text:
\section{Introduction}
The Mahler measure of a Laurent polynomial $P \in \mathbb{C}[x_1^{\pm1}, \dots, x_n^{\pm1}]$,  is defined by
\begin{eqnarray}
m(P)& := &\frac{1}{(2  \pi \ii)^n} \int_{\TT^n} \log |P(x_1, \dots, x_n) | \frac{ \dd x_1}{x_1} \dots \frac{ \dd x_n}{x_n} \\
& = & \int_0^1 \dots \int_0^1 \log |P(\e^{2 \pi \ii t_1}, \dots , \e^{2 \pi \ii t_n}) | \dd t_1 \dots \dd t_n.
\end{eqnarray}

In particular, Jensen's formula implies, for a one-variable polynomial that
\begin{equation}
m(P)= \log|a| +\sum_i \log \max \{|\alpha_i|, 1 \} \qquad \mathrm{for} \quad P(x)=a \prod_i (x-\alpha_i).
\end{equation}

In this article we use some ideas of Rodriguez-Villegas \cite{RV:ModularMahler} to generalize the definition of the Mahler measure to certain elements of group rings. Our construction yields the logarithm of the L\"uck-Fuglede-Kadison determinant as defined by L\"uck in \cite{LueckBook:L2invariants}.

We relate the Mahler measure for elements in the group ring of a finite group to the characteristic polynomial of the adjacency matrix of a (weighted) Cayley graph and also to characters of the group (Theorem \ref{thm:finitegroup}). We obtain certain
rationality results for the Mahler measure of group ring elements of the form $1-\lambda P$. In Section \ref{sect:finite_approximation} we  recover a result of L\"uck \cite{LueckBook:L2invariants} that the (classical) Mahler measure is limit of Mahler measures over finite abelian groups.
We extend this result to a more general case. In particular, we complement it in Theorem \ref{thm:generalapprox} and Corollary \ref{everything} by proving an analogous property for the infinite dihedral group, $PSL_2(\mathbb{Z})$, and other infinite groups. Finally, we study how the Mahler measure changes when we vary the base group. In particular we focus our attention to the dihedral group $D_m$ versus the abelianization $\mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ in the general case (Theorem \ref{thm:dihedral}).

\section{The general technique}
In this section we will review the techniques and one example of Rodriguez-Villegas \cite{RV:ModularMahler}. The section after that will point out the connections to the combinatorial $L^2$-torsion of L\"uck \cite{LueckBook:L2invariants}. Let $P \in \mathbb{C}[x^{\pm 1},y^{\pm 1}]$  be a polynomial, $\lambda \in \mathbb{C}$, and let $P_\lambda(x,y)= 1 - \lambda P(x,y)$. Assume that $P$ is a reciprocal polynomial i.e., $P(x,y) = \overline{P\left(x^{-1},y^{-1}\right)}$. For instance, as in \cite{RV:ModularMahler}, we may think of  $P(x,y) = x+x^{-1}+y+y^{-1}$.

We define $m(P,\lambda) :=m(P_\lambda)$. Then

\begin{eqnarray}
m(P,\lambda) &=& \frac{1}{(2\pi\ii)^2} \int_{\TT^2} \log|1-\lambda P(x,y)| \frac{\dd x}{x}\frac{\dd y }{y}.\\
&=& \int_0^1 \int_0^1 \log |P(\e^{2 \pi i t}, \e^{2 \pi i s})| \, {\dd t} \, {\dd s}.
\end{eqnarray}

Since $P(x,y)$ is a continuous function defined in the torus $\TT^2$, it is bounded and  $|\lambda P(x,y)|<1$ for $\lambda$ small enough and $x,y \in \TT^2$. For instance, it is easy to see that we need to take $|\lambda|<\frac{1}{4}$ in the example. If $\lambda$ is real, $1- \lambda P(x,y)>0$ on the torus $\TT^2$.

Let
\begin{eqnarray}
u(P,\lambda) &=& \frac{1}{(2\pi\ii)^2} \int_{\TT^2} \frac{1}{1-\lambda P(x,y)} \frac{\dd x}{x}\frac{\dd y }{y} \nonumber \\
&=& \sum_{n=0}^\infty \lambda^n \frac{1}{(2\pi\ii)^2} \int_{\TT^2} P(x,y)^n \frac{\dd x}{x} \frac{\dd y}{y}= \sum_{n=0}^\infty a_n \lambda^n,
\end{eqnarray}
where
\[\frac{1}{(2\pi\ii)^2} \int_{\TT^2} P(x,y)^n \frac{\dd x}{x} \frac{\dd y}{y}= \left[P(x,y)^n\right]_0 = a_n\]
corresponds to the constant coefficient of the $n$-th power of $P$. Now we can write
\begin{eqnarray} \label{eq:defn}
 \tilde{m}(P,\lambda) &=& \frac{1}{(2\pi\ii)^2} \int_{\TT^2} \log(1-\lambda P(x,y)) \frac{\dd x}{x}\frac{\dd y }{y}\nonumber \\
\label{MMeasureSeries}
&=&- \int_0^\lambda (u(t)-1) \frac{\dd t}{t}= - \sum_{n=1}^\infty \frac{a_n \lambda^n}{n}.
\end{eqnarray}
In general,
\begin{equation}
m(P,\lambda) = \re( \tilde{m}(P,\lambda)).
\end{equation}

In our particular example, $\tilde{m}(\lambda)$ is real. For $\lambda$ real, $a_n = 0$ for $n$ odd and
\[ a_{2m} = \sum_{j=0}^m \frac{(2m)!}{j!j!(m-j)!(m-j)!} = \binom{2m}{m}\sum_{j=0}^m \binom{m}{j}^2 = \binom{2m}{m}^2.\]

We observe that the right term in equation (\ref{eq:defn}) makes sense in more general contexts. More precisely, we can make the following:
\begin{defn} \label{def:MahlerM}
Let $ \mathbb{F}_{x_1,\dots,x_l}$ be the free group in $x_1, \dots, x_l$ and let $N$ be a normal subgroup (of relations).
Let $\Gamma = \mathbb{F}_{x_1,\dots, x_l}/N$ be the quotient group and consider the group ring $\mathbb{C} \Gamma$.

Given a
\[Q= Q(x_1, \dots x_l)=\sum_{g \in \Gamma} c_g g \in \mathbb{C}\Gamma,\] its reciprocal is
\[Q^* = \sum_{g \in \Gamma} \overline{c_g} g^{-1} \in \mathbb{C} \Gamma.\]

Let $P=P(x_1, \dots , x_l) \in \mathbb{C}\Gamma$ reciprocal,
(i.e., $P=P^*$). Let $|\lambda| < \frac{1}{k}$ where $k$ is the $l_1$-norm of the coefficients of
$P$ as an element in $\mathbb{C} \Gamma$. Then we define the Mahler measure of $P_\lambda(x_1,\dots, x_l) = 1-\lambda P(x_1,\dots x_l)$ by
\begin{equation}
m_{\Gamma}(P, \lambda) = - \sum_{n=1}^\infty \frac{a_n \lambda^n}{n},
\end{equation}
where $a_n$ is the constant coefficient of the $n$th power of $P(x_1,\dots, x_l)$, in other words,
\begin{equation}
a_n = \left[P(x_1,\dots, x_l)^n\right]_0.
\end{equation}

We will also use the notation
\[u_{\Gamma}(P, \lambda)=\sum_{n=0}^{\infty} a_n \lambda^n\]
for the generating function of the $a_n$.
\end{defn}

We will confine ourselves mainly to the two variable case and
we will keep the notation $m(P,\lambda)$ for $m_{\mathbb{Z}\times\mathbb{Z}}(P,\lambda)$.

\begin{rem}
Let us observe that we can extend this definition for any polynomial $Q(x_1,\dots,x_l) \in \mathbb{C}\Gamma$. It suffices to write

\[QQ^*=\frac{1}{\lambda}\left( 1-\left(1-\lambda QQ^*\right)\right)\]
for $\lambda$ real and positive and $1/\lambda$ larger than the length of $QQ^*$.
Define
\begin{equation}
m_\Gamma(Q) = -\frac{\log \lambda}{2} - \sum_{n=1}^\infty \frac{b_n}{2n},
\end{equation}
where
\[ b_n = \left[(1-\lambda QQ^*)^n\right]_0.\]

This sum might not necessarily converge. When it is convergent then
it is clear that $m_\Gamma(Q)$ is well-defined as it does not depend on the choice of $\lambda$: write
\[ (1-(1-\lambda \mu x))= \lambda \mu x = (1- (1- \lambda x)) \mu\]
then
\[ \log(1-(1-\lambda \mu x)) - \log(\lambda \mu) = \log(1-(1-\lambda  x)) + \log \mu -\log(\lambda \mu)\]
\[ = \log(1-(1-\lambda  x) )- \log \lambda.\]
\end{rem}

\section{L\"uck's combinatorial $L^2$-torsion.}

The approach of Rodriguez-Villegas to the Mahler measure, as explained above, has a powerful generalization
in the theory of combinatorial $L^2$-torsion of L\"uck \cite{LueckBook:L2invariants}.

For example, let $K$ be a knot and \[\Gamma=\pi_1(S^3 \setminus K)=\langle x_1, \dots, x_g \,|\, r_1, \dots, r_{g-1} \rangle\]
be a presentation of the fundamental group of the knot complement, such that no $x_j$ represents a trivial element in $\Gamma$.
For a square matrix $M$ with entries in $\mathbb{C} \Gamma$ the trace $\tr_{\mathbb{C} \Gamma}(M)$
denotes the coefficient of the unit element in the sum of the diagonal elements.
The matrix $A^*=(a^*_{j,i})$ is the transpose of $A=(a_{i,j})$ with
\[\left ( \sum_{g \in \Gamma} c_g g \right ) ^* = \sum_{g \in \Gamma} \overline {c_g} g^{-1}.\]

Let \[F=\left (
\begin{array}[pos]{ccc}
\frac{\partial r_1} {\partial x_1} & \dots & \frac{\partial r_1} {\partial x_g}\\
\vdots &\ddots & \vdots\\
\frac{\partial r_{g-1}} {\partial x_1} & \dots & \frac{\partial r_{g-1}} {\partial x_g}\\
\end{array}
\right )\]
be the Fox matrix (e.g. \cite{BZ}) of the presentation.
The matrix $F$ is a $(g-1)\times g$ matrix with entries in the group ring $\mathbb{C} \Gamma$.
We obtain a $(g-1) \times (g-1)$-matrix $A$ by deleting one of the columns of $F$.

\begin{thm}[L\"uck \cite{LueckBook:L2invariants}] \label{LuecksTheorem} Suppose $K$ is a hyperbolic knot.
Then, for $k$ sufficiently large

\begin{equation} \label{LuecksFormula}
\frac 1 {3 \pi} \mathrm{Vol}(S^3 \setminus K)= 2 (g-1) \ln (k) - \sum_{n=1}^{\infty} \frac 1 n \tr_{\mathbb{C} \Gamma} \left ( ( 1-k^{-2} A A^*)^n \right).
\end{equation}
\end{thm}

In the special case that $A$ has entries in $\mathbb{C} [t,t^{-1}]$ it follows from (\ref{MMeasureSeries}) that
the right-hand side of (\ref{LuecksFormula}) is
$2 m(\det(A))$.

\section{Spectra of weighted Cayley graphs}

It will turn out that the Mahler measure is intimately related to the spectral theory of the
Cayley graph of the base group relative to the generating set given by the terms of the polynomial.

We recall a theorem of Babai \cite{Babai:SpectraCayleyGraphs}.

For a finite group $\Gamma$ of order $m$ let $\alpha$ be a function $\alpha: \Gamma \rightarrow \mathbb{C}$ such that
$\alpha(g)=\overline{\alpha(g^{-1})}$ for all elements $g \in \Gamma$. The weighted Cayley graph of $\Gamma$ with
respect to $\alpha$ is a graph with vertices $g_1, \dots, g_m$. Two vertices $g_i$ and $g_j$ are
connected by a (directed) edge of weight $\alpha(g_i^{-1} g_j)$. Weight $0$ means that there is no edge between
$g_i$ and $g_j$. The weighted adjacent matrix $A(\Gamma, \alpha)$ of the weighted Cayley graph of $\Gamma$ with respect to
$\alpha$ is the matrix $A$ with entries $a_{i,j}=\alpha(g_i^{-1} g_j)$. Thus the matrix $A$ is hermitian: $A=A^*$.

The spectrum of the Cayley graph of $\Gamma$ with respect to $\alpha$ is the spectrum of
$A(\Gamma,\alpha)$.

The spectrum can be computed from the irreducible characters of $\Gamma$ \cite{Babai:SpectraCayleyGraphs}.
Let $\chi_1, \dots, \chi_h$ be the irreducible characters of $\Gamma$ of degrees $n_1, \dots, n_h$.

\begin{thm}[Babai \cite{Babai:SpectraCayleyGraphs}] \label{TheoremBabai}
The spectrum of $A(\Gamma,\alpha)$ can be arranged as
\[{\cal S}=\left \{\sigma_{i,j}: i=1,\dots,h; j=1, \dots,n_i \right \},\]
such that
$\sigma_{i,j}$ has multiplicity $n_i$ and
\begin{equation}
\sigma_{i,1}^t + \dots + \sigma_{i,n_i}^t = \sum_{g_1, \dots, g_t \in \Gamma} \left ( \prod_{s=1}^t \alpha(g_s) \right ) \chi_i \left (\prod_{s=1}^t g_s \right ).
\end{equation}

\end{thm}

\begin{rem}
Notice that Theorem \ref{TheoremBabai} allows us to compute the spectrum of $A$ from the irreducible
characters of the base group.

Indeed, by Theorem \ref{TheoremBabai},
\begin{eqnarray}
\tr(A^n) &=& \sum_{i=1}^h n_i \sum_{g_1, \dots, g_n \in \Gamma} \left ( \prod_{s=1}^n \alpha(g_s)\right) \chi_i\left(\prod_{s=1}^n g_s \right)\\
&=&  \sum_{i=1}^h n_i P^n\left[ \chi_i(g_1), \dots, \chi_i(g_m) \right].
\end{eqnarray}
The last term means that we first compute the $n$-th power of the element $P$ as a polynomial and then we evaluate each of the monomials in the representation $\chi_i$. This is not the usual evaluation in the algebra generated by the $x_i$, therefore we use the brackets to distinguish it.
\end{rem}

\section{The Mahler measure over finite groups}

Suppose  $P\in \mathbb{C}\Gamma$ with
\[P = \sum_i (\delta_i S_i+ \overline{\delta_i} S_i^{-1})+ \sum_j \eta_j T_j\]
for some monomials $S_i \neq S_i^{-1}$, $T_j=T_j^{-1}$, $\delta_i \in \mathbb{C}$, $\eta_j \in \mathbb{R}$, and
$S_i, T_j \in \Gamma$, $\Gamma$ a group.
For reasons of simplicity we assume that the monomials of $P$ actually generate the whole group $\Gamma$, rather than a subgroup.

The coefficients $\delta_i, \eta_j$ induce a weight on the Cayley graph of $\Gamma$ with respect to the generating set
$\{S_i, T_j\}$ of $\Gamma$.
We are interested in the computation of $a_n=\left[P^n\right]_0$.
This corresponds to counting (with weights) the number of closed paths of length $n$ in the Cayley graph of $\Gamma$ that start and end at the identity. The weight of a closed path is the product over the weights of the
edges of the path. Therefore, the value of $a_n$ can be computed via the trace of the $n$-th power of $A$.

\begin{thm}\label{thm:finitegroup}
If the group $\Gamma$ is finite then
\begin{equation}
m_{\Gamma} (P, \lambda)= \frac 1 {|\Gamma|} \log \det (I- \lambda A),
\end{equation}
where $A$ is the adjacency matrix of the Cayley graph (with weights) and $\frac 1 {\lambda}$ greater than the largest eigenvalue of the hermitian matrix $A$.

In particular $\exp(|\Gamma| m_{\Gamma}(P,\lambda))$ is a polynomial in $\lambda$.
\end{thm}
This Theorem provides an analytic continuation of $m_{\Gamma}(P,\lambda)$ as a function of $\lambda$ in the complex plane $\mathbb{C}$ minus the eigenvalues of $A$.

\medskip

\noindent {\bf PROOF.} \quad
Since the Cayley graph is vertex-transitive, this implies that
\[a_n = \frac 1 {|\Gamma|} \tr (A^n),\]
where  $\tr$ is the usual trace on matrices.

Now the Mahler measure is
\begin{eqnarray*}
m_{\Gamma} (P, \lambda) &=&  - \sum_{n=1}^{\infty} \frac {a_n \lambda^n} n\\
&=& - \sum_{n=1}^{\infty} \frac 1 {|\Gamma|}\frac {\tr( (\lambda A)^n)} n\\
&=& \frac 1 {|\Gamma|} \log \det (I- \lambda A).
\end{eqnarray*}$\Box$

\begin{rem}
In the case of an arbitrary polynomial $Q$,
\begin{equation}
m_{\Gamma}(Q)= \frac {1}{2 |\Gamma|} \log \det B,
\end{equation}
where $B$ is the adjacency matrix corresponding to $Q Q^*$.
The Mahler measure $m_\Gamma(Q)$ is defined if  and only if $B$ is nonsingular.
\end{rem}

\begin{rem}
Recall that the generating function for the $a_n$ is
\[u_{\Gamma}(P,\lambda)=\sum_{j=0}^{\infty} a_n \lambda^n\]
and satisfies:
\[-\lambda \frac {\dd}{\dd \lambda} m_{\Gamma}(P,\lambda) = u_{\Gamma}(P,\lambda)-1.\]

Let $\sigma_i$ be the eigenvalues of $A$. By Theorem \ref{thm:finitegroup} we have:
\begin{equation}\label{eq:urational}
u_{\Gamma}(P,\lambda) = 1 - \frac {\lambda}{|\Gamma|} \frac {\frac {\dd}{\dd \lambda} \det(I - \lambda A)}{\det(I-\lambda A)} = \frac{1}{|\Gamma|} \sum_i \frac{1}{1-\lambda \sigma_i}.
\end{equation}

In particular, $u_{\Gamma}(P,\lambda)$ is a rational function in $\lambda$.
\end{rem}

\subsection{An example}

We will consider the example of the group
\[\mathbb{Z}/3 \mathbb{Z} \times \mathbb{Z}/2 \mathbb{Z} = \langle x, y \,|\, x^3, y^2, [x,y]\rangle.\]
Let us compute the Mahler measure of the general polynomial
\[P_\lambda(x,y)= 1-\lambda \left(a +  b x+ \overline{b} x^{-1}+  c y  +  d y x + \overline{d} y x^{-1} \right).\]
Numbering the vertices $e, x, x^{-1}, y, yx, yx^{-1}$ as $1,\dots,6$, we get an adjacency matrix:
\[ \left(\begin{array}{cccccc}a & b & \overline{b} & c & d &\overline{d}\\\overline{b} & a & b & \overline{d} & c &d\\b& \overline{b}& a & d &\overline{d} &c\\c &d & \overline{d} & a& b &\overline{b}\\\overline{d} & c& d &\overline{b}& a &b\\d & \overline{d} & c & b & \overline{b} &a\end{array} \right) .\]
We see that the characteristic polynomial $\det(t I - A)$ is
\[((t-a+c+\re(b-d))^2-3(\im(b-d))^2)((t-a-c+\re(b+d))^2-3(\im(b+d))^2)\]
\[\cdot(t-a-2\re(b)+c+2\re(d))(t-a-2\re(b)-c-2\re(d)).\]
Thus,
\[ m_{\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}(P,\lambda)  =  \frac{1}{6}\log ((1-\lambda(a-c-\re(b-d))^2-3\lambda(\im(b-d))^2)\]
\[+ \frac{1}{6}\log ((1-\lambda(a+c-\re(b+d))^2-3\lambda(\im(b+d))^2) \]
\[+\frac{1}{6}\left(  \log\left(1-\lambda (a+2\re(b)-c-2\re(d)) \right)+ \log\left(1-\lambda(a+2\re(b)+c+2\re(d)) \right) \right).\]

For instance, if $Q=1+x+y$ then $QQ^*= 3+x+x^{-1}+2y+yx+yx^{-1}$. It is easy to see that $\det B =81$. Therefore,
\[m_{\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}(1+x+y)= \frac{1}{12} \log 81 = \frac{\log 3}{3}.\]

\section{Abelian Groups}\label{sec:abel}

Suppose $\Gamma$ is a finite abelian group
\[\Gamma = \mathbb{Z}/m_1 \mathbb{Z} \times \dots \times \mathbb{Z}/m_l \mathbb{Z}\]
of order
\[| \Gamma |=m=\prod_{j=1}^l m_j.\]

We obtain
\begin{coro}\label{coro}
\begin{equation}
m_{\Gamma}(P, \lambda) =\frac{1}{|\Gamma|}\log\left(\prod_{j_1, \dots, j_l}\left( 1- \lambda P (\xi^{j_1}_{m_1},\dots,\xi_{m_l}^{j_l}) \right) \right)
\end{equation}
where $\xi_k$ is a primitive root of unity.
\end{coro}
\noindent {\bf PROOF.} \quad This Corollary is consequence from the form of the representations for a finite abelian group. The characters are given by
\[\chi_{j_1, \dots, j_l} (s_1,\dots,s_l) = \xi_{m_1}^{j_1 s_1}\dots \xi_{m_l}^{j_ls_l}.\]
$\Box$

For the following result we recall some notation of Boyd and Lawton. For an integral vector $m=(m_1, \dots, m_l)$ let \[q(m) = \min \left\{ H(s) \, \left| \, s=(s_1, \dots, s_l)\in \mathbb{Z}^l,\, \sum_{i=1}^l m_i s_i =0 \right. \right\},\] where $H(s)= \max_{1\leq i \leq l} |s_i|$.

Thus, for a reciprocal polynomial $P \in \mathbb{C} [\mathbb{Z}^l]$ in the Laurent polynomial ring of $l$ variables, we have the following approximation.

\begin{thm}\label{thm:approx} For sufficiently small $\lambda$,
\begin{eqnarray}
\lim_{q(m) \rightarrow \infty}  m_{\mathbb{Z}/m_1\mathbb{Z} \times \dots \times \mathbb{Z}/m_l\mathbb{Z}}(P, \lambda)&=& m_{\mathbb{Z}^l} (P, \lambda).
\end{eqnarray}
\end{thm}
\noindent {\bf PROOF.} \quad Following Rodriguez-Villegas technique
\[m_{\mathbb{Z}^l} (P, \lambda)=  - \int_0^1 \dots \int_0^1 \log (1 - \lambda P(\e^{2 \pi i t_1},\dots, \e^{2 \pi i t_l})) \dd t_1 \dots \dd t_l.\]
Now the function  $\log (1 - \lambda P(\e^{2 \pi i t_1},\dots, \e^{2 \pi i t_l}))$ is continuous and with no singularities in $\TT^l$. Then we can compute the Riemann integral by taking limits on evaluations on the roots of unity. $\Box$

This Theorem should be compared to the result (for the one variable case) by L\"uck (page 478 of \cite{LueckBook:L2invariants}).

\section{Dihedral groups}

Let us now consider $\Gamma = D_m$ the dihedral group, i.e.,
\[D_m= \langle \rho, \sigma \, | \, \rho^m, \sigma^2, \sigma \rho \sigma \rho  \rangle.\]
First assume that $m$ is odd, $m=2p+1$. The character  table (see, for instance \cite{Serre3}) for this group is

\bigskip
\begin{center}
\begin{tabular}{c|c|rc}
& $\rho^k$ & $\sigma \rho^k$ & \\
\hline
$\chi_j$ & $\xi_m^{jk}+\xi_m^{-jk}$ & $0$ & $j=1, \dots p$\\
$\psi_1$ & $1$ & $1$ & \\
$\psi_2$ & $1$ & $-1$ & \\
\end{tabular}
\end{center}

\bigskip

Assume that $P$ is reciprocal. In computing the Mahler measure of $1-\lambda P$ we need to determine
\[\tr(A^n) = P^n[1,\dots, 1]+P^n[1,\dots,1,-1, \dots, -1]\]
\[ + 2 \sum_{j=1}^p P^n\left[2,\xi_m^{j}+\xi_m^{-j}, \dots, \xi_m^{j(m-1)}+\xi_m^{-j(m-1)}, 0, \dots, 0\right].\]
Note that $P$ can be thought as a polynomial in two variables $\rho$ and $\sigma$. For the first two terms we can write
$P^n(1,1)+P^n(1,-1)$, then we obtain
\[=P^n(1,1)+P^n(1,-1)\]
\[ + 2 \sum_{j=1}^p \left(P^n\left[1,\xi_m^{j}, \dots, \xi_m^{j(m-1)}, 0, \dots, 0\right] +P^n\left[1,\xi_m^{-j}, \dots, \xi_m^{-j(m-1)}, 0, \dots, 0\right]\right) \]
\[=P^n(1,1)+P^n(1,-1)\]
\[ + 2 \sum_{j=1}^{m-1} P^n\left[1,\xi_m^{j}, \dots, \xi_m^{j(m-1)}, 0, \dots, 0\right]\]
\[=P^n(1,1)+P^n(1,-1) +  \sum_{j=1}^{m-1} \left(P^n\left(\xi_m^{j}, 1\right)+ P^n\left(\xi_m^{j}, -1\right) \right)\]
\begin{equation}\label{eq:dihedral}
=\sum_{j=1}^{m} \left(P^n\left(\xi_m^{j}, 1\right)+ P^n\left(\xi_m^{j}, -1\right) \right).
\end{equation}
At this point we should be more specific about the meaning of this formula. Given $P$ we consider its $n$th-power, then we write it as a combinations of monomials $\rho^k$ and $\sigma \rho^k$, and after that we evaluate each of the variables $\rho$, $\sigma$.

When $m =2p$, the character table is

\bigskip
\begin{center}
\begin{tabular}{c|c|cc}
& $\rho^k$ & $\sigma \rho^k$ & \\
\hline
$\chi_j$ & $\xi_m^{jk}+\xi_m^{-jk}$ & $0$ & $j=1, \dots p-1$\\
$\psi_1$ & $1$ & $1$ & \\
$\psi_2$ & $1$ & $-1$ & \\
$\psi_3$ & $(-1)^k$ & $(-1)^k$ & \\
$\psi_4$ & $(-1)^k$ & $(-1)^{k+1}$ & \\
\end{tabular}
\end{center}

\bigskip
\noindent and we obtain a similar expression as in equation
(\ref{eq:dihedral}).

Hence, we have proved:
\begin{thm} Let $P \in \mathbb{C}[D_m]$ be reciprocal. Then
\begin{equation}
\tr(A^n)= \sum_{j=1}^{m} \left(P^n\left(\xi_m^{j}, 1\right)+ P^n\left(\xi_m^{j}, -1\right) \right),
\end{equation}
where $P^n$ is expressed as a sum of monomials $\rho^k$, $\sigma \rho^k$ before being evaluated.
\end{thm}

On the other hand, for $\Gamma= \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z} = \langle x, y \,|\, x^m, y^2, [x,y]\rangle $,
\[ \tr (A^n) = \sum_{j=1}^m \left(P\left(\xi_m^j,1\right)^n + P\left(\xi_m^j,-1\right)^n\right).\]

From now on, in order to compare elements in the group rings of $D_m$ and $\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ we will rename $x=\rho$ and $y=\sigma$ in $D_m$.

We have
\begin{thm}\label{thm:dihedral} Let
\[P= \sum_{k=0}^{m-1} \alpha_k x^k  + \sum_{k=0}^{m-1} \beta_k y x^k\]
with real coefficients and reciprocal in $\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ (therefore it is also reciprocal in $D_m$).
Then
\begin{equation}
m_{\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}}(P,\lambda) = m_{D_m}(P,\lambda).
\end{equation}
\end{thm}
\noindent {\bf PROOF.} \quad It is important to note that being reciprocal in $\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ is not the same as being reciprocal in $D_m$.

Being reciprocal in $\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ means that $\alpha_k=\overline{\alpha_{m-k}}$ and $\beta_k = \overline{\beta_{m-k}}$. On the other hand, being  reciprocal in $D_m$ means that
$\alpha_k=\overline{\alpha_{m-k}}$ and $\beta_k$ is real.

Then condition that $\alpha_k, \beta_k$ are real and the reciprocity in $\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ imply that and $\alpha_k=\alpha_{m-k}$, $\beta_k=\beta_{m-k}$ and reciprocity in $D_m$.

We will prove that the powers of $P$ are the same in both groups rings by induction. Assume that $P^n$ looks the same in both group rings and that it satisfies the conditions:
\[ P^n = \sum_{k=0}^{m-1} a_k x^k  + \sum_{k=0}^{m-1} b_k y x^k\]
with $a_k= a_{m-k}$ and $b_k= b_{m-k}$.
Then, in $\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$,
\[ P P^n= \sum_{k_1, k_2} \left( \alpha_{k_1} a_{k_2}+\beta_{k_1} b_{k_2}\right) x^{k_1+k_2} +
 \sum_{k_1, k_2} \left( \alpha_{k_1} b_{k_2}+\beta_{k_1} a_{k_2}\right) y x^{k_1+k_2}.\]
In $D_m$,
\[ P P^n= \sum_{k_1, k_2} \left( \alpha_{k_1} a_{k_2}+\beta_{m-k_1} b_{k_2}\right) x^{k_1+k_2} +
 \sum_{k_1, k_2} \left( \alpha_{m-k_1} b_{k_2}+\beta_{k_1} a_{k_2}\right) y x^{k_1+k_2}.\]
It is easy to see that $P^{n+1}$ satisfies the induction hypothesis.
$\Box$

\begin{rem}
The premises of Theorem \ref{thm:dihedral} are somewhat strong but necessary. First, the condition that the coefficients are real is necessary. To see this, consider \[P=3+\ii x-\ii x^{-1}+y.\]
In $\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, we obtain
\[m_{\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}}(3+\ii x-\ii x^{-1}+y) = \frac{\log 104}{6},\]
however,
\[m_{D_3}(3+\ii x-\ii x^{-1}+y) =  \frac{\log 200}{6}.\]

It would be also nice to have a result that is more general for any $Q$ that is not necessarily reciprocal. Unfortunately,
this is not possible due to the different structures of the group rings.
Let
\[Q=\sum_{k=0}^{m-1} \alpha_k x^k  + \sum_{k=0}^{m-1} \beta_k y x^k.\]
We have, for $\Gamma_1 = \mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$,
\[Q^*_1=\sum_{k=0}^{m-1} \overline{\alpha_k} x^{-k}  + \sum_{k=0}^{m-1} \overline{\beta_k} y x^{-k}.\]
For $\Gamma_2 = D_m$,
\[Q^*_2=\sum_{k=0}^{m-1} \overline{\alpha_k} x^{-k}  + \sum_{k=0}^{m-1} \overline{\beta_k} y x^k.\]

Clearly $QQ^*_1$ and $QQ^*_2$ are generally different:
\[QQ^*_1= \sum_{k_1,k_2} \left(\alpha_{k_1} \overline{\alpha_{k_2}} + \beta_{k_1} \overline{\beta_{k_2}}\right) x^{k_1-k_2} + \sum_{k_1,k_2} \left(\alpha_{k_1} \overline{\beta_{k_2}} + \beta_{k_1} \overline{\alpha_{k_2}}\right) y x^{k_1-k_2}\]

\[QQ^*_2= \sum_{k_1,k_2} \left(\alpha_{k_1} \overline{\alpha_{k_2}} + \overline{\beta_{k_1}}\beta_{k_2}\right) x^{k_1-k_2} + \sum_{k_1,k_2} \left(\beta_{k_1} \overline{\alpha_{k_2}} + \overline{\beta_{k_1}} \alpha_{k_2}\right) y x^{k_1-k_2}\]

As a counterexample for this case, let us look at $Q=x+2y$.

In $\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, $QQ^*=5+2yx+2yx^{-1}$. Then
\[m_{\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}}(x+2y) = \frac{\log 63}{6} .\]

In $D_3$, $QQ^*=5+4yx^{-1}$ and
\[m_{D_3}(x+2y) = \frac{\log 3}{2}.\]

\end{rem}

Notice that the proof of Theorem \ref{thm:dihedral} is independent of $m$. Therefore, we can conclude the following.

\begin{coro}\label{coro:dihedral}
Let $P \in \mathbb{R} \left[\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \right]$ be reciprocal. Then
\begin{equation}
m_{\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}}(P, \lambda) = m_{D_\infty}(P,\lambda),
\end{equation}
where $D_\infty = \langle \rho, \sigma \, | \, \sigma^2, \sigma \rho \sigma \rho \rangle $.
\end{coro}

\section{Dicyclic groups}

We consider the dicyclic groups with presentations
\[Dic_{m}=  \langle x, y \, | \, x^{2m},  y^2x^m, y^{-1}xyx \rangle.\]
The Mahler measure in these groups will be compared to the one in  $\mathbb{Z}/2m\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} = \langle x, y \, | \, x^{2m}, y^2, [x,y]\rangle$.

\begin{thm}
Let \[P=\sum_{k=0}^{2m} \alpha_k x^k + \sum_{k=0}^{2m} \beta_k y x^k\]
such that the coefficients $\alpha_k$ are real, and it is reciprocal in both $Dic_m$ and $\mathbb{Z}/2m\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Then
\begin{equation}
m_{\mathbb{Z}/2m\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}} (P, \lambda)= m_{Dic_m} (P, \lambda).
\end{equation}
\end{thm}
\noindent {\bf PROOF.} \quad
The proof is very similar to the one for Theorem \ref{thm:dihedral}. We just need to observe that the reciprocal condition in $Dic_m$ implies $\alpha_k=\alpha_{2m-k}$ and  $\beta_k=\overline{\beta_{m+k}}$, while in $\mathbb{Z}/2m\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ we have $\alpha_k=\alpha_{2m-k}$ and  $\beta_k=\overline{\beta_{2m-k}}$.
$\Box$

\section{Finite Approximations of the Mahler measure}
\label{sect:finite_approximation}

In this section we study a generalization of Theorem \ref{thm:approx} for families of groups that are not necessarily abelian.
Suppose that $\Gamma$ is a group and $\Gamma_m$ is a family of groups. Denote by
\[a_n^{(m)} = \left[P^n\right]_0 \qquad \mathrm{in} \quad \Gamma_m,\]
and let $a_n$ denote the corresponding number in $\Gamma$. Then

\begin{thm} \label{thm:generalapprox} Assume that there are $N_m$, depending on $m$ and $\{\Gamma_m,\Gamma\}$, with
$$\lim_{m \rightarrow \infty} N_m = \infty,$$
such that
\[ a_n^{(m)}= a_n \qquad \mathrm{for}\quad n < N_m.\]
Let $k$ be the $l_1$-norm of the polynomial $P$, i.e. the sum of the absolute values of the coefficients of the monomials in $P$.
Then, for $|\lambda| < \frac{1}{k}$,
\begin{equation}
\lim_{m \rightarrow \infty} m_{\Gamma_m}(P, \lambda) = m_{\Gamma}(P, \lambda).
\end{equation}
Moreover, the convergence is uniform in $\lambda$ for  $|\lambda|<\frac 1 k$.
\end{thm}

\noindent {\bf PROOF.} \quad Consider
\[\left| \sum_{n=1} \frac{a_n^{(m)} \lambda^n}{n} - \sum_{n=1} \frac{a_n \lambda^n}{n}\right|
\leq \left| \sum_{n=N_m} \frac{(a_n^{(m)}-a_n )\lambda^n}{n}\right|\]
for any $m$. The term in the right can be made arbitrarily small by
choosing $m$ such that $N_m$ is large, since we know that the sums
converge.
Also, since $|\lambda|$ is bounded, the convergence is uniform.

$\Box$

This Theorem provides an alternative way of seeing Theorem \ref{thm:approx} but it has other applications as well.
In the following corollary the $\Gamma_m$ are quotients of $\Gamma$:

\begin{coro}\label{everything} Let $P \in \Gamma$ reciprocal.
\begin{itemize}

\item For $\Gamma = D_\infty$, $\Gamma_m= D_m$,
\begin{equation}
\lim_{m \rightarrow \infty} m_{D_m}(P, \lambda)= m_{D_\infty} (P, \lambda).
\end{equation}

\item For $\Gamma = Dic_\infty = \langle x, y \, | \, y^2, (yx)^2 \rangle $, $\Gamma_m = Dic_m$,
\begin{equation}
\lim_{m \rightarrow \infty} m_{Dic_m}(P, \lambda)= m_{Dic_\infty} (P, \lambda).
\end{equation}

\item For $\Gamma = PSL_2(\mathbb{Z}) = \langle x, y \, |\, x^2, y^3  \rangle$, $\Gamma_m = \langle x, y \, |\, x^2, y^3, (xy)^m \rangle$,
\begin{equation}
\lim_{m \rightarrow \infty} m_{\Gamma_m}(P, \lambda)= m_{PSL_2(\mathbb{Z})} (P, \lambda).
\end{equation}

\item For $\Gamma = \mathbb{Z} * \mathbb{Z} = \langle x, y \rangle$, $\Gamma_m = \langle x, y \, |\, [x,y]^m \rangle$,
\begin{equation}
\lim_{m \rightarrow \infty} m_{\Gamma_m}(P, \lambda)= m_{\mathbb{Z}*\mathbb{Z}} (P, \lambda).
\end{equation}
Note that in this case $\Gamma_1=\mathbb{Z} \times \mathbb{Z}$.

\end{itemize}
\end{coro}

\section{Changing the base group}

In this section we fix an element in $\mathbb{Z}*\mathbb{Z}$, namely $P=x+x^{-1}+y+y^{-1}$ and we investigate $m_\Gamma(P,\lambda)$ for different choices of $\Gamma = \mathbb{Z}*\mathbb{Z}/N$.

\subsection{The free abelian group}
The computations of the first section show that for the case of $\Gamma = \mathbb{Z}\times\mathbb{Z}$, we get
\[ u_{\mathbb{Z}\times \mathbb{Z}}(P,\lambda) = \sum_{m=0}^\infty \binom{2m}{m}^2 \lambda^{2m} = F\left(\frac{1}{2},\frac{1}{2};1; 16\lambda^2\right) = \frac{2}{\pi} K(16 \lambda^2) ,\]
where $K$ is the complete elliptic function and $F$ the hypergeometric series.

Recall that if
\[u_{\mathbb{Z}\times \mathbb{Z}}(P,\lambda) = \sum_{n=0}^\infty a_n \lambda^n\]
is the generating function of the $\left[P^n\right]_0$, then
\[m_{\mathbb{Z}\times \mathbb{Z}}(P,\lambda) = -\sum_{n=1} \frac{a_n \lambda^n}{n}.\]

Therefore, for this case
\begin{equation}
 m_{\mathbb{Z}\times \mathbb{Z}}(P,\lambda) = -\sum_{m=1}^\infty \binom{2m}{m}^2 \frac{\lambda^{2m}}{2m}.
 \end{equation}

\subsection{Taking the limit over one variable}

Now assume that $\Gamma=\mathbb{Z}\times \mathbb{Z}/m\mathbb{Z}$ so that $x$ is free, $y^m=1$ and they commute. By using similar ideas to those in the proof of Theorem \ref{thm:approx}, we can see that
\begin{equation}
\lim_{n\rightarrow \infty} m_{\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}}(P,\lambda)=m_{\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}}(P,\lambda).
\end{equation}

Therefore,
\begin{equation}
m_{\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}}(P,\lambda) = \frac{1}{m}  \sum_{k=0}^{m-1} m\left(1 - \lambda \left(x+x^{-1}+\xi_m^{k}+\xi_m^{-k}\right)\right),
\end{equation}
where the term in the right corresponds to the classical Mahler measure of one-variable polynomials.
The polynomial
\[ 1 - \lambda \left(x+x^{-1}+\xi_m^{k}+\xi_m^{-k}\right)\]
has two roots given by
\begin{equation} \label{eq:mixed}
\frac{\lambda^{-1}-2 \cos \frac{2\pi k}{m} \pm \sqrt{\lambda^{-2} -4 \cos \frac{2\pi k}{m} \lambda^{-1}- 4 \sin^2    \frac{2\pi k}{m}}}{2}.
\end{equation}

Since the product is $1$, one of the roots has always absolute values larger than $1$. Assume for simplicity that $\lambda>0$, then we need to consider the root with the "$+$" sign.
Thus,
\begin{equation}
 m_{\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}}(P,\lambda) = \frac{1}{m}  \sum_{k=0}^{m-1}\log \left(\frac{1-2 \cos \frac{2\pi k}{m} \lambda + \sqrt{1 -4 \cos \frac{2\pi k}{m} \lambda- 4 \sin^2    \frac{2\pi k}{m}\lambda^2}}{2}\right).
 \end{equation}

For the case of $m=2$ the eigenvalues are $2$ and $-2$. Then we have, by equation (\ref{eq:mixed})
\[  m_{\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}(P, \lambda) =-\log 2 + \frac{1}{2}  \left( \log \left(1-2 \lambda + \sqrt{1 -4 \lambda}\right) + \log \left(1+2 \lambda + \sqrt{1 +4\lambda}\right) \right).\]

In this case the coefficients can be also computed directly.
\[\left[\left(x+x^{-1}+y+y^{-1}\right)^n\right]_0 =  \sum_{j=0}^n \binom{n}{j}\left[(x+x^{-1})^j (y+y^{-1})^{n-j}\right]_0,\]
and because there is no relation between $x$ and $y$ besides the fact that they commute,
\[\left[(x+x^{-1}+y+y^{-1})^n\right]_0 = \sum_{j=0}^n \binom{n}{j}\left[(x+x^{-1})^j\right]_0 \left[(y+y^{-1})^{n-j}\right]_0.\]

Note that
\[\left[(x+x^{-1})^j\right]_0 = \left \{ \begin{array}{cc} \binom{j}{j/2} & j \quad \mbox{even}\\ 0 & j \quad \mbox{odd} \end{array} \right . \qquad  \left[(y+y^{-1})^j\right]_0 = \frac{2^j + (-2)^j}{2}.\]
Then
\[\left[\left(x+x^{-1}+y+y^{-1}\right)^{2l}\right]_0= \sum_{j=0}^{2l} \binom{2l}{2j} \binom{2j}{j} 2^{2l-2j}\]
\[ = \left[\left(x+x^{-1}+ 2 \right)^{2l}\right]_0 = \left[\left(x^2+x^{-2}+ 2 \right)^{2l}\right]_0 =
\left[\left(x+x^{-1}\right)^{4l}\right]_0 = \binom{4l}{2l}.\]
Then we obtain
\begin{equation}
m_{\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}(P,\lambda) =  - \sum_{l=1}^\infty \binom{4l}{2l}\frac{\lambda^{2l}}{2l}.
\end{equation}

Let us remark that while we are computing a function of the form
\[\sum_{n=1}^\infty \sum_{j=0}^{n/2}\frac{1}{n}
\binom{n}{2j} \binom{2j}{j} \beta^{2j}\lambda^n,\]
a similar function was studied in \cite{Dasbach:NaturalLog}:
\[\sum_{n=1}^\infty \sum_{j=0}^{n/2}\frac{1}{n}
\binom{n}{j} \binom{2j}{j} \beta^{j}\lambda^n.\]

Finally, we remark, in the same vein as Theorem \ref{thm:approx}, that
\begin{equation}
\lim_{m \rightarrow \infty} m_{\mathbb{Z} \times \mathbb{Z}/m \mathbb{Z}}(P,\lambda) = m_{\mathbb{Z} \times \mathbb{Z}}(P,\lambda).
\end{equation}

\subsection{The free group}
We now assume that there is no relation between $x$ and $y$. Then $u_{\mathbb{Z}*\mathbb{Z}}(P,\lambda)$ corresponds to counting circuits (based in a distinguished point) in the $4$-regular tree (that is the Cayley graph of $\mathbb{Z}*\mathbb{Z}$). By a result of Bartholdi \cite{Bartholdi:Counting} $u$ is equal to
\[u_{\mathbb{Z}*\mathbb{Z}}(P,\lambda) = \frac{3}{1+2\sqrt{1-12\lambda^2}}.\]

\subsection{$PSL_2(\mathbb{Z})$}
A presentation for $PSL_2(\mathbb{Z})$ is given by $\langle x,y \,|\,x^2, y^3\rangle$. Hence, it is isomorphic to $\mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$. By a result of Bartholdi \cite{Bartholdi:Counting},
 \[u_{PSL_2(\mathbb{Z})}(x+y+y^{-1},\lambda) = \frac{(2-\lambda)\sqrt{1-2\lambda-5\lambda^2+6\lambda^3 +\lambda^4}-\lambda+\lambda^2+\lambda^3}{2(\lambda -1) (3 \lambda - 1) (2 \lambda + 1)}.\]

However, our $P$ is $2x+y+y^{-1}$. If we use Theorem 9.2 in \cite{Bartholdi:Counting} we obtain
\[u_{PSL_2(\mathbb{Z})}(P,\lambda) = \frac{(2-\lambda)\sqrt{1-2\lambda-11\lambda^2+12\lambda^3 +4\lambda^4}-\lambda+\lambda^2-2\lambda^3}{2(\lambda-1)(3\lambda+1)(4\lambda -1)}.\]

\section{Arbitrary number of variables}
There is a general result in  \cite{Bartholdi:Counting} regarding the generating function of the circuits of a $d$-regular tree which is given by
\begin{equation}\label{eq:manyvariables}
g_d(\lambda)= \frac{2(d-1)}{d-2+d\sqrt{1-4(d-1)\lambda^2}}.
\end{equation}

In the case of
\[P_{1,l}=x_1+x_1^{-1}+ \dots + x_l + x_l^{-1},\]
we can easily see that for the free group $\mathbb{F}_l$ we are  counting circuits in a $2l$-regular tree, therefore
\begin{equation}
u_{\mathbb{F}_l}(P_{1,l},\lambda) = g_{2l}(\lambda).
\end{equation}

We also consider the polynomial
\[P_{2,l}=\left(1+x_1+\dots + x_{l-1}\right)\left(1+x^{-1}_1+\dots + x^{-1}_{l-1}\right).\]
Let us  add one more variable, so we obtain
\[\left(x_0+x_1+\dots + x_{l-1}\right)\left(x_0^{-1}+x^{-1}_1+\dots + x^{-1}_{l-1}\right).\]
It is easy to see that the trace of the powers of $P_{2,l}$ is not affected by this change. But now it is clear that we are counting circuits in a $l$-regular tree, thus,
\begin{equation}
u_{\mathbb{F}_{l-1}}(P_{2,l},\lambda)= g_l(\lambda).
\end{equation}

In particular,
\begin{equation}
m_{\mathbb{F}_l}(P_{1,l}, \lambda) = m_{\mathbb{F}_{2l-1}}(P_{2,2l}, \lambda).
\end{equation}

Let us examine the abelian versions of these polynomials. By using elementary combinatorics we compute the trace of the powers of $P_{1,l}$ and $P_{2,l}$:
\[ \left[P_{1,l}^n\right]_0 = \sum_{a_1+\dots + a_l=n} \frac{(2n)!}{(a_1!)^2 \dots (a_l!)^2},\]
\[ \left[P_{2,l}^n\right]_0 = \sum_{a_1+\dots + a_l=n} \left(\frac{n!}{a_1! \dots a_l!}\right)^2.\]

For $l>2$ these expressions do not seem to simplify, in the sense that we are unable to find a closed formula that does not involve a summation. However, we make the following interesting observation:
\begin{equation}
\left[P_{1,l}^{2n}\right]_0 = \binom{2n}{n}\left[P_{2,l}^n\right]_0,
\end{equation}
where the trace in the left is taken over $\mathbb{Z}^{\times l}$ and the one in the right is taken over $\mathbb{Z}^{\times (l-1)}$.

In other words, there are relations among the Mahler measure of these two families of polynomials. But the relations depend on the base group.

\bigskip

{\bf Acknowledgements}: The authors would like to thank Neal Stoltzfus and Fernando Rodriguez-Villegas for helpful discussions. Part of this work was completed while ML was a member at the Institute for Advanced Study and during  visits to the Department of Mathematics at Louisiana State University. She is deeply grateful for their support and hospitality.

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"This needs to be fixed:" FOX6 finds a new "risk on the rails," could Milwaukee be the next Quebec?
"This needs to be fixed:" FOX6 finds a new "risk on the rails," could Milwaukee be the next Quebec?
MILWAUKEE (WITI) -- Last year, the FOX6 Investigators were the first to expose a new risk on the rails -- a steady stream of long oil trains trekking across the state from North Dakota. The crude oil they carry from what's called "The Bakken" is highly explosive. Since then, there has been growing public concern about these so-called "bomb trains" in Wisconsin. Now, there's a new concern, in a neighborhood in Milwaukee.
When the mile-long oil trains lumber by Milwaukee's Fifth Ward lofts, the cars come roller-coaster close to a renovated building. A sliver of light between brick and steel.
Fifth Ward railroad
From his fifth floor window, Brian Chiu has a front row seat.
"It's so loud," Chiu said.
But it's not the noise that concerns him. The fear is five floors down.
Fracking technology has opened an oil spigot in North Dakota.
"It's increased the amount of traffic on the railroads exponentially," Wisconsin Railroad Commissioner Jeff Plale said.
The railroad traffic has increased by several thousand percent.
Bakken crude oil has a very high vapor pressure, meaning it can easily explode. And the tank cars carrying it?
"(They) were not designed to haul crude. A lot of them were designed to haul corn syrup," Plale said.
When these trains have derailed, the cracker-thin tank cars have ruptured, with disastrous results. By far the worst incident occurred in Lac Megantic, Quebec. Forty-seven people were killed in the fireball.
Quebec train derailment
Three times this year, trains carrying crude have derailed in the United States. Last week in North Dakota, the sky turned gray with smoke.
In March, a train derailed across the border in Galena, Illinois. The wreckage burned for four days.
A week before that, a train derailed in West Virginia. Hundreds had to evacuate.
The train that derailed in North Dakota was headed toward Wisconsin. Two trains before that had just been here.
"We're kind of at the epicenter of where this stuff is coming," Plale said.
That brings us back to Brian Chiu and his Fifth Ward home -- and those oil trains just feet from the Fifth Ward lofts, going over the S. 1st Street bridge.
FOX6 first photographed the concern in February -- but it wasn't until the snow and ice melted that we saw the full extent. "I beams" that support the bridge have rusted away at the base to wafer-thin strips of steel. In some spots, entire sections are just gone.
Chris Raebel, an engineer at Milwaukee School of Engineering (MSOE) agreed to take a look at what the FOX6 Investigators found.
"My focus is on steel design -- just like the bridge," Raebel said.
Unlike most railroad bridges, which have elevated foundations, the piers on this century-old span reach right to the road -- where every winter, salt eats away at the steel.
"That's hit the base of the bridge and that`s corroding the metal," Raebel said.
In the past, some of the rusted piers supporting the bridge have been reinforced, but several columns have been corroded right through.
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FOX6 News received similar comments from other structural and civil engineers who saw the photos, but they didn't want to be identified because they may do business with the railroads. They said things like:
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And: "I would definitely report these conditions to the owner of this bridge without further delay."
Canadian Pacific Railroad should already be aware. Canadian Pacific owns the bridge and is required to inspect it each year. In a written reply to a FOX6 request for those records, the company said it "meets or exceeds all federal requirements," and that the bridge was last inspected in the winter. Canadian Pacific wouldn't tell us exactly when that was -- and whether there was snow on the ground. Canadian Pacific refused to show FOX6 News any of the inspection reports.
FOX6 asked them again earlier this months at a Common Council meeting in Milwaukee.
"We`ve given you a statement on that and we won`t have anything to add," a Canadian Pacific representative said.
Canadian Pacific had been invited to Milwaukee to answer questions about the oil trains. Canadian Pacific's brash brush off didn't sit well with some Common Council members.
"You don`t give that image to the community that your facilities are*safe. You don`t give us that confidence," Milwaukee Alderman Terry Witkowski said.
Ken Wood knows what these inspections entail.
"I'm a structural engineer. My main focus is bridges. I`ve been working with bridges for 20 years -- bridge design, bridge inspection, bridge rehabilitation," Wood said. "You`re going to be looking for fatigue cracks, and the other thing you`d look for is corrosion, certainly, on a bridge -- because corrosion is basically taking away the cross section."
If you look at the base of the "I beams" on the bridge in the Fifth Ward, you'll see layers and layers of flaking -- in some places, more than an inch thick. That doesn't happen quickly.
"It`s been some time, that`s for sure," Wood said. "What happens during corrosion is the steel expands, sometimes seven to eight times what it is, so you can see that actually happening in the base here," Wood said.
FOX6's Brad Hicks: "How do you even inspect this with that much flaking on there without removing the flaking?"
"They would have to remove flaking to see what`s underneath and take some measurements with calipers to find out how much area they perceive is left," Raebel said.
So that's what the FOX6 Investigators did.
The beam is nine-tenths of an inch thick, but at the base, only four-tenths of an inch is left. The column is just over an inch thick. Corrosion has eaten it down to less than half that.
FOX6's Brad Hicks: "The kind of thinning we`re seeing here, does that impact the load capacity of a bridge like this?"
"Yes," Raebel said. "They have a certain amount of steel they need to resist the load from above."
And that load is greater than ever.
Engines alone weigh three times what they did when the bridge was built in 1914. And a one-mile train weighs more than 25 million pounds.
"Now a two-mile long train is relatively common," Plale said.
And with trains like that moving over the bridge daily -- metal fatigue adds up.
"Is the bridge really built, with all that rust and all that corrosion, to support that kind of weight?" Chiu wonders.
Officials in the state of Wisconsin had the same question. In 2006, a study was commissioned on the impact heavier trains have on state-owned railroad bridges. That study concluded "many within the railroad industry are concerned that the aging bridge infrastructure will no longer be able to withstand the increased loadings."
One bridge engineer who examined FOX6's pictures said the problem may not be that bad, because in theory, you could cut a vertical pier in two horizontally, and it would still hold up the bridge. But that's assuming you still have inch-thick "I beams" -- not corroded columns.
The concern here isn't that the bridge will completely collapse -- but that if a column gives way and the load shifts and the train tips -- with the Fifth Ward lofts just feet away, could Milwaukee become another Quebec?
"I would encourage the owner of the bridge to seriously look at this and consider repairs. And it seems like it should be done soon," Raebel said.
To their credit, the railroads, including Canadian Pacific, have been at the forefront -- pushing the federal government for stricter tank car standards. The railroads don't actually own the tank cars -- the oil companies and third-party leasers do.
Eleven days ago, the federal government announced new cars need to be thicker, and the old ones need to be retrofitted within five years.
The federal government is the only entity that can demand the railroad turn over its inspection reports on the bridge. For two months, FOX6 News repeatedly asked the Federal Railroad Administration if it has any of Canadian Pacific's inspection audits for the S. 1st Street bridge. The agency hasn't responded.
Local municipalities like Milwaukee are pretty powerless when it comes to regulating the railroads.
On Tuesday, May 12th, the Milwaukee Common Council approved a resolution urging federal regulators to immediately inspect all tracks, bridges and crossings on which Bakken crude oil is carried -- but at the end of the day, that's simply a request.
Wisconsin rape plot, man arrested in Alabama
Watertown police: Pedestrian hit by car, killed


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by Claire Viellieux
A Volunteer Explores The Data Dashboard
The following piece was written by OAS Communications Coordinator Ryan Bower for the Snapshot Wisconsin newsletter. To subscribe to the newsletter, visit this link.
This edition of the Snapshot newsletter is focused on a new Snapshot Wisconsin product, the Data Dashboard. The Data Dashboard is an interactive tool that let's the public explore the data that Snapshot has collected in a new and exciting way. This dashboard marks an important step towards Snapshot's goal of making its data more accessible to the volunteers who helped collect it, as well as the public more broadly.
You can learn more about the development of the dashboard, as well as what features are currently available on it, in this edition of the Snapshot newsletter, or you can check out the dashboard for yourself at https://widnr-snapshotwisconsin.shinyapps.io/DataDashboard/.
Snapshot Wisconsin released its new data visualization tool to the public today. The tool is called the Data Dashboard and is a major step towards bringing the project full circle. The Data Dashboard offers volunteers and the public a new way to engage directly with this data, letting people choose which data they want to visualize.
At launch, the data for 18 animal species is available to explore, including how active species are during different times of the day and year and how the species are spatially distributed across the state. Data can be viewed for specific counties or statewide, and the data from maps and graphs can even be downloaded to share with others.
Christine Anhalt-Depies, project coordinator for Snapshot Wisconsin, described the intention and purpose of the dashboard. "The purpose of the dashboard is to close the loop – to make sure that people who are involved in collecting data have an opportunity to see the outputs of their work," said Anhalt-Depies. "This first iteration of the dashboard is focused on giving the public a chance to explore data they've helped to generate."
Exploring The Data Dashboard With A Volunteer
One of the best features of the Data Dashboard is that it lets people explore the wildlife in our state at their pace. The dashboard is open ended and can be explored in whatever order you want, but sometimes a guide is nice to follow along with. A member of the Snapshot team virtually sat down with a long-term Snapshot volunteer on Skype and let them explore the Data Dashboard.
Tim Sprain has been a volunteer with Snapshot Wisconsin for nearly 3 years. He hosts three Snapshot cameras in three separate counties. Sprain is a middle school teacher and uses Snapshot in his classroom to describe biology concepts in an interactive way. Sprain shared his screen with Sarah Cameron, a member of the Snapshot team and head of the educator engagement side of Snapshot. Here are some of the highlights from the discussion between Sprain and Cameron.
The Map Of Wisconsin – Counties and Ecological Landscapes
While Sprain was getting set up to share his screen, Cameron said, "One thing I like to remember when talking about the dashboard is the fact that we were literally sitting in a room at the DNR office about two years back, brainstorming how we can share the data that the volunteers were collecting back with them. We had this grand idea of a dashboard but had no idea how we were going to accomplish it. It's been really exciting to see the progress over the years and have a product now that we can give back to our volunteers – something they can connect to."
Sprain agreed and added, "A lot of times, people don't have the knowledge or resources to understand what animals are in their backyard. I see this dashboard as a huge resource to educate them. My goal [as an educator] is to be a catalyst for experiential learning, and Snapshot Wisconsin is all about making sure people get a chance to be involved."
"Years ago," continued Sprain, "[DNR staff] pointed me towards what the DNR offers to families and students other than hunting, fishing and trapping, and I've kept up on the email updates that the DNR sends out. When Snapshot Wisconsin came onboard, it was a natural fit." Sprain finished loading the Data Dashboard on his computer and was ready to explore the dashboard. Sprain started with the map of Wisconsin (on the left side of the dashboard). The map shows how many of the trail cameras in a given area have captured the selected species. The map can be broken into counties or ecological landscapes, such as the Northern highlands or Southwest savanna.
Sprain said, "I'm excited to share the map of Wisconsin with my students, because it has ecological landscapes. It will help them understand how humans and animals use different landscapes and that animals have adaptations to suit the habitat in their landscape. Habitats are the most important piece towards maintaining our rich diversity of animals and flora in the state, which is an indication of our health as humans. This awareness can help build the understanding that we are all in this together. It's also cool that you have the ecological zones because ecological borders are more real [to wildlife] than county lines."
Sprain hovered over the three counties he has cameras in and started clicking on different species. Sprain mentioned how he has had to use a different approach this year to incorporating Snapshot into his teaching, since Sprain's school district is starting virtually. Sprain explained, "In a normal year, [my class and I] would take a trip in the first week of school. I take them to a forested park and ask them what they notice. The kids ask me questions [about the plant and animals they see], and I introduce Snapshot Wisconsin by showing the trail camera pictures. I ask again, 'What is this,' or 'What do you see now?'"
"Often times students will recognize a deer, but they get confused when they see a coyote, fox or wolf," said Sprain. Sprain recalled a special moment of discovery he had with his students. "When they saw a fisher, it was just the most wonderful moment of learning. They didn't know what a fisher is, but [to them] it looked sort of like a wolverine. They couldn't quite figure it out." Sprain used this moment to teach his students something new about fishers. "Now, they feel this connection to knowing something that they didn't know before about this mysterious creature living in their bluffs."
A fisher captured on a Snapshot Wisconsin trail camera.
Sprain hovered over La Crosse county specifically, since that is where the camera that saw the fisher was located. Sprain was excited to discover that there was only one camera out of 13 in the county that had seen a fisher. It was his camera.
Cameron added, "It will be really cool to share with your students that the data they helped collect got that one point out of 13 in La Crosse county. I'm also glad you mentioned fisher, because that wasn't there [until recently.]" Fisher, as well as four other species, was added to the dashboard in response to the feedback volunteers gave during a soft launch to trail camera hosts.
Sprain joked, "You should have seen the look on the landowner when I shared the [fisher discovery] with the family. They were dumbfounded. They didn't even know what a fisher was!"
"Oh, that's so funny," said Cameron. "That was one of our early and exciting findings of the project, realizing that fishers were a lot farther south in the state than we expected. We've had a lot of people come to us and say, 'Wow I caught a fisher on my property! I didn't know they were down here.'"
Why Species Were Included Or Excluded
Sprain moved on from the map and started looking more closely at the Species list next to the map. There are 18 species currently available on the dashboard – some common species and some rare like wolf and elk.
"Students always ask after all the carnivores, as well as ones that are unique like porcupine," said Sprain. "I love that porcupine is on the list! The kids love porcupine." Between his three cameras and a few annual field trips, Sprain's class gets to see quite a few of our Wisconsin species. "We don't have any wolves on our cameras yet, but I see [on the dashboard] that other cameras have picked them up in my counties… so they are there."
Sprain and Cameron continued to discuss different animals that his students like seeing, and the conversation turned towards species that weren't included on the species list, namely beaver.
Cameron directed Sprain to look under the species list at the blue text about missing species. Cameron said, "We thought that would be a question that would come up quite a bit, so this popup provides some extra information [on why some species were included and other excluded.]"
This popup explains the main criteria to be included on the dashboard, having a 95% accuracy rate. An accuracy rate is a measure of how often volunteers correctly classify a given species. If photos of a species are classified correctly at least 95% of the time, then that species was added to the dashboard. There is also a table with every species that Snapshot volunteers can classify, such as the partial table below. Each species on the list has an accuracy rate and explanation if it wasn't included.
Anhalt-Depies jumped back into the conversation and said, "For species with low accuracy, we will eventually be able to provide an expert review of that species' photos, such as badger with an accuracy of .624 [which is below the .95 or 95% cutoff]. We will be reviewing those photos and including them in a later version of the dashboard."
The Snapshot team knows there are some classification errors in the data, but anything with an accuracy above .95 is acceptable for this purpose. Sprain ask Anhalt-Depies about the two cameras in Milwaukee county with snowshoe hare detections, and Anhalt-Depies explained that those are likely misclassified and within the 5% of the time the data was incorrect.
Sprain replied, "I like that there is a discussion of accuracy and human error. Science is not perfection, but to see these [accuracy rate] numbers… now I understand why a species wasn't included."
Sprain and Cameron each commented on how impressed they were overall with the accuracy of Snapshot volunteers. Cameron said, "One thing I encourage for volunteers who want to help increase their accuracy is to involve others. You [Sprain] mentioned that your students work together to classify photos. My dad calls me basically every time he checks his camera to verify that a photo is an opossum… or a racoon. Just having a couple eyes on the photos increases accuracy but also makes it a more enjoyable experience working with others."
Sprain jumped in and added, "That is what it is all about, a social network of citizens contributing to science! Working together for a much bigger purpose but also feeling great about seeing the wilderness and ecology in the state of Wisconsin – that is the goal."
Visualizing Animal Activity and Detection Rates
Soon, Sprain had moved on to the next section of the dashboard, the graph of animal activity. The graph, located on the right side of the dashboard, plots how many times a species has been seen at each hour of the day or each month of the year, depending on which option you choose.
"I like that you can see activity by hour or month," said Sprain. Sprain started making connections between activity patterns he observed and known behavior of those species. Bears, for example, become more active during the spring and summer months yet are rarely seen during the winter. "I'll ask my students the question, why are they so active during certain months? Some students think it is a difference in population size, instead of a behavioral difference like preparing for hibernation or reproduction. I get a lot of giggles in the seventh-grade classroom, but the question is very relevant to teaching animal ecology and biology."
The graph can also be adjusted to show which hours of the day a species is more active. The dashboard shows that sandhill crane, for example, are mainly active during the day but are most active between 10:00am-1:00pm. Bird watchers could increase their chance of spotting one by using the dashboard to find its peak activity times. The same could apply to any of the 18 species shown on the dashboard.
The second tab of the activity chart shows detection rates. Initially set to show the top five species detected state-wide, the detection rate chart can show much more than you may expect. If the selected species isn't among the top five species, it will appear as a sixth bar on the chart. Additionally, individual counties can be selected to show the data for both the state and selected county. Using these features together, Sprain was able to discover that bobcat, elk, porcupine, and wolf were detected slightly more often in Jackson county, where one of his cameras is located.
Overcoming Insufficient Data
Three counties don't show species data and are labelled as insufficient data. These counties have less than five cameras. If enough data isn't being pulled from cameras in a county, then the dashboard doesn't make a lot of sense. For example, let's say you live in a county with only one camera. If a rare species like the whooping crane walks across that camera, then that species now has a 100% detection rate (one camera out of one camera) in that county. The dashboard would give off the impression that whooping cranes were extremely common in that county, when the species is actually quite rare. This is why Snapshot requires a minimum of five cameras per county before the dashboard will show data for it. If you live in a county that is labeled as Insufficient Data, then consider hosting a camera or encouraging others to host one. There are a lot of opportunities, not only on private lands, but also on public lands to host Snapshot cameras.
Cameron said, "There are a lot of ways to get involved, no matter where you are or whether you have access to land. Snapshot is a really unique project in that there are so many ways to get involved. You could host a trail camera, classify photos online or just participate in some of the educator resources. There is a lot to explore and to offer!"
Even if you are just interested in learning about wildlife in the state, you can help out Snapshot by giving them feedback on the dashboard. There is a survey at the bottom of the dashboard for anyone, not just volunteers, to offer ideas for future versions of the dashboard. After all, the Data Dashboard is designed to help Snapshot come full circle, so they want to hear from you. While not every idea can be implemented, the major themes in the survey responses impact what the team focuses on moving forward.
Cameron reiterated, "The data dashboard isn't just for Snapshot volunteers. It is for anyone interested in Wisconsin wildlife. Snapshot and the Data Dashboard are special because they wouldn't be possible without our trail camera hosts and folks classifying online."
Let's Discover Our Wildlife Together
Snapshot's slogan, "Let's Discover Our Wildlife Together," isn't a mistake. Snapshot is a project about people (from Wisconsin and across the globe) working together to monitor our wildlife. "While we know so much about Wisconsin wildlife, there is still so much we don't know," said Cameron. "Having tools like this dashboard help us fill in those gaps and get a more complete picture of what is really happening in Wisconsin."
Sprain added, "[The dashboard lets] each citizen take away a piece of Wisconsin culture and have it become a part of their life. There is no denying the presence of these animals on the cameras. That is the power of this data."
Whatever your motivation for wanting to see Wisconsin wildlife, check out Snapshot Wisconsin's Data Dashboard. Version 1.0 is now available to the public.
Lastly, Anhalt-Depies offered Snapshot's hope for the future of the dashboard. "We are now at a point where the project has really good coverage across the state. Our hope is that, as the dashboard evolves, it becomes a powerful tool for decision makers," so keep an eye out for what is available in future versions of the dashboard!
You can visit the Data Dashboard at https://widnr-snapshotwisconsin.shinyapps.io/DataDashboard/.
Tags: Activity Patterns, Animal Activity, Data Dashboard, Map of Wisconsin, Snapshot Wisconsin, Volunteer


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package io.gitcafe.zhanjiashu.newzhihudialy.activity;

import android.content.Context;
import android.content.Intent;
import android.support.v7.app.ActionBar;
import android.support.v7.app.AppCompatActivity;
import android.os.Bundle;
import android.support.v7.widget.Toolbar;
import android.view.MenuItem;

import io.gitcafe.zhanjiashu.newzhihudialy.R;
import io.gitcafe.zhanjiashu.newzhihudialy.fragment.DialyFragment;
import io.gitcafe.zhanjiashu.newzhihudialy.model.DialyType;
import io.gitcafe.zhanjiashu.newzhihudialy.util.DateUtil;

public class PickerActivity extends BaseActivity {

    private static final String EXTRA_DATE = "extra_date";
    private static final String EXTRA_MILLISECONDS = "extra_milliseconds";
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        if (savedInstanceState == null) {
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                DialyFragment fragment = DialyFragment.newInstance(beforeDate, DialyType.PICK_DIALY);
                getSupportFragmentManager().beginTransaction()
                        .replace(R.id.fl_container, fragment)
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    public static void startBy(Context context, String date,  long millisenconds) {
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        intent.putExtra(EXTRA_DATE, date);
        intent.putExtra(EXTRA_MILLISECONDS, millisenconds);
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    @Override
    public boolean onOptionsItemSelected(MenuItem item) {
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        if (id == android.R.id.home) {
            finish();
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        return super.onOptionsItemSelected(item);
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}

redpajama_set_name: RedPajamaGithub


I haven't seen this film yet.
Who in here liked "The Truth about Charlie"?
Did you buy the DVD?
Charlie is out on video?
The Truth About Charlie Premiere Pics!!!!!!
The Truth About Charlie website is LIVE!! New pics!
heyy! whats this movie about?!


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Maldonado (2009) IQ MR Atkins death penalty decision: A psychometric miscarriage of justice?
I'm no longer shocked by what passes as credible psychological/psychometric evidence or testimony in some Atkins IQ MR death penalty court decisions. Another such decision (Maldonado, 2009) has come to my attention. A link to a PDF copy of the decision is now available in the Court Decisions section of this blog.
Readers can gather all the relevant background information re: the case by reading the entire decision. I intend to focus primarily on the psychological interpretation and psychometric issues in the decision that are troubling.
But first, before delving into these issues, I present a few quotes (from the Maldonado record) that capture the confusion and uncertainty surrounding many Atkins decisions, a situation that is resulting in considerable variability in the quality of psychological assessment data reported/interpreted and the common occurrence of "dueling expert witnesses".
Quotes from the record:
Because the Supreme Court did not establish a bright-line test to identify mental retardation, the Atkins inquiry has become a fact-intensive question that heavily relies on the opinions provided by mental-health experts. This case, like most involving Atkins claims, requires consideration of testimony from competing experts who disagree about the nature of mental retardation, the means by which it may be identified, the manner in which it manifests in a criminal defendant's life, and the psychological profession's role in making the legal decision of whether mental capacity precludes execution. (p.32)
A "welter of uncertainty" followed the Atkins decision because "[t]he Supreme Court neither conclusively defined mental retardation nor provided guidance on how its ruling should be applied to prisoners already convicted of capital murder." Bell v. Cockrell, 310 F.3d 330, 332 (5th Cir. 2002). Accordingly, federal courts have approached the implementation of Atkins with some trepidation. (p.36)
After reading a number of of Atkins rulings (see Court Decisions section of this blog), I could not agree more with these statements. The courts appear ill-equipped to handle the complex psychological measurement issues presented, issues that are, at times, confounded by the inclusion of data from dubious procedures, interpretations of test scores that are not grounded in any solid empirical research, and the deference to a single intelligence battery (the WAIS series) as the "gold standard" when a more appropriate instrument (or combination of WAIS-III/IV and other measures) might have been administered, but the results of the more appropriate measure are summarily dismissed based on personal opinion (and not sound theory or empirical research).
Below are some of the troubling psychological/psychometric issues I see in the Maldonado (2009) decision
Lack of English language proficiency: "A theme developed by both parties is that Maldonado's lack of English proficiency has madehis exact intellectual capacity difficult to gauge." (p.38).
As a result, the prosecution psychologist administered the WAIS-III "through a translator" (p.42). The translator administered WAIS-III resulted in Verbal, Performance, and Full Scale scores of 74, 74, and 72 respectively. A defense psychologist captures the essence of my reaction to the use of a translator-administered WAIS-III. "The accepted practice in the evaluation on Spanish-speakers is to communicate with the client in Spanish without the use of translators. In addition, tests should be scientifically translated and validated and the most appropriate norms available should be applied" (p.49). I agree. I am unaware of any professionally established and endorsed procedure for the translated administration of the English-normed WAIS-III. Such a procedure violates a fundamental backbone of the science of individual intelligence testing--standardized test administration.
Use of non-empirical clinical judgement procedures to upwardly adjust IQ scores. On page 53 of the record, it is indicated that the prosecutions psychological expert believed that the translated English-normed WAIS-III scores needed to be upwardly adjusted due to Maldonado's educational and cultural background. Additional support came from the finding of a Verbal IQ score of 83 on the WAIS Espanol (p.56). Although psychologists are appropriately trained to recognize the potential impact of such environmental variables when interpreting scores, the psychologist upwardly adjusted the scores to a specific IQ score estimate ("It's around the 80s, I guess, if you had to pin me down. Around the 80s; somewhere in there"- p.48) and this expert "conceded that only 'clinical judgment,' not any statistical formula or established methodology, informed how much to alter an IQ score because of cultural and educational factors" (p. 53).
My concern with this procedure mirrors the testimony of the defense experts in the case. Adjusting obtained IQ scores, either up or down, based on an n=1 professionals clinical judgement, in the absence of any scientifically established procedure for adjusting IQ scores, is troubling and is not consistent with accepted psychological assessment practices or standards. In fact, this IQ adjustmend procedure sounds similar to a notable empirical effort (in the late 1970s and early 1980s) to produce IQ scores that better reflected a persons social-cultural background. Jane Mercer's SOMPA (System of Multicultural Pluralistic Assessment) was a valiant effort to adjust Wechsler IQ scores for African-American individuals based on their social-cultural knowledge and history. The result was a new score called Estimated Learning Potential (ELP). From the start, SOMPA was controversial and eventually was found to be flawed for many reasons (see Hellfinger, 1987; also Jirsa, 1983). If a reasonably conceived theoretical and empirical IQ adjustment procedure (i.e, SOMPA), which was intended to account for a person's social-cultural background, was found to be flawed, how can a specific n=1 psychologist be endowed with unique insights that allow for the invoking of an unspecified personal algrorithm to make IQ score adjustments? This is indeed troubling. Also, to the best of my knowledge, SOMPA is no longer around and is not, or is seldom, used. Race-based or adjusted norms have not been recognized as an acceptable professional psychological assessment practice for over twenty years!
Dismissal of the BAT-R intelligence results. Maldonado had also been administered the Woodcock-Muñoz Bateria-R ("Bateria-R"), the Spanish-language counerpart of the Woodcock-Johnson Test of Cognitive Abilities--Revised [conflict of interest notice: I am not a co-author of the BAT-R or WJ-R, but was a paid measurement consultant on the WJ-R project and have since become a coauthor of the subsequent edition, the WJ III]. Maldonado obtained a Broad Cognitive Ability (BCA) score, which is analagous to the full-scale score from other intelligence batteries, of 61. Of all the cognitive tests administered, this is the only comprehensive intelligence battery that was administered in Maldonado's natural language and where his performance is compared against appropriate US-equated Spanish norms [note-- the WAIS Español administered was also administered in his natural language and makes comparisons against Spanish norms, but only a portion, the Verbal section, was administered]. Also, as previously noted at this blog, the WJ-R/BAT-R and WJ III/BAT III provide for the most comprehensive assessment of intellectual functioning as per the consensus model of human intelligence (CHC theory) among serious intelligence scholars. This is what I've termed the "Atkins MR death penalty IQ test-theory gap."
Why were the BAT-R scores dismissed?
The prosecution expert "testified that the AAMR has not cited the Bateria-R as a predicate test to the evaluation of mental retardation and "[i]t's not well suited for that purpose, although you can use it for that" (p. 70). He also stated that the "Bateria-R test score was especially suspect because it was inconsistent with Dr. _____'s administration of the WAIS Español in which Maldonado scored well above the range for mental retardation." (p.70). Furthermore, this expert "opined that the usefulness of the test was impaired because it 'is used by school psychologists to diagnose learning disabilities and it measures a lot of things like visual and auditory processing. It really measures very little in terms of general intelligence.' " (p.70). As a result, "the state habeas court dismissed the Bateria-R score because it "is not one of the tests the AAMR cites for mental retardation evaluation," but instead "is generally used by school psychologists to diagnose learning disabilities" and, in fact, "is not very relevant for establishing general intellectual functioning, so it is not well-suited for determination of the first prong . . . to determine mental retardation" (p.70-71).
There are many problems with the reasons given for dismissing the most culturally appropriate (for Maldonado) and comprehensive measure of intelligence (BAT-R).
First, dismissing an instrument because it is used primarily by a particular set of psychologists (school psychologists) is non-sensical, and frankly, condescending. School psychologists typically give many more intelligence tests than psychologists working in adults settings and use these instruments to diagnose mental retardation. School psychologists, in many respects, have more intimate familiarity with intelligence testing than most other professional psychologists.
Second, the previously mentioned national expert panel that examined the Dx of MR at the same cut-point as Atkin's cases (for SSA benefits) indicated that most intelligence tests would be moving towards measuring the Cattell-Horn and Carroll Gf-Gc models of intelligence (now collectively referred to as CHC theory; also see McGrew, 2009), and instruments based on this model are very relevant to the Dx of MR. The English version of the WJ-R WJ III (a CHC-based revision of the WJ-R) was listed as one of the approved instruments by the expert panel...which should also implicitly argue for use of the Spanish-language versions. The prosecution witness, who appears stuck in the land where the WAIS is the "gold standard," appeared unaware of the recent advancements in understanding the psychometric nature of human intelligence, which has converged on the CHC model of intelligence. This is particularly ironic given that both John Horn (of Cattell-Horn) and Jack Carroll served as theoretical consultants on the WJ-R...which was the foundation of the BAT-R.
Third, stating that the BAT-R (and WJ-R by implication) is not a respected measure of general intellectual functioning reflects a complete lack of awareness of the CHC-foundation of the instruments, as well as published research (including the WJ-R and BAT-R technical manuals and bulletins). If CHC theory is the consensus model of psychometric intelligence, then the only battery administered to Maldonado that measured most of the model, which in most conceptualizations has general intelligence (g) at the apex, should have been given serious weight. I, and others, heard Dr. Arthur Jensen, the most prominent pscyhometric expert on g, at an ISIR conference in Nashville, TN, state, during a discussion of a presentation in front of the entire audience, that (at the time) he considerd the WJ-R (and, thus, the BAT-R by implicit endorsement) the best available intelligence battery for measuring g. I will return to this point in future posts as I've been analyzing WAIS-III data together with the WJ batteries (as well as other accepted intelligence batteries, e.g., KAIT, K-ABC; SB-IV; SB-IV) to evalute the g-ness of each battery when jointly analyzed.
Fourth, prosecution psychologist used the WAIS Español Verbal score as evidence that the BAT-R was not accurate. The problem with this logic is that the WAIS Verbal scale is known to be an excellent measure of crystallized intelligence/comprhension-knowledge (Gc), only one of the major 7-8 domains in the CHC model of intelligence. Conversely, the BAT-R includes indicators from seven of the major CHC broad ability domains, only one of which is Gc. No attempt was made to compare the WAIS Gc score with the corresponding Gc measure(s) on the BAT-R. They might have been similar...or not. More importantly, you cannot compare a measure of Gc to one that includes Gc and fluid reasoning (Gf), visual-spatial processing (Gv), short-term memory (Gsm), long-term retrieval (Glr), auditory processing (Ga), and processing speed (Gs).
I could go on and comment on other issues, such as the use of only the Verbal scale of the Spanish WAIS-III, as well as the whole area of adaptive behavior, but these are issues for another post, possibly by a guest blogger.
Technorati Tags: psychology, forensice psychology, criminal psychology, criminal justice, IQ, IQ tests, IQ scores, Verbal IQ, WAIS-III, Wechserl tests, WJ-R, WJ III, BAT-R, BAT-3, MR, mental retardation, developmental disabilities, death penalty, capital punishment, Atkins cases
Posted by Kevin McGrew at 12:45 PM
Labels: adaptive behavior, bilingual, CHC theory, court decisions, cultural issues, general intelligence, IQ scores, IQ tests, part scores, Wechsler batteries, WJ III
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text:
Will retailers go on the road with self-driving mobile stores?
Source: Robomart
Artificial Intelligence Retail Concepts Retail Tech Robomart Robotics
by Matthew Stern
While legislators in some cities have been trying to keep autonomous robots off sidewalks, a new startup wants to enable any big box retailer to get a full wing of its operation rolling around on city streets.
Santa Clara, CA-based startup Robomart purports to be in talks with big box retailers and wholesalers to implement its mobile autonomous grocer platform, according to TechCrunch. The company plans to allow partner retailers to use its fully-autonomous, full-sized delivery vehicles to serve as auxiliary, mobile storefronts.
If the vehicles work as intended, they could allow big retailers to bring a portion of their operations into areas where they have no store presence, perhaps along a schedule with multiple stops per-day. In this fashion, the technology seems like a combination of a food truck (albeit possibly carrying non-food products) and a wheeled delivery drone. Retailer could bring along product selection from different departments on the road on different days. The service could also facilitate deliveries within a particular area.
Robomart plans to offer the platform as a white-label service, according to TechCrunch. Retailers would license the entire service, including the vehicle, as part of a two-year lease.
But there are potential hurdles that could stand in the way of the concept's success. The framework for the legality of autonomous vehicles is still being developed on the federal level, so it is not yet clear how such vehicles would be classified. Finding legal places to park, as has sometimes been a point of concern for food trucks in big cities, could be another problem. And the service would no doubt face the same concerns about theft as with any store concept operating without human oversight.
Robomart is not the only tech company working on a fully mobile mini-mart. Last year, a Swedish tech startup and a Chinese university partnered to create and beta test Moby Mart, a convenience store on wheels that customers can direct to their location via app. At the time its testing in Shanghai began, however, Moby Mart was still controlled by humans rather than being fully autonomous.
Robomart is the latest startup to try and unseat the local convenience store – TechCrunch
Congress in the driver's seat on self-driving cars – Washington Examiner
What happens when you put a c-store on wheels? – RetailWire
DISCUSSION QUESTIONS: Do you see self-driving mobile stores catching on in the U.S.? In what situations might these vehicles be most effectively deployed by retailers and brands?
"The bagel cart has officially been replaced! In areas that have ample foot traffic, these appear to be a perfect fit. "
Seth NagleSenior Marketing Manager, RW3
"This will enable the ultimate in localization of goods, allowing for supply to literally follow demand."
Shawn HarrisCustomer Partnerships & Strategy, SmartLens, NA
"The biggest hurdle will be for cities and towns to monetize parking and update zoning requirements, similar to the food truck movement..."
Kiri MastersFounder and CEO, Bobsled Marketing
15 Comments on "Will retailers go on the road with self-driving mobile stores?"
Charles Dimov
Vice President of Marketing, OrderDynamics
Absolutely brilliant! Yes — this concept of the mobile store will definitely catch on. It is a true extension of the omnichannel retail strategy. Will all shoppers be into this? No. But there will be enough that will be interested in the novelty factor, will have a passing interest (but perhaps not enough to motivate a trip to the mall) and there will be the busy rushed folk (like me) who could use the store that comes to me.
This will be an interesting new connection into order management systems, where the stores are moving around and inventory really needs real-time visibility for customers.
Another step will be an app that lets a shopper choose the brand that they would like to drop by their place of work/house/recreation at a certain time … kind of like Uber or Lyft for travel services.
It's a brave new world … isn't the future of retail exciting?
Kiri Masters
Founder and CEO, Bobsled Marketing
I can think of a few scenarios where this retail delivery system makes sense.
Vending essential items at special events. Items that people either forget or plan not to bring to an event like a marathon, festival or other event. Towels, water, a paid phone charging station, etc.
Merch. Similarly, branded merchandise for bands and artists could be made available in a just-in-time fashion, before and after these events.
Convenience items. As the article states, a mobile convenience store could roam to areas of cities or towns where there are no such stores.
Food items. Rapid service restaurants and convenience stores do most of their business in a few hours of each day. What if the store could only be open and pay "rent" during those money-making hours?
The biggest advantage is that the "store" doesn't have to pay rent during idle hours. The biggest hurdle will be for cities and towns to monetize parking and update zoning requirements, similar to the food truck movement of a few years ago.
Art Suriano
Chief Executive Officer, The TSi Company
I love new technology, but the technology that catches on usually does so because of purpose and convenience. If we already had a thriving industry of retail vehicles bringing merchandise and food items into neighborhoods the next step might be thinking about self-driving vehicles. But we don't, at least not in a big way. The reasons are because of the cost and lack of support from the communities. When was the last time the Good Humor man had a long line of customers waiting for ice cream? We don't think that way anymore because we want everything immediately. Delivery as fast as possible is the desire, and that is where technology continues to make it happen quicker and quicker. So I don't see this going forward as the next big thing. Also, before any of these self-driving vehicles catch on, the public has to become comfortable with them and that most likely won't happen until we have self-driving cars. That's not too far in the future, but until that happens, I don't see self-driving vehicles… Read more »
Yes, I think this has potential – indeed, it is possible we'll see combined delivery/pick-up here/buy-now vehicles to service the needs of urban populations. This type of model would reduce servicing and fulfilment costs for retailers and so is attractive, especially in low-margin categories like food.
Naturally, cities will need to adapt to accommodate this and other self-driving vehicles. Some will do this faster, some slower — but gradually all will move in the right direction.
Steve Montgomery
President, b2b Solutions, LLC
An idea ahead of its time. For mobile stores to work the entire autonomous vehicle issue has to be resolved. Issues will be resolved but in my opinion, not in the next five years. For a retailer to move forward before technological, legal, insurance, etc. issues are resolved would be risky.
Senior Marketing Manager, RW3
The bagel cart has officially been replaced! In areas that have ample foot traffic, these appear to be a perfect fit.
As grocers experiment with this new technology I expect a few more ideas to be spun off and wouldn't be surprised if we start seeing a deli, fresh produce, snacks and who knows what else carts popping up outside office complexes and busy street corners.
Lee Kent
Principal, Your Retail Authority, LLC
This is indeed a great concept however, why does it need to be driverless? This is an opportunity to extend the brand through its human interaction as well. To surprise and delight the customer. Imagine the relationships an individual could make if they have a steady route. Also, with no human oversight, the obstacles just increase. And that's my 2 cents.
Bob Amster
Principal, Retail Technology Group
It is coming in the next three to five years. Then we will have to legislate against overcrowding downtown streets and other venues and roads with these ambulant stores that pay no rent. While many in the country complain about government overreach and legislation, we bring it on ourselves with our own innovation.
Shawn Harris
Customer Partnerships & Strategy, SmartLens, NA
This will enable the ultimate in localization of goods, allowing for supply to literally follow demand. These bots will revolutionize supply chain design, taking on more of an Uber-esque ideal. Level 4 autonomy (autonomous in specific environments) will be adequate to get started, and would allow for material reach.
Global Retail & CPG Sales Strategist, IBM
It's inevitable and only a matter of time. Tons of regulations to write and then follow, but the opportunity is there in urban areas where parking is limited. Eventually, this will work and be cost-effective in suburban areas. The branding upside opportunity is limitless.
This is like the vending machine of the future. Why have the machines set in place when they can move to where to customers are to maximize use? This concept is definitely part of the future of commerce in some shape or form; the only question is … when?
Ricardo Belmar
Sr Director, Retail Transformation Specialist, Infovista
Yes, there is great potential in this concept of mobile stores/showroom/delivery vehicles for retailers to experiment with. The one obvious question that comes to mind, however, is why aren't we seeing this with human drivers today? Why wait for driverless vehicles? For me, the best result from a driverless vehicle solves one of two problems — provides needed driving skill for difficult scenarios where human beings may have difficulties, and allows for larger numbers of vehicles up and running quickly where otherwise you would need too many people to train successfully.
It will be interesting to see where local governments get involved and how quickly they allow these vehicles to multiply in number!
Andrea Ramirez
Manager, Consumer & Customer Market Insight
I LOVE this concept and can see it working as an additional convenience option when placed within the existing parking lots of the stores they represent as a secondary pick up/shopping point for last minute staples/meal kits/grab & go lunches, and whatnot.
Yoav Vilner
CEO, Ranky
Really brilliant idea. The most interesting aspect of these vehicles is the potential they can bring to areas with a limited number of retailers. This would be a huge advantage to customers that live in these areas as well as very good business for big-box retailers.
This is only one of the many potentials of these innovative vehicles, I have many others in mind but I'm waiting to see how this is going to develop. Certainly, this is one of the most innovative aspects of retail I've heard of lately.
Camille P. Schuster, PhD.
President, Global Collaborations, Inc.
Could be great — depends upon how they are used. Pop-up stores have been successful. Food trucks have been successful. Here is an opportunity to customize product assortment to a specific customer group and take that merchandise to them. Taking food to events, taking sports clothes to events, taking pet products to dog walking parks or related activities could be quite successful.
How likely is it that self-driving mobile stores will be in widespread usage within the next five years?
Somewhat likely
Somewhat unlikely
Very unlikely