High SNR Analysis for MIMO Broadcast Channels: Dirty Paper Coding vs. Linear Precoding

Definition 2.7 .

Let $N$ and $K$ be groups, and assume that there exists a group homomorphism $\alpha:K\rightarrow\mathrm{Aut}N$ . The semidirect product $N\rtimes_{\alpha}K$ of $N$ and $K$ with respect to $\alpha$ is defined as follows:

(1) As set, $N\rtimes_{\alpha}K$ is the Cartesian product of N and K;

(2) The multiplication is given by

(a,x)(b,y)=(a\alpha(x)(b),xy)

for any a $\in$ N and x $\in$ N.

Definition 2.8 .

Let $A$ and $H$ be two groups and $H$ act on the set $X$ . Assume that $B$ is the set of all maps from $X$ to $A$ . Define the multiplication of $B$ and the group homomorphism $\alpha$ from $H$ to Aut $B$ as follows:

(bb^{\prime})(x)=b(x)b^{\prime}(x)\hbox{ \ and }\alpha(h)(b)(x)=b(h^{-1}\cdot x)

for any $x\in X,h\in H,b,b^{\prime}\in B.$

The semidirect product $B\rtimes_{\alpha}H$ is called the (general) wreath product of $A$ and $H$ , written as $A$ wr $H$ .

Definition 2.2 .

A (small) category $C$ is left locally Garside if

(i)

It is left Noetherian.
(ii)

It has the left cancellation property, i.e. $xy=xz$ implies $y=z$ (in other words, every morphism is an epimorphism).
(iii)

Two morphisms which have some common right multiple have a right lcm.

Definition 3.16 .

The sc-smooth map $f:X\to Y$ between two M-polyfolds is called a fred-submersion if at every point $x_{0}\in X$ resp. $f(x_{0})\in Y$ there exists a chart $(U,\varphi,{\mathcal{T}}\oplus\widehat{\mathcal{T}})$ resp. $(W,\psi,{\mathcal{T}})$ satisfying $f(U)\subset W$ and

\psi\circ f\circ\varphi^{-1}(v,e^{\prime},e^{\prime\prime})=(v,e^{\prime})

and, moreover, the splicing $\widehat{\mathcal{T}}=(\widehat{\rho},E,V)$ has the special property that the projections $\widehat{\rho}_{v}$ do not depend on $v$ and project onto a finite dimensional subspace of $E$ .

Definition 2.1 .

A differential form $\omega\in\Omega^{\bullet}(\mathcal{G})$ is multiplicative if

\partial^{*}\omega=0.

(2.1b)

A symplectic groupoid is a groupoid $\Sigma$ with a multiplicative symplectic form $\omega\in\Omega^{2}(\Sigma)$ . (I usually denote a symplectic groupoid as $\Sigma$ .)

Definition 1.2 .

For every $n\geq 0$ we denote by $*n$ a strip in Nim of length $n$ . We write $0$ and $*$ as shorthand for $*0$ and $*1$ , respectively. Formally, we have

*n=\{0,*,*2,*3,\ldots,*(n-1)\}.

Definition 5.19 .

A quotient map $\Phi:\mathscr{A}\to\mathcal{Q}$ is faithful if, for all $G,H\in\mathscr{A}$ ,

\Phi(G)=\Phi(H)\Rightarrow\mathscr{G}(G)=\mathscr{G}(H).

Definition Definition 2.1

The Kac-Moody Lie algebra $\mathfrak{g}(A)$ associated with the generalized Cartan matrix $A$ is the Lie algebra generated by the elements $e_{i},\;f_{i}\;(i=1,2,\cdots,n)$ and $\mathfrak{h}$ with the defining relations

		$\displaystyle[h,h^{\prime}]=0\qquad\text{for all}\;h,h^{\prime}\in\mathfrak{h},$
		$\displaystyle[e_{i},f_{j}]=\delta_{ij}\alpha_{i}^{\vee}\qquad\text{for}\;1\leq i% ,j\leq n,$
		$\displaystyle[h,e_{i}]=\alpha_{i}(h)e_{i},\;\;[h,f_{i}]=-\alpha_{i}(h)f_{i}\;% \quad\text{for}\;i=1,2,\cdots,n,$
		$\displaystyle(ad\,e_{i})^{1-a_{ij}}(e_{j})=\;\;(ad\,f_{i})^{1-a_{ij}}(f_{j})=0% \;\quad\text{for}\;i\neq j.$

The elements of $\Pi$ (resp. $\Pi^{\vee}$ ) are called simple roots (resp. simple coroots ) of $\mathfrak{g}(A).$

5.8 Definition (Factor sets) .

Let $A$ and $C$ be (Abelian) groups. A factor set is a map $\mathpzc{f}:C\times C\to A$ such that, for every $x$ , $y$ , $z\in C$ ,

•

$\mathpzc{f}(x,y)=\mathpzc{f}(y,x)$ ;
•

$\mathpzc{f}(x,0)=\mathpzc{f}(0,x)=\mathpzc{f}(0,0)=0$ ;
•

$\mathpzc{f}(y,x)+\mathpzc{f}(x,y+z)=\mathpzc{f}(x,y)+\mathpzc{f}(x+y,z)$ .

Given such a factor set, the crossed product of $C$ and $A$ is the group $\times(C,A,\mathpzc{f})$ , whose underlying set is $C\times A$ , and whose sum is defined as follows:

(c,a)+(c^{\prime},a^{\prime})=\bigl{(}c+c^{\prime},a+a^{\prime}+\mathpzc{f}(c,% c^{\prime})\bigr{)}.

Definition 1.4 .

Let $f:[\lambda]^{<\aleph_{0}}\rightarrow\mu$ be a function.
$ID(f):=\{(a,e):(a,e)$ is an identity, and there exists a one-to-one mapping

h:a\rightarrow\lambda,\hbox{ {such that}\ }bec\Rightarrow f(h^{\prime\prime}(b% ))=f(h^{\prime\prime}(c))\}

Notice that $ID(\lambda,\mu)\subseteq\bigcap\{ID(f):f$ is a function from $[\lambda]^{<\aleph_{0}}$ into $\mu\}$ .
One of the basic tools for investigating identities is the notion of refinement. The idea is to compute the values of a coloring $c:[\lambda]^{n}\rightarrow\mu$ , with a coloring $d:[\lambda]^{m}\rightarrow\mu$ , when $m<n$ .

Definition 1 .

A group $G$ of tree automorphisms is self-similar if, for every $g$ in $G$ and a letter $x$ in $X$ there exists a letter $y$ in $X$ and an element $h$ in $G$ such that

g(xw)=yh(w),

for all words $w$ over $X$ .

Definition 3.1 .

Suppose $f(\sqrt{x})$ is a polynomial in $\sqrt{x}$ where $x$ is a formal indeterminate and $\sqrt{x}$ a formal square root. Then $f(\sqrt{x})$ can be written uniquely in the form

f(\sqrt{x})=g(x)+\sqrt{x}h(x),

where $g(x)$ and $h(x)$ are polynomials in $x$ . Define $\big{\{}f(\sqrt{x})\big{\}}:=h(x)$ . More generally, suppose $r$ is any element of any $\mathbb{C}$ -algebra. Define $\big{\{}f(\sqrt{r})\big{\}}$ to be the result of substituting $r$ for $x$ in $\big{\{}f(\sqrt{x})\big{\}}$ . There is no assumption here that $r$ has a square root.

Definition 4.4 .

Let $Y\subset X$ be a Lagrangian submanifold and $(h,\psi)$ an exact Lagrangian isotopy. The time-reversal of $(h,\psi)$ is the pair $(\widetilde{h},\widetilde{\psi})$ defined by

\widetilde{\psi}(t,x)=\psi(1-t,x),\quad\widetilde{h}(t,x)=-h(1-t,x).

Definition 5.1 .

For matrices defined over a field we define a function which is called the determinant of the matrix ,

(5.1)

\det()=1

(5.2)

\det a=\sum_{a}(-1)^{|a|+|b|}a^{b}_{a}\det a^{[a]}_{[b]}

∎

Definition 2.1 .

A self-similar group $(G,\mathsf{X})$ is a group $G$ acting faithfully on the rooted tree $\mathsf{X}^{*}$ such that for every $g\in G$ and every $x\in\mathsf{X}$ there exist $h\in G$ and $y\in\mathsf{X}$ such that

g(xw)=yh(w)

for all $w\in\mathsf{X}^{*}$ .

Definition 6.5

Denote by $\mu^{0}:X^{{\mathbf{R}}}\rightarrow{\cal C}^{*}({{\mathbf{N}}^{0}})$ the sequence measure function defined as

\mu^{0}({\bf x})(\hat{p})=\mu({\bf x})(p)

(22)

Denote by $\mu^{0}$ as well the corresponding frame measure function $\mu^{0}\circ b$ .

Definition 2 (Well-formed prefix.)

Let $M=(K,\rightarrow,\lhd,\nparallel)$ and $M^{\prime}=(K^{\prime},\mathbin{\rightarrow^{\prime}},\mathbin{\lhd{}^{\prime}% },\mathbin{\nparallel^{\prime}})$ be two multilogs. $M^{\prime}$ is a well-formed prefix of $M$ , noted $M^{\prime}\stackrel{{\scriptstyle\mathit{wf}}}{{\sqsubset}}M$ , if (i) it is a subset of $M$ , (ii) it is stable, (iii) it is left-closed for its actions, and (iv) it is closed for its constraints.

M^{\prime}\stackrel{{\scriptstyle\mathit{wf}}}{{\sqsubset}}M\stackrel{{% \scriptstyle\mathrm{def}}}{{=}}\left\{\begin{array}[]{l}M^{\prime}\subseteq M% \\ K^{\prime}=\mathit{Stable}(M^{\prime})\\ \forall{\alpha,\beta\in A},\beta\in K^{\prime}\Rightarrow\left\{\begin{array}[% ]{l}\alpha\mathbin{\rightarrow}\beta\Rightarrow\alpha\mathbin{\rightarrow^{% \prime}}\beta\\ \alpha\mathbin{\lhd}\beta\Rightarrow\alpha\mathbin{\lhd^{\prime}}\beta\\ \alpha\mathbin{\nparallel}\beta\Rightarrow\alpha\mathbin{\nparallel^{\prime}}% \beta\\ \end{array}\right.\\ \forall{\alpha,\beta\in A},(\alpha\mathbin{\rightarrow^{\prime}}\beta\lor% \alpha\mathbin{\lhd^{\prime}}\beta\lor\alpha\mathbin{\nparallel^{\prime}}\beta% )\Rightarrow\alpha,\beta\in K^{\prime}\\ \end{array}\right.

Definition 3.17 .

Let $R$ and $S$ be idempotent rings. A (surjective) Morita context $(R,S,M,N,\psi,\phi)$ between $R$ and $S$ consists of unital bimodules ${}_{R}M_{S}$ and ${}_{S}N_{R}$ , a surjective $R$ -module homomorphism $\psi:M\otimes_{S}N\to R$ , and a surjective $S$ -module homomorphism $\phi:N\otimes_{R}M\to S$ satisfying

\phi(n\otimes m)n^{\prime}=n\psi(m\otimes n^{\prime})\qquad\text{ and }\qquad m% ^{\prime}\phi(n\otimes m)=\psi(m^{\prime}\otimes n)m

for every $m,m^{\prime}\in M$ and $n,n^{\prime}\in N$ . We say that $R$ and $S$ are Morita equivalent in the case that there exists a Morita context.

Definition 2.8 .

Let $(M,\mathcal{F})$ be a filtered $R$ -module. $\mathcal{F}$ is called faithful if

\deg{(r\cdot m)}=\deg{r}+\deg{m},

for any non-zero elements $r\in R,~{}m\in M$ . In this case, $M$ is said to be a faithfully filtered $R$ -module.

Definition 2.1 .

[ VdB04 ] A double Poisson algebra is an associative algebra $A$ with a $\mathbf{k}$ -linear map $\mathopen{\{\!\!\{}\,\mathclose{\}\!\!\}}:A\otimes A\rightarrow A\otimes A$ satisfying:

(2.2)		$\displaystyle\mathopen{\{\!\!\{}a,b\mathclose{\}\!\!\}}=-(21)\mathopen{\{\!\!% \{}b,a\mathclose{\}\!\!\}},$
(2.3)		$\displaystyle\sum_{i=0}^{2}(231)^{i}\circ\mathopen{\{\!\!\{}-,\mathopen{\{\!\!% \{}-,-\mathclose{\}\!\!\}}\mathclose{\}\!\!\}}\circ(231)^{-i}=0,$
(2.4)		$\displaystyle\mathopen{\{\!\!\{}a,bc\mathclose{\}\!\!\}}=(b\otimes 1)\mathopen% {\{\!\!\{}a,c\mathclose{\}\!\!\}}+\mathopen{\{\!\!\{}a,b\mathclose{\}\!\!\}}(1% \otimes c).$