Let $X\,\longrightarrow\,S$ be a $S$ –scheme. Let $dx$ denote the image of $x$ under the canonical de Rham differentiation map $d\,:\,\mathcal{O}_{X}\,\longrightarrow\,\Omega^{1}_{X/S}$ . Let $\mathcal{F}$ be a coherent sheaf of $\mathcal{O}_{X}$ –modules over $X$ . By a $D_{X}$ – module structure on $\mathcal{F}$ we mean a $\mathcal{O}_{S}$ –linear homomorphism of sheaf of abelian groups $\nabla\,:\,\mathcal{F}\,\longrightarrow\,\mathcal{F}\otimes_{\mathcal{O}_{X}}\Omega^{1}_{X/S}$ satisfying Leibniz rule which says that

Given a left pre-Lie algebra $(L,\vartriangleright)$ , the associated symmetric brace operations $\{\#,\ldots,\#|\#\}:L^{\odot n}\otimes L\to L$ are defined recursively by (see [ 22 , 10 ] )

Let $q$ be a primitive $\ell$ -th root of unity prime) and let $T_{q}$ be the Hopf algebra generated by $g,x$ with relations and coproduct as follows:

$T_{q}$ has dimension $\ell^{2}$ and decomposes into a Radford biproduct product $T_{q}=K\rtimes A=\mathbb{C}[x]\rtimes\mathbb{C}[\mathbb{Z}_{\ell}]$ where the $A$ -action and -coaction on $K$ is given by $g.x=qx$ and $\delta(x)=g\otimes x$ . It is a self-dual Hopf algebra via the linear forms $g^{*}:g,x\mapsto q,0$ and $x^{*}:g,x\mapsto 1,1$ .

The Taft algebra appears naturally as the Borel part of the small quantum groups $u_{q^{-1/2}}(\mathfrak{sl}_{2})^{+}$ . The Drinfel’d double $\mathrm{D}T_{q}$ is generated by two isomorphic Taft algebras $g,x$ and $g^{*},x^{*}$ with relations

It has the full quantum group as quotient by the central element $gg^{*}-1$ .

A point $(\alpha,\beta)\in\Omega$ , $\xi_{1}\leq\alpha\leq\xi_{2}$ is said to lie in the domain of influence of the stable root $y=\phi(x)$ if the solution of the problem

exists and remains in $\Omega$ for all $\tau>0$ , and if it tends to $\phi(\alpha)$ , as $\tau\rightarrow+\infty$ .

Operator $p:X\times X\times X\rightarrow X$ is called Maltsev if

Operator $p:X\times X\times X\rightarrow X$ is called a majority operator if

A function $f:\mathbb{Z}^{n}\longrightarrow\mathbb{R}^{*}$ is called a quadratic form if

for $\mathbf{a,b,c}\in\mathbb{Z}^{n}$ .

Let $T_{e}$ denote the number of pairs $(a,b)$ such that

where $a,b$ are integers on the intervals $0\leq a<2^{\ell+m}$ and $0\leq b<2^{\ell}$ .

The result of mounting the directory $d^{\prime}$ at the location $p$ into the directory $d$ is given by

The symmetric measure $\mu$ on the space of perimeter 2 triangles is given by the pushforward of the uniform probability measure on the unit sphere under the map

Attacker action. The attacker’s action consists of hiding her malicious activity for a specified period of time. Such an action would restore all of the kernel address space location to their original state, thus avoid detection in case the verifier initiates a PowerAlert -porotocol attestation process. Formally, let $c:\mathcal{R}^{+}\longrightarrow\mathcal{B}$ be the function defining the state of the attacker’s malicious activity at any time $t>0$ , i.e.,

Let $\mathcal{C}$ be the state of all such state functions. We slightly abuse the notation to write $c([t_{a},t_{b}])$ to refer to the state of the attacker’s activity in the time interval $t_{a}\leq t\leq t_{b}$ .

An attacker’s action is therefore defined as a function $a_{1}:\mathcal{C}\longrightarrow\mathcal{C}$ that changes the state of the attacker’s activity for a period of time $\alpha_{1}$ . Formally,

For $x,y\in 2^{\omega}$ , the join of $x$ and $y$ is $x\oplus y$ , defined by

We make a similar definition for $\sigma\in 2^{m}$ and $\tau\in 2^{m}$ or $\tau\in 2^{m-1}$ : $\sigma\oplus\tau$ has domain $2m$ in the first case, and $2m-1$ in the second, and

If $z\in 2^{\leq\omega}$ , we can view $z$ as a join, and define its left and right parts: If $z=x\oplus y$ , then $\ell(z)=x$ and $r(z)=y$ .