SSM-Net for Plants Disease Identification in Low Data Regime

Definition 2.1 .

Let $G$ be a graph and let $\Gamma\subseteq S^{1}$ be a group. A signature for $G$ is a map $s:E^{or}\to\Gamma$ such that

s(\bar{e})=s(e)^{-1},

(2.1)

where $s(e)^{-1}$ is the inverse of $s(e)$ in $\Gamma$ . The trivial signature $s\equiv 1$ , is denoted by $s_{1}$ . For an oriented edge $e=(u,v)\in E^{or}$ , we will almost always write $s_{uv}:=s(e)$ for simplicity.

Definition 2.4 .

A Riemannian manifold $(M,g,Q)$ is called almost Einstein if the metrics $g$ and $\tilde{g}$ satisfy

\rho(x,y)=\alpha g(x,y)+\beta\tilde{g}(x,y),

where $\alpha$ and $\beta$ are smooth functions on $M$ .

Definition 2.16 .

[ S ] Let $(\mathfrak{g},\cdot)$ be a finite dimensional left symmetric algebra. A scalar product $k$ on $\mathfrak{g}$ is said to be left symmetric if it verifies

k(x\cdot y-y\cdot x,z)=k(x,y\cdot z)-k(y,x\cdot z),\qquad x,y,z\in\mathfrak{g}.

(4)

Moreover, if $\mathfrak{g}$ is a Lie algebra whose Lie bracket is given by the commutator of $\cdot$ the triple $(\mathfrak{g},\cdot,k)$ is called a pseudo-Hessian Lie algebra .

Definition A.22 .

Given the tensor product $J=G\times H$ of graphs $G$ and $H$ , define $f:V(J)\rightarrow V(G)$ so that for all $(u,v)\in V(J)$ ,

f((u,v))=u.

Define $g:V(G)\rightarrow\mathcal{P}(V(J))$ so that for all $u\in V(G)$ ,

g(u)=f^{-1}(u)=\{(u,v)\mid v\in V(H)\}.

Definition 4.1 .

[ 19 ] A 3-Lie algebra is a vector space $A$ together with a skew-symmetric linear map ( $3$ -Lie bracket) $[\;,\;,\;]:\otimes^{3}A\rightarrow A$ such that the following Fundamental Identity (FI) holds:

[[x,y,z],u,v]=[[x,u,v],y,z]+[[y,u,v],z,x]+[[z,u,v],x,y],

(32)

for $x,y,z,u,v\in A$ .

Definition 4.1 .

(Symmetrical solution) The solution $z(t,x)=(\rho,u,v)$ to 3NS is $x$ -symmetric if there exists a function $b(t)\in C^{1}(0,+\infty)$ such that

z(t,x)=z(t,2b(t)-x),\ \ \ \forall t\in[0,\infty),\ \ \ a.e.,

the function $b(t)$ is called the symmetric axis of $z(t,x)$ .

Definition 1.11 .

Let $0<p<1$ be rational. A $p$ -martingale is a function $m:2^{<\omega}\to\mathbb{R}^{\geq 0}$ such that

m(\sigma)=(1-p)m(\sigma 0)+pm(\sigma 1)

for all $\sigma$ . A $p$ -supermartingale is a function $s:2^{<\omega}\to\mathbb{R}^{\geq 0}$ with

s(\sigma)\geq(1-p)s(\sigma 0)+ps(\sigma 1)

for all $\sigma$ . A $p$ -(super)martingale $m$ succeeds on a set $X$ if $\limsup_{n\to\infty}m(X\upharpoonright n)=\infty$ . A set $X$ is $p$ -1-Random if no computably enumerable $p$ -supermartingale succeeds on it.

Definition 4.3 ( [ Pym17 , Theorem 4.5] ) .

Let $X$ be a smooth fourfold, and let $Y$ be a divisor in $X$ . A log symplectic structure $\sigma$ on $(X,Y)$ is purely elliptic if there are local holomorphic coordinates around each singular point of $Y$ in which $Y$ is given by one of the equations in Table 1 and $\sigma$ is of the form

\sigma=d\log f\wedge dw-a\dfrac{xdy\wedge dz}{\lambda f}+b\dfrac{ydx\wedge dz}% {\lambda f}-c\dfrac{zdx\wedge dy}{\lambda f}

up to scaling.

Definition 6 (Future (past) distinguishing) .

A spacetime $\langle\mathcal{M},g_{ab}\rangle$ is future distinguishing iff, for all $p,q\in\mathcal{M}$ ,

I^{+}(p)=I^{+}(q)\Rightarrow p=q

(and similarly for past distinguishing ).

Definition 3.2 (Uniqueness in law) .

We endow the space $C(\mathbb{R})$ with the $\sigma$ -algebra generated by the topology of uniform convergence. Moreover, we equip the space $C([0,T],C(\mathbb{R}))$ with the corresponding Borel $\sigma$ -algebra which we denote $\mathcal{B}(C([0,T],C(\mathbb{R})))$ .
Recall, that if $(u,v)$ is a solution of the system $\eqref{eq:SteppingStoneWithSeedbank}$ then we have for $v$ by Theorem 2.14 that

\displaystyle v(t,x)=e^{-ct}v(0,x)+\tilde{v}(t,x)

(3.7)

where $\tilde{v}$ is a stochastic process taking values in $C([0,T],C(\mathbb{R}))$ and $v(0,\cdot)\in B(\mathbb{R})$ is deterministic. We thus say that the SPDE ( 1.1 ) exhibits uniqueness in law if for any two weak solutions $((u_{1},v_{1}),W_{1},\allowdisplaybreaks\Omega_{1},(\mathcal{F}^{1}_{t})_{t% \geq 0},\mathbb{P}_{1})$ and $((u_{2},v_{2}),W_{2},\Omega_{2},({\mathcal{F}^{2}_{t}})_{t\geq 0},\mathbb{P}_{% 2})$ we have

\displaystyle\mathbb{P}_{1}((u_{1},\tilde{v}_{1})\in\Gamma)=\mathbb{P}_{2}((u_% {2},\tilde{v}_{2})\in\Gamma)

for every $\Gamma\in\mathcal{B}(C([0,T],C(\mathbb{R})))\otimes\mathcal{B}(C([0,T],C(% \mathbb{R})))$ .

Definition 1.1 (Discrete Hessian) .

Let $f:\mathbb{L}\rightarrow\mathbb{R}$ be a function defined on $\mathbb{L}$ . We define the (discrete) Hessian $\nabla^{2}(f)$ to be a function from the set $E(\mathbb{T}_{n})$ of rhombi of the form $\{a,b,c,d\}$ of side $1$ (where the order is anticlockwise, and the angle at $a$ is $\pi/3$ ) on the discrete torus to the reals, satisfying

\nabla^{2}f(\{a,b,c,d\})=-f(a)+f(b)-f(c)+f(d).

Definition 2.1 .

Let $A$ be a ring and let $M$ be an $A$ -module. The Nagata idealization $A\ltimes M$ of $M$ is the ring with support $A\times M$ and operations defined as follow:

(r,m)+(s,n)=(r+s,m+n),\,(r,m)\cdot(s,n)=(rs,sm+rn).

Definition 3.3 (Inner Product) .

If a semi-inner product space $\langle\cdot,\cdot\rangle$ satisfies

\langle x,x\rangle=0\Longrightarrow x=0

then it is called an em inner product.

Definition 5 ( [ 28 ] ) .

Let $\mathcal{F}$ be a germ of foliation on $({\mathbb{C}}^{2},0)$ generated by a holomorphic $1$ -form $\omega$ without common factors in its coefficients. Consider an invariant branch $\Gamma$ of $\mathcal{F}$ given by an irreducible equation $f=0$ . There is an expression

g\omega=hdf+f\alpha,

where $\alpha$ is a holomorphic $1$ -form and $f$ does not divide $g$ . The Camacho-Sad index $\operatorname{CS}_{0}({\mathcal{F}},\Gamma)$ of $\mathcal{F}$ with respect to $\Gamma$ is defined by

\operatorname{CS}_{0}({\mathcal{F}},\Gamma)=\frac{-1}{2\pi i}\int_{\gamma(f)}% \frac{\alpha}{h},

where $\gamma(f)$ is the homological class of the image of the standard loop $z\mapsto\exp(2\pi i)$ under a Puiseux parametrization of $\Gamma$ .

Definition 2.1 .

Let $(A,\cdot)$ be an anti-flexible algebra and $V$ be a vector space. Let $\displaystyle l,r:A\rightarrow{\rm End}(V)$ be two linear maps. If for any $x,y\in A$ ,

\displaystyle l{(x\cdot y)}-l(x)l(y)=r(x)r(y)-r({y\cdot x}),

(2.1)

\displaystyle l(x)r(y)-r(y)l(x)=l(y)r(x)-r(x)l(y),

(2.2)

then it is called a bimodule of $(A,\cdot)$ , denoted by $\displaystyle(l,r,V)$ . Two bimodules $(l_{1},r_{1},V_{1})$ and $(l_{2},r_{2},V_{2})$ of an anti-flexible algebra $A$ is called equivalent if there exists a linear isomorphism $\varphi:V_{1}\rightarrow V_{2}$ satisfying

\varphi l_{1}(x)=l_{2}(x)\varphi,\varphi r_{1}(x)=r_{2}(x)\varphi,\;\;\forall x% \in A.

(2.3)

Definition 2.1 .

A derivation on a ring ${\mathcal{R}}$ is a map $\delta:{\mathcal{R}}\rightarrow{\mathcal{R}}$ satisfying that for all $a,b\in{\mathcal{R}}$ ,

\delta(a+b)=\delta(a)+\delta(b),\,\,\delta(ab)=\delta(a)b+a\delta(b).

A ring (resp. field) equipped with a derivation is called a differential ring (resp. differential field). An ideal $I\subset{\mathcal{R}}$ is called a differential ideal if $\delta(I)\subset I$ .

Definition 3.1 .

$\mathfrak{s}l_{2}$ (respectively ${\cal U}(\mathfrak{s}l_{2})$ ) is the Lie algebra (respectively the associative algebra) over ${\mathbb{Q}}$ generated by $\{e,f,h\}$ with relations

[h,e]=2e,\,[h,f]=-2f,\,[e,f]=h.

${\cal U}_{{\mathbb{Z}}}(\mathfrak{s}l_{2})$ is the ${\mathbb{Z}}$ -subalgebra of ${\cal U}(\mathfrak{s}l_{2})$ generated by $\{e^{(k)},f^{(k)}|\;k\in{\mathbb{N}}\}$ .

Definition 6 .

For two predicates $p$ and $q$ , we say that $p$ “if” and only if $q$ when both of the following hold:

p\Rightarrow q,\qquad q\Rightarrow\mu(p)=1.

Definition 1 (Static stabilizability) .

System ( 1 ) is said to be locally (resp. globally) asymptotically stabilizable by a static state feedback if there exists a locally Lipschitz mapping $\lambda:\mathbb{R}^{n}\rightarrow\mathcal{U}$ such that

\dot{x}=f(x,\lambda(x))

(2)

is locally (resp. globally) asymptotically stable at the origin.

Definition 2 (Dynamic stabilizability) .

System ( 1 ) is said to be locally (resp. globally) asymptotically stabilizable by a dynamic state feedback if there exist $\hat{f}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathcal{U}\to\mathbb{R}^{n}$ such that $\hat{f}(\cdot,\cdot,u)\in C^{1}(\mathbb{R}^{n}\times\mathbb{R}^{n},\mathbb{R}^% {n})$ for all $u\in\mathcal{U}$ ,

$\frac{\partial\hat{f}}{\partial(x,\hat{x})}\in C^{0}(\mathbb{R}^{n}\times% \mathbb{R}^{n}\times\mathcal{U},\mathbb{R}^{n})$

and $\hat{f}(x,\hat{x},\cdot)$ is locally Lipschitz for all $(x,\hat{x})\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ and a locally Lipschitz mapping $\lambda:~{}\mathbb{R}^{n}\rightarrow\mathcal{U}$ such that

\dot{x}=f(x,\lambda(\hat{x})),\quad\dot{\hat{x}}=\hat{f}(x,\hat{x},\lambda(% \hat{x}))

(3)

is locally (resp. globally) asymptotically stable at the origin.

Definition 5 .

(PDM) Following [ 6 , Definition 4] we refer to ( 36 ) as the Payoff Dynamics Model (PDM), and assume that it recovers the static model $p=F(x)$ in steady-state, that is,

f(q,x)=0\quad\Rightarrow\quad h(q,x)=F(x).

(37)

Definition 1 (Memory equivalence) .

Two memories $m$ and $s$ are equivalent up to level $\ell$ , written $m\mathrel{\sim_{\ell}}s$ , if $\mathsf{dom}(m)=\mathsf{dom}(s)$ and it holds that for all $x\in\mathsf{dom}(m)$ ,

\mathit{lev}(x)\sqsubseteq\ell\implies m(x)=s(x)

Definition 3.3.1 .

The category $\textrm{Rep}(H_{t,k}(\nu))$ is defined as follows. The objects are given by triples $(M,x,y)$ , where $M$ is an ind-object of $\textrm{Rep}(S_{\nu})$ , $x$ is a map $x:\mathfrak{h}^{*}\otimes M\to M$ and $y$ a map $y:\mathfrak{h}\otimes M\to M$ , both of which are morphisms in $\textrm{IND}(\textrm{Rep}(S_{\nu}))$ . They also satisfy the following conditions:

x\circ(1\otimes x)-x\circ(1\otimes x)\circ(\sigma\otimes 1)=0,

as a map from $\mathfrak{h}^{*}\otimes\mathfrak{h}^{*}\otimes M$ to $M$ ;

y\circ(1\otimes y)-y\circ(1\otimes y)\circ(\sigma\otimes 1)=0,

as a map from $\mathfrak{h}\otimes\mathfrak{h}\otimes M$ to $M$ ;

y\circ(1\otimes x)-x\circ(1\otimes y)\circ(\sigma\otimes 1)=t\cdot{\rm ev}_{% \mathfrak{h}}\otimes 1-k\cdot({\rm ev}_{\mathfrak{h}}\otimes 1)\circ(\Omega^{3% }-\Omega^{1,3}),

as a map from $\mathfrak{h}\otimes\mathfrak{h}^{*}\otimes M$ to $M$ , where $\Omega$ is a central element from Definition 3.2.7 , and indices indicate the spaces on which $\Omega$ acts in the tensor product $\mathfrak{h}\otimes\mathfrak{h}^{*}\otimes M$ .

The morphisms of $\textrm{Rep}(H_{t,k}(\nu))$ are the morphisms of $\textrm{IND}(\textrm{Rep}(S_{\nu}))$ which commute with the action-maps $x$ and $y$ .

Also by $\textrm{Rep}^{\rm ext}(H_{t,k}(\nu))$ denote the similar category constructed over $\textrm{Rep}^{\rm ext}(S_{\nu})$ .

Definition

[ 8 ] Let $\mu\in\mathcal{M}$ . A continuous function $f:\mathbb{R}\longrightarrow X$ is said to be $\mu$ -pseudo almost automorphic if $f$ is written in the form:

f=g+\varphi,

where $g\in AA(\mathbb{R},X)$ and $\varphi\in\mathcal{E}(\mathbb{R},X,\mu).$
The space of all such functions is denoted by $PAA(\mathbb{R},X,\mu).$ ${}_{\blacksquare}$

Definition

[ 4 ] Let $\mu\in\mathcal{M}$ and $f:\mathbb{R}\times X\longrightarrow Y$ be such that $f(\cdot,x)\in BS^{p}(\mathbb{R},Y)$ for each $x\in X$ . The function $f$ is $\mu$ - $S^{p}$ -almost automorphic if $f$ is written as:

f=g+\varphi,

where $g\in AAS^{p}U(\mathbb{R}\times X,Y)$ , and $\varphi\in\mathcal{E}^{p}U(\mathbb{R}\times X,Y,\mu).$
The space of all such functions is denoted $PAAS^{p}U(\mathbb{R},X,\mu).$ ${}_{\blacksquare}$

Definition 5.2 ( [ Ete18 , §5] ) .

A $j$ -derivation on the field of complex numbers is a derivation $\delta:\operatorname{\mathbb{C}}\operatorname{\rightarrow}\operatorname{% \mathbb{C}}$ such that for any $z\in\operatorname{\mathbb{H}}$ we have

\delta j(z)=j^{\prime}(z)\delta(z),~{}\delta j^{\prime}(z)=j^{\prime\prime}(z)% \delta(z),~{}\delta j^{\prime\prime}(z)=j^{\prime\prime\prime}(z)\delta(z).

The space of $j$ -derivations is denoted by $j\rm{Der}(\operatorname{\mathbb{C}})$ .

Definition 4.4 .

Let $\chi$ be a Dirichlet character mod $q$ . An arithmetical function $f$ is said to be separable with respect to $\chi$ if

f(na)=f(n)\chi(a)

for any positive integers $n,a$ with $\gcd(a,q)=1$ .

Definition 3.2 .

( $\mathcal{N}(n,\omega,\varepsilon)$ -NS sets) Let $\omega>0$ and $0\leq\varepsilon<1$ . An open set $\Omega_{0}\subset\mathbb{R}^{n}$ is said $\mathcal{N}(n,\omega,\varepsilon)$ -NS set if there exists a function $v\in\mathcal{N}(n,\varepsilon)$ such that, up to a translation, its boundary can be represented in polar coordinates as

r(\xi)=\rho(1+v(\xi))\in[0,+\infty]

(3.1)

where $\xi\in\mathbb{S}^{n-1}$ and $\rho=(\omega/\omega_{n})^{\frac{1}{n}}$ is the radius of the ball having the same measure of $\Omega$ .

Definition 4.1 ().

Let ${\bm{x}}\in\mathbb{R}^{n}$ be a real-valued vector and $\pi\in S_{n}$ be a permutation of $n$ elements. A function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is called permutation-equivariant iff

f(\pi({\bm{x}}))=\pi(f({\bm{x}})).

That is, a function is permutation-equivariant if it commutes with any permutation of the input elements.

It is a trivial observation that the self-attention operation is permutation-equivariant.

Definition 2.11 .

Let $N$ and $H$ be inverse semigroups and let $\alpha\colon H\times N\to N$ be a function which we write as $\alpha(h,n)=h\cdot n$ . Then $\alpha$ is an action of inverse semigroups if the following conditions are satisfied for all $h,h^{\prime}\in H$ and $n,n^{\prime}\in N$ .

(1)

$h\cdot(nn^{\prime})=(h\cdot n)(h\cdot n^{\prime})$ ,
(2)

$hh^{\prime}\cdot n=h\cdot(h^{\prime}\cdot n)$ .

Definition 3.1 .

Let $N$ and $H$ be inverse monoids and let $\alpha\colon H\times N\to N$ be a function with application written $\alpha(h,n)=h\cdot n$ . Then $\alpha$ is an action of inverse monoids if it is an action of inverse semigroups and satisifies that for all $n\in N$

1\cdot n=n.

Definition 2.1 .

[ 27 ] A Hom-Lie algebra $(L,\alpha)$ is an ${\mathbb{F}}$ -vector space $L$ endowed with a bilinear map $[-,-]:L\times L\rightarrow L$ and a linear homomorphism $\alpha:L\rightarrow L$ satisfying that for any $x,y,z\in L$ ,

	$\displaystyle[x,y]=-[y,x],~{}~{}\text{(skew-symmetry)}$
	$\displaystyle]+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]=0.~{}~{}\text{(Hom-Jacobi % identity)}$

In particular, a Hom-Lie algebra $(L,\alpha)$ is called multiplicative , if $\alpha$ is an algebra homomorphism, i.e., $\alpha([x,y])=[\alpha(x),\alpha(y)]$ , for all $x,y\in L$ .

Definition 2.2 .

[ 27 , 34 ] For a Hom-Lie algebra $(L,\alpha)$ , a triple $(V,\rho,\beta)$ consisting of a vector space $V$ , a linear map $\rho:L\rightarrow{\rm End}(V)$ and $\beta\in{\rm End}(V)$ is said to be a representation of $(L,\alpha)$ or an $(L,\alpha)$ - module , if for all $x,y\in L$ , the following equalities are satisfied

(2.1)		$\displaystyle\beta\circ\rho(x)=\rho(\alpha(x))\circ\beta,$
(2.2)		$\displaystyle\rho([x,y])\circ\beta=\rho(\alpha(x))\circ\rho(y)-\rho(\alpha(y))% \circ\rho(x).$

For brevity of notation, we usually put $xv=\rho(x)(v),~{}\forall~{}x\in L,v\in V$ , just like the case in Lie algebras. A subspace $W$ of $V$ is called a submodule of $(V,\rho,\beta)$ if $W$ is both $\beta$ -invariant and $L$ -invariant, i.e., $\beta(W)\subset W$ and $xW\subset W,~{}\forall~{}x\in L$ .