Machine learning Applied to Star-Galaxy-QSO Classification and Stellar Effective Temperature Regression

Definition 3.3 .

An almost Kähler manifold $(M,\omega,J)$ is called Chern-Einstein if its Ricci form $\rho$ satisfies

\rho=\lambda\omega

for some constant $\lambda\in\mathbb{R}$ .

Definition 3.1 .

A commutative semiring (sometimes called a commutative rig — commutative ri n g without negative elements) $(R,0,1,+,\cdot)$ consists of a set $R$ , two distinguished elements of $R$ named $0$ and $1$ , and two binary operations $+$ and $\cdot$ , satisfying the following relations for any $a,b,c\in R$ :

\begin{array}[]{rcl}0+a&=&a\\ a+b&=&b+a\\ a+(b+c)&=&(a+b)+c\\ \\ 1\cdot a&=&a\\ a\cdot b&=&b\cdot a\\ a\cdot(b\cdot c)&=&(a\cdot b)\cdot c\\ \\ 0\cdot a&=&0\\ (a+b)\cdot c&=&(a\cdot c)+(b\cdot c)\end{array}

Definition 3.4 .

( $\sharp$ ). By recursion on $|b|$ . First define $\tau$ that maps numeric sizes to their corresponding types. We will revert to using type notation for greater clarity of this definition:

\begin{array}[]{rcl}\tau~{}(0)&=&\bot\\ \tau~{}(1+n)&=&\top\uplus\tau~{}(n)\\ \end{array}

so that we can define $\sharp\,b=\tau~{}|b|$ .

Definition 5

Let ${\mathbb{A}}$ be an associative superalgebra. A ${\mathbb{Z}}_{2}$ -graded linear map $\star:{\mathbb{A}}\longrightarrow{\mathbb{A}}$ is called a superinvolution (or ${\mathbb{Z}}_{2}$ -graded involution) if

(ab)^{\star}=(-1)^{\tau(a)\tau(b)}b^{\star}a^{\star},\qquad(a^{\star})^{\star}=a

for any elements $a,b\in{\mathbb{A}}$ . The pair $({\mathbb{A}},\star)$ is called a ${\mathbb{Z}}_{2}$ -graded $\star$ -algebra.

Definition 5.17 .

An Olšák function is a 6-ary function $o$ that satisfies

o(x,x,y,y,y,x)\approx o(x,y,x,y,x,y)\approx o(y,x,x,x,y,y).

Definition 5.24 .

A Siggers function is a 6-ary function $s$ that satisfies

s(x,y,x,z,y,z)\approx s(y,x,z,x,z,y).

Definition 2.5 (Choreographic holomorphic sphere) .

A holomorphic sphere $u$ in $\mathbb{CP}^{k}$ is choreographic if

u\circ\tau=\sigma\circ u.

Definition 1 .

Let us define

n(p,\omega,\omega^{\prime})=m(m(p,\omega)+1,\omega^{\prime}).

(24)

Using the definition of $m(p)$ in ( 15 ), $n$ is the first geometric scale after $m(p)$ that contains an open circuit.

Definition 2.5 .

Let $f:G\rightarrow G^{\prime}$ be an isogeny of $p$ -divisible groups over a base scheme $S$ . Then the kernel of $f$ is a finite group scheme of rank a power of $p$ . If the rank is $p^{i}$ for some constant $i\in\mathbb{Z}$ , then $i$ is called the height of the isogeny f, and we write:

\mathrm{ht}(f)=i.

Let $\rho:G\rightarrow G^{\prime}$ be a quasi-isogeny of $p$ -divisible groups. Given $n\in\mathbb{Z}$ such that $p^{n}\rho:G\rightarrow G^{\prime}$ is an isogeny, we define the height of the quasi-isogeny $\rho$ to be:

\mathrm{ht}(\rho)=\mathrm{ht}(p^{n}\rho)-\mathrm{ht}(p^{n}).

Definition 3.1 (Halfspaces) .

A halfspace is a function $h:\mathbb{R}^{n}\to\{\pm 1\}$ such that for some unit vector $w\in\mathbb{R}^{n}$ (called the normal vector ) and threshold $t\in\mathbb{R}$

h(x)=\operatorname{sign}(\langle w,x\rangle-t)\,.

Definition 3.13 (Halfspace Angle) .

For two halfspaces $g,h$ with normal (unit) vectors $u,v$ respectively, we will write $\alpha(g,h)$ for the angle between $u$ and $v$ :

\alpha(g,h)\vcentcolon=\cos^{-1}(\langle u,v\rangle)\,.

Definition 6 (Compositional Exponential Family of Distributions) .

The form of compositional probability distribution of exponential family is given by the probability density function

p(\bm{x},\bm{h};\bm{\theta})d\bm{x}d\bm{h}=e^{(k(\bm{h},\bm{x};\bm{\theta})-% \psi(\bm{\theta}))}d\mu(\bm{x})d\nu(\bm{h}),k(\bm{h},\bm{x};\bm{\theta})=% \langle\bm{f}(\bm{\theta};\bm{h}),\bm{g}(\bm{x})\rangle

where $\bm{x}$ is realizable values of a multivariate random variable, $k$ is a function called compositional kernel that for a given $\bm{h}$ , $k$ is the inner product between certain vector function $\bm{g}(\bm{x})$ , called sufficient statistic , (of which the component functions are linearly independent) and certain vector function $\bm{f}(\bm{\theta};\bm{h})$ , called composition function , $\psi(\bm{\theta})$ is the normalization factor, and $\mu,\nu$ ares the laws on r.v. $\bm{x},\bm{h}$ , respectively.

Definition 2.3 (Inversion map) .

We define the inversion map $i:\mathbb{C}^{*}\to\mathbb{C}^{*}$ by

i(z)=1/z

Definition 3.9.1 .

(Connection list.) For each subprogram $p(i)~{}x(i)~{}y(i)$ of the derivation program list $[p(i)~{}x(i)~{}y(i)]_{i=1}^{n}$ that is obtained from an extended program derivation is generated a list that contains the line labels of the premises used to obtain that subprogram. If $k(i)$ is the length of the premise program list of the axiom/theorem labeled $a(i)$ that is employed to infer the statement $p(i)~{}x(i)~{}y(i)$ then the associated connection list, $cl(i):\mathbf{lst}[k(i)]$ , is of the form

cl(i)=[cl(i,1)~{}\ldots~{}cl(i,k(i))]

where $1\leq cl(i,1),~{}\ldots~{},cl(i,k(i))\leq i-1$ are the line labels of the subprograms that make up the sublist

~{}[p(cl(i,j))~{}x(cl(i,j))~{}y(cl(i,j))]_{j=1}^{k(i)}

that is program and I/O equivalent to the premise program of the axiom/theorem, labeled $a(i)$ . Each derived statement of a proof is accompanied by an axiom label followed by a connection list.

Definition 9.2.1 .

(The $gen$ function.) A finite and nonempty set $t$ defined by

t=gen[a]

(9.2.1)

is the set of all irreducible extended programs that have proofs that can be derived from a set, $a$ , such that $a\subset uset$ contains no redundant elements with respect to the derivations of elements of $t$ , i.e. every element of $a$ is involved in the derivation of at least one element of $t$ . We say that $t$ is a derivable set generated by the set $a$ and refer to $a$ as a set of generators of $t$ .

Definition 2.2 ().

The demographic disparity , or Calders-Verwer gap (Calders and Verwer, 2010 ; Kamishima et al . , 2012 ) , $\delta$ , is the difference in mean group outcomes between the advantaged and disadvantaged groups:

(1)

\delta=\mu(a)-\mu(b)

Definition 2.1 .

Let

c(n,k)=e^{\gamma}b(n,k)

and

c(n)=e^{\gamma}b(n).

Definition 3 ( [ 30 ] ) .

The Cassini curve ( or the Cassini ovals ) is a quartic curve defined as the set of points in the plane such that the product of the distances ( denoted by $b^{2}$ ) to two fixed points $(a,0)$ and $(-a,0)$ is constant :

[(x-a)^{2}+y^{2}][(x+a)^{2}+y^{2}]=b^{4}.

(9)

Definition 2.1 .

$\vee:\{0,1\}^{2}\longrightarrow\{0,1\}$ with

\vee(0,0)=0

\vee(0,1)=1

\vee(1,0)=1

\vee(1,1)=1

$\wedge:\{0,1\}^{2}\longrightarrow\{0,1\}$ with

\wedge(0,0)=0

\wedge(0,1)=0

\wedge(1,0)=0

\wedge(1,1)=1

$\neg:\{0,1\}\longrightarrow\{0,1\}$ with

\neg(0)=1

\neg(1)=0

Definition 4.1 .

Let $G=\bigoplus_{\ell=0}^{s}\mathbb{Z}_{m_{\ell}}$ be an arbitrary finite abelian group. The (finite) Weyl-Heisenberg group over $\bigoplus_{\ell=0}^{s}\mathbb{Z}_{m_{\ell}}$ is the semidirect product $G\rtimes(G\times G)$ equipped with the operation

(\lambda,k,\kappa)\cdot(\tilde{\lambda},\tilde{k},\tilde{\kappa})=(\lambda+% \tilde{\lambda}+\tilde{k}\kappa-k\tilde{\kappa},k+\tilde{k},\kappa+\tilde{% \kappa}),

where $k\tilde{\kappa}\in G$ is the sum of the component-wise product of the elements in $G$ . Set $\left|m\right|=\prod_{\ell=0}^{s}m_{\ell}$ , and for each $\ell\in\{0,\ldots,s\}$ , fix a primitive $2m_{\ell}$ th root of unity $\tau_{m_{\ell}}$ where $\zeta_{m_{\ell}}=\tau_{m_{\ell}}^{2}$ is the primitive $m_{\ell}$ th root of unity used to define the modulation operator $M_{m_{\ell}}$ . A standard unitary representation of the finite Weyl-Heisenberg group is

(\lambda,k,\kappa)\mapsto\left(\prod_{\ell=0}^{s}(-\tau_{m_{\ell}})^{\lambda_{% \ell}}\right)D_{m}^{(k,\kappa)}=\left(\prod_{\ell=0}^{s}(-\tau_{m_{\ell}})^{% \lambda_{\ell}-k\kappa}\right)M_{m}^{(\kappa)}T_{m}^{(k)}\in U(\left|m\right|)\,,

We define $\sigma:G\times G\rightarrow U(\left|m\right|)$ by $\sigma(k,\kappa)=M_{m}^{(\kappa)}T^{(k)}_{m}$ . If $\left|m\right|$ is odd, we further define the mapping $\pi:G\times G\rightarrow U(\left|m\right|(\left|m\right|-1)/2)$ as

\pi(k,\kappa)=I_{(\left|m\right|-1)/2}\otimes\left(M_{m}^{(\kappa)}T^{(k)}_{m}% \right).

Definition 2.1 (Admissible trajectory-control pairs and costs) .

For every $z\in{{{\rm I\mskip-3.5mu R}}}^{n}\setminus\mathbf{C}$ , we will say that $(x,u)$ is an admissible trajectory-control pair from $z$ for the control system

(8)

\dot{x}=f(x,u),\quad x(0)=z

if there exists $T_{x}\leq+\infty$ such that $u\in L^{\infty}_{loc}([0,T_{x}),U)$ and $x$ is a Carathéodory solution of ( 8 ) in $[0,T_{x})$ corresponding to $u$ , verifying

x([0,T_{x}))\subset{{{\rm I\mskip-3.5mu R}}}^{n}\backslash\mathbf{C},\ \ % \displaystyle\text{and, if $T_{x}<+\infty$,}\ \ \lim_{t\to T^{-}_{x}}{\bf d}(x% (t))=0

(notice that such a solution might be not unique). We shall use ${\mathcal{A}}_{f}({z})$ to denote the family of admissible trajectory-control pairs $(x,u)$ from $z$ for the control system ( 8 ) . Moreover, we will call cost associated to $(x,u)\in{\mathcal{A}}_{f}({z})$ the function

x^{0}(t):=\int_{0}^{t}l(x(\tau),u(\tau))\,d\tau\ \ \forall t\in[0,T_{x}).

If $T_{x}<+\infty$ , we extend continuously $(x^{0},x)$ to $[0,+\infty)$ , by setting

(x^{0},x)(t)=\lim_{t\to T_{x}^{-}}(x^{0},x)(t)\qquad\forall t\geq T_{x}.

From now on, we will always consider admissible trajectories and associated costs defined on $[0,+\infty)$ .

Definition 3.1 ( $W$ -Feedback) .

Let $W$ be a ${p_{0}}$ -MRF with ${p_{0}}\geq 0$ for $(f,l,{\bf C})$ and fix a selection $p(x)\in D^{*}W(x)$ for any $x\in{{{\rm I\mskip-3.5mu R}}}^{n}\setminus{\bf C}$ . Let $N$ be the same as in Proposition 3.1 . We call $W$ -feedback for the control system

(21)

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

a map

K:x\mapsto K(x)\in U\cap B(0,N(W(x))

verifying

(22)

\langle p(x),f(x,K(x))\rangle+{p_{0}}\,l(x,K(x))<-\gamma(W(x))

for every $x\in{{{\rm I\mskip-3.5mu R}}}^{n}\setminus{\bf C}$ .

Definition 2.1.1

A vector space $\mathcal{E}$ over $\mathbb{C}$ is called an algebra if $\mathcal{E}$ is endowed with a multiplication map

\mu:\mathcal{E}\times\mathcal{E}\longmapsto\mathcal{E}

such that $\mu$ is bilinear and associative. The multiplication between 2 elements $x,y\in\mathcal{E}$ is usually denoted by $x.y$ , i.e. $\mu(x,y)=x.y$ . Thus for all $x,y,z\in\mathcal{E}$ and $\lambda\in\mathbb{C}$ , the conditions of an algebra are given as follows:

1.

$x.(\lambda y+z)=\lambda xy+xz$ and
2.

$(x.y).z=x.z+y.z$ .

Definition 2.1.2

Suppose there is an element $e\in\mathcal{A}$ satisfying

e.x=x.e=x

for all $x\in\mathcal{A}$ . Then $\mathcal{A}$ is called a unital algebra and the element $e$ is called identity element or unit.

It is direct from the definition of the neutral element that in case of existence, the neutral element is unique. It should be mentioned that not every algebra is a unital algebra. Indeed, the space of all real valued functions defined on $\mathbb{R}$ , whose limit at infinity is 0, is an algebra under the point wise multiplication without a unit.

Definition 2 .

Weak full extraction holds if, given $v:T\to{\bf R}$ , there exists a menu of contracts $\{c(t)\in{\bf R}^{S}:t\in T\}$ such that for each $t\in T$ ,

v(t)-\pi(t)\cdot c(t)=0

and

v(t)-\pi(t)\cdot c(s)\leq 0\ \ \forall s\not=t

for some $\pi(t)\in\Pi(t)$ .

Definition 13 .

An arrangement of tuples $(w^{(1)}_{1},\ldots,w_{q-1}^{(1)}),\ldots,(w_{1}^{(q)},\ldots,w_{q-1}^{(q)})$ (understood, depending on the context, as element of $\mathbb{Z}^{q(q-1)}$ or $\mathbb{Z}_{N}^{q(q-1)}$ ) is feasible if

	$\displaystyle\forall j=3,\ldots,q:$
	$\displaystyle\qquad w_{1}^{(j)}=\sum_{a=1}^{q-1}w_{a}^{(j-2)}-\sum_{a=2}^{q-1}% w_{a}^{(j-1)}$		(7)
	$\displaystyle\qquad\forall a=2,\ldots,q-1:w_{a}^{(j)}=-w_{a-1}^{(j-2)}+w_{a-1}% ^{(j-1)}+w_{a}^{(j-1)}$		(8)

For $q\geq 3$ and $N\geq 1$ we let $\mathcal{E}:=\mathcal{E}(q,N)\subseteq\mathbb{Z}_{N}^{q(q-1)}$ to be the set of all feasible arrangements of tuples.

Definition 4.6 .

Let $A$ be a Lie algebroid over the presymplectic manifold $(M,\omega)$ . A $D$ -momentum section $\mu\in\Gamma(M,A^{*})$ is called bracket-compatible if

(15)

(\mathbf{d}\mu)(a,b)=-\omega(\rho a,\rho b)

for all $a,b\in\Gamma(M,A)$ .

Definition 1 .

Perruquetti et al. ( 2008 ) Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a piecewise continuous function with $f(0)=0$ , $\psi(t,x_{0})$ is said to be a right-maximally defined solution of

\dot{x}=-f(x)

(1)

if $\psi(t,x_{0})$ is such that

\frac{{d\psi(t,x_{0})}}{{d}t}=-f(\psi(t,x_{0})),~{}~{}\psi(0,x_{0})=x_{0},\ \ % \ \ \forall t\in[t_{0},T_{m}(x_{0})),\forall x_{0}\in{\cal D}\setminus\{0\}

where $T_{m}(x_{0})\in(0,\infty)$ is the maximal possible real number with the above property, or plus infinity. Moreover, ( 1 ) is said to have a unique solution in forward time if for any $x_{0}\in\mathbb{R}^{n}$ and two right-maximally defined solutions of ( 1 ), $\psi(t,x_{0})$ and $\phi(t,x_{0})$ defined on $[t_{0},T_{m}^{\psi}]$ and $[t_{0},T_{m}^{\psi}]$ , respectively, there exists $t_{0}<T_{m}(x_{0})<\min(T_{m}^{\psi}(x_{0}),T_{m}^{\phi}(x_{0}))$ such that $\psi(t,x_{0})=\phi(t,x_{0})$ for all $t\in[t_{0},T_{m}(x_{0}))$ .

Definition 2.3 .

Let $A$ be a finite set equipped with a sign function; that is, a function $\mathrm{sgn}\colon A\rightarrow\{\pm 1\}$ . A sign-reversing involution $\iota\colon A\rightarrow A$ is an involution such that for every $a\in A$

\mathrm{sgn}(\iota(a))=-\mathrm{sgn}(a).

Definition 1

Let a triangle have angles $(\alpha,\beta,\gamma)$ . We define

s=2\sin(\alpha/2).

Definition 3.2 .

Let $\mathcal{X}=\mathcal{Y}=\{0,1\}^{m}_{\leq d}$ and

f(x,y)=\begin{cases}1&\text{if $\langle x,y\rangle=0$}\\ 0&\text{otherwise}\end{cases}.

Definition 1

A High General Granular Operator Space ( GFSG ) $\mathbb{S}$ shall be a structure of the form $\mathbb{S}\,=\,\left\langle\underline{\mathbb{S}},\mathcal{G},l,u,\mathbf{P},% \leq,\vee,\wedge,\bot,\top\right\rangle$ with $\underline{\mathbb{S}}$ being a set, $\mathcal{G}$ an admissible granulation (defined below) for $\mathbb{S}$ and $l,u$ being operators $:\underline{\mathbb{S}}\longmapsto\underline{\mathbb{S}}$ satisfying the following ( $\underline{\mathbb{S}}$ is replaced with $\mathbb{S}$ if clear from the context. $\vee$ and $\wedge$ are idempotent partial operations.):

	$\displaystyle(\forall a,b)a\vee b\stackrel{{\scriptstyle w}}{{=}}b\vee a\;;\;(% \forall a,b)a\wedge b\stackrel{{\scriptstyle w}}{{=}}b\wedge a$
	$\displaystyle(\forall a,b)(a\vee b)\wedge a\stackrel{{\scriptstyle w}}{{=}}a\;% ;\;(\forall a,b)(a\wedge b)\vee a\stackrel{{\scriptstyle w}}{{=}}a$
	$\displaystyle(\forall a,b,c)(a\wedge b)\vee c\stackrel{{\scriptstyle w}}{{=}}(% a\vee c)\wedge(a\vee c)$
	$\displaystyle(\forall a,b,c)(a\vee b)\wedge c\stackrel{{\scriptstyle w}}{{=}}(% a\wedge c)\vee(a\wedge c)$
	$\displaystyle(\forall a,b)(a\leq b\leftrightarrow a\vee b=b\,\&a\wedge b=a)$
	$\displaystyle(\forall a\in\mathbb{S})\,\mathbf{P}a^{l}a\,\&\,a^{ll}\,=\,a^{l}% \,\&\,\mathbf{P}a^{u}a^{uu}$
	$\displaystyle(\forall a,b\in\mathbb{S})(\mathbf{P}ab\longrightarrow\mathbf{P}a% ^{l}b^{l}\,\&\,\mathbf{P}a^{u}b^{u})$
	$\displaystyle\bot^{l}\,=\,\bot\,\&\,\bot^{u}\,=\,\bot\,\&\,\mathbf{P}\top^{l}% \top\,\&\,\mathbf{P}\top^{u}\top$
	$\displaystyle(\forall a\in\mathbb{S})\,\mathbf{P}\bot a\,\&\,\mathbf{P}a\top$

Let $\mathbb{P}$ stand for proper parthood, defined via $\mathbb{P}ab$ if and only if $\mathbf{P}ab\,\&\,\neg\mathbf{P}ba$ ) and let $t$ be a term operation formed from the weak lattice operations. A granulation is said to be admissible if the following three conditions hold:

	$\displaystyle(\forall x\exists x_{1},\ldots x_{r}\in\mathcal{G})\,t(x_{1},\,x_% {2},\ldots\,x_{r})=x^{l}$
	$\displaystyle\mathrm{and}\>(\forall x)\,(\exists x_{1},\,\ldots\,x_{r}\in% \mathcal{G})\,t(x_{1},\,x_{2},\ldots\,x_{r})=x^{u},$		(Weak RA, WRA)
	$\displaystyle{(\forall y\in\mathcal{G})(\forall{x\in\wp(\underline{S})})\,(% \mathbf{P}yx\,\longrightarrow\,\mathbf{P}yx^{l}),}$		(Lower Stability, LS)
	$\displaystyle{(\forall x,\,y\in\mathcal{G})(\exists z\in\wp(\underline{S}))\,% \mathbb{P}xz,\,\&\,\mathbb{P}yz\,\&\,z^{l}\,=\,z^{u}\,=\,z,}$		(Full Underlap, FU)

Definition 2.1 (Cayley-Dickson construction [ 6 ] , [ 1 ] ) .

Consider the vector space of the direct sum of two copies of an algebra with conjugation: $\mathcal{A}^{2}=\mathcal{A}\oplus\mathcal{A}$ . A multiplication on $\mathcal{A}^{2}$ is defined as

(a,b)(u,v)=(au-\bar{v}b,b\bar{u}+va).

It is easy to check, that relative to this multiplication the vector space $\mathcal{A}^{2}$ is an algebra of dimension $2\cdot\mathrm{dim}(\mathcal{A})$ . This is called the doubling of the algebra $\mathcal{A}$ .

Definition 2.1 .

[ 6 ] Let $A$ be an associative algebra. A bilinear map $\delta:A\times A\rightarrow A$ is called a biderivation if for all $x,y,z\in A$ ,

(i)

$\delta(xy,z)=\delta(x,z)y+x\delta(y,z)$ ,
(ii)

$\delta(x,yz)=\delta(x,y)z+y\delta(x,z)$ .

Definition 2 (Latent Discrimination) .

Classifier $h$ exhibits no latent discrimination if for every pair $(\mathbf{x}^{1},\mathbf{x}^{2})\in\mathcal{X}^{2}$ such that $\mathbf{x}^{1}_{0}=\mathbf{x}^{2}_{0}$ (i.e., pairs with similar sensitive features) we have $:$

	$\displaystyle h^{}(\mathbf{x}^{1})=h^{}(\mathbf{x}^{2})$	$\displaystyle\Rightarrow h(\mathbf{x}^{1})=h(\mathbf{x}^{2}),\ and$		(1)
	$\displaystyle h^{}(\mathbf{x}^{1})>h^{}(\mathbf{x}^{2})$	$\displaystyle\Rightarrow h(\mathbf{x}^{1})\geq h(\mathbf{x}^{2}).$		(2)

Definition 1 .

A Hilbert algebra is an algebra $H=(H,\rightarrow,1)$ of type $(2,0)$ which satisfies the following conditions for every $a,b,c\in H$ :

1)

$a\rightarrow(b\rightarrow a)=1$ ,
2)

$(a\rightarrow(b\rightarrow c))\rightarrow((a\rightarrow b)\rightarrow(a% \rightarrow c))=1$ ,
3)

if $a\rightarrow b=b\rightarrow a=1$ then $a=b$ .

Definition 3.3 .

Definition 3.1 .

Definition 3.4 .

Definition 5

Definition 5.17 .

Definition 5.24 .

Definition 2.5 (Choreographic holomorphic sphere) .

Definition 1 .

Definition 2.5 .

Definition 3.1 (Halfspaces) .

Definition 3.13 (Halfspace Angle) .

Definition 6 (Compositional Exponential Family of Distributions) .

Definition 2.3 (Inversion map) .

Definition 3.9.1 .

Definition 9.2.1 .

Definition 2.2 ().

Definition 2.1 .

Definition 3 ( [ 30 ] ) .

Definition 2.1 .

Definition 4.1 .

Definition 2.1 (Admissible trajectory-control pairs and costs) .

Definition 3.1 ( W 𝑊 W italic_W -Feedback) .

Definition 2.1.1

Definition 2.1.2

Definition 2 .

Definition 13 .

Definition 4.6 .

Definition 1 .

Definition 2.3 .

Definition 1

Definition 3.2 .

Definition 1

Definition 2.1 (Cayley-Dickson construction [ 6 ] , [ 1 ] ) .

Definition 2.1 .

Definition 2 (Latent Discrimination) .

Definition 1 .

Definition 3.1 ( $W$ -Feedback) .