text:
Home CRIMES Ghanaian Footballer arrested for murder as 3 human heads are found in his freezer.
Ghanaian Footballer arrested for murder as 3 human heads are found in his freezer.
A Ghanaian footballer and sport commentator, Richard Gyamfi, has been arrested for alleged gruesome murder of at least three persons as victims heads were found in his refrigerator.
According to reports, the incident had happened in Abesim area of Sunyami, Ghana and had triggered several outrage across the country.
Gyamfi and another suspect was said to have been arrested after the father of his latest victim, 15-year-old Louis Agyemang Junior, had reported him missing and a witness confirmed that he last saw Agyemang with Gyafi in the football park at Sunyani Abesim.
Gyamfi sparked suspicions when he denied seeing the boy. He raised further suspicions, when he said his house keys were missing and this led to his house being broken into.
Upon force entry, they found found the remains of Agyemang, on the floor, his head had been severed and kept in the fridge, in the culprit's living quarters at Alaska, near Abesim.
They also found three heads packed in the fridge of the footballer, one of which was that of late Agyemang.
Gyafi, who is 28 years old, is now accused of not just the murder of 15 year-old Agyemang, but also the murder of a 12-year-old.
His third victim was yet-to-be-identified.
Both Gyamfi, said to be a Bible reader and the second suspect arrested will help the police investigation, said Director of Public Affairs, Ghana Police Service, Alexander Kwaku Obeng.
"The remains are being moved to the Ghana Police Service hospital morgue in Accra for forensic analysis to be done to support the investigation," he added, in a report by MyJoyOnline.
He revealed that clinical psychologists and counsellors have also been deployed to support and counsel persons affected by the murder, including the families, some community members and the Police who led the search.


text:
Billiard's contributions will be remembered
For many years, when someone thought of Brooke County, they probably also thought of Paul "Bud" Billiard.
Over a 23-year coaching career, he guided the Brooke Bruins football team to 185 wins, with 15 playoff appearances, three Class AAA state championships and seven OVAC championships, bringing attention from not just the Ohio Valley or even the state, but also other parts of the country.
He also spent six years coaching wrestling at the school in addition to his role as a teacher.
For his success on the field, he was honored with the H.V. Corcoran Sr. Award, named West Virginia Coach of the Year, Upper Ohio Valley Dapper Dan Man of the Year and was inducted into multiple halls of fame.
The football field at Brooke Memorial Stadium now bears his name.
However, if you ask anyone from those years, whether a fellow teacher, coach, student or player, many of them will tell you his focus was on building for the future.
Even after his retirement from the gridiron, Billiard's thoughts were on giving back and finding ways to build up his community.
In recent years, Billiard put his energy into improvements to Brooke Hills Park and honoring local veterans, as well as serving with several other community-based groups.
Officials with the park have noted his efforts to develop upgrades for recreational vehicles, as well as the installation of disc golf and foot golf courses, and the Brooke County Veterans Memorial Park, which overlooks the paddle boat pond at Brooke Hills.
The disc golf course was named in honor of him and his wife, Carol.
No matter the role, "Bud" Billiard looked for ways to contribute for the benefit of others, and thousands of individuals have been affected in some way as a result.
There has been sadness in Brooke County in recent days because of his loss, but his memory, and the lessons he provided to generations of county residents, will last for lifetimes to come.
Over the past few weeks, nearly 5 percent of all West Virginians have received either the Pfizer or the Moderna ...
We can't please everyone
West Liberty University's concert band makes its national recording debut
West Virginia sets pace for COVID-19 vaccine distribution
Fears grow about inside attack at inauguration

redpajama_set_name: RedPajamaCommonCrawl


S.U.F Helsinki is a small, independent watch brand that produces approximately 100 timepieces each year. S.U.F actually stands for 'Sarpaneva Uhren Fabrik', and if you know your independent watch brands and/or watchmakers, that first word should immediately grab your attention. You see, S.U.F is actually the second watch brand of well-respected Finnish watchmaker Stepan Sarpaneva, and, with the launch of Vetehinen, the brand now has its first divewatch.
Lets not mince words, the Vetehinen is handsome.
As per a statement issued by luxury conglomerate Richemont, Georges Kern has resigned as head of watchmaking, marketing and digital. The surprising change takes effect immediately, and comes just several weeks after Kern assumed his new position. Kern was previously CEO of IWC for several years.
Richemont watchmaking boss resigns to become an entrepreneur.
Spider-Man: Homecoming hit cinemas on July 7 and if – like this writer – you saw the film and caught the red carpet premiere videos on YouTube, you probably noticed that Robert Downey Jr., rocked an incredibly cool wristwatch. The watch was, in fact, a stunning Urwerk UR-110 RG but the really, really cool thing is that the watch was personally picked by Robery Downey Jr. himself. In other words, this was not a product placement deal.
Robert Downey Jr. wears a Urwerk UR110 RG to the premiere of Spider-Man: Homecoming.


text:
Home Private Equity News & Analysis First Round: Heinz boss on the sauce
First Round: Heinz boss on the sauce
Lessons in leadership from Heinz CEO William Johnson
There's just so much to like about the $23 billion 3G Capital/Warren Buffett/Heinz buyout.
Of course, First Round speaks not of such quotidian things as favourable price/earnings ratios, solid free cash flows, established distribution networks or even – to utilise one of Buffett's favourite phrases – 'durable competitive advantage'. Oh no. It's thinking more about pure comedy value.
For one thing, there's a degree of Victor Kiam-esque wonderfulness about a billionaire who famously loves nothing better than a good hamburger going out and buying his favourite ketchup brand.
But First Round's favourite bit of the whole story emerged in the subsequent press conference. Heinz's chairman, president, chief executive, commander-in-chief and Supreme Leader* William Johnson revealed that when the 3G Capital team first asked for a meeting, he assumed they wanted to complain – since in their capacity as owners of Burger King (which 3G bought in 2010), they're one of his biggest ketchup customers.
But rather than whingeing about under-powered ketchup sachets – a very real First World problem, First Round is sure you'll agree – they instead offered to buy his entire company (presumably making him considerably richer in the process). "Turns out they were a very happy customer," the Heinz boss said, in admirably understated fashion.
Let's hope he was joking. After all, First Round is clearly no stranger to serious lapses of professional judgement (wearing that Borat costume to the office last Halloween was definitely a mistake, in hindsight). But this little anecdote doesn't exactly suggest that Johnson has his finger on the pulse of his biggest customers.
Then again, given that he's just managed to sell his company to one of the world's richest men at a 20 percent premium to its previous share price, it seems unlikely that his shareholders will hold it against him for long.
* two of these may be made-up


Dear incoming Coxies, welcome to the MEi:CogSci programme in Bratislava, which you have chosen as your mobility semester destination. We, your host teachers, are delighted with your choice and will do our best to assist you to make the most of your ERASMUS experience. This short material will provide you with basic introductory orientation information. Where possible, the external links point to English versions of university-level webpages, however not many of the pages exist in English. For the time being, please use Google translator or ask a Slovak speaker for help. We do apologize for any inconvenience. If you have any questions, don't hesitate to contact us!
MEi:CogSci programme in Bratislava is hosted by the Department of Applied Informatics at the Faculty of Mathematics, Physics and Informatics, Comenius University. It has been running since 2006. Although our strongest in-house specialty is computational modeling, the teachers participating in the programme provide a much wider expertise in various disciplines. The teachers you will probably encounter the most frequently are Prof. Igor Farkaš (programme coordinator) and Dr. Martin Takáč (the student tutor).
Here you can find a presentation about MEi:CogSci in Bratislava.
The lecture period lasts Sep. 24 - Dec. 14, 2018. The examination period starts after Christmas holidays and lasts Jan. 7 - Feb. 15, 2019. You should be done with all your exams and the project by the end of examination period.
You need to fill in some paperwork and register in the University information system (more info here, please ignore the obsolete dates). This is very important, because the system keeps the record of all courses you enrolled for including your project - without their existing in the system, your teachers would not be able to fill in your marks at the end of the semester. The person of contact is Mgr. Patrik Kmetek, office No. F1-120, tel. +421-2-60295212 (or just ext. 212 if calling from internal phones within the Faculty building), Patrik.Kmetek@fmph.uniba.sk. Office hours 9:00-11:30. So make sure to visit Mr. Kmetek to sort out all the paperwork.
You need to bring home 30 ECTS for mobility. Part of this is covered by compulsory courses (10 ECTS). The remaining 20 ECTS can be divided between your project and other optional courses. Make sure to get in contact with your project supervisor ASAP and negotiate how many ECTS will your project be for (it can be for 10, 15 or 20 ECTS, depending on the expected workload - 1 ECTS translates to approx 25 hours of work, so a project for 20 ECTS amounts to approx. 500 hours spent on it).
See the course schedule for the 1st year and 2nd year cognitive science courses. The schedule is in Slovak, so a better way to look for the time and place of a course you are interested in is to first look for the course in the list of courses, then enter the code of the course (e.g. 2-IKV-102) into a search bar (left of the button 'Hľadaj', which means search) of this interactive schedule application.
After you choose your courses and negotiate the ECTS for project, start working on a detailed specification for your project. This will result in filling in your learning contract (LC). The form is available on the MEi:Cogsci portal, a local copy can be found here. After your supervisor approves you the final version of the contract, send it via email to your home and host coordinators for their approval, with cc to your supervisor. Your host coordinator is Prof. Igor Farkaš. Detailed instructions can be found in the LC form.
After you have successfuly passed your exams and completed the project, you need to bring home the signed and stamped LC. At this stage, include a summary (3000-5000 characters) of your project findings in the LC form and get your supervisor to sign it and fill in the grade of the project. The local coordinator then fills the grades for the courses you took and adds his signature and stamp. When back home, deliver the LC to your home coordinator.
There is some paperwork to take home, obtainable either from the study office, or Mr. Kmetek. Visit Mr. Kmetek for details.
While our 2nd year students will be at their mobility semester, our 1st year students are here, so get in contact with them to get useful introduction to student's life in Bratislava. The first good opportunity to meet them, as well as your teachers and our last year's graduates, is our annual Cognitive beer event, that takes place in the second half of September in Patronsky pivovar pub. You will be cordially invited. Follow also our MEi:CogSci Bratislava Facebook page.
The buliding has two entrances: from the South (Physics building) and from the East (Mathematics building). Most of your classes will take place in the Informatics building which is attached to Mathematics, so you will want to use the East entrance. After entering the building and passing the reception, turn right for the Informatics building. The East entrance is open on working days bw 6:00-20:15. After these hours use the Physics entrance (both buildings are connected).
The Faculty has two school canteens and a cafeteria. One large canteen (FAYNFOOD) and the cafeteria are located in the Physics building - very close to the entrance, the other canteen (FREEFOOD) is in the Mathematics building at the very end of the long corridor that you enter if you go straight from the reception. The current menu, as well as other dining options, can be found here.
Note: the food in FREEFOOD is not free :), the cost of a lunch in the canteens is around 3,30 eur, with a possible discount on ISIC card. Besides our Faculty, there are several buffees, a cafeteria and a pizzeria in the nearby Faculty of natural sciences, and plenty others are in the student's dormitory area in Slávičie údolie.
The Faculty's library is on the basement floor of the Informatics building. Besides books on math and physics, it recently dedicated a special shelf to several hundreds of cognitive science related books.
Eduroam Wifi is available in the Faculty buildings - you should be available to log in with your home university Eduroam credentials.
Lots of other practical information about life in Bratislava can be found here.
Where possible, the external links point to English versions of university-level webpages, however not many of the pages exist in English. For the time being, please use Google translator or ask a Slovak speaker for help. We do apologize for any inconvenience.


adidas Originals Superstar 80s "Cork Pack"
Adidas Originals continues it's new iterations of the Adidas Superstar 80's silhouette and the latest concept summons cork material for the toe caps. Two colorways will be available in the Superstar 80's, both of which feature cork toe boxes that sit atop a white outsole. Look out for these cork and suede equipped adidas superstar 80's at adidas originals retailers across the globe like SNKRS.


text:
\section{Introduction}

In 1990, Bak et al. introduced a simple model for the spreading of a fire
in a forest or the spreading of
disease in a population \cite{bak90}.
The individuals (sites on a square lattice in 2 dimensions)
can be in one of three states: tree (healthy, excitable),
tree on fire (infected, excited), and ashes or empty site
(immune or dead, refractory). New individuals are 'fed' into the system
with a small rate $p$. Whether the third state is considered
as death of an individual and $p$ consequently as the birth rate
of new individuals,
or as a state of immunity and $1/p$ as time scale of the loss of that
immunity,
is a matter of interpretation.
In the following, we will use the terms
tree, fire, empty site and refer to $p$ as the tree growth rate.
The exact rules of the model \cite{bak90}
were as follows: (i) at each time step trees grow at
empty sites with a small probability $p$, (ii) trees on fire will burn down the
next time step and turn to empty sites,
(iii) the fire on a site will spread to the trees at its
nearest neighbor sites at the next time step.
Although originally claimed to be critical in the
limit $p \to 0$, the simulations in \cite{gra91,mos92} showed that the model
does not display criticality. Instead, one could observe
quasi-deterministic spiral waves of fires.

In 1992, Drossel and Schwabl introduced the self-organized critical
forest-fire model (SOC FFM) \cite{dro92}
with the additional rule: (iv) if no nearest
neighbor is burning, a tree catches fire with a small 'lightning'
probability $f$. Under the condition of a double separation of time scales
(time between two lightning strokes $1/f$ $\gg$ time scale of tree growth
$1/p$ $\gg$ time needed to burn down large tree
clusters) the model shows critical behavior over a wide range of
parameter values. The properties of this model were investigated in e.g.
\cite{hen93,gra93,chr93,cla94}.

In this paper, we shall investigate a model with the same type of interactions
while keeping constant the number of empty
sites or immune individuals. Their density is the control
parameter of the model. Parameters like $p$ or $f$ do not enter the model.
We will show that the model exhibits three phases,
two of which can be shown to display the same behavior as the abovementioned
models. The third phase shows clustering of trees with the fire
burning at the edges of the clusters.

In the 'spiral wave' phase,
which exists not only in this model, but in a large
number of excitable systems (for a review on excitable systems see e.g.
\cite{tys88,mer92}), we will point out an interesting relation to the
non-equilibrium percolation model of \cite{cla95}.
There, the tree density was the control parameter and the
following rules were iterated : (i) Lightning strikes an arbitrary site in the
system. If the site is occupied, the whole cluster of $s$ trees which is
connected to this site (by nearest-neighbor coupling) burns down, i.e. the
trees of this cluster turn to empty sites. (ii) Then $s$ new trees are grown at
randomly chosen empty sites (including the ones that have just turned empty).
The close relation between this model and the model treated in the present
paper will enable
us to identify the mechanism that determines the density distribution of
excitable constituents and, in particular, the density
immediately in front of excitation fronts in the spiral-wave phase of
excitable systems.

Furthermore, the model presented in this paper, as well as the model in
\cite{cla95}, are examples of reformulations of a SOC model
in terms of a control parameter (the density of trees or empty sites,
respectively). Both models indicate that the claim in
\cite{sor95} that all SOC models can be mapped onto ordinary critical
systems exhibiting a sub-critical phase, a critical phase with a smoothly
varying order parameter and a critical point that separates
the two phases, is not true in general. Non-equilibrium systems and
their phase transitions show a much richer behavior than equilibrium
systems, with many features that are unknown in equilibrium.

\section{The model}

The model is defined on a $d$-dimensional hyper-cubic lattice with $L^d$
sites. If not stated otherwise, we choose $d=2$ and periodic boundary
conditions in the following.
$\rho_e L^d$ sites are randomly chosen to be empty. The density of empty sites
$\rho_e$ is the control parameter
of the model. The remaining sites are randomly filled with trees and fires.
Their densities are denoted by $\rho_t$ and $\rho_f$.
It is always $\rho_t + \rho_e + \rho_f = 1$.
The exact values of $\rho_t$ and $\rho_f$ in the initial state do not
affect the stationary state, except in the vicinity of
certain points which will be discussed later.

The system is iterated as follows: (i) all trees on fire will burn down the
next time step, (ii) the fire on a site will spread to the trees at its
nearest neighbor sites in the next time step, (iii) after each time step
the same number of trees that have burnt down grow at randomly
chosen empty sites (including the ones which have just become empty),
thereby keeping $\rho_e$ fixed, and (iv) if the fire
dies out, a randomly chosen tree catches fire spontaneously.

The motivation of rule (iv) is the following:
We want to investigate the system
under the action of a vanishingly small lightning probability .
Since the process which is described by rule (iv) may then take very long in
real time, but nevertheless can be simulated in one iteration step,
rule (iv) represents an acceleration of the real process. When calculating
temporal averages of the fire density, this point has to be considered.

The reason for choosing the density of empty sites as parameter and not
the density of trees is the following: Consider a system that consists
only of trees and replace one tree by a fire. In the next step we have four
fires, but only one empty site to grow new trees. This extreme example
shows that there might be situations in which it is not possible
to keep the density of trees constant.
The density of empty sites, in turn, can always be kept constant for
arbitrary values within the interval $[0,1]$.

In the following sections, we discuss the properties of the stationary
state as function of the density of empty sites $\rho_e$.
In Sec.~\ref{secthree}, we start with a high density of empty sites
and investigate the region of vanishing fire density. In Sec.~\ref{secfour},
we lower the density of empty sites which leads to a state with spiral waves.
A detailed description of the mechanism which determines the density
distribution in the spiral state is given in Sec.~\ref{secfive}.
Sec.~\ref{secsix} investigates the more homogeneous, 'mixed' phase that can
be observed
after further decreasing the density of empty sites. Up to here, only
two-dimensional square lattices are considered.
Sec.~\ref{secseven} treats other dimensions and lattice types. In
Sec.~\ref{seceight}, we comment on the issue of mapping self-organized
criticality onto ordinary criticality. Finally, in the appendix, we
present some general properties of the order parameter curve.

\section{Region of Vanishing Fire Density and Critical Point}
\label{secthree}

For $\rho_e \lesssim 1$,
there exist only very small tree clusters, and, consequently, if one starts
a fire
by setting on fire a randomly chosen tree, it soon dies out, and one has to
start a new one. The average number of trees $\bar s$ destroyed by a fire
therefore
is finite and small, and in the thermodynamic limit $L \to \infty$ the fire
density $\rho_f$ equals zero
(taking into account that rule (iv) is an acceleration of a process which
takes infinitely long in real time).
In our simulations the maximum system size was
$6000^2$.
With decreasing $\rho_e$, $\bar s$ increases, but still remains finite, and
therefore $\rho_f = 0$. If $\rho_e$ is decreased further, we finally arrive
at a critical density $\rho_e^{c,1} \approx 59.2\%$
($\rho_t^{c,1} = 1 - \rho_e^{c,1} \approx
40.8\%$), where $\bar s$ diverges with a power law
$\bar s \propto (\rho_e - \rho_e^{c,1})^{-\delta}$, with some exponent
$\delta$.
A snapshot of the system in the vicinity
of $\rho_e^{c,1}$ is shown in Fig.~\ref{luisa}.

The critical behavior close to $\rho_e^{c,1}$
can be described by exponents which
are defined as in percolation theory \cite{sta92}. The size distribution of
tree clusters is $n(s) \propto s^{-\tau} {\cal C}(s / s_{\text{max}})$ with
a cutoff function ${\cal C}$. $s_{\text{max}}$ is the size of the
largest cluster in the system and diverges for $\rho_e \to \rho_e^{c,1}$.
The fractal dimension $\mu$
of the clusters is defined by $R(s) \propto s^{1/\mu}$,
where $R$ is the radius of gyration of a cluster.
The correlation length is given by $\xi \propto (\rho_e -
\rho_e^{c,1})^{-\nu}$.
More exponents can be defined and scaling relations between them can be
derived (see e.g. \cite{hen93,gra93,chr93,cla94}).

The critical exponents found in the simulations ($\tau = 2.14(4)$,
$\mu = 1.95(2)$, and $\nu = 0.28$) are the same as in the SOC FFM, when
appropriately redefined (for $\nu$
one has to change variables from $f/p$ to $\rho_e$ via
$\rho_e - \rho_e^{c,1} \propto (f/p)^{1/\delta}$ (see \cite{cla94})).
Also the critical density $\rho_t^{c,1} \approx 40.8\%$ remains the same.
This model displays exactly the same critical behavior as the SOC FFM and
the model of \cite{cla95}, because
in a system which is much larger than the correlation length,
neither the difference
between a globally conserved density $\rho_e$ (this model and \cite{cla95})
and a density $\rho_e$
which is only conserved on an average (SOC FFM)
nor the difference between instantaneous re-growth of trees (this model) or
delayed re-growth (SOC FFM and \cite{cla95}) can be seen on length scales
comparable to the correlation length.

In the stationary state of the SOC FFM $\bar s = p \rho_e / f (1 - \rho_e)$,
since in one time step there are $\rho_t L^d f$ lightning strokes
and $\rho_e L^d p$ new trees are growing.
Therefore, if we measure in our model $\bar s$
for a certain value of the control
parameter $\rho_e$ we know that its behavior is
that of the SOC FFM for $f/p = \rho_e/ \bar s (1 - \rho_e)$.

\section{Region of Finite Fire Density I: Spirals}
\label{secfour}

If we decrease the number of empty sites beyond the critical point
$\rho_e^{c,1}$, the size of the largest forest cluster diverges, and we would
expect the fire not to be extinguished any more. Therefore,
one might expect the fire density
$\rho_f$ to behave as an order parameter that
sets in at $\rho_e^{c,1}$ and grows smoothly from zero
to finite values, obeying some power law $\rho_f \propto
(\rho_e^{c,1} - \rho_e)^\beta$.
Instead, the following behavior can be observed in the simulations:
The system restructures, fires gathering
to form spirals and the regions of different densities
(see Fig.~\ref{luisa}) vanishing in favor of a smooth density distribution
between the spiral arms (see Fig.~\ref{carmen}).
The behavior of the spirals is quasi-deterministic.
Immediately in front of the
fire fronts the tree density is very
high and immediately behind them it is very low.
The density distribution is treated in more detail in Sec.~\ref{secfive}.
The distance $\Delta$ between two spiral arms is finite and
constant throughout the system. The fire density
for $\rho_e = \rho_e^{c,1} - 0$ is also finite.
Decreasing $\rho_e$ further, the distance
between the spiral arms becomes smaller and the fire density
becomes larger, since it is now easier for the fire to survive.
At $\rho_e = \rho_e^{c,3} \approx 54.2\%$,
the spiral state breaks down and
another restructuring to a new phase takes place, which will be treated in
Sec.~\ref{secsix}.

For $\rho_e \le \rho_e^{c,1}$ the fire is able to sustain itself and
it is no longer necessary to keep it alive by setting on fire randomly chosen
trees each time the fire has died out. The 'external field' can be set zero.
In terms of tree growth and lightning,
we have a model with tree growth rate $p$, but without lightning rate $f$.
Therefore, the behavior in the spiral state at some density $\rho_e$
is exactly the same as in the
forest-fire model of Bak et al. in \cite{bak90} for a certain
value of the tree growth probability $p$, since
in the thermodynamic limit it does not matter which of the variables is
kept fixed. The fluctuations of the densities in the Bak model vanish for
$L \to \infty$, as do the fluctuations of the number of new grown trees
per time step in this model.

In the Bak model, one observes a diverging distance between
spiral arms for $p \to 0$. If we start in our model
with a spiral state and increase
$\rho_e$ beyond $\rho_e^{c,1}$ (corresponding to a decrease of $p$),
the system
does not undergo a phase transition at $\rho_e = \rho_e^{c,1}$,
but chooses to retain the spiral state
up to $\rho_e = \rho_e^{c,2} \approx 60.8\%$.
This value of $\rho_e$ with $\Delta = \infty$
corresponds to $p = 0$ in the Bak model.
As function of $p$ the order parameter follows
the power law $\rho_f \propto p$ (explanation in Sec.~\ref{secfive}).

As already found in \cite{gra91,mos92},
the spiral state does not display criticality or scale
invariance in the sense of clusters or events on all length scales.
For a particular value of $p$ the model does not show events on all
scales, but only on the scale $1/p$, and behaves essentially
deterministic.
However, the model allows similarity transformations, since a system with
parameter $p_1$ can be obtained by rescaling a system with parameter $p_2$.

What we have found is a discontinuous
phase transition which is neither of first nor of
second order. We find hysteresis (the state of the system for a density
in the interval $[\rho_e^{c,1};\rho_e^{c,2}]$ depends on from which side one
approaches this region), an order parameter that increases smoothly from zero
at $\rho_e^{c,2}$ to finite values, and critical behavior
on only one side of the transition
($\rho_e \ge \rho_e^{c,1}$, the side with vanishing order parameter).
The order parameter curve for a two-dimensional system
is shown in Fig.~\ref{2dphasediagram}.

The reason why the critical point $\rho_e = \rho_e^{c,1}$ is not an ordinary
critical point is the same as in \cite{cla95}. From Fig.~\ref{luisa},
it can be seen that, in addition to large tree clusters, there exist also
large clusters of empty sites. In contrast to ordinary critical phenomena
(e.g. percolation), there is no homogeneously distributed set of large
clusters that could join at $\rho_e = \rho_e^{c,1}$ to form an infinite
cluster that spans the whole system. Rather,
the largest cluster has to compete with all other regions
with different densities (which, due to the very nature of the dynamics,
are nothing more than different
growth stages of the largest cluster itself) for space in the system.
Or, to put it differently, an infinite cluster like in
percolation is impossible, because the system has to provide
space not only for the infinite cluster, but also for a large
number of 'younger' copies of it.
The violation of the hyperscaling relation $d = \mu (\tau - 1)$
also indicates the inhomogeneous distribution of density in the
forest-fire model. As explained in \cite{hen93},
the violation is equivalent
to the statement that not every part of the system contains a spanning
cluster at criticality.

\section{Density Distribution in the Spiral State}
\label{secfive}

In order to understand better the density
distribution in the spiral state, we shall now investigate a
state with a single front propagating through a square system
with periodic boundary conditions (see left side of Fig.~\ref{arantza}).
Since this can be considered to be a section of a spiral state
(see right side of Fig.~\ref{arantza}), and the
spiral state, in turn, can be completely covered by such sections (if the
edge length is chosen to be equal to the distance $\Delta$
between two successive
arms of the spirals), it is sufficient to understand the density
distribution in this part of the whole system.
Instead of successive fire fronts passing through the small section of
the system, one might also think of a single fire front which repeatedly
leaves the section at one end and reenters at the other end due to
periodic boundary conditions.

The tree density immediately in front of the fire front
$\rho_t^{\text{before}}$
has to be larger than or equal to the percolation threshold for site
percolation on a square lattice $p_c \approx 0.59$,
for otherwise there wouldn't be
propagation. With increasing distance from the fire front,
the tree density smoothly decreases, until one finally arrives again on
the other side of the front, where the density takes its lowest
value $\rho_t^{\text{after}}$. If $\rho_t^{\text{before}}$ was exactly the
percolation threshold,
the fire would burn down a vanishing fraction of all trees,
and the propagation speed of the front
would be zero. The densities in front of and behind the fire front would
be equal ($\rho_t^{\text{before}} = p_c = \rho_t^{\text{after}}$),
as claimed in \cite{gra91}.
However, the simulations clearly show that this is not the case.
$\rho_t^{\text{before}}$ is higher than the percolation threshold,
the density $\rho_t^{\text{after}}$ in a region
after the fire has passed through it is very low, but not equal to zero,
and the propagation
speed of the fire fronts $v_{\text{fire}}$ is non-vanishing ($0.7 \pm 0.05$
sites per iteration step near $\rho_e^{c,2}$).
What is the mechanism that determines the density distribution and
the values of the density immediately in front of and behind the fire front?

We shall approximate the periodic 'single front' state even further
by a coarsened state with a
finite but large number $n$ of stripes of equal width and different densities
parallel to the fire front. Each stripe has homogeneous density.
Let $\rho_t^1, \ldots, \rho_t^n$ be the densities of the $n$ stripes,
starting with the highest density. $\rho_t^i$ is then the average density
of the original smooth 'single front' system in the region covered by
the $i$th stripe ($\rho_t^1 \approx \rho_t^{\text{before}}$
and $\rho_t^n \approx
\rho_t^{\text{after}}$). We also coarsen time, in that we consider the
propagation
of the fire front from the bottom of one stripe to its top as one
time step. Growing of new trees then takes place after each
(coarsened) time step.
If one sets on fire the baseline of the stripe with highest density,
the infinite cluster in that stripe will surely be set on fire,
since it has many
connections with the baseline, and the propagation of the fire front
effectively causes the removal of the infinite cluster from this stripe.

The propagation of the fire front therefore can be modeled as follows.
First, we identify the infinite cluster in the stripe with highest density
and remove it from the system. Doing this one has to respect the
boundaries of the stripes, although the infinite cluster of course
extends into the neighboring stripes. The strength of the infinite cluster
at density $\rho_t^{1}$ is denoted $P(\rho_t^1)$.
(We now define also the density
$\rho_t^{n+1}$ of the stripe which contained the infinite cluster after
its removal, i.e. $\rho_t^{n+1} = \rho_t^{1} - P(\rho_t^1)$.)
Second, the $P(\rho_t^1) \Delta^2/n$ trees of the infinite cluster
are redistributed randomly amongst the empty sites of the
whole system of size $\Delta^2$.
If the system is to be stationary, the stripes thereby just
exchange their densities, i.e. the stripe with second highest
density $\rho_t^2$ now assumes the highest density $\rho_t^1$,
and so on. From that condition, one can easily derive the equation
\[
\rho_t^{i-1} = \rho_t^i + (\rho_t^1 - \rho_t^{n + 1}) \cdot
(1 - \rho_t^i)/(n (1 - \rho_t) + \rho_t^1 - \rho_t^{n + 1})
\]
for $i = 2, \ldots, n + 1$.
The last factor on the r.h.s. represents the fraction of trees of the
infinite cluster that are re-grown in the stripe with density $\rho_t^i$.
We finally obtain
\begin{equation}
\frac{1 - \rho_t^1}{1 - \rho_t^2} = \frac{1 - \rho_t^2}{1 - \rho_t^3}
= \ldots =
\frac{1 - \rho_t^n}{1 - \rho_t^{n+1}}. \label{siegfried}
\end{equation}
Together with $\rho_t = (1/n) \sum_{i=1}^n \rho_t^i$
we have $n$ equations for $n+1$ densities.
These equations were already derived in \cite{cla95} for a related model.
The average density in a system with $n$ stripes is then:
\begin{eqnarray*}
\rho_t &=& 1 - \frac{1 - \rho_t^{n+1}}{n} \sum_{i=1}^{n}
{\left(\frac{1 - \rho_t^1}{1 - \rho_t^{n+1}}\right)}^{i/n} \\
&=& 1 - \frac{\rho_t^1 - \rho_t^{n+1}}{n(((1-\rho_t^1)/
(1-\rho_t^{n+1}))^{-1/n}-1)}
\end{eqnarray*}
In the system that we are really interested in, the density varies smoothly, so
we have to consider large $n$.
In \cite{cla95} it was argued that $\rho_t^1$ has always to be greater
than or equal to a certain constant $\rho_t^* \approx 0.625 > p_c$. If this
were not the case, a traversing fire front would leave behind large tree
clusters which would lead to inhomogeneities that prevented the next
fire front from passing through in the same quasi-deterministic way than
the first one.
Likewise, $\rho_t^{n+1}$ has always to be greater than or equal
to another constant $\rho_t^{\infty} = \rho_t^* - P(\rho_t^*) \approx 0.078$.
$\rho_t^*$ is always larger than
the corresponding percolation threshold $p_c$.
These constants
have a more fundamental significance independently of this model
and of any states with spiral waves (for details see \cite{cla95,cla96}).
The case $\rho_t^1 = \rho_t^*$ and $\rho_t^{n+1} = \rho_t^{\infty}$
corresponds to the lowest possible
overall density and therefore to the state with infinite spirals at $p = 0$
or $\rho_e = \rho_e^{c,2}$.
In this case, we have $\rho_t^{\text{before}} = \rho_t^*$ and
$\rho_t^{\text{after}} = \rho_t^{\infty}$.

The overall density in the spiral state can then be written as
\begin{eqnarray}
\rho_t &=&
1 - \frac{\rho_t^{\text{before}} - \rho_t^{\text{after}}}{\lim_{n \to \infty}
n ((\frac{1-\rho_t^{\text{before}}}{1-\rho_t^{\text{after}}})^{-1/n} - 1)}
\nonumber \\
&=& 1 - \frac{\rho_t^{\text{before}} - \rho_t^{\text{after}}}{\lim_{n \to
\infty} n (1 - \ln(\frac{1-\rho_t^{\text{before}}}{1-\rho_t^{\text{after}}})/n
\pm \ldots - 1)} \label{soledad} \\
&=& 1 - \frac{\rho^{\text{before}} - \rho_t^{\text{after}}}{\ln(\frac{1-
\rho_t^{\text{after}}}{1- \rho_t^{\text{before}}})} \nonumber
\end{eqnarray}
For $\rho_t^{\text{before}} = \rho_t^*$ and
$\rho_t^{\text{after}} = \rho_t^{\infty}$,
this density should be equal to the density $\rho_t^{c,2}$ of the state with
spirals of infinite extension.
With the values of $\rho_t^*$ and $\rho_t^\infty$ measured in \cite{cla95}
($0.625$ and $0.078$ for the square lattice, $0.533$ and $0.062$ for the
triangular lattice),
one arrives at
$\rho_t^{c,2,\text{calculated}} = 0.392 \pm 0.002$ for the square lattice
and $\rho_t^{c,2,\text{calculated}} =
0.325 \pm 0.002$ for the triangular lattice,
in excellent agreement with the values
$\rho_t^{c,2} = 0.392 \pm 0.002$ for the square lattice
and $\rho_t^{c,2} = 0.323 \pm 0.005$ for the triangular lattice,
measured in the spiral state of this model.

Since Eq.~(\ref{siegfried}) represents a geometric series, the density as
function of the distance from the fire front is given by
\begin{equation}
\rho_t(x) = 1 - (1 - \rho_t^{\text{after}})
\left(\frac{1 - \rho_t^{\text{before}}}{1 -
\rho_t^{\text{after}}}\right)^{x / \Delta},
\label{eva}
\end{equation}
with $\rho_t(0) = \rho_t^{\text{after}} \gtrsim \rho_t^\infty$
and $\rho_t(\Delta) =
\rho_t^{\text{before}} \gtrsim \rho_t^*$.

With our new knowledge about the nature of the spiral state, we can
derive an equation for the distance $\Delta$ between the fire fronts
as function of the tree growth probability $p$.
Let the speed of the fire fronts $v_{\text{fire}}(\rho_t^{\text{before}})$
be measured in sites per iteration step. Like $P$ it depends on the density
in front of the fire front $\rho_t^{\text{before}}$.
The amount of matter burnt in unit time is then $P(\rho_t^{\text{before}})
v_{\text{fire}}(\rho_t^{\text{before}}) \Delta$.
This has to be equal to the number of growing trees per unit time
$(1 - \rho_t) p \Delta^2$, leading to the relationship
\begin{equation}
\Delta = \frac{P(\rho_t^{\text{before}})
v_{\text{fire}}(\rho_t^{\text{before}})}{1 - \rho_t} \frac{1}{p}.
\label{guadalupe}
\end{equation}
This confirms the observed scaling behavior $\Delta \propto p^{-1}$
and additionally delivers the constant of proportionality, being
$0.63 \pm 0.05$
for $\rho_t^{\text{before}} = \rho_t^*$
and in good agreement with simulations.
Since the fire density is inversely proportional to the distance between the
spiral arms, we find $\rho_f \propto p$, i.e. the order parameter exponent
$\beta$ equals one. This can also be seen from the equality
$\rho_f = p \rho_e$ \cite{dro93}, which states that the number of burnt trees
has to be equal to the number of new grown trees in the stationary state.

If one chooses a fixed site behind the front and wants to know the tree
density in this region as function of the time $t$ until at $T = \Delta
/ v_{\text{fire}}$ the next front passes through, one has to replace in
Eq.~(\ref{eva}) $x$ by $v_{\text{fire}} t$ and arrives with
Eq.~(\ref{guadalupe}) at
\begin{displaymath}
\rho_t(t) = 1 - (1 - \rho_t^{\text{after}})
\left[\left(\frac{1 - \rho_t^{\text{before}}}{1 -
\rho_t^{\text{after}}}\right)^{\frac{1 -
\rho_t}{P(\rho_t^{\text{before}})}}\right]^{p t},
\end{displaymath}
which with Eq.~(\ref{soledad}) and $P(\rho_t^{\text{before}}) =
\rho_t^{\text{before}} - \rho_t^{\text{after}}$ leads to
\begin{equation}
\label{angela}
\rho_t(t) = 1 - (1 - \rho_t^{\text{after}}) e^{-p t}.
\end{equation}
Eq.~(\ref{angela}) can also be derived easily from
$\partial \rho_t / \partial t = p (1 - \rho_t)$,
first stated in \cite{gra91}. This shows that our picture is in accordance
with the basic equations describing the spiral state.

One additional point concerning Eq.~(\ref{guadalupe}) should be mentioned.
Eq.~(\ref{guadalupe}) relates $\Delta$, $\rho_t$, and $p$. If one regards
e.g. $\Delta$ as fixed, only the product $(1 - \rho_t) p$ is determined.
This reflects the fact that in a 'single front' state with fixed edge length
$\Delta$ one can have different overall densities $\rho_t$. The other
quantities adjust themselves automatically. However, in the 'real' spiral
state, fixing $\Delta$ determines invariably all other quantities. Since
the spiral state can be constructed by connecting 'single front' systems, this
seems to be a contradiction. The solution of this apparent contradiction is
that one has neglected the spiral centers. They yield a second, unfortunately
unknown, relation between $\Delta$, $\rho_t$, and $p$. This relation is brought
about by the rotation of the spiral centers. $\rho_t$ and $p$ determine the
angular velocity $\omega$ of the rotation, which, in turn, determines $\Delta$
via $\Delta = v_{\text{fire}} T = v_{\text{fire}} 2 \pi / \omega$. For the
calculation of $\rho_t^{c,2}$ with Eq.~(\ref{soledad}), using the decomposition
into 'single front' states, it is justified to neglect the spiral centers,
because in the thermodynamic limit their number density is zero. Nevertheless,
one has to be aware of the fact that, although not important for calculating
$\rho_t^{c,2}$, the spiral centers are the 'pacemakers' of the spirals and
therefore responsible for the magnitude of $\Delta$.

Another interesting point is that the density $\rho_e^{c,2}$ can also be
found from an extremum principle first stated in \cite{dro92}.
There, the extremum
principle was erroneously used to determine the critical density
$\rho_t^{c,1}$ of the SOC FFM.
The fire was believed to destroy as much trees as it can at the
critical point, but the result $\approx 39\%$ was in contradiction to
the measured value $\rho_t^{c,1} \approx 40.8\%$
\cite{hen93,gra93,chr93,cla94}.
However, the equations derived in \cite{dro92} from the extremum principle
can easily be seen to be equivalent to Eq.~(\ref{soledad})
for the spiral state. Therefore, the
principle yields the correct critical density,
but for a different, then unknown, phase of the model.

From the results found in this section one can draw some conclusions for
excitable media in general. In many excitable systems
stationary states with spiral waves can be found.
Famous examples are e.g. the Belousov-Zabhotinski (BZ) reaction \cite{ros88} or
the electro-physiological activity of heart tissue \cite{dav92}.
If the spirals are to sustain themselves in a stationary state, the
density of the excitable constituents (the 'fuel')
in a region immediately before a
fire front passes through it $\rho_t^{\text{before}}$
has always to be larger than or equal to
some threshold $\rho_t^*$, which, in turn,
is larger than the percolation threshold $p_c$
for that particular situation. The percolation threshold can in principle
be measured by preparing a homogeneous system with a certain
density of excitable constituents and 'exciting' one edge of the system.
If the resulting excitation front dies out before reaching the other end
of the system, we are still below the percolation threshold, and if it
reaches the other end in finite time we are above. At the percolation
threshold the front barely survives and needs an infinitely long time to
reach the other end. The overall density of the excitable constituents
has to be above the value $\rho_t^{c,2}$, below which no excitation
can be sustained.

A system that displays spiral waves or avalanches (concentric waves, target
patterns) depending on the density of excitable constituents
are e.g. populations of Dictyostelium Discoideum
amoebae \cite{lee96}. There, circular waves of signaling activity
emanating from pacemakers
are found for low cell densities, whereas for high cell densities one observes
spiral waves.

\section{Region of Finite Fire Density II: The Mixed Phase}
\label{secsix}

With decreasing $\rho_e$, the distance between the spiral arms becomes
smaller and the spirals finally break
up to single fronts. This change occurs continuously. The fronts, which
are more irregular than the spirals, have also been observed in the model of
Bak et al. Decreasing $\rho_e$ further, the coherence length of the fronts
soon becomes comparable with the lattice constant, and the
system reorganizes itself discontinuously
into another state, which constitutes the third phase of this model.
This happens at $\rho_e = \rho_e^{c,3} \approx 54.2\%$.
The simulations show that the trees in the new state tend to 'cluster'
(to form regions with higher tree density) with the
fire burning at their edges (see Fig.~\ref{covadonga}).
To sustain this type of structure, a certain minimum density of fires is
needed.
At $\rho_e = \rho_e^{c,3}$, $\rho_f$
jumps from $2\%$ to $10\%$.
If one lowers $\rho_e$ further, the size of the 'clusters' decreases, and
the fire density increases. The system as a whole becomes more homogeneous
(trees, fires and empty sites are 'mixed').
For $\rho_e = 0$, we observe $\rho_t = \rho_f = 1/2$ for all dimensions and
lattice types. The order parameter curve starts
linearly at $\rho_e = 0$
with a slope that depends only on the number of nearest neighbors $z$
(explanations in the appendix).
If we start with small $\rho_e$ and
traverse the phase transition in the other direction, it takes
place at different $\rho_e$ ($\rho_e^{c,4} \approx 54.7 \% > \rho_e^{c,3}$),
i.e. one has a first-order phase transition with hysteresis.
The order parameter as function of the density of empty sites
is shown in Fig.~\ref{2dphasediagram}.

\section{Dimensions other than Two and Different Lattice Types}
\label{secseven}

Since in one dimension there cannot exist spiral waves, we expect a simpler
phase diagram than in two dimensions.
For $\rho_e = 0$ we are dealing with a completely deterministic one-dimensional
cellular automaton where each site can be in one of two states (automaton
no. 54, according to the classification scheme of Wolfram \cite{wol83}).
In the stationary state each tree has at least one fire as neighbor and
vice versa. Strings of more than two trees or fires are not stable, since
sooner or later they would be invaded by fires. The
stationary state is periodic with period two.
The introduction of empty sites makes the automaton non-deterministic, because
the trees can now choose where to grow. This variant has not been investigated
so far.
In the simulations, system sizes of up to $L = 10^7$ were used.
The measured order parameter curves for the one-dimensional case
can be seen in Fig.~\ref{1dphasediagram}. In addition to the usual
nearest-neighbor interaction, we simulated also a variant where the fire
is allowed to jump over one empty site if necessary.
The curves start linearly
at $\rho_f = \rho_t = 1/2$ with different
slopes (explanation see Appendix).
For increasing $\rho_e$ one observes the same phenomena as in two
dimensions. The tree clusters (strings in 1D) become larger and are
accompanied by fire on at least one of their ends. At $\rho_e = \rho_e^{c,4}
\approx 20\%$
($\rho_e^{c,4} \approx 26\%$ for next-nearest neighbor interaction) the density
of empty sites is too high for this structure to survive, and the fire
density drops from a finite value of approximately $20\%$ to zero, since the
system does not have the possibility to rearrange itself into
a spiral wave phase. Instead, now isolated chunks of trees are ignited
and burn down in finite time. In the reverse direction, $\rho_f$ remains
zero until $\rho_e = \rho_e^{c,1} = 0$,
i.e. the hysteresis in one dimension is maximum.
In the vicinity of the point $(\rho_e = 0, \rho_f = 0)$, the critical
behavior of the SOC FFM in one dimension \cite{dro93a} is reproduced.
Since there exists no
spiral phase in one dimension, there are no $\rho_e^{c,2}$ and $\rho_e^{c,3}$.

The two-dimensional simulations were also done with a triangular lattice and
yielded the same behavior as for the square lattice (for the order parameter
curve see Fig.~\ref{2dphasediagram}).
The values of $\rho_e^{c,1,2,3,4}$ and
the slope of the order parameter curve for $\rho_e \to 0$ of course
are different (see Tab.~\ref{table1}).

In three dimensions, we could identify a sub-critical phase
as well as a mixed phase. The order parameter curve is plotted in
Fig.~\ref{3dphasediagram}.
Due to small $L$ ($L \le 300$) the critical exponents in the subcritical phase
and the critical density $\rho_e^{c,1}$ could not be measured with
sufficient accuracy.
The mixed phase in three dimensions shows analogous behavior to the
two-dimensional mixed phase and does not display new phenomena.
The fire density curve could not be measured for very small $\rho_f$
($\rho_f \lesssim 0.2\%$), due to finite size effects.
If one extrapolates the fire density curve to the point $\rho_f = 0$, one
arrives at $77.3\%$, therefore, $\rho_e^{c,4}$ lies between the last
simulated density $76.9\%$ and $77.3\%$, i.e. $\rho_e^{c,4} = 77.1\% \pm
0.2\%$.
The fact that there exists a gap between the end point of the
critical phase $\rho_e^{c,1} \approx 78.1\%$ (taken from \cite{cla94}) and the
end point of the mixed phase $\rho_e^{c,4}$ leaves open
the possibility of the existence of a third phase containing the
three-dimensional analogon of spiral waves (scroll waves). Due to the finite
size of our sample, however, we could only observe two-dimensional,
flat fire fronts in that region.

The results for all simulated lattices and dimensions can be seen in
Tab.~\ref{table1}.

\section{Mapping Self-Organized Criticality onto Criticality}
\label{seceight}

The model we have investigated in this paper is an example of a system
which is far from equilibrium. It shows a wealth of interesting
structures depending on the density of empty sites $\rho_e$.
For large $\rho_e$, we find a region of vanishing fire density
which contains the critical behavior of the SOC forest-fire model.
For lower density of empty sites we obtain the spiral waves of
the Bak et al. model. Finally, for even lower $\rho_e$,
we find a phase in which the trees show the tendency to form clusters
with the fire burning at their edges. The transition between the second and
the third phase is of first order, whereas the transition between the
first and the second phase is rather unconventional. We find hysteresis
and critical behavior on only one side of the transition.
The close relation of the spiral wave phase with the synchronized phase of
the model of \cite{cla95}
allows us to understand the mechanism which determines the density
distribution in the spiral state of not only this model, but of
arbitrary excitable systems. In particular, it yields the
density in front of the excitation fronts, the minimum density of excitable
constituents that is necessary to sustain the excitation, and the factor of
proportionality between the distance of the spiral arms
and the 'tree growth' rate $p$.

Apart from being interesting in its own right, the model presented in this
paper (like the model of \cite{cla95}) has
to be seen also in the context of the general
claim of \cite{sor95}, that the critical points of SOC models can be regarded
as ordinary
critical points of second order phase transitions. It was claimed in
\cite{sor95} that this should be possible for all SOC models. There were
also given instructions on how to achieve this goal
for some particular models including the forest-fire model.

The results in this paper and in \cite{cla95}, however, disprove this
hypothesis.
While the sub-critical side of the transition in both models
behaves as expected, the other
side (the side with a supposed non-vanishing order parameter)
always exhibits surprising features.
In the model of \cite{cla95} we found a whole critical region and lots of
first-order phase transitions, whereas the actual model shows hysteresis and no
criticality at all.

Although these 'negative' results can not
strictly rule out the possibility that
for some further slight change of the rules one might succeed in obtaining
the usual 'decent' behavior of the order parameter, the argument at the end
of Sec.~\ref{secfour},
together with the simulation results of this paper and of
\cite{cla95} make it seem very unlikely.
We consider it to be more probable that a subset of SOC models
are 'only' ordinary critical models 'in disguise'. With these models
the mapping proposed in \cite{sor95} should be possible
(the results in \cite{bag96} also seem
to point in that direction), while there also exist genuine
SOC models for which this procedure can not be carried through.
The forest-fire model seems to belong to the second class.

We suggest that similar phenomena and difficulties (critical regions,
hysteresis at the critical
point, critical behavior on only one side) will also be found
in other models of SOC, as e.g. the sandpile, earthquake, and evolution
models.

This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) under Contract
No Schw 348/7-1.

We thank B. Drossel for fruitful discussions.

\section{Appendix: Some Properties of the Order Parameter Curve}

All order parameter curves considered in the previous sections
started at the point $\rho_f = \rho_t = 1/2, \rho_e = 0$. This starting
point is independent of dimension or lattice type, which can be seen as
follows. For $\rho_e = 0$ each tree will sooner or later be set on fire by
one of its $z$ nearest neighbors.
Once having set on fire the tree, the fire jumps forever
between this site and its neighbors, because its neighbors,
after burning down, have to become trees again in the next time step
(there are no empty sites). The dynamics of these $1 + z$ sites
then is fixed. After each time step trees and fires change places. But
the fire can still propagate to other sites, so that finally the whole system
consists of such 'blinking' regions. The state is periodic with period 2 and
the average fire and tree densities have to be $1/2$. For a random initial
state the densities are $1/2$ also without averaging.

The slope of $\rho_f(\rho_e)$ for $\rho_e \to 0$ depends only on the number
of nearest neighbors $z$ and lies always within the interval $[-1;-1/2]$.
This can be seen as follows. If one takes a stationary state at $\rho_e = 0$
and inserts some empty sites by removing an equal number of trees and fires,
the system will not remain in this state (with a slope of $-1/2$),
but will adjust itself to a new stationary state with an even lower number
of fires, since the spreading conditions for the fire are now worse than before
due to the inserted empty sites. The magnitude of this effect depends only
on the coordination number $z$. The larger $z$, the smaller is the effect.
On an infinite-dimensional lattice,
the fire density will not readjust itself at all, because no fire can feel the
empty sites. The slope in this case is the maximum possible slope $-1/2$.
This can also be seen mathematically.
All fires have been trees in the preceding
time step, therefore $\rho_f \le \rho_t$ in the stationary state.
With $\rho_f + \rho_e + \rho_t = 1$ it
follows $\rho_f \le 1/2 - \rho_e/2$. Since for $\rho_e = 0$ both sides of
the last inequality are identical, one can differentiate and finds
$\partial \rho_f / \partial \rho_e |_{\rho_e = 0} \le -1/2$. $1/2$ is the upper
bound for the slope.

The lower bound for the slope equals $-1$. This value is assumed
for one dimension, as will be shown in the following
by considering the effect of introducing one empty site into the
stationary 1D state at $\rho_e = 0$. Since the one-dimensional
lattice has the smallest possible number of neighbors, its slope represents
the lower bound.
As argued in Sec.~\ref{secseven}, in the one-dimensional stationary state
at $\rho_e = 0$
no more than two neighboring sites can be in the same state.
In a symmetric state, where both neighbors of a tree are fires
and vice versa, i.e. where fires and trees are strictly alternating,
an empty site does not modify its neighborhood.
Since the effects of the empty site are only local, it is sufficient to
consider only three cases: One pair of sites in the same state (see left part
of Fig.~\ref{symm}), collision of two pairs (see middle part of
Fig.~\ref{symm}), and annihilation of two pairs
(see right part of Fig.~\ref{symm}). The possibilities of placing the
empty site in these cases can be further reduced to the cases shown
in the second line of Fig.~\ref{symm}. The reason is that,
since in each time step $n$ trees burn down and are refilled
into $n + 1$ empty sites, with probability $\approx 1$ for large $n$
the empty site at time $t$ was a fire at time $t-1$, i.e. at time $t$
the empty site occupies a place which would have been a tree in the
unperturbed state.
Therefore, we place the empty site at a tree site in the second line of
Fig.~\ref{symm}. If instead we had chosen the neighboring tree sites, we would
only have generated mirror-symmetric forms of the configurations shown.
In the third line the empty site has disappeared from our
small section of the system and has become a tree with probability
$\approx 1$ for $n \gg 1$. In the left case, the pair of sites
moves one lattice spacing due to the presence of the empty site.
In the middle case, two pairs which are separated by only
one lattice site collide due to an empty site.
In the right case, an empty site causes two neighboring pairs (which may have
collided earlier) to annihilate each other and thereby symmetrize the region
around the empty site.
Therefore, sooner or later all pairs will have annihilated,
and the entire system (with the exception of the single empty site)
is in the symmetrical state.
This evolution to the symmetrized state was also
observed in the simulations.

We can therefore restrict ourselves to consider
the consequences of the introduction of empty sites into such a highly
symmetrical state. This has the advantage that there is no need to consider the
neighboring sites, because they are not affected.
There are two possibilities to introduce
an empty site. The first one
is to remove a tree. In the next time step this tree would have
become a fire which it cannot do now. Instead, with very high probability
a new tree is grown at this empty site. During the re-growth of trees the empty
site effectively moves to another site in the system (which would have
been a 'tree site'). While there is one excess tree in one part of the system,
there is one tree
less in another part of the system, so the total number of trees
does not change. The number of fires, however, has decreased by one, since
we prevented the 'birth' of one fire. The second possibility is to remove
a fire initially. If that happens, the empty site will have moved after the
tree-growth phase to a 'tree site', and we have the first case again.
Therefore, in each case
the introduction of empty sites takes place completely at the expense of fire
sites and $\partial \rho_f / \partial \rho_e |_{\rho_e = 0} = -1$.

The measured results for the slope of the order parameter curve are shown
in Tab.~\ref{table1}. One can see that the slopes of the two-dimensional
triangular
lattice and the three-dimensional hyper-cubic lattice are the same, since they
have the same coordination number $z = 6$.

As mentioned earlier, not all tree densities between 0 and 1 are possible
in the stationary state. If one transforms the $\rho_f(\rho_e)$ curves to
$\rho_f(\rho_t)$ curves via $\rho_f + \rho_t + \rho_e = 1$, one can
read off the diagram the allowed $\rho_t$ values
(see Fig.~\ref{alldimensions}).
For these values it is possible to simulate the model with fixed $\rho_t$.
The results do not differ from the results with fixed $\rho_e$.
Also in this case, one
has to be careful with the initial conditions, since there are sometimes
two stationary
states for the same value of $\rho_t$, and the initial state determines
which of the two possible stationary states will be chosen by the system.

redpajama_set_name: RedPajamaArXiv


Born to Palermo, after experimentations in the field of the design, it attends the European Institute of Design and the School of the Comic strip of Milan. It undertakes covered the professional one just working for a period in the field of the publishing design and subsequently for various agencies of advertising like Art director, Graphic designer and illustrator.
Today it lives in the historical center of Palermo and works to wide beam in the field of the communication and of the acquired competences that concur it to space with professionality in the world of the design and the advertising.


text:
Leftist Filmmaker Shocked by Lack of People at Dem Convention: This Place is EMPTY! (VIDEO)
A liberal filmmaker named Josh Fox was so stunned by the number of empty seats at the Democratic National Convention on Tuesday night that he made a video to show all the empty seats.
The Daily Wire reports:
Leftist Filmmaker Uploads Video Showing His Shock At Just How Empty The DNC Is
If you're watching the big news networks, you'd think the Democratic National Convention is a standing-room-only smashing success with zero controversy, angry protests, and tons of American flags.
Film director, Josh Fox, best known for his Oscar-nominated anti-fracking documentary Gasland, uploaded a video (below) to his Facebook page live from inside the Wells Fargo Center, where he was utterly shocked at the amount of empty seats.
Fox tells the camera, "This is amazing, this place is empty. There is nobody left in here. I mean this whole stadium, look at this," as he pans his cellphone to show the lack of cheering Dems.
He continues in disbelief, adding, "This is not voter enthusiasm…. I can't believe my eyes. I've never seen anything like this. This is the primetime of the Democratic National Convention right after the nomination of Hillary Clinton and this place is emptied out like crazy. I'm stunned."
The media never showed the audience on Tuesday night or highlighted all the empty seats and there's a reason for that.
The Democrats are having a much tougher time than anyone is reporting.


text:
Q: Deslocamento de coluna no bootstrap Estou montando um código para colunar uma lista de produtos.
Cada linha tem 4 colunas e pula para a próxima linha.
Mas quando por exemplo, a altura da primeira coluna é maior que as outras e eu redimensiono a janela, a primeira coluna da segunda linha, se desloca pra frente.
Segue abaixo um trecho do código. Será que alguém pode me ajudar?
<% Do While not rs.Eof

Titulo

Valor

<% rs.MoveNext
Loop %>

A: Crie uma Class css para os item, e defina um height para esses itens.
Deste

Titulo

Valor

Mude para esse:

Titulo

Valor

No seu arquivos de css da pagina adicione esse stylo
.item{
  height : 130px; /*leve em consideração o tamanho que você acha adequado*/
}

redpajama_set_name: RedPajamaStackExchange


Income Taxes Who Should Pay The Most?
Can the United States Survive a Bear Market?
Posted in Daily Comments	| Comments Off on Tear Down The Statues!
The American reptilian brain says the Federal Reserve is critical for managing the economy. The larger thinking brain, if engaged, easily sees that Federal Reserve members are paid for the pretense of knowledge. Are these members the world's best economists? No, they are the best politicians among the country's set of economist. The larger thinking brain, again if commissioned to think, easily recognizes their actual role. The genuine role of the Federal Reserve is to carry out political agendas which normally focuses on transferring wealth from the bottom of the income ladder to the top.


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What Did Bush Know -
and When Did He Know It?
In March 2002, Rep. Cynthia McKinney (D-GA) shocked the American political establishment by demanding answers to the unanswered - and even unasked - questions about September 11.
McKinney first read her remarks on March 25, 2002 on KPFA's "Flashpoints" (advance to the 30 minute mark). Her remarks were published on March 28 in Truthout.org under the headline, Thoughts On Our War Against Terrorism.
On April 15, McKinney provided additional details in an editorial in the Atlanta Journal Constitution, under the headline Bush Must Answer September 11th Questions.
Since she made her remarks, McKinney has been viciously attacked in the media. We are tracking each attack in Democrats.com News.
McKinney made numerous points, but two of them were selected by the media for mass distribution - and distortion.
1. War Profiteering
"Persons close to this Administration are poised to make huge profits off America's new war. Former President Bush sits on the board of the Carlyle Group. The Los Angeles Times reports that on a single day last month, Carlyle earned $237 million selling shares in United Defense Industries, the Army's fifth-largest contractor. The stock offering was well timed: Carlyle officials say they decided to take the company public only after the Sept. 11 attacks. The stock sale cashed in on increased congressional support for hefty defense spending, including one of United Defense's cornerstone weapon programs."
These facts have been widely reported in the mainstream media, and have not been challenged. See Democrats.com .comments on the Carlyle Group.
2. September 11 Warnings
"Now is the time for our elected officials to be held accountable. Now is the time for the media to be held accountable. Why aren't the hard questions being asked? We know there were numerous warnings of the events to come on September 11. Vladimir Putin, President of Russia, delivered one such warning. Those engaged in unusual stock trades immediately before September 11 knew enough to make millions of dollars from United and American airlines, certain insurance and brokerage firms' stocks. What did this Administration know, and when did it know it about the events of September 11? Who else knew and why did they not warn the innocent people of New York who were needlessly murdered?"
The mainstream media has published numerous articles about warnings of an impending terrorist attack that went unheeded. For more details, visit Carolyn Kay's excellent MakeThemAccountable.com.
Rep. McKinney is not the only Member of Congress who wants an open and thorough investigation of the September 11 warnings that went unheeded. Senators Joe Lieberman (D-CT), John McCain (R-AZ), and Robert Torricelli (D-NJ) have all called for a thorough public investigation of the events and intelligence failures that led up to September 11. George Bush, Dick Cheney, and Trent Lott have adamantly opposed any public investigation. See Democrats.com .comments on September 11 Investigations.
The Smear Campaign
On April 11 2002, the Washington Post's Julie Eilperin took these two points and twisted them into a pretzel in order to accuse McKinney of blaming the Bush Administration for the September 11 attacks.
Democrat Implies Sept. 11 Administration Plot
As the text of Rep. McKinney's remarks shows, she did NOT imply September 11 was an "Administration Plot." She did not say it, she did not suggest it, and she did not even obliquely hint at it. In other words, Juliet Eilperin is LYING.
Rupert Murdoch's right-wing New York Post twisted the Washington Post's story even further.
She is demanding an investigation into her charge that Bush may have done nothing because his rich friends stood to profit from the military build-up, The Washington Post reported
McKinney did NOT charge "that Bush may have done nothing because his rich friends stood to profit from the military build-up." And since she did NOT make that charge, she is NOT "demanding an investigation into" that charge.
We Believe Cynthia!
The Washington Post, New York Post, AP, and NY Times have variously suggested that Rep. McKinney is a conspiracy theorist, crazy, pro-Iraqi, or an irresponsible publicity-seeker.
In fact, Rep. McKinney is a leading champion of human rights for blacks and all people of color, both at home and abroad. She is best known for working with conservative Republican Chris Smith (R-NJ) to enact a "code of conduct" for selling arms to dictatorial regimes. She is serving her fifth term in Congress, representing a district that was only 33% black in 1996.
The media's attempt to portray Cynthia McKinney as crazy is a repeat of the media's character assassination campaigns against Anita Hill and Lani Guinier. In all three cases, outspoken black women have been attacked for daring to express ideas that challenge the white male establishment. History has shown that both Anita Hill and Lani Guinier were telling the truth. History will also show that Cynthia McKinney is telling the truth. We believed Anita Hill and Lani Guinier - and we believe Cynthia McKinney too.
And we're taking action! We invite you to join our "We Believe Cynthia!" campaign.
1. Write letters to the media - especially those who smeared Cynthia McKinney - telling them "We Believe Cynthia!" Write to eilperinj@washpost.com and letters@nytimes.com. Send a copy to public@democrats.com and we'll publish the best ones here.
2. Post our campaign button (at the top of this page) on your Web site, with a link to this page (http://www.democrats.com/display.cfm?id=270).
Thanks for your support!!
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