5:42 AM
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I was thinking, specifically, of this paper, in which Zagier offers a proof of the PNT, inspired by a paper of Newman's, the Cliffs Notes version of which would be that, first, it's fairly easy to show that \left|\int_1^{\infty}\frac{\vartheta(x) - x}{x^2}dx\right| < \infty \implies \vartheta(...

1

Sketch the curve $\mathrm{f}(x)=x^{3}+Ax^{2}+B$ first in the case $A>0$ and $B>0$, and then in the case $A<0$ and $B>0.$ Show that the equation $x^{3}+ax^{2}+b=0$, where $a$ and $b$ are real, will have three distinct real roots if $27b^{2}+3a^{3}b<0,$ but will have fewer than three if $27b^{... 2 Define a monomial function$f: G \to G$to be a function that can be written in the form$f(x) = g x g'$for some$g, g' \in G$. Let$G$be a finite group and consider the set of$S$of all monomial functions from$G$to$G$that take one argument. For instance$(f(x) = abxcd) \in S$. Now if$...

7 hours later…
12:27 PM
Also among [new tags]: and . Probably both of them created in the same question.
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Could anyone help me to solve the following? $f:[0,1]\to[0,1]$ be a diffeomorphism, I need to show (1) $f'(x)>0\Rightarrow$ it has only fixed points but no periodic points. (2) $f'(x)<0\Rightarrow$ it has unique fixed point, periodic points has period two if there is any. Thanks.

2

What are the criteria such that a function $f(t)$ can serve as the correction in an iteration function of the form $g (t) = t - \lambda f (t)$ where $\lambda$ is some relaxation factor? It is almost reminiscent of Newton's iteration, without the derivative. for instance, if $f(t)=sin(t)$ then \$g...

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