The duality scalar product of a multiform $\Phi$ with a multivector $X$ is the scalar $\left\langle\Phi,X\right\rangle$ defined by the following axioms:

For all $\alpha,\beta\in\mathbb{R}:$

For all $\Phi_{p}\in{\textstyle\bigwedge\nolimits^{p}}V^{\ast}$ and $X^{p}\in{\textstyle\bigwedge\nolimits^{p}}V$ (with $1\leq p\leq n$ ) $:$

where $\left\{e_{j},\varepsilon^{j}\right\}$ is any pair of dual bases over $V.$

For all $\Phi\in{\textstyle\bigwedge}V^{\ast}$ and $X\in{\textstyle\bigwedge}V:$ if $\Phi=\Phi_{0}+\Phi_{1}+\cdots+\Phi_{n}$ and $X=X^{0}+X^{1}+\cdots+X^{n},$ then

The intensity functional of a (ST)CFMPP $\Psi$ is given by

Let $G$ and $H$ be finite groups. A (free) $(G,H)$ -biset $\Omega$ is a set endowed with a free left $H$ -action and a free right $G$ -action, which commute:

When it is not clear from context which groups act on $\Omega$ , we write ${}_{H}\Omega_{G}$ .

Equivalently, $\Omega$ is a left $(H\times G)$ -set such that the restrictions of the action to $H\times 1$ and $1\times G$ are free. This equivalence is formed by setting

Given a $(G,H)$ -biset $\Omega$ the opposite biset is the $(H,G)$ -biset $\Omega^{\mathrm{o}}$ with the same underlying set and with action defined by

If $G=H$ and $\Omega\cong\Omega^{\mathrm{o}}$ as $(G,G)$ -bisets, we say $\Omega$ is symmetric .

Denote by $A_{+}(G,H)$ the monoid of isomorphism classes of $(G,H)$ -bisets with disjoint union as addition. If $\Omega\in A_{+}(G,H)$ and $\Lambda\in A_{+}(H,K)$ , we define the $(G,K)$ -biset $\Lambda\circ\Omega$ to be $\Lambda\times_{H}\Omega$ . With $\circ$ as composition, the monoids $A_{+}(G,H)$ form the morphism sets of a category where the objects are all finite groups. This is also the reason why a $(G,H)$ -biset has $G$ acting from the right and not the left, so that the composition order of bisets $\Lambda\circ\Omega$ fits with the general convention for maps and morphisms.

The point-stabilizer of an element $\omega$ in a $(G,H)$ -biset $\Omega$ is $\operatorname{Stab}_{H\times G}(\omega)\leq H\times G$ , the subgroup consisting of all pairs $(h,g)$ such that $h\cdot\omega=\omega\cdot g$ , or equivalently $h\cdot\omega\cdot g^{-1}=\omega$ . A (injective) $(G,H)$ -pair is a pair $(K,\varphi)$ with $K\leq G$ and $\varphi\colon K\to H$ an injective group map. If $(K,\varphi)$ is a $(G,H)$ -pair, denote by $[K,\varphi]$ the $(G,H)$ -biset $H\times_{(K,\varphi)}G:=H\times G/(h,kg)\sim(h\varphi(k),g)$ . If we also denote by $(K,\varphi)$ the graph of $\varphi\colon K\to H$ :

then $[K,\varphi]\cong(H\times G)/(K,\varphi)$ as $H\times G$ -sets.

We will also refer to the graph $(K,\varphi)$ as a twisted diagonal (subgroup) . In the case that $G=H=S$ is a finite $p$ -group, $K=P\leq S$ , and $\varphi\in\mathcal{F}(P,S)$ for a given fusion system $\mathcal{F}$ on $S$ , we will refer to $(P,\varphi)$ as an $\mathcal{F}$ -twisted diagonal (subgroup) .

The $(G,H)$ -pairs $(K,\varphi)$ and $(L,\psi)$ are $(G,H)$ -conjugate if there are elements $g\in N_{G}(K,L)$ and $h\in N_{H}(\varphi(K),\psi(L))$ such that $L=\prescript{g\!}{}{}K$ and

commutes. This happens if and only if the twisted diagonals $(K,\varphi)$ and $(L,\psi)$ are conjugate as subgroups of $H\times G$ .

(i) A Hom-associative algebra is a triple $(A,\mu,\alpha)$ , in which $A$ is a linear space, $\alpha:A\rightarrow A$ and $\mu:A\otimes A\rightarrow A$ are linear maps, with notation $\mu(a\otimes a^{\prime})=aa^{\prime}$ , such that

for all $a,a^{\prime},a^{\prime\prime}\in A$ . We call $\alpha$ the structure map of $A$ .

A morphism $f:(A,\mu_{A},\alpha_{A})\rightarrow(B,\mu_{B},\alpha_{B})$ of Hom-associative algebras is a linear map $f:A\rightarrow B$ such that $\alpha_{B}\circ f=f\circ\alpha_{A}$ and $f\circ\mu_{A}=\mu_{B}\circ(f\otimes f)$ .
(ii) A Hom-coassociative coalgebra is a triple $(C,\Delta,\alpha)$ , in which $C$ is a linear space, $\alpha:C\rightarrow C$ and $\Delta:C\rightarrow C\otimes C$ are linear maps ( $\alpha$ is called the structure map of $C$ ) such that

A morphism $g:(C,\Delta_{C},\alpha_{C})\rightarrow(D,\Delta_{D},\alpha_{D})$ of Hom-coassociative coalgebras is a linear map $g:C\rightarrow D$ such that $\alpha_{D}\circ g=g\circ\alpha_{C}$ and $(g\otimes g)\circ\Delta_{C}=\Delta_{D}\circ g$ .

For any commuting units $a$ and $b$ in $\mathcal{L}/\mathcal{L}^{1}$ , the determinant invariant $d(a,b)$ is the nonzero number

In particular, $d(a,b)$ satisfies the relations in Proposition 2.4 .

We say that $f$ is a semimultiplicative function on $P$ if

for all $x,y\in P$ .

Let $Y$ be a $G$ -variety. We construct a closed subscheme $Y_{G}$ of $Y/G$ as follows. Let $\varphi\colon Y\to Y/G$ be the quotient map, and $\sigma$ the involution of the sheaf of $k$ -algebras $\varphi_{*}\mathcal{O}_{Y}$ . We consider the morphism $\operatorname{id}+\sigma\colon\varphi_{*}\mathcal{O}_{Y}\to\varphi_{*}\mathcal{O}_{Y}$ of sheaves of $k$ -vector spaces on $Y/G$ . The subsheaf $\mathcal{O}_{Y/G}$ of $\varphi_{*}\mathcal{O}_{Y}$ coincides with $\ker(\operatorname{id}+\sigma)$ by [ SGA I , V, Corollaire 1.2] , and $(\operatorname{id}+\sigma)\circ(\operatorname{id}+\sigma)=0$ . Thus $\operatorname{id}+\sigma$ induces a morphism $\partial\colon\varphi_{*}\mathcal{O}_{Y}\to\mathcal{O}_{Y/G}$ . If $U$ is an open subscheme of $Y/G$ , and $a,b\in(\varphi_{*}\mathcal{O}_{Y})(U)$ , then we have in $\mathcal{O}_{Y/G}(U)$

When $a\in\mathcal{O}_{Y/G}(U)\subset(\varphi_{*}\mathcal{O}_{Y})(U)$ , then $\partial(a)=0$ , so that $\partial(ab)=a\partial(b)$ . This proves that $\partial\colon\varphi_{*}\mathcal{O}_{Y}\to\mathcal{O}_{Y/G}$ is a morphism of $\mathcal{O}_{Y/G}$ -modules. Its image is a quasi-coherent sheaf of ideals of $\mathcal{O}_{Y/G}$ , and we define $Y_{G}$ as the corresponding closed subscheme of $Y/G$ .

Thus when $Y=\operatorname{Spec}A$ , the closed subscheme $Y_{G}$ is defined by the ideal $\operatorname{im}\partial_{A}$ of $A^{G}$ ( Notation 4.1.1 ).

Let $X$ be a set together with a binary operation denoted $(x,y)\mapsto x\lhd y$ such that for all $y\in X$ , the map $x\mapsto x\lhd y$ is bijective and for all $x,y,z\in X$ ,

Then we call $X$ a (right) rack . In case the invertibility of the maps $x\mapsto x\lhd y$ is not required, it is called a shelf .

Let $G$ be a group and $X$ be a (right) $G$ -set. Then a map $p:X\to G$ is called an augmented rack in case $p$ satisfies the augmentation identity, i.e. for all $g\in G$ and all $x\in X$

[ 4 ] A bimodule of Leibniz algebra $L$ is a vector space $M$ over $\mathbb{F}$ equipped with two bilinear compositions denoted by $ma$ and $am,$ for any $a\in L$ and $m\in M,$ satisfy

A ternary $\mathbb{K}$ -algebra is a triple $(A,\mu,\eta)$ where $A$ is a vector space over a field $\mathbb{K}$ with a multiplication $\mu:A\otimes A\otimes A\rightarrow A$ and a unit $\eta:\mathbb{K}\rightarrow A$ that are linear maps such that the following associativity identity is satisfied

and the following property of the unit is also satisfied

The triple $(A,\mu,\eta)$ defines a weak ternary $\mathbb{K}$ -algebra if, instead of identity ( 4 ), the following weak associativity identity holds

Let $M=(P,(t,h),f,g,\alpha)$ be a Schwarzschild model. We define the corresponding Schwarzschild plane $\Pi_{M}$ to be the pseudo-Riemannian manifold $(P,\zeta)$ such that

A Schwarzschild model $M=(P,(t,h),f,g,\alpha)$ will be called a unimodular model if and only if the special coordinates $(t,h)$ of $P$ were chosen such that

The concatenation product of two injective words $\pi=a_{1}\dots a_{r}$ and $\pi^{\prime}=b_{1}\dots b_{s}$ such that $c(\pi)\cap c(\pi^{\prime})=\emptyset$ is the word $\pi\pi^{\prime}=a_{1}\dots a_{r}b_{1}\dots b_{s}$ . It is extended as the bilinear operator $L(\Gamma_{n})\times L(\Gamma_{n})\rightarrow L(\Gamma_{n})$ defined on Dirac functions by

For $x,y\in L(\Gamma_{n})$ , we define

Let $f$ be $C^{1}$ with respect to $x$ , $\frac{\partial f}{\partial x}$ and $f$ be continuous with respect to $t$ . We will say that $W\subset{\mathbb{R}}\times\mathbb{R}^{n}$ is a trapping isolating segment for ( 28 ) iff for any function $g:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ , $C^{1}$ with respect to $x$ , $\frac{\partial g}{\partial x}$ and $g$ continuous with respect to $t$ , such that $g(t,x)\in[\delta]$ , the set $Z$ is a trapping isolating segment for the semiprocess induced by