.net/users/37600 | Does there exist harmonic function with that property? | Yes you can. First of all the condition $|f(0)|\geq 1$ is irrelevant: you can always multiply your function on a positive constant.
Take an entire function $F$ for which the set $\{ z:|F(z)|>1\}$ is contained in the strip
$\{ x+iy:|y|<\pi\}$. Such a function is constructed in Hayman's book Meromorphic functions, sec... | 10 | https://mathoverflow.net/users/25510 | 185426 | 92,173 |
https://mathoverflow.net/questions/185361 | 2 | Relaxation of [the second question here](https://mathoverflow.net/questions/185327/polynomial-recurrence-relation-covering-the-integers-and-then-gaussian-integers).
Let $a(n)$ be recurrence of the form $a(n)=f(n,a(n-1)\ldots(a(n-k))$
with fixed initial terms.
(Observe that it might depend on $n$).
$f$ may contain... | https://mathoverflow.net/users/12481 | Can the Gaussian integers be covered by restricted recurrences? | If you allow radicals, then yes to Q1, and depending only on $a(n-1)$. Let:
$$x=\Re( a(n-1))$$
$$y=\Im( a(n-1))$$
$$pos(r) = \frac{1}{2}\left( 1 + \frac{\sqrt{r^2}}{r}\right)$$
Note $pos(r)$ is 1 on positive reals and 0 on negative reals. With this we can enumerate the Gaussian integers in progressively larger diam... | 2 | https://mathoverflow.net/users/nan | 185436 | 92,178 |
https://mathoverflow.net/questions/185351 | 7 | Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a question about structure of his proofs:
>
> Did Nelson attempts try to, for some statement $\phi$, prove both $\phi$ a... | https://mathoverflow.net/users/30186 | Does Nelson try to prove PA inconsistent directly? | See the discussion here: <https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html>. As Monroe says, he tried to prove that PA was inconsistent, not just $\omega$-inconsistent.
| 4 | https://mathoverflow.net/users/8133 | 185460 | 92,187 |
https://mathoverflow.net/questions/185488 | 7 | Searched-for situation: ***A compact connected Lie group acts effectively on a closed Riemannian manifold by isometries, such that there is only one orbit type of dimension strictly less than that of the manifold, and the orbit projection $M\to M/G$ is not a principal $G$-bundle.***
I could not find or construct any ... | https://mathoverflow.net/users/58125 | Lie group actions with only one orbit type, but not defining a principal bundle | Let $G=O(n+1)$ act on the $n$-sphere $S^n\cong O(n+1)/O(n)$.
Let $M=S^1\times S^n$ and let $G$ only act on the second factor
The effectiveness is the problem, but this is clear, as the action on $S^{n}$ by linearity determines the action on $R^{n+1}$.The projection is not a principal $G$-bundle, as indeed no element of... | 4 | https://mathoverflow.net/users/nan | 185494 | 92,196 |
https://mathoverflow.net/questions/185495 | 0 | This question, however looks innocent, looks non-trivial to me. Suppose that $X$ and $Y$ are Banach spaces and let $\alpha$ be any reasonable cross-norm on $X\otimes Y$. *Reasonable means that* $\|x\otimes y \| = \|x\|\|y\|$ for $x\in X, y\in Y$ and $\|f\otimes g\|=\|f\|\|g\|$ for $f\in X^\*$ and $g\in Y^\*$.
Can we ... | https://mathoverflow.net/users/60968 | Complementation in tensor products | Tensor products usually make me nervous, but this seems straightforward. Fix a finite sum $\sum x\_j\otimes y\_j$, then
\begin{align\*}
\| I\_X\otimes P(\sum x\_j\otimes y\_j) \|\_\alpha &= \|\sum g(y\_j)x\_j \otimes y\_0\|\_\alpha \\
&= \|\sum g(y\_j)x\_j\|\_X \|y\_0\|\_Y \\
&=\sup\_{f\in X^\*, \|f\|\leq 1} \left|\sum... | 2 | https://mathoverflow.net/users/13360 | 185499 | 92,199 |
https://mathoverflow.net/questions/185475 | 3 | For a given simple graph $G$ with $n$ vertices $v\_1,v\_2,\dots v\_n$, the corresponding degree sequence is $d\_1,d\_2,\cdots,d\_n$. My qusetion is:
>
> How to determine whether there exist subgraphs in $G$ such that it has the degree sequence of $d\_1',d\_2'\cdots,d\_n'$ (where $d\_i'\le d\_i$)? If exist, how to g... | https://mathoverflow.net/users/32866 | Existence of subgraphs when given its degree sequence | Let $G$ have vertices $v\_1,\dots,v\_n$. Let $Z(G)$ denote the convex
hull in $\mathbb{R}^n$ of all *ordered* degree sequences
$(\deg(v\_1),\dots ,\deg(v\_n))$ of subgraphs of $G$. One way of stating
the well-known Erdős-Gallai characterization of degree sequences of
spanning subgraphs of the complete graph $K\_n$ is t... | 6 | https://mathoverflow.net/users/2807 | 185505 | 92,200 |
https://mathoverflow.net/questions/185501 | 3 | Are there any methods to "solve" large systems of boolean equations?
$$x\_{i1}\vee x\_{i2}\vee x\_{i3} = b\_i, \quad\text{for}\quad i=1,\dots,N,$$
where $x\_i, b\_i \in\{0, 1\}$
For example
$$x\_{1}\vee x\_{2}\vee x\_{5} = 0$$
$$x\_{1}\vee x\_{3}\vee x\_{4} = 1$$
$$x\_{2}\vee x\_{3}\vee x\_{5} = 1$$
$$x\_{1}\vee x\... | https://mathoverflow.net/users/15604 | System of boolean equations, Satisfiability | You can reduce the problem to MAXSAT as follows. For each equation $x\_{i1}\lor x\_{i2}\lor x\_{i3}=1$, include directly the clause $x\_{i1}\lor x\_{i2}\lor x\_{i3}$. For each equation $x\_{i1}\lor x\_{i2}\lor x\_{i3}=0$, choose a fresh variable $y\_i$, and include the four clauses
\begin{gather}
y\_i\lor\neg x\_{i1}\\... | 3 | https://mathoverflow.net/users/12705 | 185514 | 92,203 |
https://mathoverflow.net/questions/185500 | 8 | What is known about the following problem?
>
> Reconstruct a string $\sigma$ of known length $n$ over a known
> alphabet $\Sigma$ from a collection of uniformly and independently
> chosen $k$-long subsequences of $\sigma$ where $k$ is fixed between
> $0$ and $n$.
>
>
>
*Recall that a $k$-long subsequence of... | https://mathoverflow.net/users/54632 | Reconstructing a string from random samples | Given the string $\sigma$, for any word $x$ of length $k$ the probability $P\_{n,k}(\sigma, x)$
that a randomly chosen $k$-long subsequence matches $x$ can be computed, e.g. using
$$ P\_{n,k}(\sigma,x) = \dfrac{k}{n} \delta\_{\sigma\_1, x\_1} P\_{n-1,k-1}(\sigma',x') + \left(1 - \dfrac{k}{n}\right) P\_{n-1,k}(\sigma',... | 5 | https://mathoverflow.net/users/13650 | 185520 | 92,205 |
https://mathoverflow.net/questions/185486 | 16 | There exist many constructions of infinite families of knots with the same Alexander polynomial. However, alternating knots seem very special. While there are also many result on restricting the form the Alexander polynomial of an alternating knot can have, is it known whether infinitely many alternating knots can have... | https://mathoverflow.net/users/27433 | Can infinitely many alternating knots have the same Alexander polynomial? | No, there cannot exist infinitely many alternating knots with the same Alexander polynomial.
To see why, suppose for the contrary that $K$ belongs to an infinite family $\{K\_n\}\_{n\in\mathbb{Z}}$ of alternating knots with $\Delta\_{K\_n}(t)=\Delta\_K(t)$. Immediately we have
$$ \det(K\_n ) = |\Delta\_{K\_n} (−1)| ... | 24 | https://mathoverflow.net/users/45976 | 185531 | 92,210 |
https://mathoverflow.net/questions/185523 | 8 | Is the union of the complete curves on $\mathcal M\_g$ Zariski dense? ($g \gg 0$)
I know it is hard to find higher-dimensional complete subvarieties of $\mathcal M\_g$, but a quasiprojective variety can have lots of complete curves but nothing higher dimensional, e.g. the complement of a codimension 2 linear subspace... | https://mathoverflow.net/users/60982 | Locus of complete curves on $\mathcal M_g$ | Let $M\_g^S$ be the Satake compactification of $M\_g$. It is a singular projective variety. For $g \geq 3$, the codimension of $M\_g^S \setminus M\_g$ is $\geq 2$. If $p \in M\_g$ is a point, then a sufficiently general linear subspace through $p$ cuts out a complete curve not meeting the Satake boundary. Conclusion: t... | 10 | https://mathoverflow.net/users/1310 | 185539 | 92,214 |
https://mathoverflow.net/questions/185507 | 1 | Let $Q$ be a (say 4x4) unitary matrix, distributed according to the Haar distribution. Denote the upper left 2x2 submatrix of $Q$ as $Q\_{1:2,1:2}$. I am interested in the following expectation:
$E(I - Q\_{1:2,1:2}^HQ\_{1:2,1:2})^{-1}$, where $H$ is Hermitian transpose and the expectation is across $Q\_{1:2,1:2}$. An... | https://mathoverflow.net/users/37257 | Expected value of the inverse of a random, truncated Haar matrix | I will first consider the case of a $4\times 4$ matrix $Q$, and generalize to a higher dimensional $Q$ at the end.
**A. The four-by-four case.**
It is helpful to start from the polar decomposition
$$Q=
\left(\begin{array}{cc}U'&0\\ 0&V'\end{array}\right)
\left(\begin{array}{cc}\sqrt{1-T}&\sqrt{T}\\
\sqrt{T}&-\sqr... | 2 | https://mathoverflow.net/users/11260 | 185543 | 92,216 |
https://mathoverflow.net/questions/185530 | 5 | Consider the anti-bidiagonal matrix $B\_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows
$$
B\_6=\begin{bmatrix} & & & & & 6\\
& & & & 5 & -5\\
& & & 4 & -4\\
& & 3 & -3\\
& 2 & -2\\
1 & -1
\end{bmatrix}.
$$
Its eigenvalues are $\lambda(B\_6)=\{1,-2,3,-4,5,-6\}$, a fact easily verified ... | https://mathoverflow.net/users/60984 | Anti-bidiagonal matrix with main anti-diagonal {1,2,3,...} and first sub-anti-diagonal {-1,-2,-3,...} has eigenvalues lambda={1,-2,3,-4,...} | This problem is [essentially the same as this one](https://mathoverflow.net/questions/156090/eigenvectors-of-a-particular-transition-matrix/156202#156202). In particular, let $J$ be the anti-diagonal identity matrix, and $P^{-1}$ be the matrix mentioned in the link above. Then, the matrix in the current post is nothing... | 10 | https://mathoverflow.net/users/8430 | 185545 | 92,217 |
https://mathoverflow.net/questions/185549 | 4 | I'm not sure under which 'algorithm' it falls under, but here is the problem:
I need to match each person to 5 people from the opposite gender (each guy gets 5 girls, each girl gets 5 guys). Not all guy-girl matchings/'edges' are possible, and the edges are weights. Also, I'm not looking for a perfect solution - jus... | https://mathoverflow.net/users/60841 | Polygamous stable marriage/ assignment problem | You can do this as a weighted network-flow problem where each guy is a source of $5$ units, each girl is a sink of $5$ units, and each possible arc has capacity of $1$ unit. You can solve it using linear programming, and the integrality theorem guarantees that a basic optimal solution is in integers.
| 8 | https://mathoverflow.net/users/13650 | 185551 | 92,221 |
https://mathoverflow.net/questions/185553 | 2 | Let $N(-/\mathcal C)\colon \mathcal C\to \mathbf{sSet}$ be the functor sending $c\in\mathcal C$ to the nerve of the coslice category $c/\mathcal C$.
Given a functor $K\colon\mathcal{C}\to \mathcal{D}$, I'm trying to prove that
$$
\text{Lan}\_KN(-/\mathcal C)\cong N(-/K)
$$
where RHS is a functor $\mathcal D\to \mathb... | https://mathoverflow.net/users/7952 | $ \text{Lan}_KN(-/\mathcal C)\cong N(-/K) $ | I have a vague memory that there is a slick way to see this but I've forgotten. (If I had to guess, perhaps the idea is to swap the roles played by the weight $d/K(-)$ and the diagram $-/\mathcal{C}$ and then apply the coYoneda lemma. You probably have to pass to bisimplicial sets to make this work.) In any case, a dir... | 8 | https://mathoverflow.net/users/2181 | 185556 | 92,224 |
https://mathoverflow.net/questions/185557 | 6 | Let $D,n,d$ be three positive integers.
I am looking for the number of monomials of degree $D$ in $n$ variables where each variable appears with exponent at most $d$.
As a result of an application of inclusion/exclusion principle I found the following expression
\begin{equation\*}
\sum\_j (-1)^j \binom{n}{j} \binom... | https://mathoverflow.net/users/60990 | Number of monomials of deg D where each variables has low degree | What does "tractable mean"? You are looking for the coefficient of $x^D$ in
$$\left(\sum\_{i=0}^d x^i\right)^n = \left(\frac{1-x^{d+1}}{1-x}\right)^n,$$ I believe the RHS gives something like the formula you wrote down...
| 3 | https://mathoverflow.net/users/11142 | 185558 | 92,225 |
https://mathoverflow.net/questions/185552 | 4 | Let $X$ be a hadamard space and $\gamma\_1, \gamma\_2 \colon \mathbb{R}\rightarrow X$ be two geodesics. Part 2 of Coroallary 2.5 in
<http://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-Theodor_Sturm/papers/paper41.pdf>
states that $f(t):=d(\gamma\_1(t),\gamma\_2(t))$ is convex. I wonder under what conditions $f$... | https://mathoverflow.net/users/35593 | Geodesic comparison in Hadamard space | The minimum of $J$ is not unique $m=3$, $n=4$, $E=\{(1,2),(1,3),(2,4),(3,4)\}$ and $X=\mathbb R^2$.
Assume that $a\_1$, $a\_2$ and $a\_3$ are vertices of huge equilateral triangle.
Then at the minimum of
$$d^2(a\_1,u\_1)+d^2(a\_2,u\_2)+d^2(a\_3,u\_3)+\\+d(u\_1,u\_2)+d(u\_1,u\_3)+d(u\_2,u\_3)$$
the points $u\_i$ li... | 1 | https://mathoverflow.net/users/1441 | 185560 | 92,227 |
https://mathoverflow.net/questions/185563 | 0 | I wonder how to express the determinant of a block covariance matrix. For example, I have a covariance matrix
$\Sigma=\left[
\begin{array}{cc}
\Sigma\_1 & \Sigma\_{12} \\
\Sigma\_{21} & \Sigma\_2 \\
\end{array}
\right]$
where $\Sigma\_1\in\mathbb{R}^{n\_1}\times\mathbb{R}^{n\_1}$, $\Sigma\_2\in\mathbb{R}^{n\_2}... | https://mathoverflow.net/users/60995 | Determinant of block covariance matrix | A well-known block determinant formula: if $\Sigma\_1$ is invertible,
$$\det \Sigma = \det(\Sigma\_1) \det(\Sigma\_2 - \Sigma\_{21} \Sigma\_1^{-1} \Sigma\_{12})$$ (and a similar formula if $\Sigma\_2$ is invertible).
| 4 | https://mathoverflow.net/users/13650 | 185566 | 92,231 |
https://mathoverflow.net/questions/185570 | 5 | Let us consider the set of all polynomials with the following properties:
i) all coefficients are integer;
ii) the leading coefficient equals one;
iii) all zeros are real and simple and belonging to $[-1.99,1.99] $.
Is this set finite or infinite?
| https://mathoverflow.net/users/35959 | Is the set of certain polynomials finite or infinite? | It is finite. This follows by combining two separate results:
1. *If $p \in \mathbb{Z}[x]$ is monic integer having all its complex roots lying in $[-2,2]$, then all these roots are of the form $2\cos(2\pi q)$ with $q \in \mathbb{Q}$.*
For the proof of this, write $p = \prod\_{i=1}^d (x - 2\cos(2\pi \, t\_i))$ and o... | 13 | https://mathoverflow.net/users/26522 | 185573 | 92,234 |
https://mathoverflow.net/questions/185442 | 1 | We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the [following question](https://mathoverflow.net/questions/150344/symplectic-structure-of-tangent-bundle-of-mathbbsn-1). We have two questions which are n... | https://mathoverflow.net/users/36688 | Two questions related to $TS^{2}$ as a holomorphic manifold | This discussion seems to involve two holomorphic structures on the smooth manifold $T S^2$.
**Structure 1** Identify $S^2$ with $\mathbb{CP}^1$. Place a complex structure on the real tangent bundle to $\mathbb{CP}^1$ by using the complex structure on $\mathbb{CP}^1$. This is the structure Alex Degtyarev's answer refe... | 6 | https://mathoverflow.net/users/297 | 185589 | 92,238 |
https://mathoverflow.net/questions/185584 | 1 | Suppose you want to construct a perfect Lie algebra with a nonabelian solvable radical $\mathfrak{r}$, say with a commutator series of length 2. What are the conditions that guarantee the Lie algebra will be perfect? Is it sufficient to have an irreducible representation of a semisimple Lie algebra $\mathfrak{s}$ on $\... | https://mathoverflow.net/users/15482 | perfect Lie algebra with a nonabelian solvable radical | Let $K$ be a field of char. zero, $\mathfrak{g}$ a f.dim. Lie algebra over $K$. Let $\mathfrak{r}$ be its solvable radical, $\mathfrak{s}=\mathfrak{g}/\mathfrak{r}$ (which is semisimple) and $V=\mathfrak{r}/[\mathfrak{r},\mathfrak{r}]$ (which is naturally an $\mathfrak{s}$-module). Then $\mathfrak{g}$ is perfect iff $V... | 1 | https://mathoverflow.net/users/14094 | 185593 | 92,239 |
https://mathoverflow.net/questions/185538 | 15 | The questions I'm going to ask are non formal because they concern decidability of decidability, and I couldn't find any references on that after some quick searches. I hope that this thread is still "formal enough" to be productive.
So, let us suppose that we're working with an ambient logic that is classical (we be... | https://mathoverflow.net/users/60985 | Decidability of decidability | $\newcommand\Con{\text{Con}}
\newcommand\Dec{\text{Dec}}$
Let $F$ be the formal system in which the proofs are to be carried
out, when it comes to your formal assertions of the form
$\Dec(\varphi)$. So we assume that $F$ is described by some
computable axiomatization. For example, perhaps $F$ is simply the
usual firs... | 14 | https://mathoverflow.net/users/1946 | 185598 | 92,241 |
https://mathoverflow.net/questions/185601 | 9 | I know how to build a basis of the vector space of cusp forms for the congruence subgroups $\Gamma\_1 (N)$ and $\Gamma\_0 (N)$, but I couldn't find in the literature how to build a basis for $\Gamma(N)$.
Also, Sage doesn't seem to be able to do this.
For instance, for $N=6$ and weight 2, the dimension of this space... | https://mathoverflow.net/users/61018 | Cusps forms for $\Gamma (N)$ | You can do this in Sage but "in disguise". The idea is that if $f(z)$ is a cusp form for $\Gamma(N)$, then $g(z) := f(Nz)$ is a cusp form for a certain subgroup intermediate between $\Gamma\_0(N^2)$ and $\Gamma\_1(N^2)$ which Sage calls $\Gamma\_H(N^2, [N + 1])$; the $N + 1$ is here because it generates the subgroup of... | 19 | https://mathoverflow.net/users/2481 | 185611 | 92,246 |
https://mathoverflow.net/questions/185591 | 3 | Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound
$$\sum\_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$
where $c>0$ is independent from $m$.
Is there a simple way of proving this inequality with $c$ not beeing too large?
Edit: $c$ can only be independent from $m$ if $a>1$. Otherwise it has to grow with... | https://mathoverflow.net/users/40117 | Simple bound for generalized geometric series | For $a\ge 1$, the quantity $(m+k+1)^a-m^a$ is increasing w.r.to $m\ge0$ (because $x\mapsto x^a$ is convex). So for $0<b<1$, the term $b^{(m+k+1)^a-m^a} $ is decreasing w.r.to $m\ge0$, and so is $\sum\_{k\ge0}b^{(m+k+1)^a-m^a}=b^{-m^a} \sum\_{k> m}b^{k^a}$. That is, we have
the required bound with $$c=c(b):= \sum\_{k>0... | 2 | https://mathoverflow.net/users/6101 | 185622 | 92,250 |
https://mathoverflow.net/questions/185621 | 7 | Let $S\subset\mathbb R$ be a $G\_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\mathbb R\to\mathbb R$ that is continuous exactly on $S$.
I'm trying to find a published reference for this result. N... | https://mathoverflow.net/users/56364 | If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$? | The result you are looking for can be found in most advanced references in topology or analysis. Since I am away from my office at the time of this writing, I will provide you with a source readily available on the internet: Theorem 7.2 (p.30) in [Oxtoby's Measure and Category](http://math.rice.edu/%7Emichael/teaching/... | 10 | https://mathoverflow.net/users/9269 | 185626 | 92,251 |
https://mathoverflow.net/questions/185625 | 5 | So I want to make a system for computing with various classes of numbers. One of those is a class of number closed under the standard arithmetic operators ($+$, $-$, $\*$ and $/$) along with square roots. This is pretty simple: I can extend any sufficiently nice ordered ring $R$ into $R[X]/(X^2-K)$. How to calculate th... | https://mathoverflow.net/users/25684 | Can this way of comparing numbers of the form a+b sqrt(K) be generalized? | Yes, you can compute with roots of arbitrary polynomials in a real-closed field. The algorithms are nowhere near as simple to describe, but in one way or another they can be thought of as generalizing the trivial algorithms for computing in $R[\sqrt K]$. See e.g. Yap, [Fundamental problems in algorithmic algebra](http:... | 2 | https://mathoverflow.net/users/12705 | 185636 | 92,253 |
https://mathoverflow.net/questions/185339 | 5 | I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this OP.
---
**Terminology**
For each $ n \in \mathbb{N} $, let $ [n] \stackrel{\text{df}}{=} \mathbb{N}\_{\leq n} $.
... | https://mathoverflow.net/users/50614 | Some questions on analytic vectors and the integrability of Lie-algebra representations | To demonstrate the delicate nature of questions concerning the integrability of Lie-algebra representations, I would like to highlight a major error that I committed while I was formulating the OP.
The last item in the hypothesis set for each theorem defines $ D $ to be the largest invariant domain of $ T[{\frak{g}}]... | 1 | https://mathoverflow.net/users/50614 | 185638 | 92,255 |
https://mathoverflow.net/questions/185634 | 3 | Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ \gg \log(n)$. Players are BR and MA (BR moves first):
1. BR claims an unclaimed edge from $E$, adds it to $B$
2. MA claims an unclaimed edge from $E$, adds to $M$
3. Repeat
MA wins if for all edges (with corresponding v... | https://mathoverflow.net/users/58940 | Graph game minimum vertex degree | I would try the following. Every time breaker plays an edge, maker chooses an endpoint of that edge at random and plays any available edge at that vertex. Since the vertex degrees are much bigger than $\log n$, each vertex should be chosen by maker about half the number of times it gets used by breaker, so maker should... | 1 | https://mathoverflow.net/users/25485 | 185641 | 92,258 |
https://mathoverflow.net/questions/185620 | 1 | I'm trying to relate the slope decomposition of a product of linear operators to the slope decompositions with regard to each of the operators in the product.
First I'll give some background, for which I'm following section 2.3 of Urban's Eigenvarietes for reductive groups.
If $L/\mathbb{Q}\_p$ is a finite extensi... | https://mathoverflow.net/users/15566 | Slope decomposition of a product of operators | The answer to your exact question is clearly "no" even if $M$ is finite-dimensional. Any operator on a finite-dimensional space has a slope decomposition, and if $n = 2$ and $U\_2 = U\_1^{-1}$ then every element has slope $0$ for $U = U\_1 U\_2 = \operatorname{id}$, but that doesn't force every element to have slope $0... | 1 | https://mathoverflow.net/users/2481 | 185647 | 92,261 |
https://mathoverflow.net/questions/185628 | 9 | Define the "diagram unknotting number" of a knot diagram $D$ as the minimal number of crossings that need to be changed in $D$ in order to get a diagram of the trivial knot (the usual unknotting number of a knot $K$ is the minimum over the diagram unknotting numbers of its diagrams).
Can you give me an example of a d... | https://mathoverflow.net/users/16507 | Unknotting number of knot diagrams | It is a theorem of Stoimenow that there exist unknotting number one knots with minimal crossing diagrams of unknotting number greater than one. Two such examples are $14\_{36750}$ and $14\_{36760}$. See Figure 9 in the reference:
* A. Stoimenow. Some examples related to 4-genera, unknotting numbers and knot polynomia... | 14 | https://mathoverflow.net/users/45976 | 185648 | 92,262 |
https://mathoverflow.net/questions/156000 | 4 | This post was inspired by [this answer](https://mathoverflow.net/questions/155950/isomorphism-theorem-for-subfactors/155968#155968) of Dave Penneys.
In the category of (irreducible hyperfinite II$\_1$) [subfactors](http://en.wikipedia.org/wiki/Subfactor), the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usu... | https://mathoverflow.net/users/34538 | The category of subfactors extending the category of groups? | Take $C$ to be the category of dualizable $N$-$N$-bimoduls, $N$ a factor. A subfactor $N\subset M$
(or $N\_0\subset N$) with finite index and finite depth gives an algebra object $A$ in $C$, namely $A={}\_NM\_N$ (or ${}\_NL^2M\_N$ if you prefer) and conversely an algebra object (more precisely a Q-system) gives a subf... | 2 | https://mathoverflow.net/users/10718 | 185652 | 92,265 |
https://mathoverflow.net/questions/185678 | 8 | Let n points in the plane be given whose coordinates we don't know.
Assume, however, that for any triple of the points we know the angle.
Question: Can we decide whether the n points are realizable in the plane, i.e. can we determine whether there is an assignment of coordinates to the points such that the angular co... | https://mathoverflow.net/users/51596 | Are angles between points enough to decide the realizability? | I assume that the angles are nonzero and we know which angle corresponds to each triplet.
If there are zero angles (collinear points) or the lists of angles and points are unrelated, the problem is harder.
With these assumptions there is a method.
First, take any three points.
They can be realized in the plane if and... | 7 | https://mathoverflow.net/users/55893 | 185680 | 92,277 |
https://mathoverflow.net/questions/185675 | 1 | Suppose that $G$ is a profinite group with the property that every open compact subgroup is topologically finitely generated and just infinite. Suppose that $H$ is a commensurated subgroup of $G$ with countably infinite index. I am interested in whether such subgroups can exist and what is known about them, in particul... | https://mathoverflow.net/users/15482 | countably-infinite-index subgroup of a finitely generated profinite group | Let me make two comment: If $G$ is finitely generated, than every open subgroup is finitely generated (Edit: I removed the reference). If every open subgroup is just infinite, then the group is called hereditarily just infinite.
A subgroup of countable index cannot be closed in a compact group, e.g. profinite group,... | 1 | https://mathoverflow.net/users/5034 | 185681 | 92,278 |
https://mathoverflow.net/questions/185366 | 18 | Suppose I have (semi-infinite) chain complexes $$ \cdots \rightarrow A\_i \rightarrow A\_{i+1}\rightarrow \cdots$$ $$ \cdots \rightarrow B\_i \rightarrow B\_{i+1}\rightarrow \cdots$$
over an additive category, and $A\_i = B\_i = 0$ for $i>0$. Suppose that $f:A\rightarrow B$ is a chain map such that, for every $k\geq 0$... | https://mathoverflow.net/users/60893 | Is such a map null-homotopic? | Probably there's a much less artificial example, and even more probably there's a much simpler one, but ...
Let $A=k[\varepsilon]/(\varepsilon^2)$, where $k$ is a field of characteristic two (purely so I don't need to bother about signs).
Let $\mathcal{A}$ be the category whose objects are pairs $(M,N)$ of $A$-modu... | 7 | https://mathoverflow.net/users/22989 | 185682 | 92,279 |
https://mathoverflow.net/questions/185619 | 8 | **Background:** Let $S$ be a simplicial set. By freely adding degeneracies to $S$, I mean first applying the forgetfull functor from simplicial sets to semi-simplicial sets which forget the already existing degeneracies maps and then apply its left adjoint which freely add the degeneracies. One gets a new simplicial se... | https://mathoverflow.net/users/22131 | Freely adding degeneracies does not change the homotopy type | Here is a purely combinatorial and straightforward proof.
First, you can easily check that $\Delta[m]'$ is the nerve of the category $[m]'$ that is obtained from the poset $[m]$ by freely adjoining one idempotent endomorphism to each object of $[m]$. These categories $[m]'$ are contractible since we can write down a ... | 6 | https://mathoverflow.net/users/12547 | 185685 | 92,280 |
https://mathoverflow.net/questions/185692 | 4 |
>
> Let $f(z\_1,z\_2,\dots ,z\_n)$ be an analytic function in $\mathbb{C}[[z\_1,z\_2,\dots ,z\_n]]$ whose leading term defines an isolated singularity at the origin.
> If we have the following types of singularities, then it is called a simple(ADE) singularity.
>
>
> $A\_n:z\_1^{n+1}+\sum\_2^{n} z\_i^2=0$ $(n\ge ... | https://mathoverflow.net/users/61052 | How can one determine if a singularity is simple? | The easiest way is to use the Newton polyhedron (after a linear change of variables): once you get nondegenerate principal part, just compare to the list. See Arnold, Gussein-Zade, Varchenko (volume 1) for details.
| 3 | https://mathoverflow.net/users/44953 | 185694 | 92,284 |
https://mathoverflow.net/questions/160101 | 1 | Consider a simple SPDE as follows:
$\partial\_t u(t,x)=\partial\_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$,
$u(t,0)=u(t,1)=0$,
$u(0,x)=v(x)$,
where $V$ is a bounded, smooth potential function, $\{W\_t, \mathcal{F}\_t\}$ is a cylindrical Brownian motion and $\dot{W}$ stands for the standard spac... | https://mathoverflow.net/users/44590 | On the solution of a stochastic partial differential equation | The $\sigma-$algebra $\mathcal{F}\_0$ is trivial, i.e.$\forall A\in\mathcal{F}\_0$, $P(A)=0$ or $1$. So the answer is true.
I think the answer is also true if $\varphi$ is a $\mathcal{G}-$measurable function when $\mathcal{G}$ is dependent of the filtration generated by the Brownian motion.
| 1 | https://mathoverflow.net/users/61055 | 185701 | 92,286 |
https://mathoverflow.net/questions/185702 | 9 | I should preface this by saying that I am not a representation theorist, so I apologize if this can easily be found in standard sources (but sadly I cannot seem to extract it from any of the books I know of).
Let $p$ be a prime number. What are all the irreducible representations (over $\mathbb{C}$) of $\text{SL}(n,\... | https://mathoverflow.net/users/61058 | Representations of $\text{SL}(n,\mathbb{F}_p)$ and $\text{Sp}(2n,\mathbb{F}_p)$ whose dimensions are $p^k$ | A full classification of such representations (and much more) can be found here:
>
> *Prime power degree representations of
> quasi-simple groups* by Malle and Zalesskii
>
>
>
You can read this paper [here](http://www.uea.ac.uk//~h054/Malle-Z-2001.pdf). The main theorem implies that, apart from the Steinberg ... | 8 | https://mathoverflow.net/users/801 | 185704 | 92,287 |
https://mathoverflow.net/questions/185645 | 33 | I have three related questions about conventions for defining Clifford algebras.
>
> 1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining relations $v^2 = q(v)$ or $v^2 = -q(v)$?
>
>
> 2) Should $\text{Cliff}(n)$ denote the Clifford algebra generated by $n$ ... | https://mathoverflow.net/users/290 | What are the "correct" conventions for defining Clifford algebras? | This is not really an answer, but rather a meta-answer as to why there exist many conventions in the first place.
The symmetric monoidal category $\mathit{sVect}$ of super-vector spaces has a non-trivial involution $J$.
The symmetric monoidal functor $J:\mathit{sVect}\to \mathit{sVect}$ is the identity at the level o... | 46 | https://mathoverflow.net/users/5690 | 185705 | 92,288 |
https://mathoverflow.net/questions/185698 | 5 | How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?
| https://mathoverflow.net/users/37840 | Construct the best piece-wise linear continuous function fitting given curve | If the number and position of the knots are fixed, then the problem is a linear least squares problem for determining the coefficients of linear B-Splines (cf e.g <http://en.wikipedia.org/wiki/B-spline>); if the knot-positions also have to be determined, then the problem becomes non-linear, but is also studied in appro... | 4 | https://mathoverflow.net/users/31310 | 185706 | 92,289 |
https://mathoverflow.net/questions/84194 | 10 | Can it be shown that a positive fraction of the base-$b$ digits of n! are nonzero (in the limit as $n\to\infty$)?
| https://mathoverflow.net/users/6043 | Nonzero digits in n! | The bound given by F. Luca and cited in Gjergji Zaimi's answer has been recently improved, although only by a factor $\log \log \log n$. Precisely, in *C. Sanna, [On the sum of digits of the factorial](http://dx.doi.org/10.1016/j.jnt.2014.09.003), J. Number Theory 147 (2015), 836--841*, the author proved that
$$\min\{s... | 11 | https://mathoverflow.net/users/nan | 185711 | 92,291 |
https://mathoverflow.net/questions/185627 | 6 | First of all I want to say that algebraic geometry is not "my field of research" so I apologize if the notation is not standard.
---
$S$ is a smooth complex projective surface with a fibration $f$ over $\mathbb P^1(\mathbb C)$. Moreover suppose that the following properties hold for $f$:
1. $f$ has singular fib... | https://mathoverflow.net/users/59377 | Isotrivial fibrations over $\mathbb P^1$ | There are three important invariants for any relatively minimal fibration $f:\,S \to B$ from a smooth complex projective surface $S$ to a smooth curve $B$: the self-intersection $\omega\_{S/B}^2$, the degree of $f\_\*\omega\_{S/B}$, and the singular index $e\_f$ of $f$, where $\omega\_{S/B}:=\omega\_S\otimes\omega\_B^{... | 5 | https://mathoverflow.net/users/46464 | 185713 | 92,292 |
https://mathoverflow.net/questions/185697 | 0 | For $a$ and $q$ positive integers such that $a\lt q$ and $(a,q)=1$, let $\pi(x;q,a)$ be the number of primes $p\equiv a\pmod q$ below $x$. One can show that $\pi(x;q,a)\sim \dfrac{\pi(x)}{\varphi(q)}$ where $\pi(x)$ is the number of primes below $x$ and $\varphi(n)$ the number of positive integers $k$ less than $n$ suc... | https://mathoverflow.net/users/13625 | A conjectural convergence condition for a weakened Elliott-Halberstam conjecture | It is easy to see that $\theta\_s\leq 1/2$, hence $EH(\theta\_{s})$ holds by the [Bombieri-Vinogradov theorem](http://en.wikipedia.org/wiki/Bombieri%E2%80%93Vinogradov_theorem).
To see the claim $\theta\_s\leq 1/2$, assume that $1/2<\theta<1$, and let $1\leq q\leq x^\theta$. If $x$ and $q$ are fixed for a moment such... | 4 | https://mathoverflow.net/users/11919 | 185730 | 92,299 |
https://mathoverflow.net/questions/185248 | 2 | Let $G$ be a locally profinite (i.e., locally compact Hausdorff and totally disconnected) topological group, $H \le G$ a closed subgroup, and $(W, \sigma)$ a representation of $H$ over $\mathbb{C}$ such that $W$ is smooth, i.e., every $w \in W$ is fixed by a compact open subgroup $K \le H$.
One may form compact indu... | https://mathoverflow.net/users/53197 | Compact induction as a tensor product | No. Take $H$ the trivial subgroup, $W$ the trivial rep. The left hand side is
all smooth functions on $G$ with compact support. The right hand side is zero. The reason for this is that elements of $\mathbb C G$ are linear combinations of finitely
many elements of $G$, but it is easy to see that such element cannot be... | 1 | https://mathoverflow.net/users/57472 | 185744 | 92,302 |
https://mathoverflow.net/questions/185715 | 3 | Let large Latin symbols as $X$ and $Y$ denote sets of natural numbers and small symbols as $n$ and $n´$ denote natural numbers and small Greek letters stand for formulas.
Suppose $\alpha$ is $\Pi\_1^0$ or $\Sigma\_1^0$. Is ($\forall X$)($\exists n$)($\forall Y$)$\alpha(X,Y,n)$ $\Pi\_2^1$, or what?
| https://mathoverflow.net/users/37385 | A Question related to the Formula Hierarchy | There is an additional twist in the case where $\alpha$ is $\Sigma^0\_1$. Assuming $\mathsf{WKL}\_0$ ([Weak König Lemma](http://en.wikipedia.org/wiki/K%C3%B6nig's_lemma)), $\forall Y\alpha(X,Y,n)$ is equivalent to a $\Sigma^0\_1$ statement and hence so are $\exists n \forall Y \alpha(X,Y,n)$ and $\forall X\exists n \fo... | 4 | https://mathoverflow.net/users/2000 | 185751 | 92,304 |
https://mathoverflow.net/questions/185754 | 1 | A real $r$ is computable if given any $i\in \mathbb{N}$, the $i$th bit can be outputed by a Turing Machine or an algorithm. So, what is computational complexity or complexity measure of computing the real? Since there is just a Turing Machine or a program without input,
>
> what is the computational complexity or c... | https://mathoverflow.net/users/14024 | The definition of computational complexity or complexity measure of computing reals | This is interesting. I think a number could be of low complexity in terms of approximating it to within smaller and smaller $\epsilon $, while of high complexity in terms of finding its binary representation. Just imagine that its binary representation has extremely long stretches of 0s (and/or 1s) so the number is unu... | 3 | https://mathoverflow.net/users/4600 | 185757 | 92,306 |
https://mathoverflow.net/questions/185068 | 1 | It seems like the question stated [here](https://math.stackexchange.com/questions/982241/functional-representation-of-adapted-jointly-measurable-stochastic-processes) in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO.
Let $X\_t : \Omega \to E, \ t ... | https://mathoverflow.net/users/58682 | Functional representation of adapted jointly measurable stochastic processes | It is something of a folk theorem that every adapted measurable process (with values in a Polish space) admits a progressively measurable modification. This is in Dellacherie & Meyer's book, and a quick google search turned up [a recent paper](http://www.mat.ub.edu/EMIS/journals/EJP-ECP/article/download/2548/2548-13043... | 2 | https://mathoverflow.net/users/44169 | 185779 | 92,312 |
https://mathoverflow.net/questions/185774 | 1 | I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function $g\in C^{\infty}$. I found this paper:
>
> [Uniform approximation of continuous mappings by smooth mappings with no cr... | https://mathoverflow.net/users/61093 | Uniform approximation of increasing function in $C^{\infty}$ | Let $f$ be your two-piece linear function. Let $\varphi\in C^\infty\_0((-\epsilon,\epsilon))$ for some small $\epsilon$, such that
* $\varphi$ is even
* the integral $\int \varphi = 1$
* $x\varphi' \leq 0$
Then you can check that the convolution $\varphi\*f$ is increasing, smooth, and agrees with $f$ outside $(-2\... | 1 | https://mathoverflow.net/users/3948 | 185783 | 92,314 |
https://mathoverflow.net/questions/185687 | 2 | Let $G$ be a compact Lie group and $a\in\mathfrak{g}^\*$ (dual of Lie algebra of Lie group $G$). Then let $\mathcal O\_a$ be a coadjoint orbit. Then every co-adjoint orbit is Kähler manifold and also projective variety. How can we compute the Kodaira dimension of co-adjoint orbit as projective variety?
>
> Motivat... | https://mathoverflow.net/users/nan | Kodaira dimension of co-adjoint orbit | Since your manifold $X$ has positive Ricci curvature, the line bundles $K\_X^m$ are all ${ negative}$ for $m\geq 1$, i.e. they admit a smooth Hermitian metric with negative curvature. By Kodaira Vanishing, we conclude that $H^0(X,K\_X^m)=0$ for all $m\geq 1$, i.e. the Kodaira dimension is $-\infty$.
| 4 | https://mathoverflow.net/users/13168 | 185792 | 92,315 |
https://mathoverflow.net/questions/185789 | 4 | If $J\_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum\_{n=-\infty}^{\infty} J\_n^2(x)=J\_0^2(x)+2\sum\_{n=1}^{\infty} J\_n^2(x)=1.$$
What is known about
$$
\sum\_{n=-\infty}^{\infty} J\_n^4(x) ?
$$
It is smaller than $1$, but is there a nice lower bound?
| https://mathoverflow.net/users/40120 | Estimate on sum of $J_n^4$ | The paper [here](http://inside.mines.edu/~pamartin/ref-paps/R101_JPAw.pdf) mentions the integral $\displaystyle\sum\_{n = -\infty}^\infty J\_n^4(x) = \frac{2}{\pi} \int\_0^\frac{\pi}{2} J\_0^2(2x \sin \theta) d\theta$
and the asymptotic $\displaystyle \sum\_{n = -\infty}^\infty J\_n^4(x) = \frac{1}{x\pi^2}(\log x + 5 ... | 9 | https://mathoverflow.net/users/19029 | 185799 | 92,318 |
https://mathoverflow.net/questions/185768 | 0 | Fix $n$ and let $B, C$ be two $n \times n$ 0-1 matrices of full rank such that $\sum\_{i,j} b\_{i,j}^2 = \sum\_{i,j} c\_{i,j}^2$, in other words they have the same number of $0$ entries and the same number of $1$ entries. I want to find $\alpha\_n = \max ||B||/||C||$ subject to this constraint as a function of $n$.
| https://mathoverflow.net/users/8435 | Maximizing/Minimizing the Operator norms of 0-1 matrices subject to a constraint | Take $B$ to have $1$s on the main diagaonal and the first superdiagonal, then $\|B\|\leq 2$. If we take $C$ to have $1$s on the diagonal and filling the last column, then $\|C\|\geq \sqrt{n}$. (These both have full rank and $2n-1$ $1$'s.) This shows that $\alpha\_n\geq \frac{\sqrt{n}}{2}$, though from the examples in R... | 3 | https://mathoverflow.net/users/13360 | 185805 | 92,321 |
https://mathoverflow.net/questions/185177 | 9 | Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.
Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?
What if $X$ is an algebraic space (i.e., trivial stabilizers)?
Edit: I changed the old question to a different question which should be more clear. An an... | https://mathoverflow.net/users/4333 | Finite etale atlas for Deligne-Mumford stacks | It seems to me that the answer is NO if $X$ is a DM stack. If I'm not mistaken, it suffices to give a smooth finite type separated connected Deligne-Mumford stack over $\mathbb{C}$ which is simply connected (since such a thing has no non-trivial finite etale covers, let alone finite etale covers by a scheme). But [this... | 9 | https://mathoverflow.net/users/6950 | 185809 | 92,324 |
https://mathoverflow.net/questions/185781 | 2 | I am looking for an upper bound - up to constant factor - for:
$\sum\_{k=t}^{t+l} {n \choose k} \cdot 2^{-n}$ where:
The values of $t$ are between: $\frac{n}2+\sqrt{n} \leq t \leq \frac{9n}{10}$. (The previous $\frac9{10}$ can be replaced by any other constant between $\frac12$ and 1 ...)
In my configuration, the... | https://mathoverflow.net/users/61094 | Upper bound of sum of binomial coefficients | Let me write $a\approx b$ for $a=\Theta(b)$, and $t=\frac n2(1+\tau)$.
Stirling approximation gives
$$2^{-n}\binom nt=(1+o(1))\frac{n^n}{(2t)^t(2(n-t))^{n-t}}\sqrt{\frac n{2\pi t(n-t)}}
\approx\bigl((1+\tau)^{1+\tau}(1-\tau)^{1-\tau}\bigr)^{-n/2}\frac1{\sqrt{n-t}}.$$
For any $t\le k<t+l$, we have
$$\binom n{k+1}\bino... | 2 | https://mathoverflow.net/users/12705 | 185814 | 92,327 |
https://mathoverflow.net/questions/178590 | 5 | Background
==========
Let $X\_t$ be the continuous time Markov process on the state space {Working, Broken} with failure rate $\alpha$ and repair rate $\beta$. By elementary calculations [1]
$$
\begin{align\*}
P\left\{ X\_t = \text{Working}\ | \ X\_s = \text{Working} \right\} &= p + (1-p) \cdot e^{-(\alpha+\beta)(t... | https://mathoverflow.net/users/56843 | Sum of a random number of identically distributed but dependent random variables? | I [*think I have*] proved the calculation for $\mu\_Q$ and $\sigma^2\_Q$ by applying the [methods I used in a related problem](https://mathoverflow.net/questions/184280/publishing-an-elementary-proof-of-a-less-general-and-less-useful-version-of-a-cl).
I then [*also think I have*] proved that $Q$ is asymptotically nor... | 0 | https://mathoverflow.net/users/56843 | 185828 | 92,333 |
https://mathoverflow.net/questions/185843 | 0 | Let $K$ be a finite field with $p^n$ elements. Then the group of unit is cyclic of order $p^n-1$. What are the primes which are dividing $p^n-1$? What are the Sylow-subgroups of the group of units of $K$? Of course they are cyclic but of which order?
| https://mathoverflow.net/users/57804 | Sylow-subgroups of the group of units of a finite field | What kind of answer do you expect? Of course, not much about the prime factorization of $p^n-1$ can be said, for otherwise one could settle questions about Mersenne primes $2^q-1\in\mathbb P$ ($q$ necessarily a prime). One could even settle the existence question of Fermat primes $F\_k=2^{2^k}+1$, if we knew the prime ... | 5 | https://mathoverflow.net/users/18739 | 185845 | 92,338 |
https://mathoverflow.net/questions/185838 | 3 |
>
> Is there a general formula for the canonical divisor $K\_X$ of a smooth conic bundle $X$?
>
>
>
Motivation: for smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, $K\_X = \mathcal{O}\_X(d-n-1)$. But smooth conic bundles are not "naturally" embedded in some $\mathbb{P}^n$, so I was wondering if an analogue... | https://mathoverflow.net/users/61125 | Canonical divisor for conic bundles | Here is a method which should give you what you want, but I leave the details to you as I don't have time to work them out now.
Without loss of generality we may work over an algebraically closed field. Each singular fibre consists of a union of two $(-1)$-curves meeting at a single point. Contracting a choice of $(-... | 2 | https://mathoverflow.net/users/5101 | 185847 | 92,340 |
https://mathoverflow.net/questions/185791 | 12 | Let $L(p,q)$ be a 3-dimensional lens space, and let $L(p',q')$ be another. Is there any known result concerning the 3rd homotopy group of the connected sum $L(p,q)\# L(p',q')$? If not, I am interested in computing it, but I have no idea.
More generally, can we find a method to compute $\pi\_{3}(M\# N)$ for general pr... | https://mathoverflow.net/users/61101 | How to compute $\pi_{3}$ of $L(p,q)\# L(p',q')$? | Elaborating on Alex Suciu's first comment, let me offer a second way of computing $\pi\_3(M\#N)$ which works for any pair of $3$-manifolds with non-trivial fundamental groups (i.e. none of them is $S^3$, which is uninteresting since it is a unit for $\#$).
For any (CW-)space $X$ with universal cover $\tilde X$ there... | 8 | https://mathoverflow.net/users/12166 | 185854 | 92,342 |
https://mathoverflow.net/questions/185852 | 1 | I am trying to find an example of the following situation.
$G$ is a t.d.l.c. (totally disconnected locally compact) $\sigma$-compact topological group in which every compact open subgroup is topologically finitely generated. $H$ is a maximal compact open subgroup and $K$ is a subgroup abstractly isomorphic to $H$ suc... | https://mathoverflow.net/users/15482 | locally topologically finitely generated t.d.l.c. group | Consider a basis of the $\mathbf{Q}$-vector space $\mathbf{Q}\_p$ given as elements of $\mathbf{Z}\_p\smallsetminus p\mathbf{Z}\_p$, written as elements $e\_n$ for integers $n\in\mathbf{Z}$ and $f\_i$ for $i$ in some uncountable index set. Define a $\mathbf{Q}$-linear automorphism $q$ of $\mathbf{Q}\_p$ by $q(e\_n)=p^n... | 1 | https://mathoverflow.net/users/14094 | 185858 | 92,343 |
https://mathoverflow.net/questions/185801 | 1 | Probability density functions (PDF's) have inherent connections to the field of
Dynamical Systems.
The motivation for this question can be found in: <http://www.stat.cmu.edu/~cshalizi/754/2006/notes/solutions-2.pdf> for the logistic map when $x∈[0,1]$.
My **question** is:
Is it possible to define the density of... | https://mathoverflow.net/users/25947 | Is it possible to define the density of the logistic map for $x<0$? | As indicated by Anthony Quas, if you are looking for an invariant probability measure that is supported on the basin of attraction of infinity, then you will need to put a point mass at infinity and nowhere else. (This is clear because a compact set will enter any neighbourhood of infinity after a sufficiently large - ... | 2 | https://mathoverflow.net/users/3651 | 185868 | 92,348 |
https://mathoverflow.net/questions/185867 | 10 | I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).
So, I'm just curious why don't expert mathematicians use ZF+GCH
instead of ZF+AC(ZFC).
| https://mathoverflow.net/users/61136 | Difference between ZFC and ZF+GCH | In some sense $\sf GCH$ is a limiting axiom. While it solves a lot of things, it also means that certain things we are interested in become false or trivialized. And that's no fun.
For example, forcing axioms like $\sf MA$ become trivial assuming even just $\sf CH$, and stronger forcing axioms like $\sf PFA,MM$ and o... | 15 | https://mathoverflow.net/users/7206 | 185872 | 92,350 |
https://mathoverflow.net/questions/185860 | 1 | Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$
A map from $X$ to $BG$ induce a map from $H^\*(BG,Z)$ to $H^\*(X,Z)$ by pull back.
Let $GH^\*(X,Z)$ be the subgroup of $H^\*(X,Z)$ formed by the images of the above map
$H^\*(BG,Z)\to H^\*(X,Z)$ for all the maps $X\to BG$.
In this case $a \... | https://mathoverflow.net/users/17787 | Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes | There's no relations between SW, Pontrjagin, and Chern classes (within each set). On the other hand, for a $U(1)$-bundle, one has $w\_1=0$ and $w\_2=c\_1\bmod2$. Such a bundle has no (rational) Pontrjagin classes.
**More general setting**
Here is another bunch of relations; most likely these are all, but I'm not 100%... | 2 | https://mathoverflow.net/users/44953 | 185875 | 92,353 |
https://mathoverflow.net/questions/185824 | 8 | Let $P\_1$ be the finite support iteration of random forcings of length $\omega$. Let $P\_2 = \text{Random} \times \text{Random}$ and $P\_3 = \text{Random} \times \text{Cohen}$. Are $P\_i, P\_j$, for $1 \leq i < j \leq 3$, forcing isomorphic? All I know is that all of them add a $P\_3$-generic.
| https://mathoverflow.net/users/2689 | A question about three forcings | As you mentioned in a comment, in a $P\_2$-extension there are two random reals the sum of which is Cohen. However, forcing with **Cohen** over $V[\mathrm{Random}]$ doesn´t add any real that is random over $V$. So the sum of two random reals in a $P\_3$-extension cannot be Cohen, since **Random** doesn´t add Cohen real... | 3 | https://mathoverflow.net/users/17836 | 185884 | 92,357 |
https://mathoverflow.net/questions/185839 | 7 | I am trying to find the asymptotic behavior of the sum:
$$ \sum^n\_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have $|x|\leq1$ and $|y|\leq1$. I also know that $|x+y|\leq1$ and $|4xy|\leq1$.
One of the approaches I thought about was t... | https://mathoverflow.net/users/56913 | Bound on sum of complex summands involving binomial coefficients | Assuming that $|x+y|<1$ and $4|xy| \le 1$, here's a proof of the decay.
First suppose that $|x|> |y|$. The desired sum is
$$
\le \binom{2n}{n} |xy|^n \sum\_{j=0}^{n} |y/x|^j \le
\binom{2n}{n} |xy|^n \frac{1}{1-|y/x|}.
$$
Since $\binom{2n}{n}$ is of size about $4^n/\sqrt{n}$, the desired decay follows in this cas... | 10 | https://mathoverflow.net/users/38624 | 185885 | 92,358 |
https://mathoverflow.net/questions/185880 | 3 | I am reading Gutzwiller's papers on the relation between Hamiltonian flows and solution to Schrodinger equations.
In the two papers, he gave a semi-classical approximation of the Green's function to the Laplacian
Gutzwiller, M. C.: "The Phase Integral Approximation in Momentum Space and the Bound States of an Atom,... | https://mathoverflow.net/users/18261 | semi-classical Green's function | The difference between the semiclassical approximations of the full Green's function and the trace is whether or not you restrict the sum over paths to closed orbits; for a treatment of the semiclassical Green's function (the socalled Van Vleck-Gutzwiller propagator), see for example chapter 10 of David Tannor's [Intro... | 2 | https://mathoverflow.net/users/11260 | 185887 | 92,359 |
https://mathoverflow.net/questions/185896 | 13 | I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.
Let:
$\ \ E\_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 \right\}\ \ $ and $\ \ O\_n :=\left\{ k \in \left\{1,\ldots,n\right\} : \pi(k) \equiv 1 \mod 2 \right\}$.
My qu... | https://mathoverflow.net/users/8435 | Parity of the Prime Counting Function | The limits you conjecture are natural, and they are currently open. I believe the best known result is by Ping Ngai Chung and Shiyu Li, who proved that
$$
\liminf\_{n \to \infty} \frac{|E\_{n}|}{n}
$$
and
$$
\liminf\_{n \to \infty} \frac{|O\_{n}|}{n}
$$
are both $\geq \frac{1}{64}$. They can improve this to $\frac{1}... | 19 | https://mathoverflow.net/users/48142 | 185897 | 92,362 |
https://mathoverflow.net/questions/185870 | 4 | I recently came across the following function which intrigues me:
\begin{equation}
f(\alpha):=\sum\_{i=0}^\infty \frac{\alpha^{i(i+1)/2}}{i!}.
\end{equation}
For $-1\leq \alpha\leq 1$ this function is well defined. Moreover, $f(1)=e$ and $f(-1)=\cos(1)-\sin(1)$. However, I have been unable to find any literature on thi... | https://mathoverflow.net/users/57020 | What is known about this series? | The function $$f\_\alpha(z)=\sum\_{n=0}^\infty \frac{\alpha^{n(n+1)/2}z^n}{n!}$$
was much studied, though it does not have a common name. It is indeed the Borel transform
of an incomplete theta function. This function solves the functional equation
$$f'(z)=\sqrt{\alpha}f(z\sqrt{\alpha}),\quad f(0)=1.$$
Alan Sokal poste... | 7 | https://mathoverflow.net/users/25510 | 185908 | 92,367 |
https://mathoverflow.net/questions/185074 | 15 | **The game**
Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is impossible to tell its color without opening its wrapping. Lucy does this before offering any of the balls to Alic... | https://mathoverflow.net/users/60766 | Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game | Here is a sketch of an argument by my colleague Jim Roche of an $\Omega(\log n/\log \log n)$ lower bound. The basic idea is that Lucy chooses randomly between two strategies, one of which puts all the red balls early (thereby forcing Alice to open some number of the early boxes), and the other of which puts a lot of th... | 8 | https://mathoverflow.net/users/3106 | 185913 | 92,369 |
https://mathoverflow.net/questions/185888 | 17 | Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}\_p$ and let $Y\_0(p)$ be the fine moduli space representing elliptic curves + point of $p$-torsion over $\mathbb{Q}\_p$. We know that $Y\_0(p)$ has a model over $\mathbb{Z}\_p$ whose s... | https://mathoverflow.net/users/60519 | Special fiber of $X(p)$ in characteristic $p$ | A bit of mastication of Katz-Mazur Theorem 13.7.6 and the surrounding text seems to yield the following description of the special fiber of $Y(p)$:
1. It is fundamentally $p+1$ copies of $\mathbb{P}^1$ (each with a nonempty finite set of punctures corresponding to cusps) all glued together at supersingular points.
2.... | 16 | https://mathoverflow.net/users/121 | 185914 | 92,370 |
https://mathoverflow.net/questions/185911 | 4 | Let $m \ge 1$ be an integer, let $k$ be a field of characteristic $0$, and let
$$
1 \rightarrow \mathrm{GL}\_n \rightarrow E \rightarrow \mathbb{Z}/m\mathbb{Z} \rightarrow 1
$$
be an extension of $k$-group schemes. Since $E$ acts on $\mathrm{GL}\_n$ by conjugation, there is an induced $k$-group scheme homomorphism $\ma... | https://mathoverflow.net/users/5498 | Is a "central" extension of $\mathbb{Z}/m\mathbb{Z}$ by $\mathrm{GL}_n$ necessarily split? | To make my comment more precise, suppose that $k$ contains no primitive $2m$-th roots of $1$, where $m=2^j$ for some $j \ge 1$. Then we can form the *central product* of $G={\rm GL}\_n(k)$ with a cyclic group $\langle x \rangle$ of order $2^{j+1}$, where we amalgamate $-I\_n$ with the element of order $2$ in the cyclic... | 9 | https://mathoverflow.net/users/35840 | 185918 | 92,371 |
https://mathoverflow.net/questions/185926 | 8 | Given any Riemannian or [Semi-Riemannian manifold](http://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold) $(M,g)$, does there exist a Eucildean space $(E,g^\prime)$ of enough high dimension with metric $g^\prime=diag\{-1,-1,...,+1,+1,...\}$ with any n copies of "$-1$" and m "$+1$" such that $(M,g)$ is embedded in $(E... | https://mathoverflow.net/users/43941 | Is there some Riemannian manifold's version of Whitney theorem? | Yes, have a look at Robert Greene's book *Isometric Embeddings of Riemannian and Pseudo Riemannian Manifolds,* Volume 97 of Memoirs of the American Mathematical Society
Memoirs, 1970.
| 17 | https://mathoverflow.net/users/13972 | 185934 | 92,373 |
https://mathoverflow.net/questions/185937 | 5 | The following question is moved from math stackexchange. It seems that this is not a popular question, but I really want to know the answer so I moved it to here. The question reads as follows.
We know the expansion of the following product
$\displaystyle\prod\_{k=1}^n(1+x+y\_k)$
can be expressed by the formula
... | https://mathoverflow.net/users/41686 | The formula for a perhaps basic identity (move from stackexchange) | I guess the first product is expanded as
$$
\prod\_{k=1}^n(1+x+y\_k)=(1+x)^n\prod\_k(1+y\_k(1+x)^{-1})=\sum\_{k\geq 0}e\_k(y\_1,\ldots,y\_n)(1+x)^{-k+n}.$$
For the other products you can write
$$
\prod\_{j=1}^m\prod\_{i=1}^n(1+y\_it\_j)=\sum\_\lambda e\_\lambda(y\_1,\ldots,y\_n)m\_\lambda(t\_1,\ldots,t\_m)=\sum\_\lambd... | 4 | https://mathoverflow.net/users/4366 | 185945 | 92,377 |
https://mathoverflow.net/questions/185941 | 15 | [Harvey Friedman at the 1974 ICM](http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0235.0242.ocr.pdf) motivated Reverse Mathematics by the
following statement:
>
> When the theorem is proved from the right axioms, the axioms can be proved
> from the theorem.
>
>
>
Reverse Mathematics has had many successe... | https://mathoverflow.net/users/1587 | Natural examples of Reverse Mathematics outside classical analysis? | Reverse math usually means work in subsystems of arithmetic. That goes a bit beyond analysis---Simpson's book has plenty of classic results from the theory of countable groups, rings, and fields, and much of the more recent focus in the area has been countable combinatorics. But all of these are basically about sets of... | 12 | https://mathoverflow.net/users/8991 | 185946 | 92,378 |
https://mathoverflow.net/questions/185957 | 20 | An article in the
*Notices of the AMS*, Volume 61, Issue 10, 2014
([PDF download link](http://www.ams.org/notices/201410/rnoti-p1240.pdf)),
on Khot's Unique Games Conjecture, says this:
>
> Another group ... found a
> shape that in a certain sense lies halfway between
> a square and a circle (though in many mor... | https://mathoverflow.net/users/6094 | "a shape that ... lies halfway between a square and a circle" | This is work by Guy Kindler, Ryan O’Donnell, Anup Rao, and Avi Wigderson, published in [Spherical Cubes and Rounding in High Dimensions](http://www.cs.cmu.edu/~odonnell/papers/spherical-cubes-conference.pdf) (2008), and in [Spherical Cubes:
Optimal Foams from Computational Hardness Amplification](http://www.cs.cmu.edu/... | 21 | https://mathoverflow.net/users/11260 | 185962 | 92,384 |
https://mathoverflow.net/questions/185958 | 16 | Suppose that two finitely generated groups quasi-isometrically embed into each other. Does it follow that the two groups are quasi-isometric? Recall that a quasi-isometry is a quasi-isometric embedding that is quasi-surjective, see e.g. <https://www.math.ucdavis.edu/~kapovich/EPR/pc_lectures3.pdf>
| https://mathoverflow.net/users/14497 | Cantor-Bernstein for quasi-isometric embeddings? | Let $C\_n$ be a cyclic groups of order $n$. Then the wreath products $C\_2\wr\mathbf{Z}$ and $C\_3\wr\mathbf{Z}$ embed QI into each other (for the reverse direction, observe that $C\_2\wr\mathbf{Z}$ has a subgroup of index 2 isomorphic to $C\_2^2\wr\mathbf{Z}$). But $C\_2\wr\mathbf{Z}$ and $C\_3\wr\mathbf{Z}$ are not Q... | 16 | https://mathoverflow.net/users/14094 | 185964 | 92,386 |
https://mathoverflow.net/questions/185928 | 3 | Let $M$ be a monoid that acts transitively from the right on a finite set $X$.
Assume furthermore that the action of $M$ on $X$ induces for every $m \in M$ a bijection on $X \to X, x \mapsto x.m$.
Let
$$M\_X := \{ m \in M \; | \; \forall x \in X : x.m = x \}$$
be the fixer of $X$.
Question 1: Is there always a finite... | https://mathoverflow.net/users/58432 | Cosets of the fixer of an action of a monoid on a finite set | The answer is no. Map the free monoid $M$ on $\{a,b\}$ onto $\mathbb Z/2$ by $a$ maps to $1$ and $b$ maps to $0$. Then $M$ acts transitively on the right of $\mathbb Z/2$ via bijections by applying the homomorphism and doing the regular representation.
The fixer is all words with an even number of $a$'s. Consider th... | 2 | https://mathoverflow.net/users/15934 | 185968 | 92,388 |
https://mathoverflow.net/questions/185463 | 7 | This is a long shot, but ...
[The fraction of $\mathbb{Z}^2$ lattice points
visible from the origin](https://mathoverflow.net/q/151706/6094)
$1/\zeta(2)=6/\pi^2 \approx 61$%.
The fraction of $\mathbb{Z}^3$ lattice points visible
from the origin is $1/\zeta(3) \approx 83$%.
And this generalizes to arbitrary dimensions... | https://mathoverflow.net/users/6094 | Visibility interpretation of Riemann zeta zeros on the critical line? | The sort of picture that I have in my head looks like this: Whatever enumerative interpretation you give $\zeta(\sigma)$ for $\sigma >1$ of some (possibly weighted) objects, then information about $\zeta(\sigma)$ for $\sigma$ in the critical strip tells you how those objects are distributed or spaced from each other!
... | 7 | https://mathoverflow.net/users/2384 | 185973 | 92,390 |
https://mathoverflow.net/questions/185979 | 19 | I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question:
(1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ to itself such that for the continuation $\partial f$ of $f$ to the boundary $\partial \mathbb H^2$, there does not ex... | https://mathoverflow.net/users/14233 | Failure of Mostow rigidity in dimension 2 | ad (i):
First consider a Dehn twist at some simple, closed curve in a closed hyperbolic surface. It is obviously a quasi-isometry (as any smooth map between closed surfaces) but not an isometry.
Then lift this Dehn twist to a self-map of the hyperbolic plane, equivariant with respect to the cocompact group action ... | 14 | https://mathoverflow.net/users/39082 | 185981 | 92,393 |
https://mathoverflow.net/questions/185976 | 0 | Let $G$ be a connected reductive group with $G\_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$.
We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus $Z$.
Is it possible to lift this extension to a central extension of $\tilde{G}$ of $G$ by $Z$ such that $\tilde{G}... | https://mathoverflow.net/users/27398 | on lifting extensions | There is always the uninteresting example $\widetilde{G} = G \times Z$ relative to a choice of splitting of $\widetilde{T}$ as a central extension of $T$ by $Z$ (as may be chosen since you assumed the ground field to be algebraically closed), and we claim that it is the only one (up to isomorphism of central extensions... | 3 | https://mathoverflow.net/users/52824 | 186003 | 92,400 |
https://mathoverflow.net/questions/185991 | 2 | Let $A$ be an abelian variety over $\mathbb{F\_q}$ with dimension $n$. Let $q$ be a constant.
Is there polynomial algorithm of finding discrete logarithm in $A$?
UPD: really I don't undestend: can we polynomial fast calculate $P+Q$ if $P, Q \in A$ (as it is in case of elliptic curve)?
| https://mathoverflow.net/users/31356 | DL-problem on abelian variety | For the original question, which I understand to be whether there is an algorithm, which is polynomial time in $n$ that computes discrete logarithms on an abelian variety of dimension $n$ over a finite field of $q$ elements, where $q$ is fixed, the answer is no. This problem has been studied in the context of the Weil ... | 3 | https://mathoverflow.net/users/2290 | 186008 | 92,402 |
https://mathoverflow.net/questions/185326 | 8 | This is a reference request.
Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O\_A$ be a maximal order in $A$. Then $\mathrm{SL}\_2(A)$ acts transitively on the right on the set $\mathbf P^1(A)$ of one dimensional left subspaces of $A^2$.
Let $h\_A$ be the cardinal of the *set* (1) of l... | https://mathoverflow.net/users/39552 | Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra | The formula holds, and seems to be due to Krafft and Osenberg : *Eisensteinreihen für einige arithmetisch definierte Untergruppen von SL2(H)*. (German), Math. Z. 204 (1990), no. 3, 425–449.
(See here : [EuDML](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0204&DMDID=dmdlog45) (free), or here : [Spr... | 6 | https://mathoverflow.net/users/39552 | 186017 | 92,404 |
https://mathoverflow.net/questions/186026 | 8 | In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.
That is, a matrix (here 2x2 real, but it is more general)
$$
\begin{bmatrix}
u' \\ v'
\end{bmatrix}
=
M
\begin{bmatrix}
u \\ v
\end{bma... | https://mathoverflow.net/users/9093 | Partial inverse of a matrix - or does it have its own name? | It is a [principal pivot transform](http://www.math.wsu.edu/faculty/tsat/files/t2.pdf), also known as *sweep operator* or *gyration*. You can check the linked review paper.
| 9 | https://mathoverflow.net/users/1898 | 186029 | 92,409 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.