 read data of water level sensor connected to USB port of Raspberry Pi3
 Running a Python Telegram bot at startup and 24/7
 Pauli principle historySchrodinger equation history
 FTIR analysis, can you tell me what the polymer is?
 What is LSD1, …, LSD24, LSD26, …?
 Which are the radicals containing phosphorus in ATP?
 In 1 dimension, the curvature of Ψ = K.E. of wave. Is this the case for 2 dimensions and 3 dimensions also?
 draw a Newman projection organic chemistry
 Questions about calculating number of energy states using momentum space
 Calculate the fraction of Am241 in the smoke detector, after 50 years of service time of the smoke detector
 Can someone please post reactivity series of functional groups with various reducing agent as done in this answer on stackexchange
 How can I let my boss know I currently put out the effort for a different job title/salary level?
 small or mediumsized exercises and examples on the use, identification and instance declaration of Haskell's abstractions
 Bounded Quantification: Full F<: intuition
 How to reduce MaxUNSAT to MaxSAT?
 What does “thanks xinstitute for hospitality” mean in an acknowledgement?
 Examining a tech related topic  brainstorming for a bachelor thesis
 How to deal with lots of informations as a computer science student?
 Quality issue in class room lecturing
 How widespread is the use of “last author = teacher” on student papers?
Which functions are the composition of convex functions?
The composition of convex functions is not necessarily convex or concave: For example, composing $f_1(x) = x^21$ and $f_2(y) = y^2$ gives $f_2(f_1(x)) = (x^21)^2$. Or consider $f_1(x) = x^2$ and $f_2(y) = e^{y}$, which yield $f_2(f_1(x)) = e^{x^2}$. (Of course, the composition is convex if the outer function is additionally assumed to be monotonically increasing.)
The question, then, is: What are all the functions which can be expressed as the composition of convex functions?
Two slightly different ways to state this a little more formally:
Which functions can be expressed as $f_n \circ \cdots \circ f_1$, where $f_1, ..., f_n : \mathbb{R} \to \mathbb{R}$ are convex functions?
The same question, but with $f_k : I_{k1} \to I_k$, where $I_0, I_1, ..., I_n$ are (possibly infinite) intervals of the real line (i.e. convex subsets of $\mathbb{R}$).
(We could even generalize further to allow $f_1$ to be defined on, say, $\mathbb{R}^n$.)
I thought about this questi

Not a complete answer, but I can at least dispose of $h: x \mapsto x^3$. Suppose this is $f \circ g$ with $f$, $g$ convex. Since $h$ is onetoone on $\mathbb R$ we'd need $g$ to be onetoone on $\mathbb R$ and $f$ to be onetoone on $g(\mathbb R)$. Now the left and right onesided derivatives of a convex onetoone function are either strictly positive (if the function is increasing) or strictly negative (if the function is decreasing). This would make it impossible to get $h'(0) = 0$.
On the other hand, e.g. $x + x^3$ is a composition of convex functions. Take
$$ f (x) = g(x) = \cases{x & if $x \ge 0$\cr x  x^3 & if $x < 0$\cr} $$
20180615 09:27:57 
examples : (1) Consider $F(x)=\sin\ x$ which has an infinite local extremums.
If $F(x)=f_1\circ \cdots \circ f_k\ (x)$, then $F'=f_1'\cdots f_k'$.
That is, there is $f_i$ s.t. it has infinite local extremum. It is a contradiction.
(2) If $f_{2k+1}(x)=x^2,\ k\geq 0,\ f_{2}(x)=x1,\ f_4(x)=x1/2,\
f_6(x)=x1/8,\cdots $, then $f_{2k+1}\circ f_{2k}\circ \cdots f_1 \
(x)$ has at least $ 2^k$ local extremums.
20180615 10:46:05