text:
\section{How to measure information}

\subsection{Prologue}

How should one measure information?
There is no unique answer to
this. There are many different information measures, and what
measure of information is the most suitable one will in general
depend on the problem under consideration. Also, there are
different types of information. For example, the information a
reader gets from reading a book on quantum field theory is
different from the one he gets from reading Shakespeare's {\em Romeo
and Juliet}. In general one has to distinguish between elementary
and advanced information concepts. The elementary information is
just related to technical details such as, for example, the
probability to observe certain letters in a long sequence of
words. The advanced information is related to the information the
reader really gets out of reading and understanding a given text,
i.e. this concept requires coupling to a very complex system such
as the brain of a human being.

\begin{figure}
\epsfig{file=gentro-fig1.eps, width=8cm, height=7cm} \caption{There is no
obvious way to measure the information contents of given symbol
sequences. While it is relatively easy to distinguish between a
random sequence of symbols and Shakespeare's {\em Romeo and Juliet} in
terms of suitable elementary information measures, it is less
obvious how to distinguish the fact that the advanced information
contents given by Shakespeare's {\em Romeo and Juliet} is different
from the one given by a book on quantum field theory.}
\end{figure}

In physics, the missing information on the concrete state of a
system is related to the {\em entropy} of the system. Entropy is
an elementary information concept. Many different physical
definitions of entropy can be given, and what makes up a
`physically relevant entropy' is often subject to `heated'
discussions. Misunderstandings with respect to the name `entropy'
seem to be the rule rather than the exception within the past 130
years. Generally one may use the name `entropy' as a synonym for a
possible quantity to measure missing information, keeping in mind
that large classes of possible functions will potentially do the
job, depending on application.

The entire formalism of statistical
mechanics can be regarded as being based on maximizing the entropy
($=$ missing information)
of the system under consideration subject to suitable constraints,
and hence naturally the question arises how to
measure this missing information in the first place \cite{BS}.
While normally
one chooses the Shannon information measure,
in principle more general information
measures (that contain the Shannon information as a special case)
can be chosen as well. These then
formally lead to generalized versions of statistical
mechanics when they are maximized \cite{tsallis, mendes, kaniadakis, naudts, abc, chavanis2}.

In this paper we describe some
generalized information and entropy measures that are useful in this context.
We discuss their most important properties, and point out potential
physical applications.
The physical examples we
choose are the statistics of cosmic rays \cite{cosmic},
defect turbulence \cite{daniels}, and optical lattices
\cite{lutz, renzoni}, but the general techniques developed have applications for a variety
of other complex systems as well,
such as
driven nonequilibrium systems with
large-scale fluctuations of
temperature (so-called
superstatistical systems \cite{beck-cohen,BCS}), hydrodynamic turbulence
\cite{reynolds, prl}, scattering processes in particle physics
\cite{curado, e+e-}, gravitationally interacting systems \cite{plastino, chavanis}
and Hamiltonian systems with long-range interactions and metastable states
\cite{rapisarda,raptsa}. There are applications outside physics as well,
for example
in mathematical finance \cite{borland},
biology \cite{bio} and medicine \cite{chen}.

\subsection{Basic concepts}

One usually restricts the concept of an information measure to an
information that is a function of a given probability distribution
of events (and nothing else)\footnote{An exception to this rule is
the Fisher information, which depends on gradients of the
probability density but will not be discussed here.}. The basic
idea is as follows. Consider a sample set of $W$ possible events.
In physics events are often identified as possible microstates of
the system. Let the probability that event $i$ occurs be denoted
as $p_i$. One has from normalization
\begin{equation}
\sum_{i=1}^Wp_i=1.
\end{equation}
We do not know which event will occur. But suppose that one of
these events, say $j$, finally takes place. Then we have clearly
gained some information, because before the event occurred we did
not know which event would occur.

Suppose that the probability $p_j$ of that observed event $j$ is
close to 1. This means we gain very little information by the
observed occurrence of event $j$, because this event was very
likely anyway. On the other hand, if $p_j$ is close to zero, then
we gain a lot of information by the actual occurrence of event
$j$, because we did not really expect this event to happen. The
information gain due to the occurrence of a single event $j$ can
be measured by a function $h(p_j)$, which should be close to zero
for $p_j$ close to 1. For example, we could choose $h(p_j)=\log
p_j$, the logarithm to some suitable basis $a$. If this choice of
$a$ is $a=2$ then $h$ is sometimes called a 'bit-number'
\cite{BS}. But various other functions $h(p_j)$ are possible as
well, depending on the application one has in mind. In other
words, an information measure should better be regarded as a
man-made construction useful for physicists who don't fully
understand a complex system but try to do so with their limited
tools and ability. Once again we emphasize that an information
measure is not a universally fixed quantity. This fact has led to
many misunderstandings in the community.

In a long sequence of independent trials, in order to determine an
average information gain by the sequence of observed events $i$ we
have to weight the information gain associated with a single event
with the probability $p_i$ that event $i$ actually occurs. That is
to say, for a given function $h$ the average information gained
during a long sequence of trials is
\begin{equation}
I(\{p_i\})=\sum_{i=1}^Wp_ih(p_i). \label{BBB}
\end{equation}
Many information measures studied in the literature are indeed of
this simple trace form. But other forms are possible as well.
One then defines the entropy $S$ as `missing
information', i.e.\
\begin{equation}
S=-I.
\end{equation}
This means the entropy is defined as our missing information on
the actual occurrence of events, given that we only know the
probability distribution of the events.

If the probability distribution is sharply peaked around one
almost certain event $j$, we gain very little information from our
long-term experiment of independent trials: The event $j$ will
occur almost all of the time, which we already knew before.
However, if all events are uniformly distributed, i.e.\ $p_i=1/W$
for all $i$, we get a large amount of information by doing this
experiment, because before we did the experiment we had no idea
which events would actually occur, since they were all equally
likely. In this sense, it is reasonable to assume that an
(elementary) information measure should take on an extremum
(maximum or minimum, depending on sign) for the uniform
distribution. Moreover, events $i$ that cannot occur ($p_i=0$) do
not influence our gain of information in the experiment at all.
In this way we arrive at the most basic principles an information
measure should satisfy.

\subsection{The Khinchin axioms}

There is a more formal way to select suitable (elementary)
information measures, by formulating a set of axioms and then
searching for information measures that satisfy these axioms. A
priori, there is an infinite set of possible information
measures, not only of the simple form (\ref{BBB}) but of more
general forms as well, based on arbitrary functions of the entire
set of $p_i$. How can we select the most suitable ones, given
certain requirements we have in mind? Of course, what is `most
suitable' in this context will in general depend on the
application we have in mind. The most appropriate way of dealing
with this problem is to postulate some basic and essential
properties the information measure one is interested in should
have, and then to derive the functional form(s) that follows from
these postulates.

Khinchin \cite{khinchin} has formulated four axioms that describe
the properties a `classical' information measure $I$ should
have (by `classical' we mean an information measure yielding
ordinary Boltzmann-Gibbs type of statistical mechanics):

\subsubsection*{Axiom 1}
\begin{equation}
I=I(p_1,\cdots ,p_W)
\end{equation}
That is to say, the information measure $I$ only depends on the
probabilities $p_i$
of the events and nothing else.

\subsubsection*{Axiom 2}
\begin{equation}
I(W^{-1},\ldots ,W^{-1})\leq  I(p_1,\cdots ,p_W)
\end{equation}
This means the information measure $I$ takes on an absolute
minimum for the uniform distribution $(W^{-1},\ldots ,W^{-1})$,
any other probability distribution has an information contents
that is larger or equal to that of the uniform distribution.

\subsubsection*{Axiom 3}
\begin{equation}
I(p_1,\ldots,p_W)=I(p_1,\ldots,p_W,0)
\end{equation}
This means the information measure $I$ should not change if the
sample set of events is enlarged by another event that has
probability zero.

\subsubsection*{Axiom 4}
\begin{equation}
I(\{p_{ij}^{I, II}\})=I(\{p_i^I\})+\sum_i p_i^I  I(\{p^{II}(j|i)
\})
\end{equation}
This axiom is slightly more complicated and requires a longer
explanation. The axiom deals with the composition of two systems I
and II (not necessarily independent). The probabilities of the
first system are $p_i^I$, those of the second system are
$p_j^{II}$. The joint system I,II is described by the joint
probabilities $p_{ij}^{I, II}=p_i^Ip^{II}(j|i)$, where
$p^{II}(j|i)$ is the conditional probability of event $j$ in
system II under the condition that event $i$ has already occurred in
system $I$. $I(\{p^{II}(j|i)\})$ is the conditional information of
system II formed with the conditional probabilities $p^{II}(j|i)$,
i.e.\ under the condition that system I is in state $i$.

The meaning of the above axiom is that it postulates that the
information measure should be independent of the way the
information is collected. We can first collect the information in
the subsystem II, assuming a given event $i$ in system I, and then
sum the result over all possible events $i$ in system I, weighting
with the probabilities $p_i^I$.

For the special case that system I and II are independent the
probability of the joint system factorizes as
\begin{equation}
p_{ij}^{I,II}=p_i^Ip_j^{II},
\end{equation}
and only in this case, axiom 4 reduces to the rule of additivity
of information for independent subsystems:
\begin{equation}
I(\{ p_{ij}^{I,II}\})=I(\{p_i^I\})+I(\{p_j^{II}\}) \label{indep}
\end{equation}

Whereas there is no doubt about Axioms 1--3, the reader
immediately notices that Axiom 4 requires a much longer
explanation. From a physical point of view, Axiom 4 is a much less
obvious property. Why should information be independent from the
way we collect it?

To illustrate this point, we may consider a simple example of an
information-collecting system, a first-year undergraduate student
trying to understand physics. This student will learn much more
if he first attends a course on classical mechanics, collecting
all available information there, and then attends a course on
quantum mechanics. If he does it the other way round, he will
probably hardly understand anything in the course on quantum
mechanics, since he does not have the necessary prerequisites. So
attending the quantum mechanics course first leads to zero
information gain.
Apparently, the order in which the information of the two courses
(the two subsystems) is collected is very important and leads to
different results in the achieved knowledge.

In general complex systems, the order in which information is
collected can be very relevant. This is a kind of information
hysteresis phenomenon. In these cases we have situations where
the replacement of Axiom 4 by something more general makes
physical sense. We will come back to this in section 3.

\subsection{The Shannon entropy}

It is easy to verify that the celebrated Shannon entropy, defined
by
\begin{equation}
S=-k \sum_{i=1}^Wp_i\ln p_i
\end{equation}
satisfies all four of the Khinchin axioms. Indeed, up to an
arbitrary multiplicative constant, one can easily show (see, e.g.,
\cite{BS}) that this is the only entropic form that satisfies all
four Khinchin axions, and that it follows uniquely (up to a
multiplicative constant) from these postulates. $k$ denotes the
Boltzmann constant, which in the remaining sections will be set
equal to 1. For the uniform distribution, $p_i=1/W$, the Shannon
entropy takes on its maximum value
\begin{equation}
S=k\ln W,
\end{equation}
which is Boltzmann's famous formula, carved on his grave in
Vienna.
\begin{figure}
\epsfig{file=gentro-fig2.eps, width=8cm, height=9cm} \caption{The grave
of Boltzmann in Vienna. On top of the gravestone the formula $S=k
\log W$ is engraved. Boltzmann laid the foundations for
statistical mechanics, but his ideas were not widely accepted
during his time. He comitted suicide in 1906.}
\end{figure}
 Maximizing the Shannon entropy subject to suitable
constraints leads to ordinary statistical mechanics (see section
4.2). In thermodynamic equilibrium, the Shannon entropy can be
identified as the `physical' entropy of the system, with the
usual thermodynamic relations. Generally, the Shannon entropy has
an enormous range of applications not only in equilibrium
statistical mechanics but also in coding theory, computer
science, etc.

It is easy to verify that $S$ is a concave function of the
probabilities $p_i$, which is an important property to formulate
statistical mechanics. Remember that concavity of a differentiable
function $f(x)$ means $f''(x)\leq 0$ for all $x$. For the Shannon entropy
one has
\begin{eqnarray}
\frac{\partial}{\partial p_i}S&=&-\ln p_i -1 \\
\frac{\partial^2}{\partial p_i\partial
p_j}S&=&-\frac{1}{p_i}\delta_{ij}\leq 0,
\end{eqnarray}
and hence, as a sum of concave functions of the single probabilities $p_i$,
the Shannon entropy $S$ is a concave function.

In classical mechanics, one often has a continuous variable $u$
with some probability density $p(u)$, rather than discrete
microstates $i$ with probabilities $p_i$. In this case the
normalization condition reads $\int_{-\infty}^\infty p(u)du=1$,
and the Shannon entropy associated with this probability density
is defined as
\begin{equation}
S=-\int_{-\infty}^{\infty}du p(u)\ln (\sigma p(u)), \label{14}
\end{equation}
where $\sigma$ is a scale parameter that has the same dimension as
the variable $u$. For example, if $u$ is a velocity (measured in
units of $m/s$), then $p(u)$, as a probability density of velocities,
has the dimension $s/m$, since $p(u)du$ is a dimensionless quantity.
As a consequence, one needs to introduce the scale parameter $\sigma$
in eq.~(\ref{14}) to make the argument of the logarithm dimensionless.

Besides the Shannon information, there are lots of other
information measures. We will discuss some of the most important
examples in the next section. Some information measures are more
suitable than others for the description of various types of
complex systems. We will discuss the axiomatic foundations that
lead to certain classes of information measures. Important
properties to check for a given information measure are convexity,
additivity, composability, and stability. These properties can
help to select the most suitable generalized information measure
to describe a given class of complex systems.

\section{More general information measures}

\subsection{The R\'{e}nyi entropies}

We may replace axiom 4 by the less stringent condition
(\ref{indep}), which just states that the entropy of independent
systems should be additive. In this case one ends up with other
information measures which are called the {\em R\'{e}nyi entropies}.
\cite{renyi}. These are defined for an arbitrary real parameter
$q$ as
\begin{equation}
S_q^{(R)} =\frac{1}{q-1}\ln \sum_i p_i^q.\label{renyi}
\end{equation}
The summation is over all events $i$ with $p_i\not=0$. The R\'{e}nyi
entropies satisfy the Khinchin axioms 1--3 and the additivity
condition (\ref{indep}). Indeed, they follow uniquely from these
conditions, up to a multiplicative constant. For $q\to 1$ they
reduce to the Shannon entropy:
\begin{equation}
\lim_{q\to 1}S_q^{(R)}=S,
\end{equation}
as can be easily derived by
setting $q=1+\epsilon$ and doing a perturbative expansion in the small
parameter $\epsilon$ in eq.~(\ref{renyi}).

The R\'{e}nyi information measures are important for the
characterization of multifractal sets (i.e., fractals with a
probability measure on their support \cite{BS}), as well as for
certain types of applications in computer science. But do they
provide a good information measure to develop a generalized
statistical mechanics for complex systems?

At first sight it looks nice that the R\'{e}nyi entropies are additive
for independent subsystems for general $q$, just as the Shannon
entropy is for $q=1$. But for non-independent subsystems I and II
this simplicity vanishes immediately: There is no simple formula
of expressing the total R\'{e}nyi entropy of a joint system as a
simple function of the R\'{e}nyi entropies of the interacting
subsystems.

Does it still make sense to generalize statistical mechanics using
the R\'{e}nyi entropies? Another problem arises if one checks whether
the R\'{e}nyi entropies are a convex function of the probabilities.
The R\'{e}nyi entropies do not possess a definite convexity---the
second derivative with respect to the $p_i$ can be positive or
negative. For formulating a generalized statistical mechanics,
this poses a serious problem. Other generalized information
measures are better candidates--we will describe some of those
in the following.

\subsection{The Tsallis entropies}
The {\em Tsallis entropies} (also called $q$-entropies) are given by the
following expression \cite{tsallis}:
\begin{equation}\label{s}
S_q^{(T)} =  \frac{1}{q-1}\left(1-\sum_{i=1}^W p_i^q\right).
\label{tsallis-entro}
\end{equation}
One finds definitions similar to eq.~(\ref{tsallis-entro}) already
in earlier papers such as e.g.\ \cite{havrda}, but it was Tsallis
in his seminal paper \cite{tsallis} who for the first time
suggested to generalize statistical mechanics using these
entropic forms. Again $q\in\cal{R}$ is a real parameter, the
entropic index. As the reader immediately sees, the Tsallis
entropies are different from the R\'{e}nyi entropies: There is no
logarithm anymore. A relation between R\'{e}nyi and Tsallis entropies
is easily derived by writing
\begin{equation}
\sum_ip_i^q=1-(q-1)S_q^{(T)}=e^{(q-1)S_q^{(R)}}
\end{equation}
which implies
\begin{equation}
S_q^{(T)}=\frac{1}{q-1}(1-e^{(q-1) S_q^{(R)}}). \label{tsallis-e}
\end{equation}
Apparently the Tsallis entropy is a monotonous function of the
R\'{e}nyi entropy, so any maximum of the Tsallis entropy will also be
a maximum of the R\'{e}nyi entropy and vice versa. But still, Tsallis
entropies have many distinguished properties that make them a
better candidate for generalizing statistical mechanics than,
say, the R\'{e}nyi entropies.

One such property is concavity. One easily verifies that
\begin{eqnarray}
\frac{\partial}{\partial p_i}S_q^{(T)}&=&-\frac{q}{q-1}p_i^{q-1} \\
\frac{\partial^2}{\partial p_i\partial p_j}S_q^{(T)}&=&
-qp_i^{q-2}\delta_{ij}.
\end{eqnarray}
This means that, as a sum of concave functions, $S_q^{(T)}$ is concave for all $q>0$ (convex for all
$q<0$). This property is missing for the R\'{e}nyi entropies. Another
such property is the so-called Lesche-stability, which is
satisfied for the Tsallis entropies but not satisfied by the
R\'{e}nyi entropies (see section 3.3 for more details).

The Tsallis entropies also contain the Shannon entropy
\begin{equation}
S= -\sum_{i=1}^Wp_i\ln p_i\label{s1}
\end{equation}
as a special case. Letting $q\to 1$ we have
 \begin{equation}
S_1^{(T)}=\lim_{q\to 1}S_q^{(T)} =S
\end{equation}

As expected from a good information measure, the Tsallis entropies
take on their extremum for the uniform distribution $p_i =
1/W\;\;\forall{i}$. This extremum is given by
\begin{equation}
S_q^{(T)}= \frac{W^{1-q}-1}{1-q}
\end{equation}
which, in the limit $q \rightarrow 1$, reproduces Boltzmann's
celebrated formula $S= \ln W$.

It is also useful to write down the definition of Tsallis
entropies for a continuous probability density $p(u)$ with
$\int_{-\infty}^\infty p(u)du=1$, rather than a discrete set of
probabilities $p_i$ with $\sum_i p_i=1$. In this case one defines
\begin{equation}
S_q^{(T)}=\frac{1}{q-1}\left(
1-\int_{-\infty}^{+\infty}\frac{du}{\sigma} (\sigma p(u))^q\right)
, \label{25}
\end{equation}
where again $\sigma$ is a scale parameter that has the same dimension as
the variable $u$. It is introduced
for a similar reason as before, namely to make the
integral in eq.~(\ref{25}) dimensionless
so that it can be substracted from 1.
For $q\to 1$ eq.~(\ref{25}) reduces to the Shannon entropy
\begin{equation}
S_1^{(T)}=S=-\int_{-\infty}^{\infty}du p(u) \ln (\sigma p(u)).
\end{equation}

A fundamental property of the Tsallis entropies is the fact that
they are not additive for independent subsystems. In fact, they
have no chance to do so, since they are different from the R\'{e}nyi
entropies, the only solution to eq.~(\ref{indep}).

To investigate this in more detail, let us consider two
independent subsystems I and II with probabilities $p_i^I$ and
$p_j^{II}$, respectively. The probabilities of joint events $i,j$
for the combined system I,II are $p_{ij}=p_i^Ip_j^{II}$. We may
then consider the Tsallis entropy for the first system, denoted as
$S_q^I$, that of the second system, denoted as $S_q^{II}$, and
that of the joint system, denoted as $S_q^{I,II}$. One has
\begin{equation}
S_q^{I, II}=S_q^I +S_q^{II}-(q-1) S_q^{I}S_q^{II}.\label{pa}
\end{equation}

 {\small
{\bf Proof of eq.~(\ref{pa}):} We may write
\begin{eqnarray}
\sum_i p_i^{Iq} &=&1-(q-1)S_q^I \label{aa}\\ \sum_j p_j^{IIq}
&=&1-(q-1)S_q^{II} \label{bb}
\\
\sum_{i,j} p_{ij}^q&=&
\sum_i(p_i^{I})^q\sum_j(p_j^{II})^q  \nonumber
\\ \,&=&
1-(q-1)S_q^{I, II}.\label{there}
\end{eqnarray}
From eqs.~(\ref{aa}) and (\ref{bb}) it also follows that
\begin{eqnarray}
\sum_i(p_i^{I})^q\sum_j(p_j^{II})^q &=&1-(q-1)S_q^I-(q-1)S_q^{II}
\nonumber \\
&+&(q-1)^2 S_q^IS_q^{II}. \label{here}
\end{eqnarray}
Combining eqs.~(\ref{there}) and (\ref{here}) one ends up with
eq.~(\ref{pa}), q.e.d.}

Apparently, if we put together two independent subsystems then the
Tsallis entropy is not additive but there is a correction term
proportional to $q-1$, which vanishes for $q=1$ only, i.e.\ for
the case where the Tsallis entropy reduces to the Shannon entropy.
Eq.~(\ref{pa}) is sometimes called the `pseudo-additivity'
property.

\begin{figure}
\epsfig{file=gentro-fig3.eps, width=8cm, height=8cm}
\caption{If the nonadditive entropies $S_q$ are used to measure
information, then
the information contents of
two systems I, II (blue) that are put together is not equal
to the sum of the
information contents of the isolated single systems.
In other words, there is
always an interaction between the subsystems (red).}
\end{figure}

Eq.~(\ref{pa}) has given rise to the name {\em nonextensive
statistical mechanics}. If we formulate
a generalized statistical mechanics based on maximizing Tsallis
entropies, then the (Tsallis) entropy of {\em independent} systems
is not additive. However, it turns out that for special types of
{\em correlated} subsystems, the Tsallis entropies do become
additive if the subsystems are put together \cite{gell-mann}.
This means, for these types of correlated complex systems a
description in terms of Tsallis entropies  in fact can make
things simpler as compared to using the Shannon entropy, which is
non-additive for correlated subsystems.

\subsection{Landsberg-Vedral entropy}

Let us continue with a few other examples of generalized
information measures. Consider
\begin{equation}
S_q^{(L)}=\frac{1}{q-1}\left(\frac{1}{\sum_{i=1}^Wp_i^q}-1\right).
\end{equation}
This measure was studied by Landsberg and Vedral
\cite{landsberg-vedral}. One immediately sees that the
Landsberg-Vedral entropy is related to the Tsallis entropy
$S_q^{(T)}$ by
\begin{equation}
S_q^{(L)}=\frac{S_q^{(T)}}{\sum_{i=1}^Wp_i^q},
\end{equation}
and hence $S_q^{(L)}$ is sometimes also called {\em normalized Tsallis
entropy}. $S_q^{(L)}$ also contains the Shannon entropy as a
special case
\begin{equation}
\lim_{q\to 1}S_q^{(L)}=S_1
\end{equation}
and one readily verifies that it also satisfies a
pseudo-additivity condition for independent systems, namely
\begin{equation}
S_q^{(L)I,II}=S_q^{(L)I}+S_q^{(L)II}+(q-1)S_q^{(L)I}S_q^{(L)II}
.
\end{equation}
This means that in the pseudo-additivity relation (\ref{pa}) the role of
$(q-1)$ and $-(q-1)$ is exchanged.

\subsection{Abe entropy}

Abe \cite{abe-entropy} introduced a kind of symmetric modification
of the Tsallis entropy, which is invariant under the exchange $q
\longleftrightarrow q^{-1}$. This is given by
\begin{equation}
S_q^{Abe}=-\sum_i \frac{p_i^q-p_i^{q^{-1}}}{q-q^{-1}} \label{sabe}
\end{equation}
This symmetric choice in $q$ and $q^{-1}$ is inspired by the
theory of quantum groups which often exhibits invariance under
the `duality transformation' $q\to q^{-1}$. Like Tsallis entropy,
the Abe entropy is also concave. In fact, it is related to the
Tsallis entropy $S_q^T$ by
\begin{equation}
S_q^{Abe}= \frac{(q-1)S_q^T-(q^{-1}-1)S_{q^{-1}}^T}{q-q^{-1}}.
\end{equation}
Clearly the relevant  range of $q$ is now just the unit
interval$(0,1]$, due to the symmetry $q \longleftrightarrow
q^{-1}$: Replacing $q$ by $q^{-1}$ in eq.~(\ref{sabe}) does not
change anything.

\subsection{Kaniadakis entropy}
The {\em Kaniadakis entropy} (also called $\kappa$-entropy) is defined
by the following expression \cite{kaniadakis}
\begin{equation}
S_\kappa = \sum_i \frac{p_i^{1+\kappa}-p_i^{1-\kappa}}{2\kappa}
\end{equation}
Again this is a kind of deformed Shannon entropy, which reduces
to the original Shannon entropy for $\kappa =0$. We also note that
for small $\kappa$, and by  writing $q=1+\kappa$, $q^{-1} \approx
1-\kappa$, the Kaniadakis entropy approaches the Abe entropy.
Kaniadakis was motivated to introduce this entropic form by
special relativity: The relativistic sum of two velocities of
particles of mass $m$ in special relativity satisfies a similar
relation as the Kaniadakis entropy does, identifying
$\kappa=1/mc$.
Kaniadakis entropies are also concave and Lesche stable (see
section 3.3).

\subsection{Sharma-Mittal entropies}

These are two-parameter families of entropic forms
\cite{sharma-mittal}. They can be written in the form
\begin{equation}
S_{\kappa ,r}= -\sum_i p_i^r\left( \frac{p_i^\kappa -p_i^{-\kappa}}{2 \kappa} \right)
\end{equation}
Interestingly, they contain many of the entropies mentioned so far as special cases.
The Tsallis entropy is obtained for $r=\kappa$ and $q=1-2\kappa$. The Kaniadakis entropy
is obtained for $r=0$. The Abe entropy is obtained for $\kappa =\frac{1}{2} (q-q^{-1})$
and $r=\frac{1}{2}(q+q^{-1})-1$. The Sharma-Mittal entropes are concave and Lesche stable.

\section{Selecting a suitable information measure}

\subsection{Axiomatic foundations}

The Khinchin axioms apparently are the right axioms to obtain the
Shannon entropy in a unique way, but this concept may be too
narrow-minded if one wants to describe general complex systems.
In physics, for example, one may be interested in nonequilibrium
systems with a stationary state, glassy systems, long transient
behaviour in systems with long-range interactions, systems with
multifractal phase space structure etc. In all these cases one
should be open-minded to allow for generalizations of axiom 4,
since it is this axiom that is least obvious in the given
circumstances.

Abe \cite{abe-axiom} has shown that the Tsallis entropy follows
uniquely (up to an arbitrary multiplicative constant) from the
following generalized version of the Khinchin axioms. Axioms 1--3
are kept, and Axiom 4 is replaced by the following more general
version:

\subsubsection*{New Axiom 4}
\begin{equation}
S_q^{I, II}=S_q^I+S_q^{II|I}-(q-1)S_q^IS_q^{II|I}
\end{equation}
Here $S_q^{II|I}$ is the conditional entropy formed with the
conditional probabilities $p(j|i)$ and averaged over all states
$i$ using the so-called escort distributions $P_i$:
\begin{equation}
S_q^{II|I}=\sum_i P_i S_q(\{ p(j|i) \}).
\end{equation}
Escort distributions $P_i$ were introduced quite generally in \cite{BS}
and are defined for any given probability distribution $p_i$ by
\begin{equation}
P_i= \frac{p_i^q}{\sum_i p_i^q}.
\end{equation}
For $q=1$, the new axiom 4 reduces to the old Khinchin axiom 4,
i.e. $S_q^{I, II}=S_q^I+S_q^{II|I}$. For independent systems I and
II, the new axiom 4 reduces to the pseudo-additivity property
(\ref{pa}).

The meaning of the new axiom 4 is quite clear. It is a kind of
minimal extension of the old axiom 4: If we collect information
from two subsystems, the total information should be the sum of
the information collected from system I and the conditional
information from system II, plus a correction term. This
correction term can a priori be anything, but we want to restrict
ourselves to information measures where
\begin{equation}
S^{I,II}=S^I+S^{II|I}+g(S^I,S^{II|I}),
\end{equation}
where $g(x,y)$ is some function. The property that the entropy of
the composed system can be expressed as a function of the
entropies of the single systems is sometimes referred to as the
{\em composability} property. Clearly, the function $g$ must depend
on the entropies of {\em both} subsystems, for symmetry reasons.
The simplest form one can imagine is that it is given by
\begin{equation}
g(x,y)=\mbox{const} \cdot xy,
\end{equation}
i.e.\ it is proportional to both the entropy of the first system
and that of the second system. Calling the proportionality
constant $q-1$, we end up with the new axiom 4.

It should, however, be noted that we may well formulate other
axioms, which then lead to other types of information measures.
The above generalization is perhaps the one that requires least
modifications as compared to the Shannon entropy case. But
clearly, depending on the class of complex systems considered, and
depending on what properties we want to describe, other axioms
may turn out to be more useful. For example, Wada and Suyari
\cite{wada} have suggested a set of axioms that uniquely lead to
the Sharma-Mittal entropy.

\subsection{Composability} Suppose we have a given complex system
which consists of subsystems that interact in a complicated way.
Let us first analyze two subsystems I and II in an isolated way
and then put these two dependent systems I and II together. Can
we then express the generalized information we have on the total
system as a simple function of the information we have on the
single systems? This question is sometimes referred to as the
composability problem.

The Tsallis entropies are composable in a very simple way. Suppose
the two systems I and II are not independent. In this case
one can still write the joint probability $p_{ij}$ as a product
of the single probability $p_i$ and conditional probability
$p(j|i)$, i.e.\ the probability of event $j$ under the condition
that event $i$ has already occurred is
\begin{equation}
p_{ij}=p(i|j)p_j.
\end{equation}
The conditional Tsallis entropy associated with system II (under
the condition that system I is in state $i$) is given by
\begin{equation}
S_q^{II|i}=\frac{1}{q-1}(1-\sum_j p(j|i)^q).
\end{equation}
One readily verifies the relation
\begin{equation}
S_q^I+\sum_ip_i^qS_q^{II|i}=S_q^{I, II}. \label{rela}
\end{equation}
This equation is very similar to that satisfied by the Shannon
entropy in Axiom 4. In fact, the only difference is that there is
now an exponent $q$ that wasn't there before. It means our
collection of information is biased: Instead of weighting the
events in system I with $p_i$ we weight them with $p_i^q$. For
$q=1$ the above equation of course reduces to the fourth of the
Khinchin axioms, but only in this case. Hence, for general
$q\not=1$, the Tsallis information is {\em not} independent of the
way it is collected for the various subsystems.

To appreciate the simple composability property of the Tsallis
entropy, let us compare with other entropy-like functions, for
example the R\'{e}nyi entropy. For the R\'{e}nyi entropy there is no
simple composability property similar to eq.~(\ref{rela}). Only
the exponential of the Renyi entropy satisfies a relatively simple
equation, namely
\begin{equation}
\exp \left((q-1)S_q^{(R)I,II}\right)=\sum_ip_i^q\exp \left( (q-1)
S_q^{(R)II|i}\right) . \label{corres}
\end{equation}
However, by taking the exponential one clearly removes the
logarithm in the definition of the R\'{e}nyi entropies in
eq.~(\ref{renyi}). This means one is effectively back to the
Tsallis entropies.

\subsection{Lesche stability}

Physical systems contain noise. A necessary requirement for a
generalized entropic form $S[p]$ to make physical sense is that
it must be stable under small perturbations. This means a small
perturbation of the set of probabilities $p:=\{p_i\}$ to a new
set $p'=\{p_i'\}$  should have only a small effect on the value
$S_{max}$ of $S_q[p]$ in the thermodynamic state that maximizes
the entropy. This should in particular be true in the limit $W\to
\infty$ (recall that $W$ denotes the number of microstates). The
stability condition can be mathematically expressed as follows
\cite{lesche}:

\subsubsection*{Stability condition}
For every $\epsilon >0$ there is a $\delta >0$ such that
\begin{equation}
||p-p'||_1\leq \delta \Longrightarrow \left|
\frac{S[p]-S[p']}{S_{max}}\right| < \epsilon \label{stability}
\end{equation}
for arbitrarily large $W$. Here $||A||_1=\sum_{i=1}^W|A_i|$
denotes the $L_1$ norm of an observable $A$.

Abe \cite{abe-pre} has proved that the Tsallis entropies are
Lesche-stable, i.e.\ they satisfy eq.~(\ref{stability}) for all
$q$, whereas the R\'{e}nyi entropies and the Landsberg entropies are
not stable for any $q\not= 1$ (for a discrete
set of probabilities $p_i$ with $W\to \infty$).
This is an important criterion to
single out generalized entropies that may have potential physical
meaning. According to the stability criterion, the Tsallis
entropies are stable and thus may be associated with physical
states, whereas the other two examples of entropic forms have a
stability problem.
Kaniadakis entropies and Sharma Mittal
entropies are also Lesche-stable. Only entropies that are
Lesche-stable are good candidates for physically relevant
information measures. For a recent re-investigation of
these types of stability problems in a physical setting, see \cite{thurner}.

\section{Maximizing generalized entropies}

\subsection{A rigorous derivation of statistical mechanics?}

The rigorous foundations of statistical mechanics are a kind of
miracle. There is little progress in rigorously deriving
statistical mechanics from the microscopic classical Hamiltonian
equations of motion, neither there is a rigorous derivation
starting from quantum mechanics or quantum field theory.
It is
almost surprising how well statistical mechanics
works in daily life, given its lack of rigorous
derivation from other microscopic theories.

The problem is that for a rigorous derivation of statistical mechanics
from dynamical systems theory one needs
the underlying dynamical
system to be {\em ergodic}, and even that is not enough: It
should have the
stronger property of {\em mixing}.
Ergodicity essentially means
that
typical trajectories fill out the entire phase space
(which implies that for typical trajectories the time average
is equal to the ensemble average) and mixing
means asymptotic independence, i.e.\ the correlation function
of distant events decays to zero if the time difference
between the events goes to infinity. For strongly
chaotic dynamical systems
(i.e.\
those exhibiting exponential sensitive dependence
on the initial conditions) one normally expects the mixing
property to hold (though there are some mathematical subtleties
here).
From a mathematical point of view,
the mixing property is the theoretical ingredient
that is needed to guarantee the approach to
an equilibrium state in statistical mechanics.

Unfortunately,
ergodicity and mixing can only be rigorously
proved for simple toy examples
of dynamical systems, for example the discrete-time map $x_{n+1}=1-2x_n^2$
with initial values in the interval $[-1,1]$ or other
very simple toy models (see, e.g. \cite{BS}). For realistic systems
of physical relevance, such as the Hamiltonian equations of a large
number of weakly or strongly interacting particles, a rigorous
mathematical proof of the mixing property does
not exist, and the deeper reason why
statistical mechanics works
so well in typical situations remains a miracle.

\subsection{Jaynes' information theory}
In view of the fact that there are no rigorous foundations of
statistical mechanics, one usually sticks to some simple principle such
as the maximum entropy principle in order to `derive' it.
Jaynes has given a simple and plausible interpretation of the
maximum entropy principle \cite{jaynes}. His interpretation is
purely based on concepts from information theory, and applicable
to many problems, not only to equilibrium statistical mechanics.

In simple words, the idea is as follows. Assume we have only
limited information on a system containing many particles or
constituents. We may know the mean values of some observables
$M^\sigma$, $\sigma=1,\cdots, s$ but nothing else. For example,
we may just know one such quantity, the mean energy of all
particles and nothing else ($s=1$). What probability
distributions $p_i$ should we now assume, given that we have such
limited information on the system?

Suppose we measure information with some information measure
$I(\{p_i\})=:I[p]$. Among all distributions possible that lead to
the above known mean values $M^\sigma$ we should select those that
do not contain any unjustified prejudices. In other words, our
information measure for the relevant probabilities should take on
a minimum, or the entropy ($=$ negative information) should take a
maximum, given the constraints. For, if the information associated
with the selected probability distribution does not take on a
minimum, we have more information than the minimum information,
but this means we are pre-occupied by a certain belief or
additional information, which we should have entered as a
condition of constraint in the first place.

Of course, if we have no knowledge on the system at all ($s=0$),
the principle yields the uniform distribution $p_i=1/W,
\;i=1,\ldots W$ of events. For this to happen, the information
measure $I[p]$ must only satisfy the second Khinchin axiom,
nothing else. In statistical mechanics, the corresponding ensemble
is the microcanonical ensemble.

If some constraints are given, we have to minimize the information
($=$ maximize the entropy) subject to the given constraints. A
constraint means that we know that some observable $\tilde{M}$ of
the system, which takes on the values $M_i$ in the microstates
$i$, takes on the fixed mean value $M$. In total, there can be $s$
such constraints, corresponding to $s$ different observables
$\tilde{M}^\sigma$:
\begin{equation}
\sum_i p_i M_i^\sigma =M^\sigma  \;\;\;\;(\sigma=1, \ldots , s).
\end{equation}
For example, for the canonical ensemble of equilibrium statistical
mechanics one has the constraint that the mean value $U$ of the
energies $E_i$ in the various microstates is fixed:
\begin{equation}
\sum_i p_i E_i =U
\end{equation}
We may also regard the fact that the probabilities $p_i$ are
always normalized as a constraint obtained for the special choice
$\tilde{M}=1$:
\begin{equation}
\sum_i p_i =1.
\end{equation}

To find the distributions that maximize the entropy under the
given constraints
one can use the method of Lagrange
multipliers. One simply defines a function $\Psi [p]$ which is the
information measure under consideration plus the condition of
constraints multiplied by some constants $\beta_\sigma$ (the
Lagrange multipliers):
\begin{equation}
\Psi[p]=I[p]+\sum_\sigma \beta_\sigma (\sum_i p_iM_i^\sigma).
\label{psi}
\end{equation}
One then looks for the minimum of this function in the space of
all possible probabilities $p_i$. In practice, these distributions
$p_i$ are easily obtained by evaluating the condition
\begin{equation}
\frac{\partial}{\partial p_i} \Psi[p]=0 \;\;\;(i=1,\ldots ,W),
\end{equation}
which means that $\Psi$ has a local extremum. We obtain
\begin{equation}
\frac{\partial}{\partial p_i} I[p] +\sum_\sigma \beta_\sigma
M_i^\sigma = 0 \label{nursery}
\end{equation}
All this is true for {\em any} information measure $I[p]$, it need not be
the Shannon information.
At this point we see why it is important that the information
measure $I[p]$ is convex: We need a well-defined inverse function
of $\frac{\partial}{\partial p_i}I[p]$, in order to uniquely solve
eq.~(\ref{nursery}) for the $p_i$. This means
$\frac{\partial}{\partial p_i}I[p]$ should be a monotonous
function, which means that $I[p]$ must be convex.

Note that Jaynes' principle is (in principle) applicable to all
kinds of complex systems, many different types of observables, and
various types of information measures. There is no reason to
restrict it to equilibrium statistical mechanics only. It's
generally applicable to all kinds of problems where one has
missing information on the actual microscopic state of the system
and wants to make a good (unbiased) guess of what is happening
and what should be done. The concept of avoiding unjustified
prejudices applies in quite a general way. An important question
is which information measure is relevant for which system.
Clearly, the Shannon entropy is the right information measure to
analyse standard type of systems in equilibrium statistical
mechanics. But other systems of more complex nature can
potentially be described more effectively if one uses different
information measures, for examples those introduced in the
previous section.

\subsection{Ordinary statistical mechanics}

For ordinary statistical mechanics, one has $I[p]=\sum_i p_i\ln
p_i$ and $\frac{\partial}{\partial p_i} I[p]=1+\ln p_i$. For the
example of a canonical ensemble eq.~(\ref{psi}) reads
\begin{equation}
\Psi[p] =\sum_i p_i\ln p_i +\alpha \sum_i p_i + \beta \sum_i p_i
E_i
\end{equation}
and eq.~(\ref{nursery}) leads to
\begin{equation}
\ln p_i +1+\alpha +\beta E_i =0.
\end{equation}
Hence the maximum entropy principle leads to the canonical
distributions
\begin{equation}
p_i=\frac{1}{Z}e^{-\beta E_i}.
\end{equation}
The partition function $Z$ is related to the Lagrange multiplier
$\alpha$ by
\begin{equation}
Z:=\sum_i e^{-\beta E_i} =e^{1+\alpha}.
\end{equation}

\subsection{Generalized statistical mechanics}

More generally we may start from a generalized information
measure of the trace form
\begin{equation}
I[p]=-S[p]=\sum_i p_i h(p_i)
\end{equation}
where $h$ is some suitable function, as introduced before. Tsallis
entropy, Abe entropy, Kaniadakis entropy, Sharma-Mittal entropy
and Shannon entropy are examples that can all be cast into this
general form, with different functions $h$ of course. Again let
us consider the canonical ensemble (the extension to further
constraints/other ensembles is straightforward). The functional
to be maximized is then
\begin{equation}
\Psi [p]= \sum_i p_i h(p_i) +\alpha \sum_i p_i +\beta \sum_i p_i
E_i
\end{equation}
By evaluating the condition
\begin{equation}
\frac{\partial}{\partial p_i} \Psi [p]=0
\end{equation}
we obtain
\begin{equation}
h(p_i)+p_ih'(p_i)+\alpha +\beta E_i =0
\end{equation}
Defining a function $g$ by
\begin{equation}
g(p_i):=h(p_i)+p_ih'(p_i)
\end{equation}
we end up with
\begin{equation}
g(p_i)=-\alpha -\beta E_i
\end{equation}
Hence, if a unique inverse function $g^{-1}$ exists, we have
\begin{equation}
p_i=g^{-1} (-\alpha -\beta E_i)
\end{equation}
and this is the generalized canonical distribution of the
generalized statistical mechanics.

Let us consider a few examples of interesting functions functions $h$.
For the Shannon entropy one has of course
\begin{equation}
h(p_i)=\ln p_i.
\end{equation}
For the Tsallis entropy,
\begin{equation}
h(p_i)= \frac{p_i^{q-1}-1}{q-1}=: \log_{2-q} (p_i).
\end{equation}
This is like a deformed logarithm that approaches the ordinary
logarithm for $q\to 1$. In fact, a useful definition commonly
used in the field is the so-called $q$-logarithm defined by
\begin{equation}
\log_q (x):= \frac{x^{1-q}-1}{1-q}.
\end{equation}
Its inverse function is the $q$-exponential
\begin{equation}
e_q^x:= (1+(1-q)x)^{\frac{1}{1-q}}.
\end{equation}
For the Kaniadakis entropy one has
\begin{equation}
h(p_i)= \frac{p_i^\kappa -p_i^{-\kappa}}{2\kappa} =: \ln_\kappa (x),
\end{equation}
where the $\kappa$-logarithm is defined as
\begin{equation}
\ln_\kappa (x) =\frac{x^\kappa -x^{-\kappa}}{2 \kappa}.
\end{equation}
Its inverse is the $\kappa$-exponential
\begin{equation}
exp_\kappa (x) = ( \sqrt{1+\kappa^2 x^2}+\kappa x)^{\frac{1}{\kappa}}
\end{equation}
Essentially, the generalized canonical distributions obtained by
maximizing Tsallis entropies are given by $q$- exponentials
of the energy $E_i$ and
those by maximizing Kaniadakis entropies are
$\kappa$-exponentials. Both decay with a power law for large
values of the energy $E_i$.

\subsection{Nonextensive statistical me\-chanics}

Let us consider in somewhat more detail a generalized statistical
mechanics based on Tsallis entropies. If we start from the
Tsallis entropies $S_q^{(T)}$ and maximize those subject to
suitable constraint, the corresponding formalism is called
{\em nonextensive statistical mechanics}. We have
\begin{equation}
I_q^{(T)}[p]=-S_q^{(T)}[p]=\frac{1}{q-1} (1-\sum_i p_i^q),
\end{equation}
thus
\begin{equation}
\frac{\partial}{\partial p_i} I_q^{(T)}[p]=
\frac{q}{q-1}p_i^{q-1}.
\end{equation}
For a canonical ensemble eq.~(\ref{nursery}) leads to
\begin{equation}
\frac{q}{q-1} p_i^{q-1}+\alpha +\beta E_i =0.
\end{equation}
Thus the maximum entropy principle leads to generalized canonical
distributions of the form
\begin{equation}
p_i=\frac{1}{Z_q}(1-\beta (q-1)E_i)^{\frac{1}{q-1}},
\end{equation}
where $Z_q$ is a normalization constant. This is the original
formula Tsallis introduced in his paper \cite{tsallis}. These
days, however, the convention has become to replace the parameter
$q$ by
 $q'=2-q$
and then rename $q' \to q$. That is to say, the generalized
canonical distributions in nonextensive statistical mechanics are
given by the following $q$-exponentials:
\begin{equation}
p_i=\frac{1}{Z_q}(1+\beta (q-1)E_i)^{\frac{-1}{q-1}}.
\end{equation}
They live on a bounded support for $q < 1$ and exhibit power-law
decays for $q>1$.

Starting from such a $q$-generalized approach, one can easily
derive formal $q$-generalized thermodynamic relations. The details
depend a bit how the constraints on energy taken into account
\cite{mendes}. All relevant thermodynamic quantities now get an
index $q$. Typical examples of such formulas are
\begin{equation}
1/T= \beta=\partial S_q^{(T)} /\partial U_q,\;\;\forall q
\end{equation}
with
\begin{equation}
\sum_{i=1}^W (p_i)^q=(\bar{Z}_q)^{1-q},
\end{equation}
\begin{equation}
F_q \equiv U_q - T S_q = -\frac{1}{\beta}
\frac{(Z_q)^{1-q}-1}{1-q}
\end{equation}
and
\begin{equation}
U_q = -\frac{\partial}{\partial \beta}\frac{(Z_q)^{1-q}-1}{1-q},
\end{equation}
where
\begin{equation}
\frac{(Z_q)^{1-q}-1}{1-q}=\frac{(\bar{Z}_q)^{1-q}-1}{1-q}-\beta
U_q.
\end{equation}
and
\begin{equation}
\sum_{i=1}^WP_i E_i= \frac{\sum_{i=1}^W p_i^q E_i }{ \sum_{i=1}^W
p_i^q }= U_q
\end{equation}
There are some ambiguities on how to take into account the constraints,
using for example the original $p_i$ or the escort distributions $P_i$,
but we will not comment on these technicalities here.

\section{Some physical examples}

\subsection{Making contact with experimental data}

It should be clear that a direct physical measurements of
generalized entropy measures is impossible since these are
basically man-made information-theoretic tools. However, what can
be measured is the stationary probability distribution of certain
observables of a given complex system, as well as possibly some
correlations between subsystems. As we have illustrated before,
measured probability densities in some complex system that
deviate from the usual Boltzmann factor $e^{-\beta E}$ can then
be formally interpreted as being due to the maximization of a
more general information measure that is suitable as an effective
description for the system under consideration.

In this approach one regards the complex system as a kind of
`black box'. Indeed many phenomena in physics, biology, economics,
social sciences, etc.
 are so complicated that there is not a simple
equation describing them, or at least we do not know this
equation. A priori we do not know what is the most suitable way to
measure information for any output that we get from our black
box. But if a distribution $p_i$ of some observable output is
experimentally measured, we can indirectly construct a
generalized entropic form that takes a maximum for this
particular observed distribution. This allows us to make contact
with experimental measurements, make some predictions e.g. on
correlations of subsystems and translate the rather abstract
information theoretical concepts into physical reality.

\subsection{Statistics of cosmic rays}
Our first example of making contact to concrete measurements is
cosmic ray statistics \cite{cosmic}. The earth is constantly
bombarded with highly energetic particles, cosmic rays.
Experimental data of the measured cosmic ray energy spectrum are
shown in Fig.~4.
\begin{figure}
\epsfig{file=gentro-fig4.eps, width=8cm, height=7cm}
\caption{Measured energy spectrum of cosmic rays and a fit by
eq.~(\ref{can}) with $q=1.215$.
The `knee' and `ankle' are structures that go
beyond the simple model considered here.}
\end{figure}
It has been known for a long time that the observed distribution of cosmic rays
with a given energy $E$ exhibits strongly pronounced power laws
rather than exponential decay. It turns out that the observed
distribution is very well fitted over a very large range of
energies by the formula
\begin{equation}
p(E)=C \cdot \frac{E^2}{(1+b(q-1)E)^{1/(q-1)}}. \label{can}
\end{equation}
Here $E$ is the energy of the cosmic ray particles,
\begin{equation}
E=\sqrt{c^2p_x^2+c^2p_y^2+c^2p_z^2+m^2c^4},
\end{equation}
$b=(k\tilde{T})^{-1}$ is an effective inverse temperature
variable, and $C$ is a constant representing the total flux rate.
For highly relativistic particles the rest mass $m$ can be neglected and
one has $E\approx c |\vec{p}|$. The reader immediately recognizes
the distribution (\ref{can}) as a $q$-generalized relativistic
Maxwell-Boltzmann distribution, which maximizes the Tsallis
entropy. The factor $E^2$ takes into account the available phase
space volume. As seen in Fig.~4, the cosmic ray spectrum is very
well fitted by the distribution (\ref{can}) if the entropic index
$q$ is chosen as $q=1.215$ and if the effective temperature
parameter is given by $k\tilde{T}=b^{-1}=107$ MeV.
Hence the measured cosmic ray spectrum effectively
maximizes the Tsallis entropy.

The deeper reason why this is so could be
temperature fluctuations during the production process
of the primary cosmic ray particles \cite{cosmic}. Consider quite generally
a superposition of ordinary Maxwell-Boltzmann
distributions with different inverse temperatures $\beta$:
\begin{equation}
p(E) \sim \int f(\beta) E^2 e^{-\beta E} d\beta \label{superpo}
\end{equation}
Here $f(\beta)$ is the probability density to observe
a given inverse temperature $\beta$.
If $f(\beta)$ is a Gamma distribution,
then the integration in eq.~(\ref{superpo})
can be performed and one ends up with eq.~(\ref{can})
(see \cite{cosmic} for
more details). This is the basic idea underlying
so-called superstatistical
models \cite{beck-cohen}: One does a kind
of generalized statistical mechanics where the inverse temperature $\beta$
is a random variable as well.

The effective temperature parameter $\tilde{T}$
(a kind of average temperature in the above superstatistical model)
is of the same order of magnitude
as the so-called Hagedorn temperature $T_H$ \cite{hage}, an
effective temperature well known from collider experiments. The
fact that we get from the fits something of the order of the
Hagedorn temperature is encouraging. The Hagedorn temperature is
much smaller than the center-of-mass energy $E_{CMS}$ of a typical
collision process and represents a kind of `boiling temperature'
of nuclear matter at the confinement phase transition. It is a
kind of maximum temperature that can be reached in a collision
experiment. Even largest $E_{CMS}$ cannot produce a larger
average temperature than $T_H$ due to the fact that the number of
possible particle states grows exponentially.

Similar predictions derived from nonextensive statistical
mechanics also fit measured differential cross sections in
$e^+e^-$ annihilation processes and other scattering data very
well (see e.g. \cite{curado, e+e-} for more details). The hadronic cascade
process underlying these scattering data is not well understood,
though it can be simulated by Monte Carlo simulations. If we
don't have any better theory, then the simplest model to
reproduce the measured cross sections is indeed a generalized
Hagedorn theory where the Shannon entropy is replaced by Tsallis
entropy \cite{e+e-}.

\subsection{Defect turbulence}

Our next example is so-called `defect turbulence'. Defect
turbulence shares with ordinary turbulence only the name as
otherwise it is very different. It is a phenomenon related to
convection and has nothing to do with fully developed
hydrodynamic turbulence. Consider a Rayleigh-B\'{e}nard convection
experiment: A liquid is heated from below and cooled from above.
For large enough temperature differences, interesting convection
patterns start to evolve. An inclined layer convection experiment
\cite{daniels} is a kind of Rayleigh-B\'{e}nard experiment where the
apparatus is tilted by an angle (say 30 degrees), moreover the
liquid is confined between two very narrow plates. For large
temperature differences, the convection rolls evolve chaotically.
Of particular interest are the defects in this pattern, i.e.\
points where two convection rolls merge into one (see Fig.~5).
\begin{figure}
\epsfig{file=gentro-fig5.eps, width=8cm, height=7cm}
\caption{Convection rolls and defects (black and white boxes) as
observed in the experiment of Daniels et al. \cite{daniels}}.
\end{figure}
These defects behave very much like particles. They have a
well-defined position and velocity, they are created and
annihilated in pairs, and one can even formally attribute a
`charge' to them: There are positive and negative defects, as
indicated by the black and white boxes in Fig.~5. But the theory
underlying these highly nonlinear excitations is pretty unclear,
they are like a `black box' complex system whose measured output is
velocity.

The probability density of defect velocities has been
experimentally measured with high statistics \cite{daniels}. As
shown in Fig.~6, the measured distribution is well fitted by a
$q$-Gaussian with $q \approx 1.45$.
\begin{figure}
\epsfig{file=gentro-fig6.eps, width=8cm, height=7cm}
\caption{Measured probability density of defect velocities and
fit with a $q$-Gaussian with $q=1.45$.}
\end{figure}
The defects are also observed to exhibit anomalous diffusion.
Their position $X(t)$ roughly obeys an anomalous diffusion law of
the type
\begin{equation}
\langle X^2(t) \rangle \sim t^\alpha,
\end{equation}
with $\alpha \approx 1.3$. The relation $\alpha \approx 2/(3-q)$
can be theoretically derived \cite{daniels}.

Apparently defects are a very complicated nonlinear system with
complicated interactions in a nonequilibrium environment. Their
dynamics is not fully understood so far. But we see that
effectively they seem to behave like a gas of nonextensive
statistical mechanics that leads to $q$-exponential Boltzmann
factors rather than ordinary Boltzmann factors.

\subsection{Optical lattices}

Optical lattices are standing-wave potentials obtained by
superpositions of counter-propagating laser beams. One obtains
easily tunable periodic potentials in which atoms can perform
normal and anomalous quantum transport processes. If the
potential is very deep, there is diffusive motion. If it is very
shallow, there is ballistic motion. In between, there is a regime
with anomalous diffusion that is of interest here.

Optical lattices can be theoretically described by a nonlinear
Fokker-Planck equation for the Wigner function $W(p,t)$ (the
Wigner function is an important statistical tool for the quantum
mechanical description in the phase space). The above
Fokker-Planck equation admits Tsallis statistics as a stationary
solution. This was pointed out by Lutz \cite{lutz}. The equation
is given by
\begin{equation}
\frac{\partial W}{\partial t}= - \frac{\partial}{\partial p}
[K(p)W] +\frac{\partial}{\partial p} \left[ D(p) \frac{\partial
W}{\partial p} \right]
\end{equation}
where
\begin{equation}
K(p)=-\frac{\alpha p}{1+(p/p_c)^2}
\end{equation}
is a momentum-dependent drift force and
\begin{equation}
D(p)=D_0+\frac{D_1}{1+(p/p_c)^2}
\end{equation}
a momentum-dependent diffusion constant. The stationary solution
is
\begin{equation}
W(p)= C \frac{1}{(1+\beta (q-1) E)^{\frac{1}{q-1}}}
\end{equation}
where
\begin{eqnarray}
E= &=& \frac{1}{2} p^2 \\
\beta &=& \frac{\alpha}{D_0+D_1} \\
q &=& 1+\frac{2D_0}{\alpha p_c^2}
\end{eqnarray}
So the optical lattice effectively maximizes Tsallis entropy in
its nonequilibrim stationary state.
Another way to express the entropic index in terms
of physical parameters is the formula
\begin{equation}
q=1+ \frac{44E_R}{U_0}
\end{equation}
where $E_R$ is the so-called recoil energy and $U_0$ the potential
depth. These types of $q$-exponential predictions have been
experimentally confirmed \cite{renzoni}. Lutz' microscopic theory
thus yields a theory of the relevant entropic index $q$ in terms
of system parameters.

\subsection{Epilogue}

There are many other examples of physical systems where
generalized entropies yield a useful tool to effectively describe
the complex system under consideration. Important examples
include Hamiltonian systems with long-range interactions
that exhibit metastable states
\cite{rapisarda, raptsa} as well as driven nonequilibrium systems with
large-scale fluctuations of temperature or energy dissipation, i.e.
superstatistical systems \cite{beck-cohen, wilk, prl01}. The best
way to define generalized entropies for superstatistical systems
is still subject of current research \cite{abc,
straeten,souza}. Superstatistical turbulence models yield excellent
agreement with experimental data \cite{BCS, reynolds, prl}.
Generalized statistical mechanics methods have also applications
outside physics, for example in mathematical finance
\cite{borland,bouchard}, for traffic delay statistics \cite{briggs} or in
the  medical \cite{chen} and biological sciences \cite{chavanis2}. It is often in these types of
complex systems that one does not have a concrete equation of
motion and hence is forced to do certain `unbiased guesses' on
the behaviour of the system---which for sufficiently complex
systems may lead to other entropic forms than the usual Shannon
entropy that are effectively maximized. The beauty of the
formalism is that it can be applied to a large variety of complex
systems from different subject areas, without knowing the details
of the dynamics.

redpajama_set_name: RedPajamaArXiv


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Speech by Mr S Iswaran, Minister for Communications and Information, at Buddhist Fellowship's Vesak Celebration
Type: Speeches
Mr Lim Phang Hong, President of Buddhist Fellowship,
Mr Jerry Ong, Deputy President of Buddhist Fellowship,
Ajahn Brahmali, senior monk from Bodhinyana Monastery in Perth,
Ladies and gentlemen, good morning.
I want to start by wishing you a very Happy Vesak Day and thank you very much for inviting me to this very auspicious event. I am very glad to join you for this Vesak Day. The theme of your celebration today - Discovering the Buddha, Walking His Path – reminds us of the Buddha's own wise observation that: Just as a candle cannot burn without fire, men cannot live without a spiritual life." That is the essence of the human existence. So it is an opportune moment for us to reflect on the values that Buddha sought hard to promote: Tolerance, harmony, and compassion, values that are relevant to all of us, regardless of race or religion.
We cannot take our social and religious harmony for granted
2 These values are now, more than ever before, important amidst the rising intolerance we see in the world today. Many take for granted the multi-racial and multi-religious character of our nation. But many of you here know well enough that we have had to work very hard over a long time, over many decades, as a people to build trust and understanding across racial and religious boundaries. This is work that has been done painstakingly, week in week out, on the ground by community leaders, by religious organisations – like all of you working with your Reverend in order to promote that mutual understanding.
3 Today, the internet and social media, and all the latest info-comm technology have made it possible for news of religious conflict and hate speech in one part of the world to be instantaneously accessed anywhere in the world, stoking anger and hatred in the process. This was seen in both the attacks in Christchurch and Sri Lanka, which were instigated by the spread of religious violence and hate speech online. Christchurch and Sri Lanka are a sombre reminder to us all of the consequences we face if and when we let our guard down and allow irresponsible speech to spread hate and violence in Singapore and in the rest of the world.
4 At the same time, we also face the challenge posed by the spread of fake news and irresponsible speech that seeks to divide communities. Falsehoods, including hate speech, have been weaponised to instigate violence and anger between religious groups. Just last year, Sri Lanka saw widespread ethnic and religious violence due to online rumours that the Muslim minority in Sri Lanka were trying to sterilise the Sinhalese majority. In the case of the recent bombings in April, the Sri Lankan government had to resort to extreme measures by shutting down access altogether to all major social media networks. This is a measure of the challenge that we face in this digitalised world.
5 To prevent such incidents from occurring in Singapore, we recently passed the Protection from Online Falsehoods and Manipulation Act in Parliament a few weeks ago. This Act is one of the measures we are taking to help curb the spread of online falsehoods by holding online news sources and platforms accountable. What we mean by this is when there is something that is being fabricated and that poses a threat to public interest, because it can disrupt our social harmony and create distrust in communities; then there is a reason for us to act. What we want to do is shine a torch on them, let the light from them come through, by having the truth next to it so that people can see, assess and understand for themselves what is the truth, and the conclusions they can draw about the site that is perpetuating the falsehoods. This is the crux of the intent, it is not targeting individuals and communities – it is really aimed at the kind of messages that pose a threat to our public interest. We want to ensure that our society is protected from these falsehoods and we want to ensure that Singaporeans, our citizens who are active on the internet, are well-equipped to deal with it.
6 But I want to emphasise that legislation alone is not enough. We have emphasised this time and again. Yes, we need laws so that we can take action when it is required. But in fact, we need a larger whole-of-government, all of nation effort. We need public education, we need fact checking and we need efforts like what you are undertaking to build understanding between the communities and to remain strong and steadfast in our commitment to multiracial harmony. And it also means that each and every one of us, as users of the internet have a responsibility – a responsibility to be vigilant, a responsibility to ensure that when something comes to us, we make the effort to ascertain whether it is true or false, and also resist the temptation to propagate something without first ascertaining its authenticity.
Social harmony is our shared responsibility
7 So I am very grateful that religious organisations in Singapore, such as the Buddhist Fellowship, place importance on developing and strengthening bonds between different groups. Over the past 30 years, the Buddhist Fellowship has done excellent work in building up religious harmony in Singapore. Besides being an active member of the Telok Blangah and Bedok Reservoir-Punggol Inter-Racial and Religious Confidence Circle (IRCC), Buddhist Fellowship has shown its commitment to strengthening our social fabric by organising various interfaith forums and developing relationships with different religious leaders. The IRCCs provide a local platform for religious organisations in each constituency to network and engage one another.
8 In 2018, the Buddhist Fellowship supported the launch of the Buddhist Youth Network's SAY-IM (Singapore Action for Youth Interfaith Movement). The event encourages youths of different faith groups to come together to better understand one another. Interfaith forums and organisations like IRCCs provide opportunities for each of us as citizens to get to know each other and understand our differences. The bonds and trust that develops out of these interactions go a long way in preventing misunderstanding and divisions between groups, especially in light of the spread of online falsehoods and disinformation.
9 I want to congratulate the Buddhist Fellowship on the excellent work that you have been doing – at the national level, as well as the local level. I think we now face a challenge not just in the real world but in the virtual world and your efforts in helping us deal with the challenge in the virtual world – whether it is online falsehoods or hate speech or other falsehoods that use digital technologies to tear societies apart – these are other aspects that you need to deal with. The work you are doing in the real world can be extended to the virtual world and how we can be better prepared to resist such forces. So I want to take this opportunity as we celebrate Vesak Day also to congratulate you on your 30th anniversary just a few months back in December 2018 Three decades is a long time, well done.
10 So if I may just add that this point I made about online falsehoods is something I want to leave with you as a parting thought, because all of us may take it for granted when we are using technology. I think all of us here have smartphones, and use it to access information to do our work, to have entertainment, to communicate with family and friends. All of this is positive, all of this is the kind of benefits that we derive from technologies. At the same time, we must recognise that it does pose challenges. Whether we are students, young adults, parents or senior citizens – all of us have the responsibility to acquire the skills and awareness to discern these threats, and to be able to deal with them, even as we go about using the internet and social media responsibly.
11 So I hope all of you will join us in this effort, and organisations like the Buddhist Fellowship have a good foothold in that regard. Let us work together to ensure that Singapore will continue to be able to navigate the challenges and build the right future for our people. Thank you very much.
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text:
A report on Hoarse voice
When the voice involuntarily sounds breathy, raspy, or strained, or is softer in volume or lower in pitch.
- Hoarse voice
11 related topics with Alpha
Organ in the top of the neck involved in breathing, producing sound and protecting the trachea against food aspiration.
Injury to one of the recurrent laryngeal nerves produces hoarseness, if both are damaged the voice may or may not be preserved, but breathing becomes difficult.
Inflammation of the larynx .
Symptoms often include a hoarse voice and may include fever, cough, pain in the front of the neck, and trouble swallowing.
Vocal cord nodule
Vocal cord nodules are bilaterally symmetrical benign white masses that form at the midpoint of the vocal folds.
As such, the major perceptual signs of vocal fold nodules include vocal hoarseness and breathiness.
Muscle tension dysphonia
Muscle tension dysphonia (MTD) was originally coined in 1983 by Morrison and describes a dysphonia caused by increased muscle tension of the muscles surrounding the voice box: the laryngeal and paralaryngeal muscles.
Laryngopharyngeal reflux
Retrograde flow of gastric contents into the larynx, oropharynx and/or the nasopharynx.
LPR causes respiratory symptoms such as cough and wheezing and is often associated with head and neck complaints such as dysphonia, globus pharyngis, and dysphagia.
In humans, vocal cords, also known as vocal folds or voice reeds, are folds of throat tissues that are key in creating sounds through vocalization.
Patients suffering from vocal fold scar complain about increased phonatory effort, vocal fatigue, breathlessness, and dysphonia.
Inflammation of the back of the throat, known as the pharynx.
Other symptoms may include a runny nose, cough, headache, difficulty swallowing, swollen lymph nodes, and a hoarse voice.
Voice therapy consists of techniques and procedures that target vocal parameters, such as vocal fold closure, pitch, volume, and quality.
This technique can be used to treat stuttering, breathing, dysprosody, dysphonia, and to increase control of breathing, phrasing, and rhythm.
Vocal cord paresis
Injury to one or both recurrent laryngeal nerves , which control all intrinsic muscles of the larynx except for the cricothyroid muscle.
A unilateral injury of the nerve typically results in hoarseness caused by a reduced mobility of one of the vocal folds.
Spasmodic dysphonia
Disorder in which the muscles that generate a person's voice go into periods of spasm.
Gradual onset can begin with the manifestation of a hoarse voice quality, which may later transform into a voice quality described as strained with breaks in phonation.


text:
Wee Geordie Throws a Hammer!
Bill Travers as the adult Geordie.
Young Geordie MacTaggert doesn't like to be called "wee' by the other lads in his rural Scottish community. Yet, it's accurate to say that he's decidedly short for his age. It's a sore point, though, and comes to a head when he and childhood playmate Jean visit an eagle's nest. Jean is tall enough to see the baby birds, but Geordie is neither tall enough nor strong enough to view the nest.
That night, he sees a newspaper ad that will change his life. In the advertisement, bodybuilder Henry Samson asks: "Are you undersized? Let me make a different man of you!" Geordie sends off for Samson's exercise program and soon becomes obsessed with physical fitness. He eventually grows into a 6' 6" muscular young man! (As one character notes, the exercises can't have accounted for his growth spurt.)
Unfortunately, Geordie's focus on building his muscles has come with a cost. Jean, now an attractive young woman, feels ignored. The situation doesn't improve when Samson recommends that Geordie take up a sport like hammer throwing--at which he excels. Indeed, his hammer throwing attracts the interest of officials organizing Britain's team for the 1956 Olympics in Melbourne.
Norah Gorsen as Jean and Travers.
Made in 1955, Geordie (aka Wee Geordie) is a heartfelt film brimming with Scottish charm. Despite a handful of rear projection shots, it's one of those movies that will make you want to move to the Scottish glen--or at least take a vacation there. In the title role, Bill Travers makes a charming, reluctant hero who has to be convinced to participate in the Olympics. His Geordie has no desire to leave his beloved home and see the rest of the world. Why would he--when everything he loves is right there in the glen?
Indeed, Geordie works best when staying in Scotland and focusing on the Geordie-Jean relationship. One of the best scenes has Geordie floundering in his first hammer throwing competition until he hears Jean calling out to him from a nearby hill. Later, when the plot relocates to Melbourne, it becomes a conventional fish-out-of-water story.
Alastair Sim as the Laird.
Travers gets wonderful support from Norah Gorsen as Jean, Paul Young as the young Geordie, and Alastair Sim as The Laird. While it's true that Sim frequently portrayed quirky British gentlemen, that doesn't take away from his typical amusing performance. Paul Young, who made his film debut in Geordie, had a long television career (that's still ongoing). In contrast, the fresh-faced Norah Gorsen retired from acting in the mid-1960s.
Upon its release in Great Britain, Geordie quickly became a box office hit. Hollywood took notice of the ruggedly handsome Bill Travers and cast him opposite established stars in movies like Footsteps in the Fog (1955) and The Barretts of Wimpole Street (1957). His biggest success, though, didn't come until 1967 when he appeared with his wife Virginia McKenna in Born Free (1966). That film and Ring of Bright Water (about an otter) transformed the couple into animal rights activists. It was a passion that Travers pursued until his death in 1994.
Labels: alastair sim, bill travers, geordie, rick29 (author), wee geordie
Caftan Woman June 5, 2020 at 7:26 AM
Wee Geordie! Now, this is a movie that I have heard about but not had the luck or timing to see yet. When My dad mentioned it, in relation to Mr. Sim and Born Free playing on the radio, he gave it the air of legend.
I enjoyed this review and I know I will enjoy the movie.
Silver Screenings June 6, 2020 at 6:24 PM
The only Bill Travers films I've seen are Born Free and Ring of Bright Water, which I watched numerous times as a kid (thanks to a local television station and my wacky anti-TV parents who thought these films would be educational). I'll search for Geordie because it sounds wonderful.
Hal June 7, 2020 at 11:40 PM
An Elephant Called Slowly might well have been the most important film in Travers and McKenna becoming animal rights activists. Pole Pole, their elephant co-star, later died at an early age after isolation in a zoo for many years.
Doris Day in Hitchcock and Hitchcock-Lite
The Five Best Greer Garson Performances
Burt Lancaster and Ossie Davis Take on The Scalphu...
Alastair MacLean's The Guns of Navarone
Seven Things to Know About Robert Lansing
The Alternate TV Series Title Game (British Edition)
Rodgers and Hammerstein's Flower Drum Song


text:
MPR News Intelligence on higher education
Notes in the Margins
Notes in the Margins: Legacies, last-dollar scholarships and the unionization vote
Alex Friedrich April 25, 2014, 8:27 AM Apr 25, 2014
Northwestern University football players to vote today on union amid debate over labor rights The outcome of the secret-ballot vote among Northwestern's football players will not be known for some time as the board considers a case with broad ramifications not only for intercollegiate athletics but also for how colleges operate. (The Washington Post)
The Political Attractiveness of "Last-Dollar" Scholarships There is a growing number of "last-dollar" scholarships, in which private entities or state/local governments agree to cover students' remaining tuition and fees after all federal grants have been provided. (Note that it does not cover room and board or living expenses—an important component of the total cost of attendance.) (Washington Monthly)
End College Legacy Preferences Because of legacy admissions, elite colleges look almost nothing like America. (The New York Times)
5 Bold Predictions For The Future Of Higher Education What, where, and how will we learn? (Fast Company via University Business)
Job Market for College Grads Better But Still Weak The Labor Department on Tuesday said the unemployment rate for 2013 college graduates—defined as those ages 20 to 29 who earned a four-year or advanced degree—was 10.9 percent. That was down from 13.3 percent in 2012 and was the lowest since 7.7 percent in 2007. The drop reflects the steady recovery in overall U.S. economic growth and hiring. (Associated Press via NAICU)
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Alex Friedrich
afriedrich@mpr.org
Alex Friedrich reports on higher education issues for MPR News. Among the stories he has covered: the fall of the Berlin Wall, aftermath of Hurricane Katrina, collapse of the I-35W bridge in Minneapolis, 2003 Moscow suicide bombing and 2004 presidential elections in the Republic of Georgia. He holds a bachelor's degree in journalism from the University of Georgia and a master's in European political economy from the London School of Economics.
How Augsburg is going all Mark Twain
How past U of M presidents view higher-ed today
Metro State data breach affected 160,000 students


WARNING: When I first saw the pictures that you'll see below, I felt sick to my stomach. It wasn't a physical sickness but a spiritual sense of such palpable evil that my soul felt ill. Whether you're Christian or not — especially if you're not a Christian — please bless yourself with the sign of the cross and say a prayer for God's protection before you read further. I recommend the powerful Prayer to St. Michael, HERE. Please do it even if you don't believe — if only just to humor a fruitcake, me.
Anyone who's been on the Internet cannot avoid coming across dark warnings about the Illuminati — a purported conspiratorial organization that acts as a shadowy "power behind the throne" or "the power that be" (TPTB), a modern incarnation or continuation of the Enlightenment-era Bavarian Illuminati, a secret society founded on May 1, 1776. It is said that the Illuminati are the masterminds who actually control world affairs through governments and corporations, eventuating in the establishment of a one-world government — the New World Order.
The Rothschild family (known as The House of Rothschild, or more simply as the Rothschilds) is a European family of German [Ashkenazi] Jewish origin that established European banking and finance houses from the late eighteenth century. Five lines of the Austrian branch of the family were elevated into the Austrian nobility, being given hereditary baronies of the Habsburg Empire by Emperor Francis II in 1816. The British branch of the family was elevated into the British nobility at the request of Queen Victoria. It has been argued that during the 19th century, the family possessed by far the largest private fortune in the world, and by far the largest fortune in modern history.
At this point you're probably rolling your eyes and making the "ding-ding ding-ding ding-ding" Twilight Zone sound. Admittedly, while this mother of all conspiracy theories makes for interesting reading, I'd always retained my skepticism.
For one, conspiracy theories by their very nature are difficult, if not impossible, to prove/verify or disconfirm –" It's secret 'cause it's a conspiracy!" If you ask a conspiracy theorist (CT), "If it's such a conspiracy, then how do you know all this?", the CT either says "I just know" or "Well, TPTB leak clues and leave hints of their plans because they are so arrogant and cocky." If you bring up a counter-argument such as "But how do you account for there being these powerful wealthy groups that don't agree with each other?," the CT replies, "You see, there are contending factions within TPTB." And so on and so forth. In other words, the CT is always right no matter what, which makes the Illuminati conspiracy theory air- and water-tight, impervious to counter-evidence or counter-arguments.
The flawed epistemology of conspiracy theories such as the one about the Illuminati makes me all the more skeptical until I saw pictures of a Rothschild — the Baroness Philippine de Rothschild — in an article on Makow's website.
The Baroness Philippine Mathilde Camille de Rothschild (born Paris, France, 22 November 1933) is the only daughter of the vintner Baron Philippe de Rothschild, a member of the Rothschild banking dynasty and owner of the French winery Château Mouton Rothschild.
Rothschild entered her father's wine business in the late 1970s. When Philippe died in 1988, Philippine inherited three winery estates in Bordeaux. At the time of her father's death, the company sold 1.3 million cases of wine a year. By 2000, sales had almost doubled to 2.1 million cases. In 1999, sales amounted to around $155 million. Her personal wealth has been estimated at €190 million by Le Nouvel Economiste.
Below are photos of the Baroness. Note that in every picture she wears a heavy chainlinked gold necklace with a pendant. The pendant in the first pic below is a huge image of Baphomet, a pagan deity that, since the 19th century, has become a figure or synonym of Satan.
Remember that this woman is one of the wealthiest people on Earth who can wear any piece of jewelry she desires, but again and again she chooses to wear these big ugly Satanic pendants on big, heavy, gold chain necklaces.
I leave you to draw your own conclusions….


Terhes Dénes Korzó Design Studio Kft.
Grabner Balázs Korzó Design Studio Kft.
Dr. András Reith András ABUD Mérnökiroda Kft.
The spatial structure of South Buda Hospital Center (DBC) is a neutral structure that consists of various hospital-specific areas. In this way, the areas of nursing function, spaces of hospital technology and therapy, and service areas are adequately separated. These separate spaces are incorporated into an integrated entity by the twin cores that provide vertical connections.
The spatial structure - the creation of a clear and coherent space structure as a primary architectural value - had a fundamental role in determining the design. Architecture is primarily the art of space organization, and in our case space organization follows motion or flux. The size of the various spaces was based on the intensity of movement: starting from the generous entrance hall, through the wide space (magistral) with the most intense traffic, to the entry points of the various units and then thinning into the interior spaces.
The flux - the architectural philosophy of the hospital center is based on the flux, which is reflected not only in the internal space organization, but distinctly so in the external mass. The characteristics of the flux are likewise echoed in the plum-shaped nursing blocks and the amoeba-shaped childcare facilities. The latter functions of the building are seemingly floating above the two-story hospital building by the insertion of the retracted engineering level.
The role of light - ensuring the flow of natural light was a top priority in design. An important element of the spatial structure we created is the series of internal conservatories and cooling climate courtyards. These courtyards vary in level and depth, following the functional requirements in a flexible way, and provide access to natural light wherever possible.


More homeowners plan to spruce up their main home and their investment properties/rentals this year,but a lot of things can go wrong—particularly financially—when setting out to do a home remodel. More than a quarter of home renovation projects go over budget.
To help avoid costly mistakes, make sure they do plenty of prep work, take a full assessment of the home and its walls, and consult a professional when necessary.
In 2018, most homeowners planned to tackle at least one home improvement project.
The three r's start with: 51% were going to renovate.
What's next? 48% were going to redecorate.
This is the painful one (were going to be the keyword) 34% were going to do repairs.
Did you know? 1 in every 4 home renovation projects goes over budget.
So not only are you stressed out by the task at hand but now your wallet is uncomfortable too.
How do you avoid costly mistakes?
Architect before making structural changes on your home.
Interior designer to help select finishes and make design choices.
Contractor when a renovation is too big, or too difficult to do it yourself (DIY).
Home inspector to find problems before your renovation begins.
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The King's Buildings site contains most of the Schools within the College of Science and Engineering, and is located approximately two miles from the Central Area.
The King's Buildings site is located on the south side of the city, and within walking distance of the city's main transport hubs and the Central Area.
The King's Buildings site is located on the south side of the city and is extremely well served by bus. The University also provides a shuttle bus between the King's Buildings and the Central Area during term time.
The King's Buildings site is located to the south of the city, approximately three miles from Edinburgh Waverley Train Station, the city's main train station.
University staff and students can join City Car Club for business or personal use at a significantly discounted rate. A City Car Club car is based at the King's Buildings, at the entrance to the Ashworth Laboratories.
The King's Buildings site is located on the south side of the city, approximately three miles from the centre of Edinburgh and can be accessed by car.