text:
\section{Introduction}
The concept of ``filter bubble'' has crossed
the boundaries of the academic world reaching public discourse
and mainstream media. This reflects the realization that online social
media (OSM) have a tremendous impact on how people share information and
form their opinions. For this reason they may constitute not only a great
opportunity for the diffusion of knowledge, but also a great threat for
the stability of social fabric and the functioning of democracy.
A filter bubble occurs when a user is selectively exposed
predominantly to content that tends to reinforce his/her
current opinion/belief/state, while suppressing other alternatives \cite{pariser2011filter, dillahunt2015detecting, nagulendra2014understanding}.
In OSM this typically happens because of personalized
recommender systems, which leverage information on past user activity
to provide suggestions which, aiming at maximizing user satisfaction,
tend to be very similar to what the user has already shown to prefer \cite{nguyen2014exploring, bryant2020youtube, o2013extreme}.
Together with the ``echo chamber'' effect~\cite{Cinelli2021, Cota2019, barbera2015tweeting},
filter bubbles are thought to be at the heart of the overall increase of polarization
and radicalization that is observed in many social contexts~\cite{Pew2017, chitra2020analyzing, maes2015will}.

A great deal of activity has been devoted in the last years to
the goal of understanding what are the basic microscopic mechanisms
underlying the rise of polarization and how phenomena
observed at population scale are linked to
them~\cite{galam1997rational, crokidakis2013role, Ciampaglia2018,Perra2019,Sirbu2019, freitas2020imperfect, Baumann2020,Peralta2021,Baron2021}
These efforts follow the line of research aimed
at understanding how basic mechanisms underlying
the interaction of individuals give rise to collective consensus
phenomena~\cite{Castellano2009,Sen2014}.
The voter model~\cite{Clifford1973,Frachebourg1996,Rednerbook} played
an important role in this activity, because of its extremely simple
nature amenable to exact analytical treatment. Its dynamics describes
the evolution of a population of agents which have to choose between
two perfectly equivalent alternatives and do it by selecting at
random a peer and copying their selection.
Starting from a disordered state, clusters of individuals sharing the
same opinion form and grow over time. For any structure of the interaction
pattern among individuals, consensus (i.e. all agents having the same
opinion) is invariably reached.
A very natural generalization is the Multistate
Voter Model (MVM), where the number of available equivalent
options is a fixed value $M>2$~\cite{Starnini2012,Pickering2016,Peralta2020}.
In mean-field, MVM reaches consensus
for any value of $M$, and the average time required for it depends on $M$ only weakly.
In this paper we investigate the behavior of MVM in the presence
of an additional interaction mechanism which biases the opinion of an
agent toward the state the agent has chosen most frequently in the past.
This interaction mimics the filter bubble
effect of personalized recommenders in OSM, which present to users
suggestions based on their previous behavior.

The effect of personalized information on binary voter model dynamics
has been investigated in a previous publication~\cite{DeMarzo2020}.
In a homogeneous mean-field framework it was shown that for
sufficiently strong coupling with the agents' past, consensus is no
longer reached and the system remains stuck in a polarized state with
coexistence of both opinions.  Here we generalize the work in
Ref.~\cite{DeMarzo2020} by studying both analytically and numerically
the effect of personalized information in the context of the MVM.  The
goal is to understand whether, depending on the number of agents $N$,
the number of possible opinions $M$ and the strength of the
personalized information ($\lambda$, to be defined below) consensus is
reached or not.  Depending on the scaling of $M$ with respect to $N$
we identify three different regimes and in each of them we compute the
threshold $\lambda_c$ separating consensus from polarized
states. Numerical simulations are in good agreement with theory in all
cases expect for $M=N$ and they also show that the transition from
consensus to polarization is continuous and characterized by a power
law distribution of opinions numerosity at the threshold. Remarkably,
for $N\to\infty$ the threshold goes to zero in all three regimes and
this implies that in large systems even a very small form of
personalized information always breaks consensus, leading to opinion
polarization.

\section{Multistate voter model with personalized information}\label{secII}
\begin{figure*}
	\centering
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{{fig_1a_N=10000_M=10000_L=0.001_C=nan_Mst}.pdf}
		\caption{}\label{sfig:large_c_1}
	\end{subfigure}\hfill%
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{{fig_1b_N=10000_M=10000_L=0.04_C=nan_Mst}.pdf}
		\caption{}\label{sfig:large_c_2}
	\end{subfigure}
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{{fig_1c_N=10000_M=10000_L=0.05_C=nan_Mst}.pdf}
		\caption{}\label{sfig:large_c_3}
	\end{subfigure}\hfill%
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{{fig_1d_N=10000_M=10000_L=0.08_C=nan_Mst}.pdf}
		\caption{}\label{sfig:large_c_4}
	\end{subfigure}
	\caption{Evolution of the number of surviving opinions for 100
          realizations of the dynamical process with various values of
          $\lambda$. (a) Very small $\lambda=0.001$; (b) Slightly
          subcritical $\lambda=0.04$; (c) Critical $\lambda=0.05$; (d)
          Supercritical $\lambda=0.08$. In all panels $N=M=10000$,
          $c=\infty$.  For convenience we plot $t+1$ along the
          horizontal axis.}
 \label{fig:large_c}
\end{figure*}
\subsection{Definition of the model}
Many real life situations are characterized by the
presence of more than two possible opinions or
factions, as in the case of political elections or
football clubs, and so the voter model with binary
opinions is not suitable for schematizing the dynamics
of such systems. However this difficulty can be easily
bypassed by considering the usual voter dynamics and
allowing the states of the $N$ agents to vary among
$M$ distinct opinions instead of only two, so to
obtain the so called multistate voter model.
Denoting by $\sigma_i(t)$ the opinion of agent $i$ at time $t$ we have
$\sigma_i(t)=1\ldots M$ and the update rule reads
\begin{equation}
  \sigma_i(t+\delta t) = \sigma_j(t) \ \text{with prob.} \ \frac{A_{ij}}{\sum_j A_{ij}}.
  \label{eq:update_MVM}
\end{equation}
Here $A_{ij}$ is the binary adjacency matrix of the undirected network
over which the dynamics take place, while $\delta t=1/N$. In the
following we focus on the mean field case, meaning that the underlying
network is a complete graph and it holds $A_{ij}=1$ for $i\neq j$; in
this case the update rule Eq.~\eqref{eq:update_MVM} can be written as
\begin{equation}
  \sigma_i(t+\delta t) = k \ \text{with prob.} \ \frac{N_k(t)}{N},
\end{equation}
where $N_k(t)$ is the number of agents with opinion $k$ at time $t$.
In order to endow this system with personalized information (PI) we consider
also $N$ PI fields $e_i$, each coupled with
the corresponding standard voter agent $\sigma_i$.
The state of PI fields is a random variable ranging from $1$ to $M$
and which assumes the value $k$ with
probability $P\qua*{e_i(t)=k}$. This probability varies from agent to
agent and over time, depending on the history of the corresponding
voter: the more a voter has chosen a given opinion in the past, the higher
the probability that its corresponding PI field suggests that opinion.
In order to quantify this reinforcement process we generalize the expression
for $P\qua*{e_i(t)=k}$ proposed in \cite{DeMarzo2020} to the case of $M$
distinct opinions, more precisely
\begin{equation}
  P\qua*{e_i(t)=k}=P\qua*{n^{(k)}_i(t)}=\frac{c^{n_i^{(k)}}(t)}{\sum_{j=1}^Mc^{n_i^{(j)}}(t)},
  \label{eq:prob_PI}
\end{equation}
where $n_i^{(k)}(t)$ is the number of times agent $i$
has chosen (or also confirmed) opinion $k$ up to time
$t$; in the following we will often write $n_i^{(k)}$,
keeping the time dependence implicit. The state of the
system is thus described by $N(M+1)$ variables, namely
$\{(\sigma_i,n_i^{(k)})\}$, and the dynamics takes
place as in the voter model with personalized
information~\cite{DeMarzo2020}. Initially each opinion
$\sigma_i$ is set equal to a random value; at each
time step a given agent $i$ is selected uniformly at random and
with probability $1-\lambda$ it follows the usual
voter dynamics, while with probability $\lambda$ the agent
copies the state $e_i(t)$ of the corresponding PI field.
More explicitly
\begin{equation}
  \sigma_i(t+\delta t)=
  \begin{cases}
    e_i(t) \ \text{with prob. }\lambda\\
    k\ \text{with prob. }(1-\lambda)\frac{N_k}{N}
  \end{cases}
  \label{eq:update_vmpi}
\end{equation}
As in~\cite{DeMarzo2020}, the parameter $\lambda$ sets the strength
of the personalized information with respect to the interaction with
other individuals, while $c$ determines how fast personalized
information adapts to the preferences of agents.

\subsection{Phenomenology of the multistate voter model with personalized information}
\begin{figure}
  \centering
  \includegraphics[width=\columnwidth]{fig_2-3}
  \caption{Temporal evolution of the number of surviving
    opinions for $N=1000$, $M=1000$, $c=1.2$ and various
    values of $\lambda$. The inset shows the same plot
    for $\lambda=0.35$ and various values of
    $c$.}\label{fig:small_c}
\end{figure}
Let us illustrate the overall qualitative phenomenlogy of the model,
common for generic values of $M$, $N$ and $c$.
A crucial observable is the number of surviving
opinions $M_s(t)$ as function of time, which plays the role of the order
parameter of the system. In the absence of personalized information (that
is for $\lambda=0$), for a homogeneous initial condition with
$M_s(0)=M$ states, the average value of $M_s(t)$ satisfies
for $t \ll N$~\cite{Starnini2012}
\begin{equation}
  M_s(t)=\frac{M}{1+\frac{M}{N}t}.
  \label{eq:M_s_starnini}
\end{equation}
For times of the order of  $N$, $M_s$ reaches the value $M_s=1$,
implying an ordered configuration (consensus) in which all
agents share the same opinion, as also shown in Fig.~\ref{fig:large_c}a.
Notice also that there is an intrinsic timescale $t_0=N/M$
in Eq.~\eqref{eq:M_s_starnini} such that
$M_s$ remains constant and equal to $M_s(0)$ for times up to $t_0$.
As $\lambda$ increases deviations from
Eq.~\eqref{eq:M_s_starnini} start to appear, but the system keeps
reaching the ordered state after a sufficiently large amount of time, see
Fig.~\ref{fig:large_c}b. The behavior changes as $\lambda$ approaches
the threshold $\lambda_c$; in this case consensus can still be
reached, but in some realizations the system remains trapped in a
stable disordered state (also denoted as polarized state) characterized by the
presence of more than one opinion (Fig.~\ref{fig:large_c}c).
Finally, for $\lambda$ substantially larger
than $\lambda_c$, the system never reaches consensus and the
asymptotic state is always the disordered one (Fig.~\ref{fig:large_c}d).
This qualitative phenomenology is observed
independently of the value of $c$ (provided that $c>1$). This parameter
only determines the possible presence of an initial transient dominated by
the randomness of personalized information in the first time
steps, as shown Fig.~\ref{fig:small_c}. Indeed for $c=1+\delta$ with
$\delta\ll1$ the probability of PI fields, Eq.~\eqref{eq:prob_PI},
can be approximated as
\[
P\qua*{n^{(k)}_i}=\frac{(1+\delta)^{n_i^{(k)}}}{\sum_{j=1}^M(1+\delta)^{n_i^{(j)}}}\approx \frac{1+n_i^{(k)}\delta}{M+\delta\sum_{j=1}^Mn_i^{(j)}}.
\]
Exploiting the fact that $\sum_{j=1}^Mn_i^{(j)}$ is nothing but the
number of times agent $i$ has been updated and that going from $t$ to
$t+1$ each agent is on average updated once we have
$\sum_{j=1}^Mn_i^{(j)}\approx t$ so that
\[
P\qua*{n^{(k)}_i}\approx \frac{1+n_i^{(k)}\delta}{M+t \delta}\approx\frac{1}{M}+\frac{\delta}{M}\ton*{n_i^{(k)}-\frac{t}{M}}.
\]
Since $0<n_i^{(k)}<t$ this implies
\[
\frac{1}{M}-\delta\frac{t}{M^2}<P\qua*{n^{(k)}_i}<\frac{1}{M}+\delta\frac{t}{M}\ton*{1-\frac{1}{M}}.
\]
As a consequence $P\qua*{n^{(k)}_i}=\frac{1}{M}+O(\delta t)$ and so
for $\delta$ small the dynamics is initially equal to that of a
multistate voter model with external random field. This produces the
observed transient. A detailed analysis of the multistate voter model in presence of random external information is beyond the scope of this work, the interested reader can find further details in \cite{herrerias2019consensus}.
In order to determine an upper bound of the $c_{th}$ above which no transient is
observed, let us consider the first
update of agent $i$ and let us suppose that opinion $m$ is selected.
As a consequence the probability of the corresponding PI field for
the successive update is
\[
P\qua*{e_i=k}=
\begin{dcases}
  \frac{c}{c+M-1}\ \text{for } k=m\\
  \frac{1}{c+M-1}\ \text{for } k\neq m.
\end{dcases}
\]
If the probability of the PI to be in state $m$ is larger than the
probability of being in any other state, a reinforcing loop gets
established and the PI gets more and more polarized along opinion $m$.
This implies that a sufficient condition for observing a polarized PI
already after the first step is
\[
P\qua*{e_i(\delta t)=m}>\sum_{k\neq m}P\qua*{e_i(\delta t)=k} ,
\]
which implies
\[
\frac{c}{c+M-1}>\frac{M-1}{c+M-1}
\]
yielding a threshold value
\begin{equation}
c_{th}=M-1.
\end{equation}

Fig.~\ref{fig:small_c} shows that this estimate is an upper bound of the real $c_{th}$. The duration of the transient governed by a random external field gets
shorter as $c$ is increased and it is completely absent for $c \approx c_{th}=999$.
Since $c$ plays only a marginal role, in the rest of the paper we focus
on the case $c > c_{th}$ so as to remove the initial transient.

Finally, we illustrate the nature of the transition observed
as $\lambda$ is varied.
As evident from Fig.~\ref{fig:large_c}, around the transition
different realizations of the process lead to different outcomes:
either consensus or a stationary state.
The transition is characterized by the variation, as a function
of $\lambda$, of the fraction $P_s$ of runs reaching a
stationary state (Fig.~\ref{fig:Ms_vs_lambda},inset). As it is possible to see, the larger is $N$, the sharper the transition becomes. Moreover, since for large $N$ the critical threshold $\lambda_c$ goes to zero (see Subsec.~\ref{subsec:transition_point} and figures therein) and the transition gets very sharp, this implies that in large systems even an infinitesimal amount of personalized information is sufficient to make the reaching of consensus impossible.
The main panel of Fig.~\ref{fig:Ms_vs_lambda} displays how the fraction
of surviving opinions in the stationary state $M_s(t\to\infty)/M=M_s^{\infty}/M$,
averaged only over surviving runs, varies as a function of $\lambda$.
The quantity $M_s^{\infty}/M$ grows in a continuous fashion, starting
from a finite value decreasing with $N$.
This suggests that in the large $N$ limit the transition is continuous,
as also confirmed by inspecting the distribution of $S_k(t\to\infty)=S_k^{\infty}$,
the number of agents polarized along opinion $k$ in the stationary
state. When $c>c_{th}$ the number of polarized agents $S_k$ along opinion $k$ is given by the number of agents such that $n_i^{(k)}>n_i^{(j)}$ for any $j\neq k$, since, in this case, the personalized information suggest the most chosen opinion. Indeed, by looking at Fig.~\ref{fig:power_law}
it is clear that such a distribution decays as a power-law for
$\lambda=\lambda_c$, with a nontrivial exponent approximately equal to 2.8 (for $N=M$).

\begin{figure*}
  \centering
  \begin{subfigure}[b]{0.49\textwidth}
    \centering
    \includegraphics[width=\textwidth]{fig_4b_statMs}
    \caption{}
    \label{fig:Ms_vs_lambda}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.49\textwidth}
    \centering
    \includegraphics[width=\textwidth]{fig_4c_powerlaw}
    \caption{}
    \label{fig:power_law}
  \end{subfigure}

  \caption{a) Main: Stationary value of the fraction $M_s$ of
    opinions, for $M=N$, averaged only over surviving
    runs. Inset: Probability
    that consensus is reached as a function of $\lambda$ for $N=10^2$, $10^4$, $10^6$ and $M=20$. As $N$ increases the transition between consensus and polarization becomes sharper and sharper. b) Probability distribution of the number of agents $S_k$
    polarized along opinion $k$ at criticality for $N=M$. This has been obtained by letting the system evolve toward the stationary state and then performing a binning over the $S_k$ so to obtain their histogram. Note that such an histogram corresponds to a single realization of the system.}
  \label{fig:continuous_transition}
\end{figure*}

\section{Analytical approach}
As shown above, the phenomenology of the model is only
marginally influenced by the parameter $c$.
Therefore consider the case $c\gg c_{th}$
which can be more easily handled analytically.
Indeed in such a situation the probability of the
PI field $e_i$ is strongly peaked on the opinion more frequently
held by agent $i$ and
Eq.~\eqref{eq:prob_PI} reduces to
\begin{equation}
  P\qua*{e_i=k}=\delta_{k,m},
  \label{Prob_PI_large_c}
\end{equation}
where $m$ is the opinion which satisfies
\[
m=\underset{k}{\text{argmax}}\qua*{n_i^{(k)}}.
\]
In other words the PI suggests only the favorite opinion
in the past.
Our goal is to derive, under the assumption of large $c$,
analytical estimates of the critical value $\lambda_c$
for generic large values of $N$ and $M$.

\subsection{Stability of polarized states}
\begin{figure*}
	\centering
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{{fig_5a_N=10000_M=200_L=0.07_C=nan_NAOk}.pdf}
		\caption{}
	\end{subfigure}\hfill%
	\begin{subfigure}{.45\textwidth}
		\includegraphics[width=\textwidth]{{fig_5b_N=10000_M=200_L=0.08_C=nan_NAOk}.pdf}
		\caption{}
	\end{subfigure}
	\caption{Temporal evolution of $N_{\max}-N/M$, compared with the critical value $N m_c(\lambda)$
          for $N=10^4$, $M=200$: a) $\lambda=0.07$; b) $\lambda=0.08$. Different colors correspond to different simulations.}
        \label{fig:m_c}
\end{figure*}
Let us consider an agent $i$ in state $\sigma_i$ and let
us assume that its PI is polarized along opinion $m$. Combining
Eqs.~\eqref{eq:update_vmpi} and \eqref{Prob_PI_large_c} we can write
the transition probability of this agent~\footnote{Note that these
transition probabilities do not depend on the state $\sigma_i$ of
the agent} as
\begin{equation}
W\ton*{\sigma_i \to k}=
\begin{dcases}
  \frac{1}{N}\qua*{\lambda+(1-\lambda)\frac{N_k}{N}} &\text{ for }k=m\\
  \frac{1}{N}(1-\lambda)\frac{N_k}{N}&\text{ for }k\neq m.
\end{dcases}
\label{W}
\end{equation}
For the overall stability of the polarized state, the PI field $e_i$ should
remain polarized on opinion $m$ so as to keep the agent we are
considering fixed (on average) on this same opinion. This requires
the transition probability to opinion $m$ to be larger than the
transition probability to any other opinion $l$, otherwise $n_i^{(l)}$
would grow faster than $n_i^{(m)}$ and the PI would eventually
depolarize and then polarize along opinion $l$.
As a consequence, in order for the polarized state to be stable it must be
\[
W\ton*{\sigma_i \to m}>W\ton*{\sigma_i \to l}~~~~~~\forall l \neq m,
\]
which yields, using Eq.~(\ref{W})
\begin{equation}
  \frac{N_m-N_l}{N}>-\frac{\lambda}{1-\lambda}=-m_c(\lambda).
  \label{eq:stability_preliminary}
\end{equation}
Here, in analogy with~\cite{DeMarzo2020}, we defined the critical
magnetization $m_c(\lambda)$ as
\begin{equation}
  m_c(\lambda) = \frac{\lambda}{1-\lambda}.
  \label{eq:m_c}
\end{equation}
We can then repeat this same reasoning but considering an agent $j$
whose PI field $e_j$ is polarized along opinion $l$. This leads to
\[
\frac{N_m-N_l}{N}<m_c(\lambda).
\]
Combining this expression with
Eq.~\eqref{eq:stability_preliminary} we obtain the following necessary
condition for a polarized state to be stable
\begin{equation}
  \left | \frac{N_m-N_l}{N} \right|<m_c(\lambda) ~~~~\forall(l, m).
  \label{eq:condition_stability_N}
\end{equation}

By introducing the partial
magnetization $m_k$ along opinion $k$,
we can also rewrite this constraint as

\[
m_k=\frac{N_k-\sum_{l\neq k}N_k}{N}=\frac{2N_k-N}{N}.
\]
In this way Eq.~\eqref{eq:condition_stability_N} becomes
\begin{equation}
  \left|\frac{m_m-m_l}{2}\right|<m_c.
  \label{eq:condition_stability_m}
\end{equation}
The conclusion if the difference between any two magnetizations is smaller than the threshold $m_c$ then the polarized state is stable. Note, however, that this condition is very strict and a polarized state can be stable on average even if there are some opinions held by a small number of agents not fulfilling it. Indeed in such a situation these opinions will be absorbed by the others, but still consensus will not be reached. As a consequence what really matters is that \eqref{eq:condition_stability_N} is fulfilled when considering the most common opinion whose size is $N_{max}$ and the average opinion size given by $N/M$. The condition ensuring the system not to reach consensus is thus
\begin{equation}
  N_{max}-\frac{N}{M} <Nm_c(\lambda) .
  \label{eq:condition_stability_N_average}
\end{equation}

Fig.~\ref{fig:m_c} checks and confirms the validity of this argument in numerical simulations.
Note that $m_c$ is larger than 1 for $\lambda > 1/2$ and so, as in
the voter model with personalized information, above $\lambda=1/2$ an
opinion can survive even with only a single agent supporting it.

\subsection{Polarization time}
\label{subsec:polarization_time}
\begin{figure}
    \includegraphics[width=\columnwidth]{fig_6_Qnik}
    \caption{Temporal evolution of the $Q(n_i^{(k)})$ distribution as
      a function of time.  Data are for $c=400,$ $N=10^6$, $M=20$,
      $\lambda=0.07$.  The component with high mean value represents
      agents polarized along opinion $k$; the component with low mean
      value represents agents unpolarized or polarized along other
      opinions.}
  \label{fig:splitting}
\end{figure}
In deriving Eq.~\eqref{eq:condition_stability_m} we assumed all PI
fields to be polarized on a certain opinion, meaning that for any $i$
there is only one $k$ such that $n_i^{(k)}$ is maximum. However
initially $n_i^{(k)}=0$ for all pairs $(i, k)$ and the randomness of voter
updates implies the occurrence of several ties between the various
$n_i^{(k)}$.
The assumption that PI fields are polarized starts to be valid
only after a polarization time $t^*>0$. In order to compute this
time we write down the master equation for the distribution of
the $n_i^{(k)}$ values, $Q\ton{n_i^{(k)}}$, that reads
\begin{equation}
  \frac{dQ\ton*{n_i^{(k)}}}{dt}=NR\ton*{n_i^{(k)}-1}Q\ton*{n_i^{(k)}-1}-NR\ton*{n_i^{(k)}}Q\ton*{n_i^{(k)}},
  \label{eq:master_equation_n_general}
\end{equation}
where we introduced the single transition probability
\begin{align}
  NR\ton*{n_i^{(k)}}&=\ton*{1-\lambda}\frac{N_k}{N}+\lambda P\ton*{e_i=k} \nonumber \\
  &\approx \ton*{1-\lambda}\frac{1}{M_s(t)}+\lambda P\ton*{e_i=k}.
  \label{eq:transition_R_general}
\end{align}
Here we assumed that all surviving opinions share
approximately the same number of agents and so we can make the
approximation ${N_k}/{N}\approx{1}/{M_s(t)}$. Now, considering
opinion $k$, only a fraction $S_k/N$ of the agents is polarized along
this opinion and thus we assume that the distribution
$Q\ton{n_i^{(k)}}$ is formed by two normalized components $Q_1$ and
$Q_2$ corresponding to the two types of agents: those polarized on
opinion $k$ (component $Q_2$) and those polarized on any other
opinion or not polarized at all (component $Q_1$).
We can then write $Q$ as a bimodal distribution
\[
Q\ton*{n_i^{(k)}}=\ton*{1-\frac{S_k}{N}}Q_1\ton*{n_i^{(k)}}+\frac{S_k}{N}Q_2\ton*{n_i^{(k)}}.
\]
Both components evolve according to the same general master equation
Eq.~\eqref{eq:master_equation_n_general}, but the transition rates are
different. For what concerns the polarized component $Q_2$ we have
\[
NR\ton*{n_i^{(k)}}=\ton*{1-\lambda}\frac{1}{M_s(t)}+\lambda
\]
and using this expression and Eq.~\eqref{eq:master_equation_n_general}
we can compute both the mean value $n_2$ and the variance $\sigma_2^2$
of the distribution $Q_2$ for polarized agents, finding
\begin{align}
  n_2 = \sigma_2^2 = \int_0^t dt' \ton*{\frac{1-\lambda}{M_s(t')}+\lambda},
  \label{eq:mean_variance_2}
\end{align}
Analogously, the transition rate of the component $Q_1$, corresponding
to agents not polarized on opinion $k$, is
\[
NR\ton*{n_i^{(k)}}=\ton*{1-\lambda}\frac{1}{M_s(t)}
\]
and so the mean value $n_1$ and the variance $\sigma_1^2$ are
\begin{align}
  n_1 = \sigma_1^2 = \int_0^t dt' \ton*{\frac{1-\lambda}{M_s(t')}}.
  \label{eq:mean_variance_1}
\end{align}
Detailed computations are reported in
Appendix~\ref{apx:Master equation, transition rates and moments}.

Looking at Eqs.~\eqref{eq:mean_variance_2}
and \eqref{eq:mean_variance_1} it is clear that initially the two
components are indistinguishable since they both have null mean and
variance. However, having different velocities they
tend to separate over time.
This behavior is shown in Fig.~\ref{fig:splitting}.
Until $Q_1$ and $Q_2$ are superposed there is
no real distinction between polarized and unpolarized agents, as all agents
can change their preferred opinion in one or few voter updates.
Consequently we identify the polarization time $t^*$ introduced above
as the time when the two components split for the first time.
This can be readily determined by imposing the distance between the
two peaks to be equal to their widths, that is
\[
n_2(t^*)-n_1(t^*)=2\qua*{\sqrt{\sigma_1^2(t^*)}+\sqrt{\sigma_2^2(t^*)}},
\]
where the factor $2$ is somewhat arbitrary and does not influence the
scaling of $t^*$. Substituting Eqs.~\eqref{eq:mean_variance_2} and
\eqref{eq:mean_variance_1} into this last expression we come up with
an implicit integral expression for the polarization time
$t^*$
\begin{equation}
  t^*=\frac{2}{\lambda}\gra*{\sqrt{\int_0^{t^*}\qua*{\frac{1-\lambda}{M_s(t)}+\lambda}dt'}+\sqrt{\int_0^{t^*}\frac{1-\lambda}{M_s(t)}dt'}}
  \label{eq:polarization_time}
\end{equation}

\subsection{The transition point}
\label{subsec:transition_point}

Once the polarization time has been obtained we can turn to
the determination of the critical parameter $\lambda_c$.
The idea is the following: In the initial stage of the dynamics,
the numbers of agents holding the different opinions, $N_l$, tend
to fluctuate and their differences tend to grow over time.
If this growth is slow enough, at the polarization time $t^*$
the condition for the stability of polarized states,
Eq.~\eqref{eq:condition_stability_N_average} is satisfied. In such a case the disordered state with multiple
coexisting opinions is stable and persists forever.
Conversely, if at $t^*$ Eq.~\eqref{eq:condition_stability_N_average}
is violated, the system will eventually evolve toward the consensus state. The condition for $\lambda_c$ thus reads
\begin{equation}
  x_{\max}(t^*)=N_{\max}(t^*)-\frac{N}{M}=N\frac{\lambda_c}{1-\lambda_c}.
  \label{eq:condition_lambda_c_begin}
\end{equation}
By means of numerical simulations and a simple scaling argument
(see Appendix~\ref{apx:Scaling of x_max}) we determine that for $M \ll N$ $x_{\max}(t)$ scales as
\begin{equation}
  x_{\max}(t) \sim \gamma\frac{N^{1/2}}{M}t^{3/2},
  \label{eq:x_max(t)}
\end{equation}
where $\gamma\approx1/4$ is a numerical constant. We also derived an
equation for the time growth of $N_k-N/M$ and we checked that the
temporal scaling of $x_{\max}(t)$ is consistent with this expression,
see Appendix~\ref{apx:evolution_xk_yk}.  Replacing this last
expression into Eq.~\eqref{eq:condition_lambda_c_begin} we obtain an
equation for $\lambda_c$
\[
\gamma\frac{N^{1/2}}{M}{t^*}^{3/2}=N\frac{\lambda_c}{1-\lambda_c},
\]
implying
\begin{equation}
  \lambda_c = \frac{\gamma \frac{N^{1/2}}{M}\qua*{t^*(\lambda_c)}^{3/2}}{N+\gamma \frac{N^{1/2}}{M}\qua*{t^*(\lambda_c)}^{3/2}}.
  \label{eq:condition_lambda_c_final}
\end{equation}

Eq.~\eqref{eq:condition_lambda_c_final}, together with
Eq.~\eqref{eq:polarization_time} for the polarization time,
provide a closed system of equations in $\lambda_c$ and $t^*$
\begin{equation}
	\begin{dcases}
    t^*=\frac{2}{\lambda_c}\gra*{\sqrt{\int_0^{t^*}\qua*{\frac{1-\lambda_c}{M_s(t')}+\lambda_c}dt'}+\sqrt{\int_0^{t^*}\frac{1-\lambda_c}{M_s(t')}dt'}}\\
    \lambda_c = \frac{\gamma \frac{N^{1/2}}{M} (t^*)^{3/2}}{N+\gamma \frac{N^{1/2}}{M} (t^*)^{3/2}}.
  \end{dcases}
  \label{eq:system}
\end{equation}

In order to solve this system we need an explicit expression for the
number $M_s(t)$ of surviving opinions.
We consider two possible assumptions, which are expected to be fairly
accurate for different values of $M$ and $N$.
Here we limit ourselves to report the
main results, the interested reader can find detailed
calculations in Appendix~\ref{apx:scaling_regimes}.

\subsection{$\mathbf{M \ll N}$}

  If the number of opinions in much smaller
  than the number of agents, each opinion is initially shared by
  a large number of individuals. As a consequence during a first time
  interval no opinion disappears and it is reasonable to make the
  approximation
  \[
  M_s(t)=M
  \]
  In this way Eq.~\eqref{eq:system} can be rewritten as
  \begin{equation}
  \begin{dcases}
    t^*=\frac{4}{\lambda_c^2}\qua*{\sqrt{\frac{1-\lambda_c}{M}+\lambda_c}+\sqrt{\frac{1-\lambda_c}{M}}}^2\\
    \lambda_c = \frac{\gamma \frac{N^{1/2}}{M}(t^*)^{3/2}}{N+\gamma \frac{N^{1/2}}{M}(t^*)^{3/2}}.
  \end{dcases}
  \label{eq:system_M<<N}
  \end{equation}
  Depending on how large $M$ is with respect to $N$,
  the solutions of this system scale in different ways.

  \begin{itemize}
  \item $\mathbf{M \ll N^{1/3}}$\\
    \begin{equation}
      \begin{cases}
	t^* \approx 4(MN)^{1/4}\\
	\lambda_c \approx 2\ton*{M^5N}^{-1/8}.
      \end{cases}
      \label{eq:M<<N^1/3}
    \end{equation}
  \item $\mathbf{N^{1/3}\ll M\ll N}$\\
    \begin{equation}
      \begin{cases}
	t^* \approx 2^{8/5}\ton*{NM^2}^{1/5}\\
	\lambda_c \approx 2^{2/5}\ton*{NM^2}^{-1/5}.
      \end{cases}
      \label{eq:M<<N}
    \end{equation}
  \end{itemize}

\subsection{$\mathbf{M=N}$}
If $M=N$ each agent has initially a different
opinion. Hence, even during the first time steps, some opinions
disappear by chance.
In this case we can approximate the surviving opinions
$M_s(t)$ with the expression valid for the simple multistate voter
model, Eq.~\eqref{eq:M_s_starnini}, that is for $M=N$,
\begin{equation}
  M_s(t)=\frac{N}{1+t}.
  \label{Mstarnini}
\end{equation}
Substituting this expression in Eq.~\eqref{eq:system} and imposing
$M=N\gg 1$, $\lambda_c\ll 1$ we obtain the following system
\begin{equation}
  \begin{dcases}
    t^*=\frac{32N^{1/2}+8\lambda_cN}{2\ton*{\lambda_c^2N-16}}\\
    \lambda_c = \frac{\gamma N^{-1/2}\qua*{t^*(\lambda_c)}^{3/2}}{N+\gamma N^{-1/2}\qua*{t^*(\lambda_c)}^{3/2}},
  \end{dcases}
  \label{eq:system_M=N}
  \end{equation}
whose solution scales as
\begin{equation}
  \begin{cases}
    t^* \approx 2^{4/3}N^{2/3}\\
    \lambda_c \approx 4N^{-1/2}
  \end{cases}
  \label{eq:M=N}
\end{equation}

\begin{figure*}
  \centering
  \begin{subfigure}[t]{.45\linewidth}
    \caption{$M=20 \ll N$.}
    \includegraphics[width=\textwidth]{fig_7_compMllN}
    \label{fig:theory_vs_simulations}
  \end{subfigure}\hfill%
  \begin{subfigure}[t]{.45\linewidth}
    \caption{$M=200 \ll N$.}
    \includegraphics[width=\textwidth]{fig_8_compMllN2}
    \label{fig:theory_vs_simulations2}
  \end{subfigure}\hfill%
  \begin{subfigure}[t]{.45\linewidth}
    \caption{$M=N$.}
    \includegraphics[width=\textwidth]{fig_9_compMN}
      \label{fig:theory_vs_simulations3}
  \end{subfigure}
  \caption{Comparison of analytical predictions for the
    threshold $\lambda_c$ and numerical estimates of it}
  \label{}
\end{figure*}

\section{Numerical simulations}

We now compare theoretical predictions with numerical simulations.
We iterate MVM dynamics for various values of $M$ and $N$ up to $t=2000$
and we determine the fraction of times $P_s(\lambda)$ the system
has reached consensus.
We take as numerical estimate of $\lambda_c$ the values such that
$P_s(\lambda_c)=1/2$.

For fixed value of $M$ and increasing size $N$, the analytical
results predict initially a scaling regime given by Eq.~\eqref{eq:M<<N}
as long as $N$ is much larger than $M$ but much smaller than $M^3$,
followed by a scaling given by Eq.~\eqref{eq:M<<N^1/3}.
For $M=20$, the first regime spans a short interval of $N$ values.
In Fig.~\ref{fig:theory_vs_simulations}
we see only a hint of the associated scaling, while for larger values of $N$
the agreement between theory and numerics is very good.
For $M=200$ instead (see Fig.~\ref{fig:theory_vs_simulations2}), the first regime
extends to much larger values of $N$, so that feasible values of $N$ lie only
in this regime.
In this case $\lambda_c$ nicely scales as $N^{-1/5}$, as predicted by
Eq.~\eqref{eq:M<<N}, although the prefactor is not predicted exactly.

For the case $M=N$ instead, Figure~\ref{fig:theory_vs_simulations3}
show that the theory is not able to catch the correct scaling.
This mismatch has various potential origins.
Indeed using Eq.~\eqref{Mstarnini} for the temporal dependence of $M_s(t)$
is a quite rough approximation, as the number of different opinions
actually decays much more slowly over time. But also the use
of Eq.~\eqref{eq:condition_lambda_c_begin} for $x_{\max}(t)$ is not warranted
for $M=N$. A deeper understanding of the phenomenology of the case $M=N$
remains an interesting open question.

\section{Conclusions}

In this paper we have introduced and analyzed a multistate voter model
where the coupling with an external history-dependent individual field
mimics the effect of personalized recommendation algorithms in online
social media.  A population of agents, initially having different
opinions, reaches consensus on a single opinion or remains polarized
on multiple different opinions, depending on the strength of
personalized information.  The phenomenology is governed by the
competition between the fluctuations induced by voter dynamics and the
tendency, due to personalized information, to bind agents to the
opinion they adopted most frequently in the past.  By means of
arguments based on this physical picture, we estimated analytically
the critical threshold between the two regimes, obtaining a reasonable
agreement with simulations for $M \ll N$. Conversely, the dynamics
when the initial number of opinions is comparable with the number of
agents seems to elude our approach.

From a more general point of view, our study indicates how difficult
reaching consensus is, in the presence of personalized
recommendations.  For any number $M$ of initial opinions the threshold
$\lambda_c$ tends to vanish when the number of agents $N$
diverge. This means that no matter how weak is the coupling with the
personalized information, if the system is large enough polarization
unavoidably arises.  In this respect, note that voter dynamics is
extremely favorable to the establishment of consensus: for any
interaction pattern, consensus is necessarily reached for any finite
number of interacting individuals.  The addition of a personalized
recommendation completely changes this picture, at least for large
systems.  For a finite number of interacting agents instead consensus
is still reached if the strength of personalized information is small
enough. The transition between the two regimes exhibits nontrivial
features that may be the focus of future activity along with
generalizations to include nontrivial interaction patterns or a
different functional dependence between the probability distribution
of the personalized information and the number of times an opinion has
been selected in the past.

A natural extension of this work would be to analyze more realistic
recommendation algorithms. With respect to this, a promising
possibility is to consider collaborative filtering algorithms
\cite{schafer2007collaborative}, which are based on similarities among
the history of different users or opinions instead of considering only
the history of the user itself. This typology of algorithm is used in
real life applications \cite{linden2003amazon, smith2017two}, but
still it is sufficiently simple to try an analytical study of its
effects on opinion dynamics models.

redpajama_set_name: RedPajamaArXiv


I've found that the analogy works because not only do execs understand what a mutual fund is, most also own them. And since mutual funds have both investments and returns, we can always calculate a return on investment (ROI) from them, right?
In order to calculate the ROI of a mutual fund, we need a minimum of four pieces of data: the purchase price, purchase date, sale price, and sale date. Without all four of these values, we can't calculate ROI. Period.
Let's take a look at three scenarios for calculating the ROI of a mutual fund investment: buy-and-sell, buy-and-hold, and dollar-cost-averaging.
If we buy a mutual fund at $25 per share on January 1st and sell it for $50 per share on December 31st, our return on investment is a simple $25 gain, or a 100% annual return on our investment. Whenever someone asks the social media ROI question, this is the information that they are looking for.
But, what if December rolls around and we decide to hold the investment rather than selling it? What's the ROI then? By definition, we can't calculate an ROI, because we only have half of the required information–the purchase price and date. Sure, we can estimate the value of the investment, but estimates aren't the same as cash. We can't buy groceries with estimates. And so, since we can't calculate the ROI of a mutual fund without selling the stock, does that mean mutual funds aren't worth investing in?
Investors understand that their stock portfolios consist of financial assets as opposed to cash. They understand that assets fluctuate in value due to other variables such as the economy, competition and technological advances. Investors constantly balance these facts with their personal tolerance for risk when deciding whether to increase, decrease or liquidate their positions in these assets.
Dollar-cost-averaging is a common investment technique for retirement savings. By directing a fixed-percentage of each paycheck into purchasing shares of mutual funds, we exploit the power of time to smooth short-term market fluctuations. Since share purchases are executed every pay-cycle, we add to our positions at different price points–making for interesting ROI scenarios. For example, if our mutual fund's share price has increased from $25 to $50 in one year and we've been purchasing additional shares twice per month through dollar-cost-averaging, what's the ROI if we decide to take some profits by selling half of our position on December 31st? Although there's an acceptable accounting calculation for it, in the end, the "true" answer is a still a little fuzzy.
ROI requires a start and end time– like a buy and a sell investment strategy.
Social Media is an ongoing effort, similar to a buy and hold or a dollar-cost-averaging investment strategy which makes measuring ROI fuzzy.
The audiences that companies build through publishing relevant, online content is an asset to manage.
Mutual fund assets can be converted into cash returns by selling shares. Audiences can be converted into cash returns by asking them to do things for you–like buying your products and services!
But even when distributions are taken from assets, the ROI of the transaction can still be a little fuzzy.
What is the value of my investment into these audience assets?
How can I take distributions from my audience asset?
Does it make sense to keep making these investments?
I followed this analogy up until you talked about selling shares. When you share a mutual fund share it's no longer part of your ownership. So if the audience is the asset and you sell part of the asset then its as if you are selling members of your audience or converting them into a one time cash transaction. I like to think of the return that your audience provides as cash that is in your account that is available to pay back as dividends to the shareholders. An audience member (aka "follower") may purchase more than once from you. The value of the "shares" goes up as you add more followers. So in the end, the ROI can still be "fuzzy".
As always, thanks for your insight, Bob!


EL PASO, Texas (CBS NEWS) — Looking to get yourself a present this Valentine's Day? The El Paso Zoo has you covered. It will name a cockroach after your ex and then feed it to a meerkat live on camera.
The El Paso Zoo isn't the only zoo offering a non-traditional Valentine's Day promotion. The Hemsley Conservation Center in Kent, England, will name a cockroach after your ex in exchange for a donation. You can also name a roach at the Bronx Zoo, which calls them "eternal" and "timeless" gifts.
Misplaced priorities of Ghanaian slay queens in the digital age. The sorry case of Moesha Boduong and Co.


The newly designed recreation center, conveniently located on the fourth floor at Coral Beach, includes 8 AMF lanes of bowling, exciting arcade games, a snack bar, ice cream shop and beverages for all ages. It's just part of what makes the Coral Beach one of Myrtle Beach's best family resorts. Take a break from the sun and sand and check it out!
The on-site Entertainment Zone is fun for the whole family! The video game arcade features classics like Pac-Man, pinball, air hockey, pool, and much more. And, as one of Myrtle Beach's only hotels with an on-site bowling alley, your family can enjoy endless hours of fun without having to leave the resort!
The Chill & Grill Zone is our on-site snack bar and ice cream shop. Refuel with popular grill items including a juicy burger, hot dogs, wings and more! And if you're trying to beat the heat, tempt your taste buds with a delightful hand-scooped ice cream treat. The adjacent Beverage Zone serves a variety of refreshing beverages guaranteed to quench your thirst and energize you for even more entertainment fun – including beer, wine and cocktails for our adult guests!


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redpajama_set_name: RedPajamaCommonCrawl


text:
Alumni Spotlight: Jared Baker '92
Home ▸ Alumni Affairs ▸ Alumni Spotlight: Jared Baker '92
Jared Baker, IASMH Class of 1992
After graduating from the Academy, Jared Baker (IASMH '92) moved to Los Angeles and attended the University of Southern California, where he received a Bachelor of Arts in Film and Television Production.
Jared came to The Academy from South Bend, where he was raised as a Fighting Irish fan. Although Jared still loves to root for the Irish, he has to be true to his alma mater and root for the Trojans when they meet on the gridiron every fall.
Jared had always been interested in theatre and filmmaking, and his experiences at The Academy helped further those ambitions. He served his community service in the Burris Media Center under Fred Shears, who encouraged Jared to utilize all of the available resources to learn and create. Jared spent a great deal of his senior year at The Academy with either a video camera in his hands, (yes, one of those old school full size VHS cameras) or in the editing bay at the media center. Jared created a video yearbook, and with the help of an entourage of classmates, a video of Dante's Inferno, a class project for Christine Ney's world literature class. Jared doubts that he would have been accepted into USC's film school without attending the Academy, and is grateful for how The Academy prepared him for college.
Once his time at USC was concluded, Jared had a brief stint at an independent production company. Feeling that office work wasn't really his forte, Jared landed a job at a special effects shop where he spent a few years learning about fabricating, hydraulics, pneumatics, pyrotechnics and weather effects. Jared now works as a freelance special effects technician for film and television. Jared works very hard at fighting the stereotype that FX guys don't know how to spell. Jared has worked on a number of award-winning television shows including Six Feet Under, True Blood, Big Love, The Mentalist, Las Vegas, The Forgotten (he appeared in an episode) and the Indiana-based show Close To Home (he had to inform an actor on the proper pronunciation of Notre Dame). Jared has also worked on several feature films including A Single Man, The Informant!, and most recently Scouts vs. Zombies, The Gambler starring Mark Walhberg, and Gone Girl directed by David Fincher.
Jared is currently finishing a script in hopes of aspiring to his true goal of directing.
Jared lives in Simi Valley, California with his wife, fellow Indiana transplant and USC Trojan, Dr. Shannon Johnson, and their two very spoiled hound dogs.
Other Alumni Spotlights
Stephen Taller (2009)
Dr. James M. LeBeau (2002)
Michael Hughes (2008)
Dr. Kian Karimi (1998)
Connor Caudill (2008)
Jordan Dauble (2001)
Derek Sprunger (2009)
Laura Vanderkam (1997)
Tim Katamba (2010)
Dina Rabadi (1992)
Stephanie Kemery (2003)
Prachi Murarka (2006)
Wanda Collins (1998)
Kara Claudy (2007)
Nathan Strange (1993)
Jason Irwin (1993)
Natasha Browder (2011)
Emily Kao (2010)
Ruth (Rivers) Bowers (1998)
Ryan Shelley (2007)
Jared Baker (1992)
Lori (Powell) Craig (1994)
Dr. Kristen K. Williams (2000)
Jen Carlson (2000)
Robert Hero (2009)
Hillary (Spike) Close (1996)
Miranda Summers Lowe (2001)
Jordan (2004) and Mason (2005) Smith
Blake Johnson (2005)
Dr. Laura Morgan Roberts (1992)
Ryan (2007) and Valerie (Sargent, 2006) Smith
Aliguma Young (1994)
Andie Redwine (1992)
Drew Ramsey, M.D. (1992)
Rachel (Brack) Sharp (1993)
Home Counties of Alumni
Colleges Attended by Alumni
Request Dual-Credit/BSU Transcript
https://academy.bsu.edu/alumni/spotlight-jared-baker-92/


text:
2 Unlimited
Get Ready for This
Get Ready (Rapversion Edit)
No Limit (Rap Version)
Get Ready (Orchestral Mix)
Real Things
No Limit - EP
Get Ready (Steve Aoki Radio Mixes) - Single
Get Ready (Including Steve Aoki Remixes) - EP
Unlimited Hits & Remixes
2 Brothers On the 4th Floor
Dr. Alban
About 2 Unlimited
The Netherlands techno/dance-pop phenomenon 2 Unlimited was one of the most popular groups in Europe during the early '90s, selling 18 million records and charting number one singles in every European country. (In England, 11 consecutive singles topped the charts.) America, however, remained unaffected, bringing only "Get Ready for This" to number 38 in late 1994, over two years after it was first released.
In 1990, producers Phil Wilde and Jean-Paul DeCoster had previously gained success with AB Logic, and were looking for another vehicle for their songs. 2 Unlimited formed when Wilde and DeCoster were introduced to rapper Ray Slijngaard and vocalist Anita Doth by Marvin D., who had featured both in his rap group in the past. In 1991, debut single "Get Ready for This" reached number two on several European charts; the following year Get Ready was released.
The modest success of the first few singles prepared the way for 2 Unlimited's 1993 sophomore album No Limits. The title track hit number one in 15 European countries plus Canada, and other singles from the album also did well, including "Faces," "Maximum Overdrive," and "Let the Beat Control Your Body." Third album Real Things and the singles "The Real One," "No One," and "Here I Go" continued the hitmaking. The best-of album Hits Unlimited was the group's last release, as Slijngaard left to manage his X-Ray label and Doth moved into TV and radio talk shows (though she has recorded with Mad Cobra). ~ John Bush


Celebrate Oktoberfest all year long with authentic Germany food and traditional Bavarian music performed by Oktoberfest Musikanten. This Bavarian festival has been taking place annually in Munich, Germany since October 1810, a celebration to honour the marriage of Prince Ludwig and Therese of Saxe-Hildburghausen. Here you can experience this lively festival every day in Epcot.
Tables are long and communal, seating 8 guests so small parties will be seated with others, giving it an authentic Germany feel.
Oktoberfest Musikanten's music is available for purchase in the Germany pavilion stores.
Sing along to Ein Prosit with everyone else. Even better, get up and dance.


Our Student Saver ticket range offers great money-saving deals for students, for a term up to an academic year. There are lots of benefits and they're so easy to buy on your smart phone or online.
Please note that valid Student ID is required for discounted fares. This may be provided by your school or college, or alternatively you can purchase an International Student Identity Card here. Buy an ISIC card.
Our Student Saver tickets are available for students studying at further education establishments in places with Arriva bus services across the UK. Details vary for each area, but all offer great savings over the cost of buying individual tickets every day.
Check out our local deals including tickets from a term at a time up to the whole academic year.
Becoming a student represents a major change in the life of young people and their families. It may mean leaving home and living independently for the first time, perhaps moving to a new and unfamiliar area. Living to a budget may well be a new experience for many, and bus travel may not feature very highly on priority lists.
We recognise this at Arriva, and know that parents at least will be happier knowing that things like getting to and from lectures are covered, the ability to go out and shop for food and being able to make the most of all the new activities and experiences that will on offer to the new student. Some parents buy their fledgling student's first tickets to give them peace of mind that at least the basic essential of travel is taken care of.
Some people worry about the risk of losing paper tickets if they buy for a lengthy period. However it's probably less likely that they'll lose their smart phone, so Student Saver tickets can be bought on iPhone or Android smart phones and delivered instantly to the phone, or bought online and then delivered to the phone. There is also a postal delivery option for tickets bought online.


text:
const { assert, skip, test, module: describe, only } = require('qunit');
const { GPU } = require('../../../../../../src');
const { greenCanvas } = require('../../../../../browser-test-utils');

describe('feature: to-string unsigned precision arguments HTMLCanvas');

function testArgument(mode, done) {
  const canvasInput1 = greenCanvas(mode, 1, 1);
  const canvasInput2 = greenCanvas(mode, 1, 1);
  const gpu = new GPU({mode});
  const originalKernel = gpu.createKernel(function (canvas1, canvas2) {
    const pixel1 = canvas1[this.thread.y][this.thread.x];
    const pixel2 = canvas2[this.thread.y][this.thread.x];
    return pixel1[1] + pixel2[1];
  }, {
    output: [1],
    precision: 'unsigned',
    argumentTypes: ['HTMLCanvas', 'HTMLCanvas'],
  });
  const canvas = originalKernel.canvas;
  const context = originalKernel.context;
  assert.deepEqual(originalKernel(canvasInput1, canvasInput2)[0], 2);
  const kernelString = originalKernel.toString(canvasInput1, canvasInput2);
  const newKernel = new Function('return ' + kernelString)()({context, canvas});
  const canvasInput3 = greenCanvas(mode, 1, 1);
  const canvasInput4 = greenCanvas(mode, 1, 1);
  assert.deepEqual(newKernel(canvasInput3, canvasInput4)[0], 2);
  gpu.destroy();
}

(GPU.isSinglePrecisionSupported && GPU.isWebGLSupported ? test : skip)('webgl', () => {
  testArgument('webgl');
});

(GPU.isSinglePrecisionSupported && GPU.isWebGL2Supported ? test : skip)('webgl2', () => {
  testArgument('webgl2');
});

redpajama_set_name: RedPajamaGithub


is to nurture children in a Christ centred learning environment in order to develop their full academic, creative, personal, physical, moral and spiritual potential.
Torrens Valley Christian School is built on faith in Jesus Christ, providing excellence in education and teaching servant leadership, compassion, respect and responsibility for the world in which we live.
Gleeson College is a Catholic Co-Educational Secondary College situated on the Golden Grove Secondary Schools Campus in the North Eastern suburbs of Adelaide. With an enrolment maximum of 700 students, Gleeson College is particularly well resourced and is complemented by our outstanding shared facilities.
Gleeson College was established in 1989 and is named after the late Emeritus Archbishop of Adelaide, the very Reverend James Gleeson. Archbishop Gleeson had a strong personal commitment to the pursuit of excellence, justice and peace, and we believe that the students of Gleeson College continue to strive for those goals in our Catholic tradition.
We are a faith-filled community, providing our students and staff with the opportunity to discover, develop and use their God-given talents in the service of others. We aim to foster and encourage all students to strive to be the best they can academically, socially and as a person for the world.
Our mission is to always be a College of quality learning and quality teaching. We also aim to work in close partnership with our students and parents, in order to achieve success for all. Pursuit of excellence is a core value of Gleeson College, and is demonstrated by our goal of continuous improvement throughout an evidenced-based approach to 21st Century education.
Gleeson College is underpinned by the values of the Gleeson 10, Choice Theory and Brain-Based Learning. The College operates a Vertical House Structure (VHS) of Pastoral Care, with House Home Groups including both Year 8 and 9 students in the Middle School, and Year 10-12 students in the Senior School. Since its introduction in 2010, we are continually observing the positive outcomes of the VHS implementation, including increased College Spirit and improved student relationships and sense of belonging over all year levels.
Gleeson College is a Catholic Co- Educational Secondary College situated in Golden Grove in Adelaide's picturesque north-eastern suburbs. With an enrolment maximum of 700, Gleeson College is particularly well-resourced, offers a broad curriculum and is complemented by the extended learning opportunities on offer at our One+ (Opportunities North East) shared learning complex.
Established in 1989, and named after the late Emeritus Archbishop of Adelaide, the very Reverend James Gleeson, Gleeson College is a faith-filled community, providing our students and staff with the opportunity to discover, develop and use their God-given talents in the service of others.
We aim to foster and encourage all students to strive to be the best they can academically, socially and as a person for the world.
Our mission is to be a College of quality learning and quality teaching. We aim to work in close partnership with our students and parents, in order to achieve success for all.
'Pursuit of Excellence' is a core value of Gleeson College, and is demonstrated by our goal of continuous improvement through an evidenced-based approach to 21st century education.
As a values-rich Anglican and Uniting Church School, students are provided with a vast range of opportunities to fully develop their potential. With an environment committed to promoting excellence in academic achievement, the results of our students speak for themselves with consistently high scores in Year 12 studies.
But it's not just our particularly gifted students who are inspired and supported to perform at such high levels. Our classes at all levels regularly perform well above not just state, but national averages. Our 'Share | Belong | Succeed' philosophy manifests in all that we do. We prepare our students and instil in them life values including compassion, excellence, faith and integrity. We light the way through example and instruction and give each individual student the chance to truly shine.


text:
Michael Barenboim, Daniel Barenboim et Kian Soltani © Harald Hoffmann
Beethoven Trios / Daniel Barenboim
Michael Barenboim - Kian Soltani
Home page / Calendar / Beethoven Trios / Daniel Barenboim
Variations Kakadu
Trio n°6
Trio n°7 "À l'Archiduc"
Daniel Barenboim , piano
Michael Barenboim , violin
Kian Soltani , cello
Under the direction of Daniel Barenboim, this second concert of Beethoven's trios is dominated by his 'Archiduc' Trio. From the grandeur and poetic flair of the variations in the slow movement to the brilliance of the finale, this prodigious work is strikingly moving.
Coproduction Piano****, Philharmonie de Paris
Thursday, 9 June 2022 — 8:30 pm
Accessibility informations
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Gianandrea Noseda conducts the London Symphony Orchestra, of which he is principal guest conductor, in Shostokovich's…
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Daniel Barenboim returns to a score he knows well, Smetana's Ma Vlást, with the West-Eastern Divan Orchestra, which he…