A unit speed nonconstant geodesic $\gamma:I\to M$ called an $f$ -geodesic if

$$f(\gamma(t))-f(\gamma(s))=t-s$$

(2.8)

for any $s,t\in I,$ where $I$ denotes an interval.

We say that a linear functional $b:\operatorname{Lip}(X,\mathsf{d})\to L^{0}(X,\mathfrak{m})$ is a derivation if it satisfies the Leibniz rule, that is

$$b(fg)=b(f)g+fb(g),$$

(1.12)

for any $f,g\in\operatorname{Lip}(X,\mathsf{d})$ .

Given a derivation $b$ we will write $\left\lvert b\right\rvert\in L^{p}$ if there exist some function $g\in L^{p}(X,\mathfrak{m})$ such that

$$b(f)\leq g\left\lvert\nabla f\right\rvert\quad\text{$\mathfrak{m}$-a.e. on $X$,}$$

(1.13)

for any $f\in\operatorname{Lip}(X,\mathsf{d})$ and we will denote by $\left\lvert b\right\rvert$ the minimal (in the $\mathfrak{m}$ -a.e. sense) $g$ with such property.

[ 11 ] Consider the nonlinear system

(1.3)

$$\dot{x}=g(t,x)$$

with $g(t,0)=0$ for any $t\geq 0.$ The equilibrium point $x=0$ is uniformly asymptotically stable if there exist a $\mathcal{KL}$ function $\beta\colon\mathbb{R}^{+}\times\mathbb{R}^{+}\to\mathbb{R}^{+}$ and $c>0$ with

$$|x(t)|\leq\beta(|x(t_{0})|,t-t_{0})\quad\forall t\geq t_{0}\geq 0,\quad\textnormal{for all}\quad|x(t_{0})|<c,$$

where $c$ is independent of $t_{0}$ . We recall that $\beta\in\mathcal{KL}$ if $s\mapsto\beta(s,r)$ is strictly increasing with $\beta(0,r)=0,$ for any fixed $r\geq 0$ and $r\mapsto\beta(s,r)$ is decreasing and tends to zero when $r\to\infty$ for any fixed $s$ .

Let $f:\mathbb{N}_{0}\times\mathbb{N}_{0}\to\mathbb{Z}$ be a function defined as follows. Set:

$$f(0,0)=0,f(0,1)=f(1,0)=-1,f(1,1)=2.$$

In all other points $f$ is defined by the following recurrent relations:

$\displaystyle f(a,b)=\begin{cases}8f({a\over 2},{b\over 2})\ \text{if }a\equiv 0\ (\mathrm{mod}\ 2)\text{ and }b\equiv 0\ (\mathrm{mod}\ 2),\\ 4\left(f({a-1\over 2},{b\over 2})+f({a+1\over 2},{b\over 2})\right)+3\ \text{if}\ a\equiv 1\ (\mathrm{mod}\ 2)\text{ and }b\equiv 0\ (\mathrm{mod}\ 2),\\ 4\left(f({a\over 2},{b-1\over 2})+f({a\over 2},{b+1\over 2})\right)+3\ \text{if }a\equiv 0\ (\mathrm{mod}\ 2)\ \text{and }b\equiv 1\ (\mathrm{mod}\ 2),\\ 2\left(f({a-1\over 2},{b-1\over 2})+f({a-1\over 2},{b+1\over 2})+f({a+1\over 2},{b-1\over 2})+f({a+1\over 2},{b+1\over 2})\right)+2,&\\ \text{if}\ a\equiv 1\ (\mathrm{mod}\ 2)\text{ and}\ b\equiv 1\ (\mathrm{mod}\ 2).\end{cases}$

(6.3)

A holomorphic Courant algebroid $(Q,\langle\cdot,\cdot\rangle,[\cdot,\cdot],\pi)$ over $X$ consists of a holomorphic vector bundle $Q\to X$ , with sheaf of sections $\mathcal{O}_{Q}$ , together with a holomorphic non-degenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$ , a holomorphic vector bundle morphism $\pi:Q\to TX$ , and an homomorphism of sheaves of $\underline{{\mathbb{C}}}$ -modules

$$[\cdot,\cdot]\colon\mathcal{O}_{Q}\otimes_{\underline{{\mathbb{C}}}}\mathcal{O}_{Q}\to\mathcal{O}_{X},$$

satisfying the identities, for $e,e^{\prime},e^{\prime\prime}\in\mathcal{O}_{Q}$ and $\phi\in\mathcal{O}_{X}$ ,

• •

  $[e,[e^{\prime},e^{\prime\prime}]]=[[e,e^{\prime}],e^{\prime\prime}]+[e^{\prime},[e,e^{\prime\prime}]]$ ,

• •

  $\pi([e,e^{\prime}])=[\pi(e),\pi(e^{\prime})]$ ,

• •

  $[e,\phi e^{\prime}]=\pi(e)(\phi)e^{\prime}+\phi[e,e^{\prime}]$ ,

• •

  $\pi(e)\langle e^{\prime},e^{\prime\prime}\rangle=\langle[e,e^{\prime}],e^{\prime\prime}\rangle+\langle e^{\prime},[e,e^{\prime\prime}]\rangle$ ,

• •

  $[e,e^{\prime}]+[e^{\prime},e]=2\pi^{*}d\langle e,e^{\prime}\rangle$ .

Let $(\mathscr{E},j)$ be an elementary ideal topos $\mathscr{E}$ equipped with a Lawvere-Tierney topology $j$ . A sheaf for $j$ on $\mathscr{E}$ is an element $P$ of $\mathscr{E}$ such that for every monic $m$ in $\mathscr{E}$ , and for every $f:\bar{m}\to P$ in $\mathscr{E}$ , if $m$ is dense, there exists a unique $g\in\mathscr{E}$ such that

$$g\cdot m=f.$$

An almost contact manifold is a (2n+1)-dimensional smooth manifold $M$ endowed with structure tensors $(\phi,\xi,\eta)$ , such that $\phi\in{\text{End}}(TM)$ , $\xi\in\Gamma(TM)$ and $\eta\in\Omega^{1}(M)$ satisfy

(2.1)

$$\phi\circ\phi=-{\rm{id}}+\eta\otimes\xi,\ \ \eta(\xi)=1.$$

A linear map $d:\,L\rightarrow L$ of a Leibniz algebra $L$ is a derivation if for all $x,\,y\in L$

$$d([x,y])=[d(x),y]+[x,d(y)].$$

A Hom-associative superalgebra is a triple $(A,\mu,\alpha)$ where $A$ is a linear superspace, $\mu:A\times A\rightarrow A$ is an even bilinear map, and $\alpha:A\rightarrow A$ is an even linear map such that

$$\mu(\alpha(x),\mu(y,z))=\mu(\mu(x,y),\alpha(z)).$$

Let $\dot{t}$ be a $k$ -ary star-tuple, and $n\geq k$ . Then $\dot{\mathsf{h}}^{n}(\dot{t})$ , the $n$ -expansion of $\dot{t}$ , is the $n$ -ary star-tuple $\dot{u}$ , where

$$\dot{u}(i)=\left\{\begin{array}[]{ll}\dot{t}(i)&\mbox{\em if }i\in\{1,\ldots,k\}\\ *&\mbox{\em if }i\in\{k+1,\ldots,n\}\\ \dot{t}(k+1)&\mbox{\em if }i=n+1,\end{array}\right.$$

For a stored relation $\dot{R}$ and stored database $\dot{\mathbf{R}}$ we have

	$\displaystyle\dot{\mathsf{h}}^{n}(\dot{R})$	$\displaystyle=$	$\displaystyle\{\dot{\mathsf{h}}^{n}(\dot{t})\;:\;\dot{t}\in\dot{R}\}$
	$\displaystyle\dot{\mathsf{h}}^{n}(\dot{\mathbf{R}})$	$\displaystyle=$	$\displaystyle(\dot{\mathsf{h}}^{n}(\dot{R}_{1}),\ldots,\dot{\mathsf{h}}^{n}(\dot{R}_{m}),\dot{\mathsf{h}}^{n}(\{(a,a):a\in\mathbb{D}\})).$

A (right) Hom-Leibniz algebra, [ 28 ] , is a triple $(L,[\cdot,\cdot],\alpha)$ where $L$ is a vector space, with a bracket $[\cdot,\cdot]:L\otimes L\longrightarrow L$ and a linear map $\alpha:L\longrightarrow L$ satisfying

(1.1)

$$[[x,y],\alpha(z)]=[[x,z],\alpha(y)]+[\alpha(x),[y,z]].$$

The Reissner-Nordstrom manifold is defined by $M^{3}=(s_{0},\infty)\times\mathbb{S}^{2},$ with the metric

$$\langle\cdot,\cdot\rangle=\dfrac{1}{1-mr^{-1}+q^{2}r^{-2}}dr^{2}+r^{2}d\omega^{2},$$

where $m>2q>0$ and $s_{0}=\frac{2q^{2}}{m-\sqrt{m^{2}-4q^{2}}}$ is the larger of the two solutions of $1-mr^{-1}+q^{2}r^{-2}=0.$ In order to write the metric in the form ( 1.2 ), define $F:[s_{0},\infty)\rightarrow\mathbb{R}$ by

$$F^{\prime}(r)=\dfrac{1}{\sqrt{1-mr^{-1}+q^{2}r^{-2}}},\ F(s_{0})=0.$$

Taking $t=F(r),$ we can write $\langle\cdot,\cdot\rangle=dt^{2}+h(t)^{2}d\omega^{2},$ where $h:[0,\infty)\rightarrow[s_{0},\infty)$ denotes the inverse function of $F.$ The function $h(t)$ clearly satisfies

(1.4)

$$h^{\prime}(t)=\sqrt{1-mh(t)^{-1}+q^{2}h(t)^{-2}},\ h(0)=s_{0},\ \mbox{and}\ h^{\prime}(0)=0.$$

A double Stone algebra (DSA) $<L,\wedge,\vee,*,+,0,1>$ is an algebra of type $<2,2,1,1,0,0>$ such that:

1. (1)

   $<L,\vee,\wedge,0,1>$ is a bounded distributive lattice

2. (2)

   $x^{*}$ is the pseudocomplement of $x$ i.e $y\leq x^{*}\leftrightarrow y\wedge x=0$

3. (3)

   $x^{+}$ is the dual pseudocomplement of $x$ i.e $y\geq x^{+}\leftrightarrow y\vee x=1$

4. (4)

   $x^{*}\vee x^{**}=1$ , $x^{+}\wedge x^{++}=0$ , e.g. the Stone identities.
   Condition 4. is what distinguishes a double Stone algebra from a double p-algebra and conditions 2. and 3. are equivalent to the equations

   • •

     $x\wedge(x\wedge y)^{*}=x\wedge y^{*}$ , $x\vee(x\vee y)^{+}=x\vee y^{+}$

   • •

     $x\wedge 0^{*}=x$ , $x\vee 1^{+}=x$

   • •

     $0^{**}=0$ , $1^{++}=1$

   so that DSA is an equational class. A double Stone algebra L is called regular, (RDSA), if it additionally satisfies

5. (5)

   $x\wedge x^{+}\leq y\vee y^{*}$

   • •

     this is equivalent to $x^{+}=y^{+}$ and $x^{*}=y^{*}\rightarrow x=y$

Given a propositional signature $\Sigma$ and two $\Sigma$ -models $\nu,\nu^{\prime}:\Sigma\to\{0,1\}$ , we note $\nu\cap\nu^{\prime}:\Sigma\to\{0,1\}$ the $\Sigma$ -model defined by:

$$p\mapsto\left\{\begin{array}[]{ll}1&\mbox{if $\nu(p)=\nu^{\prime}(p)=1$}\\ 0&\mbox{otherwise}\end{array}\right.$$

Given a set of $\Sigma$ -models $\mathcal{S}$ , we note

$$cl_{\cap}(\mathcal{S})=\mathcal{S}\cup\{\nu\cap\nu^{\prime}\mid\nu,\nu^{\prime}\in\mathcal{S}\}$$

which is the closure of $\mathcal{S}$ under intersection of positive atoms.

Let $X$ and $Y$ be locally compact Hausdorff spaces, $\sigma$ a local homeomorphism from an open subset $U_{\sigma}$ of $X$ to an open subset $V_{\sigma}$ of $X$ , and $\tau$ a local homeomorphism from an open subset $U_{\tau}$ of $Y$ to an open subset $V_{\tau}$ of $Y$ . We say that $(X,\sigma)$ and $(Y,\tau)$ are continuous orbit equivalent if there exist a homeomorphism $h:X\to Y$ and continuous maps $k,l:U_{\sigma}\to\mathbb{N}_{0}$ and $k^{\prime},l^{\prime}:U_{\tau}\to\mathbb{N}_{0}$ such that

$$\tau^{l(x)}(h(x))=\tau^{k(x)}(h(\sigma(x)))\quad\text{ and }\quad\sigma^{l^{\prime}(y)}(h^{-1}(y))=\sigma^{k^{\prime}(y)}(h^{-1}(\tau(y)))$$

for all $x,y$ . We call $(h,l,k,l^{\prime},k^{\prime})$ a continuous orbit equivalence and we call $h$ the underlying homeomorphism . We say that $(h,l,k,l^{\prime},k^{\prime})$ preserves stabilisers if $\operatorname{Stab}_{\min}(h(x))<\infty\iff\operatorname{Stab}_{\min}(x)<\infty$ , and

	$\displaystyle\bigg{|}\sum_{n=0}^{\operatorname{Stab}_{\min}(x)-1}l(\sigma^{n}(x))-k(\sigma^{n}(x))\bigg{|}=\operatorname{Stab}_{\min}(h(x))\text{ and}$
	$\displaystyle\bigg{|}\sum_{n=0}^{\operatorname{Stab}_{\min}(y)-1}l^{\prime}(\tau^{n}(y))-k^{\prime}(\tau^{n}(y))\bigg{|}=\operatorname{Stab}_{\min}(h^{-1}(y))$

whenever $\operatorname{Stab}(x),\operatorname{Stab}(y)$ are nontrivial, $\sigma^{\operatorname{Stab}_{\min}(x)}(x)=x$ , and $\tau^{\operatorname{Stab}_{\min}(y)}(y)=y$ .