论文标题
部分可观测时空混沌系统的无模型预测
Generating non-jumps from a known one
论文作者
论文摘要
令$ r \ ge 2 $为整数。如果存在常数$ c> 0 $，则实际数字$α\是$ r $的跳跃，使得对于任何$ε> 0 $和任何整数$ m \ geq r $，都存在一个整数$ n_0（ε，m），$ r $ - $ r $ - 均匀的$ n \ ge n \ ge n _ $ n_0（+n_0（+ge n_0）+dis+ε，m），m）包含一个带有$ M $顶点的子图和密度至少$α+c $。 Erdős，Stone和Simonovits的结果表明，[0,1）$中的每个$α\都是$ r = 2 $的跳跃。埃尔德（Erd）问$ r \ ge 3 $是否相同。 FranklandRödl通过表明$ 1- \ frac {1} {l^{r-1}} $给出了负面答案，如果$ r $ \ ge 3 $和$ l> 2r $，则不是$ r $的跳跃。之后，使用Frankl和Rödl方法发现了更多的非跳跃。在本说明中，我们展示了一种构建地图$ f \ colon [0,1] \ to [0,1] $的方法，该方法可以保留非跳跃，如果$α$是FranklandRödl方法给出的$ r $的非跳跃，那么$ f（α）$也是$ r $ $ r $的非jump。我们使用这些地图研究了Turán的高图密度，并回答了Grosu提出的问题。
Let $r\ge 2$ be an integer. The real number $α\in [0,1]$ is a jump for $r$ if there exists a constant $c > 0$ such that for any $ε>0$ and any integer $m \geq r$, there exists an integer $n_0(ε, m)$ satisfying any $r$-uniform graph with $n\ge n_0(ε, m)$ vertices and density at least $α+ε$ contains a subgraph with $m$ vertices and density at least $α+c$. A result of Erdős, Stone and Simonovits implies that every $α\in [0,1)$ is a jump for $r=2$. Erdős asked whether the same is true for $r\ge 3$. Frankl and Rödl gave a negative answer by showing that $1-\frac{1}{l^{r-1}}$ is not a jump for $r$ if $r\ge 3$ and $l>2r$. After that, more non-jumps are found using a method of Frankl and Rödl. In this note, we show a method to construct maps $f \colon [0,1] \to [0,1]$ that preserve non-jumps, if $α$ is a non-jump for $r$ given by the method of Frankl and Rödl, then $f(α)$ is also a non-jump for $r$. We use these maps to study hypergraph Turán densities and answer a question posed by Grosu.
