Directoids (join) are groupoids of the form $H=\left\langle\underline{H},\cdot\right\rangle$ that satisfy the following conditions:

	$\displaystyle aa=a$		(dir1)
	$\displaystyle(ab)a=ab$		(dir2)
	$\displaystyle b(ab)=ab$		(dir3)
	$\displaystyle a((ab)c)=(ab)c$		(dir4)

For $\psi\in BC(\mathbb{G})$ , define inductively:

$$u^{\varepsilon}(0,p)=\psi(p)$$

and for $t\in((k_{t}-1)\varepsilon^{2},k_{t}\varepsilon^{2}]$ :

$$u^{\varepsilon}(t,p)=\frac{1}{1+\mu\varepsilon^{2}}\inf_{\eta,\mathcal{X}}\sup_{q}\left\{u^{\varepsilon}(t-\varepsilon^{2},p\cdot\delta_{\varepsilon}(q))+R^{\varepsilon}(t,p,q,\eta,\mathcal{X})\right\}.$$

When $k_{t}=1$ , $u^{\varepsilon}(t-\varepsilon^{2},p\cdot\delta_{\varepsilon}(q))$ is understood as $u^{\varepsilon}(0,p)$ .

A variation of $(u,\eta,\boldsymbol{\lambda})\in\mathit{HP}_{\mathbf{Z}}$ is a curve $\{(u^{\epsilon},\eta^{\epsilon},\boldsymbol{\lambda}^{\epsilon})\}_{\epsilon\in(-1,1)}\subset\mathit{HP}_{\mathbf{Z}}$ of the form

$$(u^{\epsilon},\eta^{\epsilon},\boldsymbol{\lambda}^{\epsilon})=(u+\epsilon\boldsymbol{\delta}u,\psi^{\epsilon}\circ\eta,\boldsymbol{\lambda}+\epsilon\boldsymbol{\delta}\lambda),$$

where $\psi\in C^{\infty}([-1,1]\times[0,T];\operatorname{Diff}_{C^{\infty}})$ is defined to be the flow (in the $t$ -variable) given by

$$\partial_{t}\psi^{\epsilon}_{t}X=\epsilon\partial_{t}\boldsymbol{\delta}w_{t}(\psi^{\epsilon}_{t}X),\quad\psi^{\epsilon}_{0}X=X\in M,$$

for arbitrarily chosen $(\boldsymbol{\delta}u,\boldsymbol{\delta}w,\boldsymbol{\delta}\lambda)\in C^{\infty}_{T}(\mathfrak{X}_{C^{\infty}}\times\mathfrak{A}_{C^{\infty}}\times\mathfrak{A}^{\vee}_{C^{\infty}})$ which vanishes at $t=0$ and $t=T$ ,

HCT2015 Let $\ast$ be a binary operation on $I$ , $\vartriangle$ be a $t$ -norm on $I$ , and $\triangledown$ be a $t$ -conorm on $I$ . Define the binary operations $\curlywedge$ and $\curlyvee:\mathbf{M}^{2}\rightarrow\mathbf{M}$ as follows: for $f,g\in\mathbf{M}$ ,

$$(f\curlywedge g)(x)=\sup\left\{f(y)\ast g(z):y\vartriangle z=x\right\},$$

(1.1)

and

$$(f\curlyvee g)(x)=\sup\left\{f(y)\ast g(z):y\ \triangledown\ z=x\right\}.$$

(1.2)

Define a binary operation $\barwedge:\mathbf{M}^{2}\longrightarrow\mathbf{M}$ as follows: for $f,g\in\mathbf{M}$ ,

$$(f\barwedge g)(x)=\begin{cases}(f\sqcap g)(x),&x\in[0,1),\\ 0,&x=1.\end{cases}$$

Let $G$ be a group of order $v$ . A collection $D_{1},D_{2},\ldots,D_{f}$ of $f$ subsets of $G$ with $|D_{i}|=k_{i}$ is called a supplementary difference set in $G$ denoted by $f$ -SDS( $v;k_{1},\ldots,$ $k_{f};\lambda$ ) if for each $\alpha\in G\setminus\{e\}$ , the constraint

$$\alpha=xy^{-1},$$

where $x,y\in D_{i}$ for some $i\in\{1,2,\ldots,f\}$ , has exactly $\lambda$ solutions.

If $x,y\in\mathbb{N}$ we define the pairing function $\langle x,y\rangle$ to be the following term in $V^{0}$ :

$$\langle x,y\rangle=(x+y)(x+y+1)+2y$$

(10)

Since the formula for pairing function is just a term in standard vocabulary for the theory $V^{0}$ , it is obvious that $V^{0}$ proves the condition ( 7 ). It is also easy to prove in $V^{0}$ that pairing function is a one-one function, that is:

$$V^{0}\vdash\forall x_{1},x_{2},y_{1},y_{2}\ \langle x_{1},y_{1}\rangle=\langle x_{2},y_{2}\rangle\to x_{1}=x_{2}\wedge y_{1}=y_{2}$$

(11)

The “discrete Lorentz transformation” expresses the ${F^{\prime}}$ coordinates in terms of the $F$ coordinates as:

$$t^{\prime}=\left(\cosh\left(\rho\right)\right)t-\left(\sinh\left(\rho\right)\right)\mathbf{x}\cdot\mathbf{u},$$

(43)

where ${\cdot}$ denotes the standard (“dot”) inner product of vectors in ${\mathbb{Z}^{n}}$ .

$h$ is a step function taking the value $\alpha:=0.971239$ between $[0,x)$ and the value $\beta:=0.873362$ between $[x,1]$ where $x:=0.236901$ (Figure 1 ), i.e.

$\displaystyle h(\sigma)=\begin{cases*}0.971239&if $\sigma\in[0,0.236901)$\\ 0.873362&otherwise.\end{cases*}$

[ 8 ] A function $f:[a,b]\to\mathbb{R}^{\alpha}$ is said to be symmetric with respect to $\frac{a+b}{2}\in[a,b]$ , if

$$f(x)=f(a+b-x)$$

for all $x\in[a,b]$ .

A poset $(L,\geq)$ is said to be:
i) bounded if $L$ contains a minimum 0 and a maximum ${\mathbb{I}}$ ;
ii) orthocomplemented if $L$ is equipped with a map $\neg:L\rightarrow L$ , called orthocomplementation , satisfying:

a) $\neg(\neg p)=p$ for any $p\in L$ ,

b) $p\geq q\Rightarrow\neg q\geq\neg p$ for any $p,q\in L$ ,

c) the greatest lower bound $p\wedge\neg p$ and the least upper bound $p\vee\neg p$ exist in $L$ and

$p\wedge\neg p=0$ , $p\vee\neg p={\mathbb{I}}$ ;
iii) $\sigma$ -orthocomplete if every countable set $\{p_{i}\}_{i\in{\mathbb{N}}}$ made by orthogonal elements, i.e. $\neg p_{i}\geq p_{j}$ (written $p_{i}\perp p_{j}$ ) for $i\not=j$ , admits least upper bound $\vee_{i\in{\mathbb{N}}}p_{i}\in L$ .
iv) orthomodular if $L$ is orthocomplemented and $q\geq p$ implies $q=p\vee(\neg p\wedge q)$ .
Two elements $p,q\in L$ are called compatible if $p=r_{1}\vee r_{3}$ and $q=r_{2}\vee r_{3}$ with $r_{i}\perp r_{j}$ for $i\not=j$ .
A poset $(L,\geq)$ is a lattice if for any $p,q\in L$ the greatest lower bound $p\wedge q$ and the least upper bound $p\vee q$ exist.
A lattice $(L,\geq)$ is said to be distributive if

$\displaystyle p\vee(q\wedge r)=(p\vee q)\wedge(p\vee r)\qquad\mbox{and}\qquad p\wedge(q\vee r)=(p\wedge q)\vee(p\wedge r)$

(5)

for any $p,q,r\in L$ .
A Boolean algebra is a lattice that is distributive, bounded, orthocomplemented (hence orthomodular). A Boolean $\sigma$ -algebra is a Boolean algebra such that any countable subset admits least upper bound.

The pair $(A,\triangleright)$ is a pre-Lie algebra if and only if for any $x,y\in A$ , the identity $[L_{x},L_{y}]_{\circ}=L_{[x,y]_{\triangleright}}$ holds, which is equivalent to the (left) pre-Lie relation

$$x\triangleright(y\triangleright z)-(x\triangleright y)\triangleright z=y\triangleright(x\triangleright z)-(y\triangleright x)\triangleright z.$$

( cf. [ 15 ] ) Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthomodular poset. Then $\mathbf{P}$ is called weakly Boolean if the following condition holds:

• •

  If $x\wedge y=x\wedge y^{\prime}=0$ then $x=0$

( $x,y\in P$ ) . Further, $\mathbf{P}$ is said to have the property of maximality if for all $x,y\in P$ the set $L(x,y)$ has a maximal element.

Let $G$ be a group action on the sets $X$ , $Y$ . A function $f:X\to Y$ is called $G$ -linear if

$$f(gx)=gf(x)$$

holds for all $x\in X$ and $g\in G$ . If $G$ acts trivially on $Y$ , we call $f$ $G$ -invariant .

A message set $\mathcal{M}$ is consistent if

$$\text{for any two value-path pairs}~{}(x,p),(x^{\prime},p^{\prime})\in\mathcal{M},~{}~{}~{}~{}init(p)=init(p^{\prime})~{}\Rightarrow~{}x=x^{\prime}$$

A function $f\colon\mathbb{F}_{2^{n}}\to\mathbb{F}_{2^{n}}$ is called almost perfect nonlinear (APN) if the equation

$$f(x+a)+f(x)=b$$

has exactly $0$ or $2$ solutions for any $b\in\mathbb{F}_{2^{n}}$ and any nonzero $a\in\mathbb{F}_{2^{n}}$ .

A $k$ -algebra $J$ satisfying the identities

1. (1)

   $xy=yx$ ,

2. (2)

   $(x^{2}y)x=x^{2}(yx)$

for all $x,y\in J$ is called a Jordan algebra.

A vector space $V$ over a field $k$ containg $\tfrac{1}{2}$ is called a Jordan triple system if it admits a trilinear product $\{\ ,\ ,\}:V^{3}\rightarrow V$ that is symmetric in the outer variables, while satisfying the identities

1. (1)

   $\{a,b,\{a,c,a\}\}=\{a,\{b,a,c\},a\}$ ,

2. (2)

   $\{\{a,b,a\},b,c\}=\{a,\{b,a,b\},c\}$ ,

3. (3)

   $\{a,\{b,\{a,c,a\},b\},a\}=\{\{a,b,a\},c,\{a,b,a\}\}$

for any $a,b,c\in V$ .