VLBI phase–reference observations of the gravitational lens JVAS B0218+357

Definition .

A function $f:\mathbb{N}\to\mathbb{C}$ is called multiplicative if

f(mn)=f(m)f(n)

whenever $(m,n)=1$ .

Definition 10

We call $f\in\mathcal{A}_{1}^{q}\left(F\right)\backslash\left\{0\right\}$ a quadratic form if $\deg\left(f\right)=2$ . In this case it can be written as in equation 4 for some $a,b,c,d,e,k\in F$ with at least one of $a,b,c$ nonzero.

f\left(x,y\right)=ax^{2}+bxy+cy^{2}+dx+ey+k

(4)

The quantum discriminant of $f$ is $\Delta_{q}\left(f\right)=b^{2}-4acq$ .

Definition 3.6 .

For $x\in{\mathbb{R}}$ and $m\in{\mathbb{Z}}$ , define

x^{(m)}=1+mx-\lceil mx\rceil.

Note that this operation induces a selfmap of the half-open interval $(0,1]$ . For $x\in{\mathbb{R}}^{d}$ and $m\in{\mathbb{Z}}$ , define $x^{(m)}\in{\mathbb{R}}^{d}$ componentwise.

Definition 5.1 .

Let $k$ be a representative of the fundamental class $\kappa$ . The evaluation map $\operatorname{ev}\co P\to\mathbb{Z}/2$ is defined by

\operatorname{ev}(p)=\langle p,k\rangle

on $P^{1}$ and is extended to the rest of $P$ by $0$ .

The cap product map $\phi\co P\to Q$ is defined by

\phi=(1\otimes\operatorname{ev})\Phi_{QP}.

Definition 2.3 .

A homotopy operator between two chain complexes $f,g:X_{\bullet}\rightarrow Y_{\bullet}$ is a linear map $h:X_{\bullet}\rightarrow Y_{\bullet+1}$ such that

h\partial+\partial h=f-g.\quad(\star)

In that case, $f$ and $g$ are called chain homotopic and we denote it by $f\simeq g$ .
Two chain maps $f:X_{\bullet}\rightarrow Y_{\bullet}$ and $g:Y_{\bullet}\rightarrow X_{\bullet}$ are called homotopy inverse if $g\circ f\simeq id_{X}$ and $f\circ g\simeq id_{Y}$ are both homotopic to the identity. If $f:X_{\bullet}\rightarrow Y_{\bullet}$ admits a homotopy inverse, it is called a homotopy equivalence. In particular, every homotopy equivalence is a quasi-isomorphism.

Definition 3 .

Let $H$ be a group which is equipped with an involutive automorphism $\theta$ and let $g^{\ast}:=\theta(g)^{-1}$ . A $\ast$ -representation of $H$ is a representation of $H$ given by operators $\tau(g):V\rightarrow V$ where $V$ is a finite-dimensional real or complex vector space which is equipped with a nondegenerate bilinear or sesquilinear form $\langle\,,\,\rangle$ such that

\langle v,\tau(g)w\rangle=\langle\tau(g^{\ast})v,w\rangle.

(2.39)

Definition 1 (from [ 17 ] )

A Boolean algebra $(B;\wedge,\vee,\lnot,0,1)$ is a set $B$ , together with operations on the set which satisfy certain laws. We will denote the operations by ‘ $\wedge$ ’, ‘ $\vee$ ’, and ‘ $\lnot$ ’ and they will be called ‘meet’, ‘join’ and ‘complement’ respectively. There are two distinguished and distinct elements of $B$ , denoted by $0$ and $1$ that are subject to the following laws.

1.

Idempotence: $a\wedge a=a,$ and $a\vee a=a.$
2.

Complement laws: $a\vee\lnot a=1,$ $a\wedge\lnot a=0,$ and $\lnot\lnot a=a.$
3.

Commutativity: $a\wedge b=b\wedge a,$ and $a\vee b=b\vee a.$
4.

Associativity: $a\wedge(b\wedge c)=(a\wedge b)\wedge c,$ and $a\vee(b\vee c)=(a\vee b)\vee c.$
5.

Distributivity: $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c),$ and $a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c).$
6.

Property of 1: $a\wedge 1=a.$
7.

Property of 0: $a\vee 0=a.$
8.

De Morgan’s laws: $\lnot(a\wedge b)=\lnot a\vee\lnot b,$ and $\lnot(a\vee b)=\lnot a\wedge\lnot b.$

Definition 6 (from [ 6 ] )

Let $L$ be a lattice. The elements $a,b,c\in L$ form a distributive triple if

1.

$a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$
2.

$a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c)$

with the other four equalities being obtained by cyclical permutation of $a$ , $b$ and $c$ .

2.5 Definition .

Suppose that $\sigma$ is a reflection on a lattice $L$ . A strict marking for $\sigma$ is a pair $(b,\beta)$ , where $b\in L$ and $\beta\co L\to\mathbb{Z}$ is a homomorphism such that for any $x\in L$

\sigma(x)=x+\beta(x)b.

(2.6)

Two strict markings $(b,\beta)$ and $(b^{\prime},\beta^{\prime})$ are equivalent if $(b,\beta)=\pm(b^{\prime},\beta^{\prime})$ . A marking for $\sigma$ is an equivalence class of strict markings.