The Red Giant Branch in Near-Infrared Colour-Magnitude Diagrams. II: The luminosity of the Bump and the Tip Based on data taken at the ESO-MPI 2.2m Telescope equipped with the near-IR camera IRAC2-ESO, La Silla (Chile).

Definition 13.18

Let $X$ be any set. A function $f\colon\mathbb{Z}\longrightarrow X$ is called weakly periodic with period $s\in\mathbb{R}$ if for all integers $k$ , $0\leq k<s$ , and for all non-negative integers $j$ , either

1.

$f(k)=f(k+\lfloor js\rfloor)$ , or
2.

$f(k)=f(k+\lceil js\rceil)$ .

For brevity we will write $f(k)=f(k+\left[js\right])$ to indicate that one of the above conditions is satisfied. The satisfied condition may vary with $k$ and $j$ . ${}_{\blacksquare}$

Definition 2.1 .

A 1-form $\theta\in\Lambda^{1}(Z)$ is said to be a contact 1-form whenever

(j^{1}\phi)^{*}\theta=0

for every section $\phi$ of $\pi$ .

Definition 4.2 .

A space of Cauchy data is the manifold of embeddings $\gamma:M\to Z$ such that there exists a section $\phi$ of $\pi_{XY}$ satisfying

\gamma=(j^{1}\phi)\circ\tau

where $\tau:=\pi_{XZ}\circ\gamma\in\tilde{X}$ , and $\gamma(\partial M)\subseteq B$ .
The space of such embeddings shall be denoted by $\tilde{Z}$ , and we shall denote by $\pi_{\tilde{X}\tilde{Z}}$ the projection $\pi_{\tilde{X}\tilde{Z}}(\gamma)=\pi_{XZ}\circ\gamma$ . We shall also require this projection to be a locally trivial fibration.

Definition 2.1 .

Let $\mathcal{A}$ , $\mathcal{A}^{\prime}$ be a C*-algebras, $\mathcal{M}$ a Hilbert $\mathcal{A}$ -bimodule, $\mathcal{N}$ a Hilbert $\mathcal{A}^{\prime}$ -bimodule. A covariant morphism from $\mathcal{M}$ into $\mathcal{N}$ is a pair $(\beta,\eta)$ , where $\beta:\mathcal{M}\rightarrow\mathcal{N}$ is a Banach space map, $\eta:\mathcal{A}\rightarrow\mathcal{A}^{\prime}$ is a C*-algebra morphism, and the following properties are satisfied for $a\in\mathcal{A}$ , $\psi,\psi^{\prime}\in\mathcal{M}$ :

\beta(a\psi)=\eta(a)\beta(\psi)\ \ ,\ \ \beta(\psi a)=\beta(\psi)\eta(a)\ \ ,% \ \ \left\langle\beta(\psi),\beta(\psi^{\prime})\right\rangle=\eta\left\langle% \psi,\psi^{\prime}\right\rangle\ .

Definition 2.2 .

Let $G$ be a (countable discrete) group. If there exists a mean $f$ on the algebra $l^{\infty}(G-\{e\})$ , invariant under the action

(gf)(h)=f(g^{-1}h)

then $G$ is amenable. If the action is taken with respect to conjugation then $G$ is inner amenable.

Definition 1.1 .

A vector space $J$ over a field $F$ , endowed with a trilinear operation $J\times J\times J\rightarrow J$ , $(x,y,z)\mapsto xyz$ , is said to be a generalized Jordan triple system (GJTS for short) if it satisfies the identity:

uv(xyz)=(uvx)yz-x(vuy)z+xy(uvz)

(1.2)

for any $u,v,x,y,z\in J$ .

Definition 4 .

Let $(V,\rho)$ and $(W,\sigma)$ be representations of the Hopf algebra $\mathrm{A}$ . A linear map $f\in Hom(V,W)$ is a homomorphism of representations $(V,\pi)$ and $(W,\rho)$ if

f\circ\pi(\mathrm{a})=\rho(\mathrm{a})\circ f.

for any $\mathrm{a}\in$ A. The subspace of the algebra homomorphisms will be denoted $Hom_{\mathrm{A}}(V,W)$ .

Definition 4.5 .

For $a\in\mathbb{C}(q,t)$ , we denote by $\zeta(a)\in\mathbb{Z}$ the multiplicity of factor $1-t^{2}q$ in $a$ . Namely, we have

a=(1-t^{2}q)^{\zeta(a)}a^{\prime},

(39)

where the factor $a^{\prime}$ has neither pole nor zero at $t^{2}q=1$ .

Definition 46 .

(Variable-idempotence.) A substitution $\sigma\in\mathrm{RSubst}$ is said to be (strongly) variable-idempotent if and only if for all $t\in\mathord{\mathrm{HTerms}}$ we have

\mathop{\mathrm{vars}}\nolimits(t\sigma\sigma)=\mathop{\mathrm{vars}}\nolimits% (t\sigma).

The set of variable-idempotent substitutions is denoted $\mathrm{VSubst}$ .