Let $f$ and $g$ be smooth and let $\boldsymbol{\mathcal{S}}\subseteq[0,\infty)\times M$ be the topological subspace consisting of all pairs $(\lambda,p)$ such that the maximal solution of the following Cauchy problem:

(3)

$$\left\{\begin{array}[]{l}\dot{x}=g(x)+\lambda f(t,x),\\ x(0)=p,\end{array}\right.$$

is $T$ -periodic. The elements of $\boldsymbol{\mathcal{S}}$ are called starting points . If $(\lambda,p)\in\boldsymbol{\mathcal{S}}$ the point $p$ will be informally referred to as an initial point of a $T$ -periodic solution.

Let $X$ be a Hilbert $A$ -bimodule. The conjugate of $X$ , denoted $X^{*}$ , is the complex conjugate of the vector space $X$ , equipped with left $A$ -module structure defined by

$$a\cdot x^{*}=(xa^{*})^{*}.$$

Here $x^{*}\in X^{*}$ denotes the image of $x\in X$ under the obvious conjugate $\mathbb{C}$ -linear isomorphism $X\to X^{*}$ .

Let $A$ and $B$ be $C^{*}$ -algebras and let $F$ be a Hilbert $A$ - $B$ -bimodule. The conjugate of $F$ is the Hilbert module conjugate $F^{*}$ , as in Definition 2.6 , equipped with right $A$ -module structure defined by

$$f^{*}\cdot a=(a^{*}f)^{*}.$$

Let $f:[a,b]\rightarrow\mathbb{R}$ . Then its dual function, denoted by $f^{\star}$ , is defined as $f^{\star}:[-b,-a]\rightarrow\mathbb{R}$ by

$$f^{\star}(x)=f(-x).$$

We say that $h\in W^{1,p}(\mathbb{R}^{m},N)$ is a homogeneous function wrt the origin if for a.e. $\lambda>0$ and $x\in\mathbb{R}^{m}$ :

$$\displaystyle h(\lambda x)=h(x)\,,$$

(27)

or equivalently if $\frac{\partial h}{\partial n}=0$ . We say that $h$ is a $k$ -symmetric function if $h$ is homogeneous and there exists a subspace $V$ of $\mathbb{R}^{m}$ of dimension $k$ such that

$$\displaystyle h(x+y)=h(x)$$

(28)

for a.e. $x\in\mathbb{R}^{m}$ and $y\in V$ . Thus $0$ -symmetric and homogeneous are synonyms in this paper.

Define a cocycle-like function $\sigma\colon{\mathcal{G}}\times{\mathcal{S}}\rightarrow{\mathcal{P}}$ by

$$\sigma(v,s)=j(q(v)s)^{\dagger}vj(s)=j(s^{\dagger}q(v^{\dagger}))vj(s).$$

Since

$$q(\sigma(v,s))=s^{\dagger}q(v^{\dagger}v)s\in{\mathcal{E}}({\mathcal{S}}),$$

$\sigma(v,s)\in{\mathcal{P}}$ . Thus $\sigma$ indeed maps ${\mathcal{G}}\times{\mathcal{S}}$ into ${\mathcal{P}}$ .

A map $\sigma\colon R^{(2)}\to\mathbb{T}$ is a $2$ -cocycle on $R$ if

$$\sigma(x,y,z)\sigma(x,z,w)=\sigma(x,y,w)\sigma(y,z,w)$$

for all $(x,y,z,w)\in R^{(3)}$ . We say $\sigma$ is normalised if $\sigma(x,y,z)=1$ whenever two of $x$ , $y$ and $z$ are equal. By [ 7 , Proposition 7.8] , any normalised $2$ -cocycle $\sigma$ is skew-symmetric : for every permutation $\pi$ on three elements,

$$\sigma(\pi(x,y,z))=\begin{cases}\sigma(x,y,z)&\text{if $\pi$ is even},\\ \sigma(x,y,z)^{-1}&\text{if $\pi$ is odd}.\end{cases}$$

Given $\alpha\in L^{\infty}(X,\mu)$ , let $d(\alpha)\colon R\to\mathbb{C}$ be given by

$$d(\alpha)(x,y)=\begin{cases}\alpha(x)&\text{if~{}$x=y$,}\\ 0&\text{otherwise}.\end{cases}$$

Clearly $d(\alpha)\in\Sigma_{0}$ . We write $D(\alpha)=L(d(\alpha))\in\mathcal{M}$ , and we define the Cartan masa of $R$ to be

$$\mathcal{A}=\mathcal{A}(R)=\{D(\alpha)\colon\alpha\in L^{\infty}(X,\mu)\}.$$

By [ 8 ] , $\mathcal{A}(R)$ is a Cartan masa in the von Neumann algebra $\mathcal{M}(R,\sigma)$ .

Note that if $\xi\in H$ and $(x,y)\in R$ , then

	$\displaystyle D(\alpha)\xi(x,y)$	$\displaystyle=$	$\displaystyle\sum_{z\sim x}d(\alpha)(x,z)\xi(z,y)\sigma(x,z,y)=\alpha(x)\xi(x,y)\sigma(x,x,y)$
		$\displaystyle=$	$\displaystyle\alpha(x)\xi(x,y).$

Since this does not depend on the normalised $2$ -cocycle $\sigma$ , this shows that $\mathcal{A}(R)$ does not depend on $\sigma$ .

We say that a segment $\llbracket z,z^{\prime}\rrbracket$ is an optimal ray for $\bar{\phi}$ if

$$\bar{\phi}(z^{\prime})-{\bar{\phi}}(z)=\bar{\mathtt{c}}(z^{\prime}-z).$$

We say that a segment $\llbracket z,z^{\prime}\rrbracket$ is a maximal optimal ray if it is maximal with respect to set inclusion.

Let $\mathbf{\Sigma}=(N,\Sigma,\beta)$ be a stacky fan and $\boldsymbol{\rho}=\{\rho_{1},\ldots,\rho_{r}\}\subset\Sigma(1)$ a set of rays. For each $\rho_{i}\in\boldsymbol{\rho}$ , we associate a weight $d_{i}$ , which is a positive integer. Denote the function taking each ray to its weight by $\boldsymbol{d}$ . Consider the group homomorphism $\beta^{\prime}\colon\mathbb{Z}^{\Sigma(1)}\to N$ defined by

$$\beta^{\prime}(\rho)=\left\{\begin{array}[]{ll}\boldsymbol{d}(\rho)\beta(\rho)&\text{if }\rho\in\boldsymbol{\rho}\\ \beta(\rho)&\text{otherwise.}\end{array}\right.$$

We denote the stacky fan given by the triple $(N,\Sigma,\beta^{\prime})$ by $\mathbf{\Sigma}_{\boldsymbol{d}^{-1}\boldsymbol{\rho}}$ . The natural morphism $\mathbf{\Sigma}_{\boldsymbol{d}^{-1}\boldsymbol{\rho}}\to\mathbf{\Sigma}$ of stacky fans, which is the identity map on the underlying group $N$ , is called the root construction of $\mathbf{\Sigma}$ with respect to the rays in $\boldsymbol{\rho}$ with weights $\boldsymbol{d}$ .

Let $G$ be a finite group. A map $\varphi:G\rightarrow E$ satisfying the condition

$$\varphi(hgh^{-1})=\varphi(g)$$

for all $g,h\in G$ , is called class function .

If $\rho:G\rightarrow\operatorname{GL}(V)$ is a representation of $G$ on a finite dimensional $E$ -vector space, then $\chi_{\rho}:G\rightarrow E$ , $\chi_{\rho}(g):=\chi_{V}(g):=\operatorname{Tr}(\rho(g))$ is a class function; it is called the character of $\rho$ .

If $\rho$ is irreducible, then its character is also called irreducible.

A probability algebra is a pair $(\mathcal{A},\mu)$ , where $\mathcal{A}$ is a boolean algebra and $\mu:A\rightarrow[0,1]$ is a function satisfying the following properties:

1. i)

   $\mu(1)=1$ ;

2. ii)

   $\mu(a\vee b)+\mu(a\wedge b)=\mu(a)+\mu(b)$ ;

3. iii)

   $\mu(a)=0\text{ iff }a=0$ .

Let $\{a,b\}$ be an edge in $\widetilde{\pazocal{B}}$ and assume without loss of generality that $a$ lies in $s_{0}$ and $b$ lies in $s_{1}$ . We say $\{a,b\}$ is

• •

  Z-type if $\operatorname{suc}\{a,b\}=\{\operatorname{suc}(a),b\}$ and $\operatorname{suc}^{-1}\{a,b\}=\{a,\operatorname{suc}^{-1}(b)\}$ ,

• •

  S-type if $\operatorname{suc}\{a,b\}=\{a,\operatorname{suc}(b)\}$ and $\operatorname{suc}^{-1}\{a,b\}=\{\operatorname{suc}^{-1}(a),b\}$ .

(See Figure 6 .) Let $\widetilde{\pazocal{Z}}$ be the edges in $\widetilde{\pazocal{B}}$ that are Z-type and $\widetilde{\pazocal{S}}$ be the edges in $\widetilde{\pazocal{B}}$ that are S-type. Since $\widetilde{\pazocal{Z}}$ and $\widetilde{\pazocal{S}}$ are $\langle X\rangle$ -invariant, we can define $\pazocal{Z}:=\widetilde{\pazocal{Z}}/\langle X\rangle$ and $\pazocal{S}:=\widetilde{\pazocal{S}}/\langle X\rangle$ .

On appelle état produit à nombre infini de particules une suite d’opérateurs à trace $\gamma^{(n)}\in\mathfrak{S}^{1}(\mathfrak{H}_{s}^{n})$ avec

$$\gamma^{(n)}=\gamma^{\otimes n},$$

(3.15)

pour tout $n\geq 0$ où $\gamma$ est un état à une particule. Un état produit bosonique est nécessairement de la forme

$$\gamma^{(n)}=|u^{\otimes n}\rangle\langle u^{\otimes n}|=\left(|u\rangle\langle u|\right)^{\otimes n},$$

(3.16)

avec $u\in S\mathfrak{H}$ . ∎

On appelle état produit abstrait un état abstrait à nombre infini de particules avec

$$\omega^{(n)}=\omega^{\otimes n}$$

(3.40)

pour tout $n\in\mathbb{N}$ , où $\omega\in(\mathfrak{B}(\mathfrak{H}))^{*}$ est un état abstait à une particule (en particulier $\omega\geq 0$ et $\omega({\mathds{1}}_{\mathfrak{H}})=1$ ). ∎

The Invert interpolated operator $I^{(x)}$ , with parameter $x\in\mathbb{R}$ , transforms any sequence $a$ , having ordinary generating function $a(t)$ , into a sequence $b=I^{(x)}(a)$ having ordinary generating function

$$b(t)=\cfrac{a(t)}{1-xta(t)}\quad.$$

For each isolated fixed point $p$ of a tangentially stably complex $G$ -manifold, the sign of $p$ is given by

$$\sigma(p)\mathbin{:\!\raisebox{-0.36pt}{=}}\,\begin{cases}+1,&\text{if the two orientations coincide};\\ -1,&\text{if the two orientations differ}.\end{cases}$$

A locally conformal symplectic structure on a manifold $M$ is given by a pair $(\omega,\theta)$ , where $\omega$ is a non-degenerate 2-form and $\theta$ is a closed 1-form on $M$ satisfying the relation

$$d\omega+\theta\wedge\omega=0.{}$$

(2.1)

The form $\theta$ is called the Lee form of $\omega$ . If $\dim M\geq 4$ then $\omega\wedge-:\Omega^{1}(M)\to\Omega^{3}(M)$ is injective because of the non-degeneracy of $\omega$ . In this case, $\theta$ is uniquely determined by the relation ( 2.1 ).