Analysis of the picture cube puzzle

Definition 4.1 .

For $\chi\in{\rm Irr}(G)$ , let $K$ be the largest factorial subgroup contained in $\ker(\chi)$ . We define the base of $\chi$ to be the factorial complement of $K$ in $G$ , and the weight of $\chi$ by

{\rm wt}(\chi)={\rm rank}({\rm base}(\chi)).

Definition 5.2 .

A differential graded Lie algebra (DGLA) is a GLA $\mathfrak{g}$ together with a differential $d:\mathfrak{g}\rightarrow\mathfrak{g}$ , i.e. a linear operator of degree 1 which satisfies the Leibnitz rule

d[a,b]=[da,b]+(-1)^{\alpha\beta}[a,db]\qquad a\in\mathfrak{g}^{\alpha},\quad b% \in\mathfrak{g}^{\beta}

(30)

and $d^{2}=0$ .

Definition 1 .

( Semimetric ) Let $\mathcal{Z}$ be a non-empty set and let $\rho:\mathcal{Z}\times\mathcal{Z}\to[0,\infty)$ be a function such that $\forall z,z^{\prime}\in\mathcal{Z}$ ,

1.

$\rho(z,z^{\prime})=0$ if and only if $z=z^{\prime}$ , and
2.

$\rho(z,z^{\prime})=\rho(z^{\prime},z)$ .

Then $(\mathcal{Z},\rho)$ is said to be a semimetric space and $\rho$ is called a semimetric on $\mathcal{Z}$ .

2.2 Definition

A (Jordan) triple derivation is a linear map $d$ on a Jordan triple satisfying

d\{x,y,z\}=\{dx,y,z\}+\{x,dy,z\}+\{x,y,dz\}.

Definition 1.18 .

The spectral mean $a\mu b$ of $(a,b)\in\mathcal{C}$ is $\mathop{\text{P}}(a^{-1}\#b)^{1/2}a$ . It is the unique solution in $J$ of the equation

(a^{-1}\#b)^{1/2}=a^{-1}\#x.

Definition 3.1 .

If $\nu\leq\rho$ are cardinals, then write

{\rho}^{[{\nu}]}={\rho}

iff there is a family ${\mathcal{B}}\subset\bigl{[}{\rho}\bigr{]}^{\leq{\nu}}$ of size ${\rho}$ such that for all $u\in\bigl{[}{\rho}\bigr{]}^{\nu}$ there is ${\mathcal{P}}\in\bigl{[}{{\mathcal{B}}}\bigr{]}^{<{\nu}}$ such that $u\subset\cup{\mathcal{P}}$ .

Definition 3.8 .

Two Freudenthal triple systems $(V,b,t)$ , $(V^{\prime},b^{\prime},t^{\prime})$ over a field $K$ are similar if there exists a $K$ -vector space isomorphism $\psi:V\rightarrow V^{\prime}$ and $\lambda\in K^{*}$ such that

t^{\prime}(\psi(x),\psi(y),\psi(z))=\lambda\psi(t(x,y,z)).

In [ F , Lemma 6.6] it is proven that this condition is equivalent with

\begin{cases}b^{\prime}(\psi(x),\psi(y))=\lambda b(x,y)\ \text{ and }\\ b^{\prime}(\psi(x),t^{\prime}(\psi(x),\psi(x),\psi(x)))=\lambda^{2}b(x,t(x,x,x% )).\end{cases}

The map $\psi$ is then called a similarity with multiplier $\lambda$ . We say that two Freudenthal triple systems are isometric if they are similar with $\lambda=1$ ; in this case $\psi$ is called an isometry .

Definition 3.5 .

Let $A$ be a random operator mapping each element of the probability space $(\Omega,\mathcal{A},\mathbb{P})$ to an linear operator on the Hilbert space $\mathcal{H}$ . Then $A$ is called metrically transitive , if there exists a group $\mathcal{T}$ of measure preserving automorphisms of $(\Omega,\mathcal{A},\mathbb{P})$ , a group of unitary operators $\mathcal{U}:=\{U_{T}\mid T\in\mathcal{T}\}$ on $\mathcal{H}$ and a homomorphism from $\mathcal{T}$ to $\mathcal{U}$ such that

B\in\mathcal{A}\text{ such that }TB=B\text{ for all }T\in\mathcal{T}\quad% \Rightarrow\quad\mathbb{P}(B)\in\{0,1\}

(3.8)

and one has for all $\omega\in\Omega$ and all $T\in\mathcal{T}$ the relation

A(T\omega)=U_{T}A(\omega)U_{T}^{-1}.

(3.9)

Definition 13 (The bracket) .

For any Lyndon word $l$ , the bracketed form $[l]$ , is defined as follows:

[l]=[[u],[v]]

with $[x_{0}]=x_{0}$ , $[x_{1}]=x_{1}$ and $[x_{0},x_{1}]=x_{0}x_{1}-x_{1}x_{0}$ and for $l=uv$ , where $u,v$ are Lyndon words and $v$ is the longest Lyndon word such that $l=uv$ .

Definition 2 .

A path is a list of element names $n.n^{\prime}.n^{\prime\prime}.\cdots$ . The predicate $exists(p,S)$ for a path $p$ and a file system $S$ is defined recursively as follow:

		$\displaystyle exists(\varnothing,c)=true$
$\displaystyle type(c)=directory\wedge\exists(n^{\prime},c^{\prime})\in c$	$\displaystyle\implies$	$\displaystyle exists(n^{\prime}.p,(n,c))=exists(p,c^{\prime})$
$\displaystyle else$	$\displaystyle\implies$	$\displaystyle exists(n^{\prime}.p,(n,c))=false$

The function $content(p,S)$ returns the content of the element at path $p$ in $S$ . The predicate $prefix(p^{\prime},p)$ is true if and only if the list $p^{\prime}$ is a non-strict prefix of the list $p$ .

Definition 1

Given $g:\mathcal{X}\times\mathcal{Y}\to\mathcal{X}$ , the system

\displaystyle\hat{x}^{+}=g(\hat{x},\,y)

is said to be a deadbeat observer for the system ( 3 ) if there exists an integer $p\geq 1$ such that, for all initial conditions, the solutions of the array ( 3.1 ) satisfy $\hat{x}(k)=x(k)$ for all $k\geq p$ . The integer $p$ then is called a deadbeat horizon .

Definition 3.1 .

Let $Q$ be a graph. The Brauer monoid ${\rm BrM}(Q)$ is the monoid generated by the symbols $R_{i}$ and $E_{i}$ , for each node $i$ of $Q$ and $\delta$ , $\delta^{-1}$ subject to the following relation, where $\sim$ denotes adjacency between nodes of $Q$ .

\delta\delta^{-1}=1

(3.1)

R_{i}^{2}=1

(3.2)

R_{i}E_{i}=E_{i}R_{i}=E_{i}

(3.3)

E_{i}^{2}=\delta E_{i}

(3.4)

R_{i}R_{j}=R_{j}R_{i},\,\,\mbox{for}\,\it{i\nsim j}

(3.5)

E_{i}R_{j}=R_{j}E_{i},\,\,\mbox{for}\,\it{i\nsim j}

(3.6)

E_{i}E_{j}=E_{j}E_{i},\,\,\mbox{for}\,\it{i\nsim j}

(3.7)

R_{i}R_{j}R_{i}=R_{j}R_{i}R_{j},\,\,\mbox{for}\,\it{i\sim j}

(3.8)

R_{j}R_{i}E_{j}=E_{i}E_{j},\,\,\mbox{for}\,\it{i\sim j}

(3.9)

R_{i}E_{j}R_{i}=R_{j}E_{i}R_{j},\,\,\mbox{for}\,\it{i\sim j}

(3.10)

The Brauer algebra ${\rm Br}(Q)$ is the the free $\mathbb{Z}$ -algebra for Brauer monoid ${\rm BrM}(Q)$ .

Definition 3.13 .

Let us now define a product $y\cdot x$ by

\left[\begin{array}[]{cc}y\cdot x&0\\ 0&0\end{array}\right]=2\left\{\left[\begin{array}[]{cc}x&0\\ 0&0\end{array}\right]\left[\begin{array}[]{cc}0&v\\ 0&0\end{array}\right]\left[\begin{array}[]{cc}0&y\\ 0&0\end{array}\right]\right\}

and denote the corresponding matrix product by $X\cdot Y$ . That is, if $X=[x_{ij}]$ and $Y=[y_{ij}]$ , then $X\cdot Y=[z_{ij}]$ where

z_{ij}=\sum_{k}x_{ik}\cdot y_{kj}.

Definition 2.1 (pinning) .

Let $f:[q]^{d}\rightarrow\mathbb{F}$ be a $d$ -ary symmetric function. Let $0\leq k\leq d$ and $\tau\in[q]^{k}$ . We define that $\mbox{{Pin}}_{{\tau}}\left({f}\right)=g$ where $g:[q]^{d-k}\rightarrow\mathbb{F}$ is a $(d-k)$ -ary symmetric function such that

\forall\sigma\in[q]^{d-k},\quad g(\sigma)=f(\sigma(1),\ldots,\sigma(d-k),\tau(% 1),\ldots,\tau(k)).

Specifically, when $k=0$ the resulting function $g=f$ ; and when $k=d$ , the resulting function $g$ is a trivial function $f(\sigma)$ .

Definition 1 .

Let $L=\mathbb{F}_{q}$ , with $q=p^{n}$ for some positive integer $n$ . A function $f:L\rightarrow L$ is said to be almost perfect nonlinear (APN) on $L$ if for all $a,b\in L$ , $a\neq 0$ , the equation

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

(1)

has at most 2 solutions.

Definition 1 .

A bilinear map $\mu\colon U\times H\to H$ is called a composition of $\varphi$ and $\lambda$ if for any $u\in U$ and any $h\in H$ the equality

(4)

\varphi(\mu(u,h))=\lambda(u)\varphi(h)

holds.

Definition 4.11 .

For $\boldsymbol{\lambda}=(\lambda^{(1)},\lambda^{(2)},\cdots,\lambda^{(\ell)})\in% \Pi^{\ell}$ , we define $\widetilde{\boldsymbol{\lambda}}$ , $\check{\boldsymbol{\lambda}}\in\Pi^{\ell}$ by

\displaystyle\widetilde{\boldsymbol{\lambda}}=(\widetilde{\lambda^{(1)}},% \widetilde{\lambda^{(2)}},\cdots,\widetilde{\lambda^{(\ell)}})\quad,\quad% \check{\boldsymbol{\lambda}}=(\check{\lambda^{(1)}},\check{\lambda^{(2)}},% \cdots,\check{\lambda^{(\ell)}})\quad,

where $\lambda^{(i)}=\widetilde{\lambda^{(i)}}+n\check{\lambda^{(i)}}$ and $\widetilde{\lambda^{(i)}}$ is $n$ -restricted. (See Definition 3.5 .) And we define

S_{\boldsymbol{\lambda}}=\prod_{i=1}^{\ell}S_{\lambda^{(i)}}[i]=S_{\lambda^{(1% )}}[1]S_{\lambda^{(2)}}[2]\cdots S_{\lambda^{(\ell)}}[\ell].

Definition 3.3 .

We say that a polynomial mapping $F:\mathbb{A}^{n}\to\mathbb{A}^{k}$ is Newton non-degenerate at infinity , resp. Newton strongly non-degenerate at infinity, if for any vector $\mathbf{p}\in{\mathbb{Z}}^{n}\setminus\{0\}$ with $p<0$ and such that $\mathrm{N}_{\mathbf{p}}\not=\emptyset$ , the following condition is satisfied:

(*) Sing F_{△_{𝐩}} \cap {x \in 𝔸^{n} ∣ f_{△_{𝐩}^{j}} (x) = 0, \forall j \in N_{𝐩}} \cap {(𝔸^{*})}^{n} = \emptyset,

respectively

(* *) Sing F_{△_{𝐩}} \cap {(𝔸^{*})}^{n} = \emptyset .

Définition 6 .

Soit ${\mathcal{H}}$ une algèbre de Hopf et ${\mathcal{A}}$ une algèbre commutative. On dit que $\varphi:{\mathcal{H}}\rightarrow{\mathcal{A}}$ est un caractère si $\varphi(\hbox{\bf 1})=\hbox{\bf 1}$ et pour tout $x$ , $y\in{\mathcal{H}}$ on a : $\varphi(xy)=\varphi(x)\varphi(y)$ , et on dit que $\varphi:{\mathcal{H}}\rightarrow{\mathcal{A}}$ est un caractère infinitésimal si $\varphi(\hbox{\bf 1})=0$ et pour tout $x$ , $y\in{\mathcal{H}}$ on a :

\varphi(xy)=\varphi(x)e(y)+e(x)\varphi(y),

où $e=u\circ\varepsilon$ .