Resonant Andreev reflections in superconductor-carbon-nanotube devices

Definition 3

A linear operator $\partial:R\to R$ is called a divided difference operator if it is not identically trivial and satisfies the following identity

\partial(xy)=(\partial x)y+x(\partial y)-\alpha(\partial x)(\partial y)

(12)

for all $x,y\in R$ , where $\alpha\in R$ is not invertible in $R$ .

Definition 4

An algebra over $R$ additively isomorphic to $R[t]$ is called an Ore extension if $R\subset R[t]$ and

ta=\phi(a)t+\delta(a).

(19)

The operator $\phi$ from ( 19 ) is called Ore’s $\phi$ -derivation.

Definition 1

A trialgebra $(A,*,\Delta,\cdot)$ with $*$ and $\cdot$ associative products on $A$ (where $*$ may be partially defined, only) and $\Delta$ a coassociative coproduct on $A$ is given if both $(A,*,\Delta)$ and $(A,\cdot,\Delta)$ are bialgebras and the following compatibility condition between the products is satisfied for arbitrary elements $a,b,c,d\in A$ :

(a*b)\cdot(c*d)=(a\cdot c)*(b\cdot d)

whenever both sides are defined.

Definition 5.4 .

For two mapping classes $f,h\in{\cal M}_{g,b,n}$ , we let

i(f,h)=i({\sigma}(f),{\sigma}(h)).

Definition 4.5 .

Let $b\in H$ , where $H$ is canonically identified with $H^{**}$ , be the left modular function defined by

(29)

b=\eta\circ\nu^{*}

where $\eta$ is the counit of $H^{*}$ defined by $\eta(f)=f(1)$ for every $f\in H^{*}$ .