Math

Acceleration via smoothing

Is the following approach to accelerating the rate of convergence of $(1+1/2+\dots+1/n)- \ln n$ (with $n=1,2,3,\dots$), and other sequences like it, in the literature? Let $f(t)=(\sum_{1 \leq n \leq t} 1/n) - \ln t$ for $t \geq 0$, so that $f(t)$ tends to Euler's constant $\gamma$ as $t \rightarrow \infty$. ....

Sadie Sadie: 12 hours ago

Math

Analog to the Chinese Remainder Theorem in groups other than Z_n.

The idea hit me when I was in my Elliptic Curve Cryptography class. $Z_n \leftrightarrow Z_{f_1} \times Z_{f_2} \times ...$ where $f_1 \times f_2 \times ... = n$ and $\{f_1, f_2, ...\}$ are pairwise coprime. Applications of this Chinese Remainder Theorem not only include computational speedups (in the case of ....

Autumn Autumn: 13 hours ago

Math

Group action: 'Minimal' subgroup generating an orbit

Let $G$ be a symmetric group on a finite set acting on another finite set $X$ through a natural action $\alpha:G \times X \to X$, $\alpha(g,x)=gx$. Let $x \in X$ and consider the orbit $G \cdot x := \{gx: g \in G\}$. Assuming that $|G\cdot x| <|G|$, can we always ....

Ryan Ryan: 13 hours ago

Math

An elementary inequality for graph Laplacians

Let $G$ be an arbitrary graph on $n$ vertices and $\mathcal L$ be its Laplacian. I need to show that \begin{equation}\tag{$*$} \langle \mathcal Lx,\mathcal L(|x|^{p-2}x)\rangle_{\mathbb R^n}\ge 0\qquad \hbox{for all }x\in \mathbb R^n \end{equation} and small $p\in (2,\infty)$ (my guess is that $p\approx 3$ will do), where $|x|^{p-2}x$ is the coordinatewise ....

Miles Miles: 13 hours ago

Math

Do the signs in Puppe sequences matter?

A basic construction in homotopy is Puppe sequences (http://en.wikipedia.org/wiki/Puppe_sequence). Given a map $A \stackrel{f}{\to} X$, its homotopy cofiber is the map $X\to X/A=X \cup_f CA$ from $X$ to the mapping cone of $f$. If we then take the cofiber again, something remarkable happens: $(X/A)/X$ is naturally homotopy equivalent to the ....

Savannah Savannah: 15 hours ago

Math

Lovasz local lemma for the edge model

In order to successfully apply the Lovasz local lemma, one needs the events to be relatively independent. This (sometimes) works well in the $G(n,p)$ model of random graphs, where the presence or absence of edges doesn't affect the events they are not involved in. However, in the $G(n,m)$ model (where ....

Gabriella Gabriella: 15 hours ago

Math

examples of Kähler manifolds with trivial Hodge numbers and first Chern classes

Yesterday I asked the following question to which abx has given a positive answer. examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes (https://mathoverflow.net/questions/168029/examples-of-kahler-manifolds-with-trivial-odd-betti-numbers-and-first-chern-clas) But I suddenly realized that those Fano manifolds I mentioned in the quesion above (flag manifolds $G/P$ and Fano contact manifolds) not ....

Aurora Aurora: 15 hours ago

Math

$| f_n |^p - | f |^p - | f_n-f |^p$ converges in distribution sense if $f_k$ converges almost everywhere and weakly to $f$?

Let $1<p<\infty$ and $f_n$ be a sequence in $L^p(S^1)$ that converges weakly to some $f$. Here $S^1$ is the circle so we are dealing with periodic functions. Let us see if $| f_n |^p - | f |^p - | f_n-f |^p$ converges to zero in distribution sense. This means ....

Amelia Amelia: 16 hours ago

Math

Global sections of determinant bundle of symmetric powers

Let $E$ be a globally generated vector bundle of rank $r$ on a normal irreducible projective variety $X$. Suppose that $E$ induces a finite map $$X \to \mathbb{G}r (H^0(E), r)$$ to the Grassmannian of rank r quotients. Can we say that a symmetric power of $S^mE$ for $m>>0$ induces an ....

Jace Jace: 16 hours ago

Math

Splitting the determinant polynomial into linear factors - a Dedekind problem

Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial $\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left$ in $n^2$ indeterminates $X_{i,j}$. This is known to be irreducible over $\mathbb Z$, but is there a "nice" ring in which $\mathbb Z$ embeds and where this ....

Amir Amir: 17 hours ago