A function $\mathcal{A}\colon\hat{\omega}^{\omega}\to\mathcal{Q}$ is conciliatory if

$$(\forall X,Y\in{\hat{\omega}}^{\omega})\;[X{}^{{\sf p}}=Y{}^{{\sf p}}\;\Longrightarrow\;\mathcal{A}(X)=\mathcal{A}(Y)].$$

A function $\Psi\colon\hat{\omega}^{\omega}\to\hat{\omega}^{\omega}$ is conciliatory if

$$(\forall X,Y\in{\hat{\omega}}^{\omega})\;[X{}^{{\sf p}}=Y{}^{{\sf p}}\;\Longrightarrow\;\Psi(X){}^{{\sf p}}=\Psi(Y){}^{{\sf p}}].$$

We call the vertex operator $\mathcal{V}_{g}=\mathcal{O}_{g}$ a twist field when $g$ lies in the normalizer of Cartan $\mathfrak{h}\subset\mathfrak{g}$ , i.e. $g\in\mathrm{N}_{G}(\mathfrak{h})$ iff

$$\begin{gathered}\displaystyle g\mathfrak{h}g^{-1}=\mathfrak{h}\end{gathered}$$

(3.12)

We say that a triple $(\ell;\alpha,\beta)\in{\mathbb{N}}\times{\mathbb{R}}^{2}$ is admissible, if for any fixed $\varepsilon>0$ and

$$x=m^{\alpha+\varepsilon}\qquad\mbox{and}\qquad y=m^{\beta+\varepsilon}$$

the congruence ( 1.3 ) has a solution for any reduced residue class $a$ modulo $m$ , provided that $m$ is large enough, and we denote by ${\mathfrak{A}}_{3}$ the set of admissible triples.

We say that a pair $(k;\alpha)\in{\mathbb{N}}\times{\mathbb{R}}$ is admissible, if for any fixed $\varepsilon>0$ and

$$x=m^{\alpha+\varepsilon}$$

the congruence ( 1.5 ) has a solution for any reduced residue class $a$ modulo $m$ , provided that $m$ is large enough, and we denote by ${\mathfrak{A}}_{2}$ the set of admissible pairs.

We say that a pair $(k;\alpha)\in{\mathbb{N}}\times{\mathbb{R}}$ is admissible for primes, if for any fixed $\varepsilon>0$ and

$$x=m^{\alpha+\varepsilon}$$

the congruence ( 1.5 ) has a solution for any reduced residue class $a$ modulo $m$ , provided that $m$ is prime and large enough, and we denote by ${\mathfrak{A}}_{2}^{\sharp}$ the set of admissible for primes pairs.

We say that a quadruple $(k,\ell;\alpha,\beta)\in{\mathbb{N}}^{2}\times{\mathbb{R}}^{2}$ is admissible, if for any fixed $\varepsilon>0$ and

$$x=m^{\alpha+\varepsilon}\qquad\mbox{and}\qquad y=m^{\beta+\varepsilon}$$

the congruence ( 1.3 ) has a solution for any reduced residue class $a$ modulo $m$ , provided that $m$ is large enough, and we denote by ${\mathfrak{A}}_{4}$ the set of admissible quadruples.

Let $\lambda\in\mathcal{SP}(n)$ be a shifted partition.The content of a box $(i,j)$ contained in a shifted Young diagram $\lambda$ is its distance from the main diagonal.

$$\operatorname{cont}(i,j)=j-i$$

Let $X\to\operatorname{Spec}\mathfrak{o}_{\mathfrak{p}}$ be a smooth finite type morphism of relative dimension $n$ and let $D\subset X$ be an irreducible divisor which is flat over $\mathfrak{o}_{\mathfrak{p}}$ . Let $x\in X(\mathfrak{o}_{\mathfrak{p}})$ be such that $x\notin D$ and let $t=0$ be a local equation for $D\subset X$ on some affine patch $U\subset X$ containing $x$ . We define the intersection multiplicity of $x$ and $D$ above $\mathfrak{p}$ to be the integer $\iota$ which satisfies

$$x^{*}t=\varpi^{\iota},$$

where $\varpi$ denotes a uniformising parameter of $\mathfrak{o}_{\mathfrak{p}}$ and $x^{*}t$ is the pull-back of $t$ via $x:\operatorname{Spec}\mathfrak{o}_{\mathfrak{p}}\to U$ . We say that $x$ and $D$ meet transversely above $\mathfrak{p}$ if $\iota=1$ .

Let $\mu$ be a reconciliation map from $(T;t,\sigma)$ to $S$ . The auxiliary graph $A$ is defined as a directed graph with a vertex set $V(A)=V(S)\cup V(T)$ and an edge-set $E(A)$ that is constructed as follows

(A1)

For each $(u,v)\in E(T)$ we have $(u^{\prime},v^{\prime})\in E(A)$ , where

$$u^{\prime}=\begin{cases}\mu(u)&\text{if }t(u)\in\{\odot,\bullet\}\\ u&\text{otherwise}\end{cases},\ v^{\prime}=\begin{cases}\mu(v)&\text{if }t(v)\in\{\odot,\bullet\}\\ v&\text{otherwise}\end{cases},$$

(A2)

For each $(x,y)\in E(S)$ we have $(x,y)\in E(A)$ .

(A3)

For each $u\in V(T)$ with $t(u)\in\{\square,\triangle\}$ we have $(u,lca_{S}(\sigma_{T_{\mathcal{\overline{E}}}}(u)))\in E(A)$ .

(A4)

For each $(u,v)\in\mathcal{E}$ we have $(lca_{S}(\sigma_{T_{\mathcal{\overline{E}}}}(u)\cup\sigma_{T_{\mathcal{\overline{E}}}}(v)),u)\in E(A)$

(A5)

For each $u\in V(T)$ with $t(u)\in\{\triangle,\square\}$ and $\mu(u)=(x,y)\in E(S)$ we have $(x,u)\in E(A)$ and $(u,y)\in E(A)$ .

We define $A_{1}$ and $A_{2}$ as the subgraphs of $A$ that contain only the edges defined by (A1), (A2), (A5) and (A1), (A2), (A3), (A4), respectively.

Let $\Omega(t)$ be a domain in $\mathbb{R}^{3}$ depending on time $t\in[0,T)$ for some $T\in(0,\infty]$ , and $\tilde{x}={}^{t}(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3})\in[C^{3}(\mathbb{R}^{4})]^{3}$ . We call $\tilde{x}=\tilde{x}(\xi,t)$ a flow map in $\Omega(t)$ if the three properties hold:
$(\mathrm{i})$ for every $\xi\in\Omega(0)$

$$\tilde{x}(\xi,0)=\xi,$$

$(\mathrm{ii})$ for all $\xi\in\Omega(0)$ and $0\leq t<T$

$$\tilde{x}(\xi,t)\in\Omega(t),$$

$(\mathrm{iii})$ for each $0\leq t<T$

$$\tilde{x}(\cdot,t):\Omega(0)\to\Omega(t)\text{ is bijective}.$$

For $-1<\varepsilon<1$ , let $\Omega^{\varepsilon}(t)$ be a variation of $\Omega(t)$ . Let $\tilde{x}^{\varepsilon}={}^{t}(\tilde{x}^{\varepsilon}_{1},\tilde{x}^{\varepsilon}_{2},\tilde{x}^{\varepsilon}_{3})\in[C^{3}(\mathbb{R}^{4})]^{3}$ . We call $\tilde{x}^{\varepsilon}=\tilde{x}^{\varepsilon}(\xi,t)$ a flow map in $\Omega^{\varepsilon}(t)$ if the three properties hold:
$(\mathrm{i})$ for every $\xi\in\Omega(0)(=\Omega(0))$

$$\tilde{x}^{\varepsilon}(\xi,0)=\xi,$$

$(\mathrm{ii})$ for all $\xi\in\Omega(0)$ and $0\leq t<T$

$$\tilde{x}^{\varepsilon}(\xi,t)\in\Omega^{\varepsilon}(t),$$

$(\mathrm{iii})$ for each $0\leq t<T$

$$\tilde{x}^{\varepsilon}(\cdot,t):\Omega(0)\to\Omega^{\varepsilon}(t)\text{ is bijective}.$$

Let $\Gamma(t)$ be an evolving $2$ -dimensional surface in $\mathbb{R}^{3}$ on $[0,T)$ for some $T\in(0,\infty]$ . Let $x={}^{t}(x_{1},x_{2},x_{3})\in[C^{\infty}(\mathbb{R}^{4})]^{3}$ . We call $x=\hat{x}(\xi,t)$ a flow map on $\Gamma(t)$ if the three properties hold:
$(\mathrm{i})$ for every $\xi\in\Gamma(0)$

$$\hat{x}(\xi,0)=\xi,$$

$(\mathrm{ii})$ for all $\xi\in\Gamma(0)$ and $0\leq t<T$

$$\hat{x}(\xi,t)\in\Gamma(t),$$

$(\mathrm{iii})$ for each $0\leq t<T$

$$\hat{x}(\cdot,t):\Gamma(0)\to\Gamma(t)\text{ is bijective}.$$

The mapping $\xi\mapsto\hat{x}(\xi,t)$ is called a flow map on $\Gamma(t)$ , while the mapping $t\mapsto\hat{x}(\xi,t)$ is called an orbit starting from $\xi$ .

Let $\Gamma(t)$ be an evolving $2$ -dimensional surface in $\mathbb{R}^{3}$ on $[0,T)$ for some $T\in(0,\infty]$ . For $-1<\varepsilon<1$ , let $\Gamma^{\varepsilon}(t)$ be a variation of $\Gamma(t)$ . Let $\hat{x}^{\varepsilon}={}^{t}(x_{1}^{\varepsilon},x_{2}^{\varepsilon},x_{3}^{\varepsilon})\in[C^{\infty}(\mathbb{R}^{4})]^{3}$ . We call $\hat{x}^{\varepsilon}=\hat{x}^{\varepsilon}(\xi,t)$ a flow map on $\Gamma^{\varepsilon}(t)$ if the three properties hold:
$(\mathrm{i})$ for every $\xi\in\Gamma_{0}(=\Gamma(0))$

$$\hat{x}^{\varepsilon}(\xi,0)=\xi,$$

$(\mathrm{ii})$ for all $\xi\in\Gamma_{0}$ and $0\leq t<T$

$$\hat{x}^{\varepsilon}(\xi,t)\in\Gamma^{\varepsilon}(t),$$

$(\mathrm{iii})$ for each $0\leq t<T$

$$\hat{x}^{\varepsilon}(\cdot,t):\Gamma_{0}\to\Gamma^{\varepsilon}(t)\text{ is bijective}.$$

A self-similarity structure for a semigroup $G$ is the data of a set $X$ and a map $\Phi\colon X\times G\to G\times X$ satisfying

$$\Phi(x,{\mathbb{1}})=({\mathbb{1}},x),\quad\Phi(x,g)=(h,y)\wedge\Phi(y,g^{\prime})=(h^{\prime},z)\Rightarrow\Phi(x,gg^{\prime})=(hh^{\prime},z).$$