text:
\section{Introduction}

Many times, in Physics we need to extend some fundamental law and
their application \cite{Maxwell1864}, or just produce new models relating
experimental facts slightly different from original \cite{Whestphal2021},
like Perturbation Theory \cite{Merlin2021}, Energy Conservation \cite{Landau1980},
Gauss Divergence Theorem \cite{Basil2011}, Stokes Curl Theorem \cite{Arfken2012},
Reynolds Theorem \cite{Reynolds1902}, each one with their better applicability
to singular context.

Observing some new materials applications \cite{Abayev2006}, we can see
that distributions plays a central role to modeling this system \cite{Pierantoni2021}.
Sometimes some modification in theoretical model due to experimental
requirement is needed as theoretical proposition \cite{Vollath2019}.
We feel the urgency to some principle to obtain distributions quickly,
as the main goal of our work. We name it as \textbf{Dimensionless
Fluctuation Balance Principle}.

Our ``\textit{Ansatz}'' starts when we observe Boltzmann distribution
\cite{Rowlinson2005}, which was applied to Plank's hypothesis resulting
their Radiation Law \cite{Agudelo2010}. In the ultraviolet catastrophe
case as in \cite{Rayleigh1900}, with $\left(E=kT\right)$\footnote{where $k$ is the Boltzmann's constant},
we can see a pair of quantities energy and temperature $\left(E,T\right)$,
and this one takes a form of Boltzmann distribution like $u\left(E,T\right)\propto e^{-E/kT}$
\cite{Trevena2010}.

Our research starts from Dimensionless Fluctuations of $\left(E,T\right)$
pair and we verify how to obtain a Partial Differential Equation (PDE),
which the solution is desired distribution. In this case, Boltzmann
distribution $u\left(E,T\right)\propto e^{-E/kT}$, after this we
try to apply the same procedures to other distributions searching
for a fundamental principle, our tests results was verified
with success to principal Physics Distributions.

This work shows in completely new way, how to obtain statics distributions
from fluctuations balance, theses fluctuations relate fractions from
domain dimensionless terms. A special point about our analysis is
that with this one we can obtain all fundamentals distributions (including
Boltzmann Entropy Law) without probability as a required initial concept.
This achievement doesn't neglect statistical theory in measurement
of states, because we can think any mean value as inner product between
quantity to be measured as distribution on Hilbert Spaces.

\section{Small Fluctuations Balance}

In many cases, when we study a control volume conservation by Reynolds
Transport Theorem \cite{Reynolds1902}, we just observe net rate exchange
of property per unit time. As a classical example, the continuity
equation case when balancing charges in some region of space which
can decrease when some current leaves the volume across boundary surface
\cite{Marion1980}.

Our effort here is to study localized phenomenon, as simple case,
gas inside a closed box or some portion of fluid inside a thermal-mechanical
set. To reach this goal, instead of work with classical balance like
continuity equation, we will study small portions of a property inside
a fraction of material.

As illustrative case study, being a given group of molecules with
the same characteristics, the summation over $N_{i}$\footnote{$N_{i}$ also represent a group of molecules of gas.}
gives the total number of molecules $N$, then

\begin{figure}[ht]

\begin{centering}

\begin{tabular}{ccc}
\includegraphics[scale=0.30]{colado1} & \includegraphics[scale=0.30]{colado2} & \includegraphics[scale=0.30]{colado3}\tabularnewline
\textbf{(a)} & \textbf{(b)} & \textbf{(c)}\tabularnewline
\end{tabular}

\end{centering}

\caption{
(a) Representation of molecules arrangement like cluster, fractioning domain in many partitions inspire equation (\ref{eq:1}) which treat about conservation the number of molecules distributed in partitions of cluster. (b) If the cluster is large with many particles, and divided in many partitions, a single partition will retain a small portion of total $\delta N$. In a special limit, $dN=\protect\underset{N\rightarrow\infty}{\lim}\delta N$, this is the context of equation (\ref{eq:2}). (c) The same as (b) in the sense of large cluster and many partitions, but we extend from $N$ which is total number of particles, to two properties $A$ and $B$ which can represents any physical quantity, in general sense $A$ or $B$ can be \{molecules number, mass, energy, temperature, etc\} in the context that one property can be exchanged to another as physical phenomenon, this concepts are relateds to equations (\ref{eq:3})-(\ref{eq:9}). }

\end{figure}

\begin{equation}
\sum_{i=1}^{^{n}}\left(\frac{N_{i}}{N}\right)=1.\label{eq:1}
\end{equation}
For simplicity we can see that a portion instead of sum of all contributions,
can be denoted as,

\begin{equation}
\left(\frac{\delta N}{N}\right)=c_{N}.\label{eq:2}
\end{equation}

From equation (\ref{eq:1}) we can see that the sum of all portions
of molecules number reach the maximum value which is the total of
molecules amount. Otherwise, a small portion of molecules number presented
in (\ref{eq:2}) gives us the fluctuation as small fraction of molecules
amounts much less than one.

Can we extend this concept of fluctuations to more than one property?
Does this extension give us some new aspects? To reach a concise response
to this question, let us relate more than one property in some balance
around frontier of a surrounded control volume.

Let us consider two different properties $A$ and $B$, and a region
in the space like a control volume or membrane, at which this two
properties can exchange one to another inside this boundaries region.
So considering this,
\begin{equation}
\left(\frac{\delta A}{A}\right)=c_{A},\label{eq:3}
\end{equation}
and
\begin{equation}
\left(\frac{\delta B}{B}\right)=c_{B}.\label{eq:4}
\end{equation}
Here $c_{A}$ and $c_{B}$ are fractions of a total amount, their
total summation reaches one. Then, summing all portions over complete
region from (\ref{eq:3}) and (\ref{eq:4}) we obtain,
\begin{equation}
\sum_{n}\left(\frac{\delta A}{A}\right)=1,\label{eq:5}
\end{equation}
and
\begin{equation}
\sum_{n}\left(\frac{\delta B}{B}\right)=1.\label{eq:6}
\end{equation}
Observing (\ref{eq:5}) and (\ref{eq:6}) we can see as first impression
a balance condition as,
\begin{equation}
\sum_{n}\left(\frac{\delta A}{A}\right)=1=\sum_{n}\left(\frac{\delta B}{B}\right).\label{eq:7}
\end{equation}

But the question doesn't close so easily, because in physical cases
we have some law relating $A$ and $B$. As a good choice we point
here to relate fraction of quantities as their appear in (\ref{eq:3})
and (\ref{eq:4}) which gives,

\begin{equation}
\left(\frac{\delta A}{A}\right)=\frac{c_{A}}{c_{B}}\left(\frac{\delta B}{B}\right),\label{eq:8}
\end{equation}
that can be simplified to initial form as,

\begin{equation}
\left(\frac{\delta A}{A}\right)\propto\left(\frac{\delta B}{B}\right).\label{eq:9}
\end{equation}

As example, our first case study starts from temperature, considering
it as important parameter of a gas, and for this reason let's analyze
the fluctuation of temperature, searching for new useful relations
for balance of properties in a gas.

\section{Boltzmann Distribution PDE}

\subsection{A Closer View to First Law of Thermodynamics }

Let our thermodynamics system with internal energy $U$, $Q$ as energy
that external agent delivery to the system and $-W$ the work done
by system. The portion $dQ$ is delivered by heating system at each
cycle and some portion of energy that we deliver to the system will
be spent with mechanical work $-dW$. Therefore, the internal energy
of system, which is a \textbf{balance} of \textbf{energy delivered}
to the system\textbf{ $dQ$} with discount of the \textbf{mechanical
work done} by the system $-dW$, is given by,

\begin{equation}
dU=dQ-dW,\label{eq:10}
\end{equation}
which is the First Law of Thermodynamics \cite{Mortimer2000}.

For simple analysis, with the aim of verify temperature fluctuations,
we can put the equation (\ref{eq:10}) in another terms: changing
the heating energy of system $dQ$ to material components temperature
increase as $CdT$, where $C$ is thermal capacity and $T$ is temperature,
and thinking as a work done $-dW$ like expansion of system. We get

\begin{equation}
dU=CdT-pdV.\label{eq:11}
\end{equation}
Isolating small fluctuation of temperature, we can see that,

\begin{equation}
\frac{dT}{T}=\frac{dU+pdV}{CT}.\label{eq:12}
\end{equation}
Equation (\ref{eq:12}) shows that temperature fluctuation is related
to internal energy modification and work done by system.

In next steps we will consider how these laws can contribute to understand
gas dynamics.

\subsection{Small Fluctuation Balance to Gas System}

As we can see at Right Hand Side (RHS) of equation (\ref{eq:12}),
the internal energy term and work term are divided by thermal energy
amount, expressing energy fluctuations and this fact explains the
fundamental relation between temperature fluctuation and energy fluctuation.
As we discuss at equation (\ref{eq:9}), a first appointment to this
energy-temperature balance of fluctuations can be done as,

\begin{equation}
\frac{\delta E}{E}\propto\frac{\delta T}{T}.\label{eq:13}
\end{equation}
This relation is based on thermal system, which has a balance between:
input energy, work done and internal energy of system. The continuation
of the analysis that improve equation (\ref{eq:13}), will be made
for a gas inside a box as our object of study and the most important
thing to study this gas dynamic system. The analysis consists of a
small virtual work\footnote{As default in Classical Mechanics to obtain motion equations.}
considering the box frontiers slightly moving from initial state to
most outside or inside position. This analysis will give us how dependency
between energy and temperature fluctuations increase or decrease.

We start this analysis first considering a gas inside a box with $T$
as gas temperature and $E$ as energy, as two fundamental quantities\footnote{In this context, temperature $T$ is a macroscopic value and energy
$E$ is a microscopic value related to the first one.}. If we think this gas can do small displacement pushing box walls
to outside, like a small virtual work, we need to balance where the
Energy fluctuation will be related to work done by gas, which will
cost a loss of thermal energy of the gas and as effect temperature
will decrease\footnote{The work done by gas costs internal energy decreasing the temperature
of a gas.}. In mathematical terms equation (\ref{eq:13}) became,

\begin{equation}
\frac{\delta E}{E}\propto-\frac{\delta T}{T}.\label{eq:14}
\end{equation}

Another change that must be done is related to dimensional analysis,
because Left Hand Side (LHS) is a fluctuation of Energy, and RHS is
related to temperature.

At first glance the thermal energy $kT$ could be a good proposition,
considering that a gas inside a box is well represented by temperature
of gas in insulated system. Then,

\begin{equation}
\frac{\delta E}{E}=-\frac{\delta kT}{kT},\label{eq:15}
\end{equation}
as we can see, a natural simplification of constant $k$ occurs. This
is a strong aspect of fluctuation analysis, because all fluctuation
terms are dimensionless in essence.

In the next steps let's consider this fractions $\delta E$ and $\delta T$
so small that we can assume them as differentials quantities $dE$
and $dT$. So,
\begin{equation}
\frac{dE}{E}=-\frac{dT}{T}.\label{eq:16}
\end{equation}
Considering that we can have some distribution function like $u\left(E,T\right)$
and other dependencies of interest, a partial differential approach
is most affordable to this purpose, then we can change (\ref{eq:16})
to,

\begin{equation}
\frac{\partial E}{\partial T}=-\frac{E}{T}.\label{eq:17}
\end{equation}
Using chain rule below, with the goal to include distribution function
$u$$\left(E,T\right)$, given by

\begin{equation}
\frac{\partial E}{\partial T}=\frac{\partial u}{\partial T}\frac{\partial E}{\partial u}.\label{eq:18}
\end{equation}
Replacing (\ref{eq:18}) in (\ref{eq:17}), we get a partial differential
equation in the form,

\begin{equation}
\frac{\partial u}{\partial T}=-\frac{E}{T}\frac{\partial u}{\partial E}.\label{eq:19}
\end{equation}
This is a new form to study distribution, from fluctuation analysis.
\textbf{Here, we present the Boltzmann Distribution PDE}. Now, to
obtain the Boltzmann distribution, we only need to solve Partial Differential
Equation (\ref{eq:19}).

We point here a function that depends on two properties which we made
fluctuation analysis. For simplicity, we use a fashion of exponents
$a$ and $b$ like we do in dimensional analysis,
\begin{equation}
u\equiv e^{-(kT)^{a}\cdot(E)^{b}}.\label{eq:20}
\end{equation}
Replacing (\ref{eq:20}) in (\ref{eq:19}), we get,
\begin{equation}
\frac{\partial u}{\partial T}=-k^{a}E^{b}aT^{a-1}e^{-(kT)^{a}\cdot(E)^{b}},\label{eq:21}
\end{equation}

\begin{equation}
\frac{\partial u}{\partial E}=-k^{a}T^{a}bE^{b-1}e^{-(kT)^{a}\cdot(E)^{b}}.\label{eq:22}
\end{equation}
To relate (\ref{eq:21}) and (\ref{eq:22}), we just divide booth
terms and simplify in order to analyse our general solution. Then,

\begin{equation}
\frac{\partial u}{\partial T}=\frac{a}{b}\frac{E}{T}\frac{\partial u}{\partial E}.\label{eq:23}
\end{equation}

The way between (\ref{eq:20}) to (\ref{eq:23}) show us that $a$
and $b$ are parameters that we can choose to satisfy the PDE. Finally,
comparing (\ref{eq:23}) to (\ref{eq:19}) we can see that choosing
$a=-b$ set PDE in (\ref{eq:23}) as the same as Boltzmann PDE in
(\ref{eq:19}), with the same sort this choice implies in tentative
solution of equation (\ref{eq:20}) to the form,

\begin{equation}
u=e^{-\left(E/kT\right)^{b}}.\label{eq:24}
\end{equation}
At last step, if we choose $b=1$, we find the Boltzmann Distribution
function\footnote{Note that we start solving a PDE using a general solution method inspired
by exponentials combinations, like when we try find some law of Physics
using Dimensional Analysis. As expected, this method gives us free
to choose some parameters to get result, but it's a strong point of
this method, not a weakness.},
\begin{equation}
u\propto e^{-E/kT}.\label{eq:25}
\end{equation}

Next step we study application of this distribution in Planck hypothesis
context, which refers to using Boltzmann Distribution function to
get Planck's Radiation Law.

\section{Planck's Law as Boltzmann Distribution PDE Solution}

\subsection{Planck hypothesis}

Planck when solving ultraviolet catastrophe explains two points. First
one is their achievement foundations related to Boltzmann Distribution
and second that he finds their formula heuristically searching for
a curve that satisfy experimental facts.

In our Appendix A session, we treat aspects related to Planck \textit{Ansatz
}which consist proposing that number of states like a photon gas is
related to Boltzmann distribution, and we point additionally to particular
regime $h\nu\ll kT$ as an experimental foundation too. Boltzmann
distribution is given by\footnote{Another solution to Boltzmann Distribution PDE (\ref{eq:19}) have
positive signal.}

\begin{equation}
u\propto e^{+E/kT}.\label{eq:26}
\end{equation}
Replacing $E=h\nu$ in (\ref{eq:26}), the initial form of distribution
stay

\begin{equation}
u=e^{h\nu/kT}.\label{eq:27}
\end{equation}
Planck solves breaking region in ultraviolet zone, which occurs because
$kT$ term is much greater than $h\nu$, or just, $kT\gg h\nu$.

At same fashion as Einstein explains easily mass-energy equivalence,
expanding kinetic energy because of velocity $v$ is much less than
$c$ \cite{Einstein2012}, let's expand Boltzmann distribution using Planck
hypothesis, with Taylor series at variable $\nu$ around ultraviolet
zone limit. So,

\begin{equation}
e^{h\nu/kT}=\lim_{h\nu\ll kT}e^{h\nu/kT}\left[1+\frac{h\nu}{kT}+\frac{1}{2}\left(\frac{h\nu}{kT}\right)^{2}+...+\sum_{n=3}\frac{1}{n!}\left(\frac{h\nu}{kT}\right)^{n}\right].\label{eq:28}
\end{equation}
Considering $h\nu\ll kT$ as in (\ref{eq:28}), let's truncate terms
to get an approximation, for all terms with $n$ equal two or higher,
resulting,

\begin{equation}
e^{h\nu/kT}\simeq1+\frac{h\nu}{kT}.\label{eq:29}
\end{equation}
Isolating $kT$ term,

\begin{equation}
kT\simeq\frac{h\nu}{e^{h\nu/kT}-1},\label{eq:30}
\end{equation}
that is a thermal energy related to a photon in the gas at state $\mu$.
Or just $E_{\mu}$,

\begin{equation}
E_{\mu}=\frac{h\nu}{e^{h\nu/kT_{\mu}}-1}.\label{eq:31}
\end{equation}
This equation is the Planck's Law of Radiation, given energy per photon
state $\mu$ in the gas, valid across ultraviolet zone. Planck shows
by his law a mathematical relation that attends phenomenology not
solved before by the classical formulation.

The essence of Planck proposition is that his \textit{Ansatz }consists
of relating radiation as gas of photons and applying Boltzmann distribution
as hypothesis to study this gas.

\subsection{Planck's Law as Boltzmann's Differential Equation Solution }

When Planck starting from Boltzmann distribution and obtain their
radiation law, he defines the case study as a photon gas system which
has compatibility with Boltzmann gas. In this perspective that Planck's
law was originated from Boltzmann Distribution, let's investigate
if Planck's law satisfies Boltzmann PDE equation (\ref{eq:19}),

\begin{equation}
\frac{\partial u}{\partial T}=-\frac{E}{T}\frac{\partial u}{\partial E},\label{eq:32}
\end{equation}
or just,

\begin{equation}
\frac{\partial u}{\partial T}\left(\frac{\partial u}{\partial E}\right)^{-1}=-\frac{E}{T}.\label{eq:33}
\end{equation}
Starting from equation (\ref{eq:31}),

\begin{equation}
E_{\mu}\equiv u=\frac{1}{e^{h\nu/kT_{\mu}}-1}.\label{eq:34}
\end{equation}
Without loss of generality let's introduce two free parameters $a$
and $b$ in (\ref{eq:34}). Then,

\begin{equation}
u=\left(e^{h\nu/kT_{\mu}}+a\right)^{b}.\label{eq:35}
\end{equation}
Applying (\ref{eq:35}) in (\ref{eq:33}), we get

\begin{equation}
\frac{\partial u}{\partial T}=-b\left(e^{h\nu/kT_{\mu}}+a\right)^{b-1}\frac{h\nu}{kT_{\mu}^{2}}e^{h\nu/kT_{\mu}},\label{eq:36}
\end{equation}
and,

\begin{equation}
\frac{\partial u}{\partial E}=b\left(e^{h\nu/kT}+a\right)^{b-1}\frac{1}{kT_{\mu}}e^{h\nu/kT_{\mu}}.\label{eq:37}
\end{equation}
Relating (\ref{eq:36}) and (\ref{eq:37}) at same fashion we relate
(\ref{eq:21}) and (\ref{eq:22}), gives

\begin{equation}
\frac{\partial u}{\partial T}=-\frac{h\nu}{T_{\mu}}\frac{\partial u}{\partial E}.\label{eq:38}
\end{equation}
So, we can see that Planck's Radiation law satisfies Boltzmann equation
with $h\nu$ as energy $E$.

Another question that we can investigate, is related to modeling thermodynamic
systems. Macroscopic variables like temperature, pressure and volume
can be measured by experimentation. Otherwise, we know that these
ones are manifestation of micro states configuration, as the case
of temperature which is the average molecular kinetic energy. The
entropy is a quantity that turns possible to relate micro states to
macro states \cite{Endrew1984}.

Planck interpreted radiation as a photon gas and obtain their law
with success. In next session let's investigate how to relate entropy
to distribution of microstates using Boltzmann PDE.

\section{Boltzmann Entropy Law}

In previous sections we have success when obtaining some known distributions.
Considering their close relationship with Statistical Physics, let
we show how to obtain Boltzmann's Entropy Law using a distribution.

Supposing some gas volume and their free expansion, when volume grow
entropy increases, but growing entropy by increasing volume costs
thermal energy of gas which will decay. So,

\begin{equation}
\frac{\delta\left(E_{S}\right)}{E_{S}}=-\frac{\delta\left(E_{T}\right)}{E_{T}}.\label{eq:39}
\end{equation}
At proper limit we can adjust this equation to,

\begin{equation}
\frac{dE_{S}}{E_{S}}=-\frac{dE_{T}}{E_{T}}.\label{eq:40}
\end{equation}
Let we set a distribution $\Omega$ which will depends on energy displaced
to entropy $E_{S}$ when volume increases and the thermal energy costs
to volume increase $E_{T}$. Given $\Omega\equiv\Omega\left(E_{S},E_{T}\right)$,
we must change from total to partial differential equation for this
purpose,

\begin{equation}
\frac{\partial E_{S}}{\partial E_{T}}=-\frac{E_{S}}{E_{T}}.\label{eq:41}
\end{equation}
Applying the chain rule like below,
\begin{equation}
\frac{\partial E_{S}}{\partial E_{T}}=\frac{\partial\Omega}{\partial E_{T}}\frac{\partial E_{S}}{\partial\Omega}.\label{eq:42}
\end{equation}
Replacing (\ref{eq:42}) in (\ref{eq:41}) we get a final form of
partial differential equation which we name as \textbf{Boltzmann's
Entropy Distribution PDE},

\begin{equation}
\frac{\partial\Omega}{\partial E_{T}}=-\frac{E_{S}}{E_{T}}\frac{\partial\Omega}{\partial E_{S}}.\label{eq:43}
\end{equation}
Our Ansatz to solve this equation consists in use the tentative solution
below,

\begin{equation}
\Omega\left(E_{S},E_{T}\right)\equiv e^{E_{S}^{a}E_{T}^{b}}.\label{eq:44}
\end{equation}
To obtain the PDE solution let we derive our tentative from (\ref{eq:44}).
First in terms of $E_{T}$ given $\frac{\partial\Omega}{\partial E_{T}}=bE_{S}^{a}E_{T}^{b-1}\Omega$
and after in terms of $E_{S}$, $\frac{\partial\Omega}{\partial E_{S}}=aE_{S}^{a-1}E_{T}^{b}\Omega$.
Relating both derivatives we can equate,
\begin{equation}
\frac{\partial\Omega}{\partial E_{T}}\left(\frac{\partial\Omega}{\partial E_{S}}\right)^{-1}=\frac{bE_{S}^{a}E_{T}^{b-1}\Omega}{aE_{S}^{a-1}E_{T}^{b}\Omega}=\frac{b}{a}\frac{E_{S}}{E_{T}}.\label{eq:45}
\end{equation}
To satisfy Boltzmann's Entropy Distribution PDE we can set $b/a=-1$,
where our particular choice is $b=-a$. So, finally we obtain the
solution of PDE in (\ref{eq:43}),

\begin{equation}
\Omega\left(E_{S},E_{T}\right)=e^{E_{S}/E_{T}}.\label{eq:46}
\end{equation}

At next steps let we do a connection between this distribution to
Boltzmann's Entropy law. Using as first principles dimensional analysis
and thermodynamics laws. From entropy fundamental concept defined
in Thermodynamics $dQ=TdS$ from who we can see that $\left[dQ\right]=dE_{S}$
establishing relation between heat energy and entropy energy, and
as macroscopic version we can set dimensional dependence of entropy
energy as $E_{S}\propto\left[T\right]^{a}\left[S\right]^{b}$ which
dimensional solution points to $E_{S}=TS$. Another demand is to thermal
energy $E_{T}$ which we get as $E_{T}=kT$. So,

\begin{equation}
\Omega\left(E_{S},E_{T}\right)=e^{TS/kT}=e^{S/k},\label{eq:47}
\end{equation}
where, to isolate entropy $S$ in the final form, let we assume the
inverse function from (\ref{eq:47}). Wich gives

\begin{equation}
S=k\ln\left|\Omega\right|.\label{eq:48}
\end{equation}

\section{Fermi-Dirac and Bose-Einstein Distributions}

Now, we will verify how to Fermi-Dirac and Bose-Einstein are related
to a similar format as that obtained in equation of Planck's law (\ref{eq:35}).
So,

\begin{equation}
u=\left(e^{E/kT}+a\right)^{b}.\label{eq:49}
\end{equation}
Here, our emphasis consists in using the Boltzmann's partial differential
equation to inspect how close Fermi-Dirac and Bose-Einstein are to
Plank's law format. With a little modification in energy from (\ref{eq:49}),
we get

\begin{equation}
u=\left(e^{\left(\varepsilon-\mu\right)/kT}+a\right)^{b}.\label{eq:50}
\end{equation}
 And the analogous form to (\ref{eq:33}), is

\begin{equation}
\frac{\partial u}{\partial T}\left(\frac{\partial u}{\partial\varepsilon}\right)^{-1}=-\frac{\left(\varepsilon-\mu\right)}{T}.\label{eq:51}
\end{equation}
Next steps let's apply (\ref{eq:50}) in (\ref{eq:51}) to study both
Fermi-Dirac and Bose-Einstein particular cases.

\subsection{Fermi-Dirac and Bose-Einstein Distributions Analysis}

Comparing general format from (\ref{eq:50}) with Fermi-Dirac distribution
equation in \cite{Crawford1988} we see that $a$ and $b$ must assume,

\begin{equation}
\left(a,b\right)\equiv\left(+1,-1\right).\label{eq:52}
\end{equation}
This choice gives us,

\begin{equation}
u=\frac{1}{e^{\left(\varepsilon-\mu\right)/kT}+1}.\label{eq:53}
\end{equation}
Derivating in temperature and energy we have their respective equations,

\begin{equation}
\frac{\partial u}{\partial T}=\left[e^{\left(\varepsilon-\mu\right)/kT}+1\right]^{-2}e^{\left(\varepsilon-\mu\right)/kT}\frac{\left(\varepsilon-\mu\right)}{kT^{2}},\label{eq:54}
\end{equation}
and

\begin{equation}
\frac{\partial u}{\partial\varepsilon}=-\left[e^{\left(\varepsilon-\mu\right)/kT}+1\right]^{-2}e^{\left(\varepsilon-\mu\right)/kT}\frac{1}{kT}.\label{eq:55}
\end{equation}
Now relating both (\ref{eq:54}) and (\ref{eq:55}), we finally see
that,

\begin{equation}
\frac{\partial u}{\partial T}\left(\frac{\partial u}{\partial\varepsilon}\right)^{-1}=-\frac{\left(\varepsilon-\mu\right)}{T},\label{eq:56}
\end{equation}
Fermi-Dirac's distribution satisfy general form of Boltzmann's distribution
PDE. At same fashion, comparing general format from (\ref{eq:50})
with Bose-Einstein distibution equation in \cite{Barford1976} we see
that $a$ and $b$ must assume,

\begin{equation}
\left(a,b\right)\equiv\left(-1,-1\right).\label{eq:57}
\end{equation}
This choice gives us,

\begin{equation}
u=\frac{1}{e^{\left(\varepsilon-\mu\right)/kT}-1}.\label{eq:58}
\end{equation}
Which verification results will led us to, at same way that obtained
in (\ref{eq:56}), concluding that Bose-Einstein's distribution satisfy
general form of Boltzmann's distribution PDE too.

So, when we treat Plank's law of radiation, we see that same PDE which
gives us Boltzmann's Distribution is satisfied to Planck's law. We
extend this analysis using a general form of distribution and conclude
that also Fermi-Dirac's and Bose-Einstein's distributions emerges
as Planck's law from the same Boltzmann's PDE.

\section{Schrödinger's Distribution}

The uncertainty principle emerges from experimental and Gaussian widths
relations \cite{Nairz2002}. We use uncertainty as strong fundamental
to point our dimensionless fluctuation and obtain a fundamental distribution
that we can use to recover Schrödinger equation.

To obtain Schrödinger distribution, we must recover the most
fundamental aspect, which lies over uncertainty relations \cite{Heisenberg1927},

\begin{equation}
\delta x\delta p\geq\frac{\hbar}{2},\label{eq:59}
\end{equation}
and

\begin{equation}
\delta E\delta t\geq\frac{\hbar}{2}.\label{eq:60}
\end{equation}
Another association that we can make consist in show interrelation
between booth (\ref{eq:59}) and (\ref{eq:60}), starting from energy
relation $\delta E=m\delta(v)^{2}/2=mv\delta v$. Replacing it in
(\ref{eq:60}) we get $v\delta t\cdot m\delta v\geq\hbar/2$, and
remember that $\delta x=v\delta t$ and $\delta p=m\delta v$ we recover
the other uncertainty relation $\delta x\delta p\geq\hbar/2$ as in
refference \cite{Heisenberg1927}.

Now, we are supposing a system with some initial energy amount $E$
and a free particle with momentum $p$ and an isolated system. In
terms of dimensionless fluctuation analysis, if the particle momentum
$p$ increase a quantity $\delta p$, it will cost to system a decrease
amount of energy $-\delta E$. So, we can write

\begin{equation}
\frac{\delta p}{p}\propto-\frac{\delta E}{E}.\label{eq:61}
\end{equation}

As next steps, we start replacing uncertainty relations from (\ref{eq:59})
and (\ref{eq:60}), in the limit that we equate both. So,

\begin{equation}
\delta E\delta t=\delta x\delta p.\label{eq:62}
\end{equation}
Multiplying (\ref{eq:62}) by inverse of $E$ and, after a little
algebra, we get

\begin{equation}
\frac{\delta E}{E}=2\frac{\delta p}{p}.\label{eq:63}
\end{equation}

Now comparing this format of uncertainty relation presented in (\ref{eq:63})
to dimensionless fluctuation relation of (\ref{eq:61}), we can see
that instead of proportionality factor assume value $1$ as in many
physical cases \cite{Barenblat1987}, a factor $2$ will be needed, given

\begin{equation}
\frac{\delta E}{E}=-2\frac{\delta p}{p}.\label{eq:64}
\end{equation}
After a little algebra, and considering the proper limit when $\delta E$
and $\delta p$ can be assumed as differentials, we get

\begin{equation}
\frac{dE}{dp}=-2\frac{E}{p}.\label{eq:65}
\end{equation}
Assuming that we have some distribution function, $\Psi\equiv\Psi\left(p,E\right)$,
the partial derivatives turn (\ref{eq:65}) to a most interesting
format,

\begin{equation}
\frac{\partial E}{\partial p}=-v.\label{eq:66}
\end{equation}
Assuming a chain rule using $\Psi$, relating $E$ and $p$ we get,
\begin{equation}
\frac{\partial E}{\partial p}=\frac{\partial\Psi}{\partial p}\frac{\partial E}{\partial\Psi}.\label{eq:67}
\end{equation}
Replacing (\ref{eq:67}) in (\ref{eq:66}), we finally get the \textbf{Schrödinger
distribution equation PDE},

\begin{equation}
\frac{\partial\Psi}{\partial p}=-v\frac{\partial\Psi}{\partial E}.\label{eq:68}
\end{equation}
We can easily see that a simple solution is given by,

\begin{equation}
\Psi=e^{ipx/\hbar}e^{-iEt/\hbar}.\label{eq:69}
\end{equation}
This distribution equation is very important, because from this one
we can obtain all properties of the system, including schrödinger
Equation. From expected value as statistical concept $\left\langle p\right\rangle =\int_{-\infty}^{+\infty}p\Psi\left(x,t\right)dx$,
we can see that distribution play a central role. Because of complex
variable format at (\ref{eq:69}) we extend this to square integrate
as $\left\langle p\right\rangle =\int_{-\infty}^{+\infty}\Psi^{*}\left(x,t\right)\hat{p}\Psi\left(x,t\right)dx$.
At same fashion for $\left\langle H\right\rangle $ and $\left\langle V\right\rangle $
and after putting it all together in Hamiltonian energy conservative
form of square integrable expected values, we easily recover Schrödinger
equation.

\section{Concluding Remarks}

After our tests results was verified with success to principal Physics
Distributions, we find a collection of procedures as registered in
this paper, which we can summarize in a single principle: ``\textit{Given
some fundamental relations between two or more quantities, the Dimensionless
Fluctuation Analysis point to a Partial Differential Equation which
solution gives desired distribution}''. Which we name as\textbf{
Dimensionless Fluctuation Balance Principle.} At least this work shows
a new way to relate two quantities using dimensionless number that
enable us to obtain distributions as differential equations solutions.
This is not only a practical tool, but also turns possible to understand
some meaning beyond distributions. As example, we can see that when
some particle or energy or another system entity was confined submitted
to some compact region like localized phenomena or when required to
interact with some other entities not in large space field, distributions
emerges as natural imposition of space region to a group of entities
when the system size is so small as needed to impose statistical reality
to every entity within it.

\section{Appendix A}

Observing black line in the figure (\ref{fig:2}), winch is a curve
for classical theory to $5000K$, this curve diverges when compared
with Planckian blue curve to $5000K$.

In classical curve, a breaking point appears in classical curve near
$\lambda=1\times10^{-6}m$ but we can observe this problem starts
at the utraviolet region, because other curves like red $3000K$,
green $4000K$ and blue $5000K$, which is expected by experiments,
cross ultraviolet region without blow up.
\noindent \begin{center}
\begin{figure}[ht]
\noindent \begin{centering}
\includegraphics[scale=0.60]{colado4}
\par\end{centering}
\caption{
\label{fig:2}Classical radiation distribution, evidencing temperature
of $5000K$ and a breaking point near wavelength $1\mu m$. Graphic
units: wavelenght $\mu m$ and Spectral Radiance $kW\cdot sr^{-1}\cdot m^{-2}\cdot nm^{-1}$.}
\end{figure}
\par\end{center}

In other words we need to investigate $\lambda=0.375\times10^{-6}m$,
which is the ultraviolet region, where the classical curve problem
begin.

Considering $c=\lambda\nu$, we easily get $\nu=8\times10^{14}Hz$
as frequency in the point that wavelength is $\lambda=0.375\times10^{-6}m$.

To understand this limit when classical curve blow up at ultraviolet
region, let's investigate the relation $\left(h\nu/kT\right)$, which
appears in equation (\ref{eq:28}), using ultraviolet frequency $\nu=8\times10^{14}Hz$
and $T=5\times10^{3}K$ as in figure (\ref{fig:2}). Then,
\begin{equation}
\frac{h\nu}{kT}=\frac{\left(6.63\times10^{-34}m^{2}\cdot Kg\cdot s^{-1}\right)\left(8\times10^{14}s^{-1}\right)}{\left(1.38\times10^{-23}m^{2}\cdot Kg\cdot s^{-2}\cdot K^{-1}\right)\left(5\times10^{3}K\right)}=0.77\times10^{-43}.\label{eq:70}
\end{equation}

Equation (\ref{eq:41}) tell us that breaking region in ultraviolet
zone occurs because $kT$ term is much greater than $h\nu$. Or just,

\begin{equation}
h\nu\ll kT.\label{eq:71}
\end{equation}

\section{Appendix B}

In this session, we just recover Schrödinger equation in a quick way,
starting from \textbf{Schrödinger distribution obtained as PDE solution}
(\ref{eq:69}) as a fundamental to discover operators from mean value
concept, the aim of this session is show a close relation between
a distribution that became from experimental measurable parameters
like uncertainty relations and schorödinger quantum mechanics.

Starting from Hamiltonian of expected values $\left\langle H\right\rangle =\left\langle p\right\rangle ^{2}/2m+\left\langle V\right\rangle $
in square integrable system, we easily can see that,

\begin{equation}
\int_{-\infty}^{+\infty}\Psi^{*}\left(x,t\right)\hat{H}\Psi\left(x,t\right)dx=\frac{1}{2m}\int_{-\infty}^{+\infty}\Psi^{*}\left(x,t\right)\hat{p}^{2}\Psi\left(x,t\right)dx+\int_{-\infty}^{+\infty}\Psi^{*}\left(x,t\right)\hat{V}\Psi\left(x,t\right)dx.\label{eq:72}
\end{equation}
Another useful concept is that $\hat{p}$ operator must extract eigenvalue
from distribution, as in the form $\hat{p}^{2}\Psi\left(x,t\right)=p^{2}\Psi\left(x,t\right)$.
Because of format of distribution we get in (\ref{eq:69}) only a
specific $\hat{p}$ operator can extract $p$ eigenvalue. So,

\begin{equation}
\Psi\left(x,t\right)=e^{ipx/\hbar}e^{-iEt/\hbar}\Leftrightarrow\hat{p}\equiv-i\hbar\frac{\partial}{\partial x}.\label{eq:73}
\end{equation}
At same fashion, we get $\hat{H}$ operator (that extract energy $E$)
and $\hat{V}$ operator (that impose potential energy $V$) in (\ref{eq:72})
to get the Schrödinger equation from expected values and Schrödinger
distribution. Which gives,

\begin{equation}
\int_{-\infty}^{+\infty}\Psi^{*}\left(x,t\right)\left[i\hbar\frac{\partial}{\partial t}+\frac{\hbar}{2m}^{2}\frac{\partial{{}^2}}{\partial x{{}^2}}-V\left(x,t\right)\right]\Psi\left(x,t\right)dx=0,\label{eq:74}
\end{equation}
or just,

\begin{equation}
i\hbar\frac{\partial}{\partial t}\Psi\left(x,t\right)=-\frac{\hbar}{2m}^{2}\frac{\partial{{}^2}}{\partial x{{}^2}}\Psi\left(x,t\right)+V\left(x,t\right)\Psi\left(x,t\right).\label{eq:75}
\end{equation}
Here, we present the \textbf{Schrödinger equation as Heisenberg uncertainty
relations Mechanics}. The strong aspect relies in distribution, because
from this one operators assume their form. We show by unique way with
fluctuations analysis that Heisenberg uncertainty principle is determinant
to obtain distribution that will lead us to find Schrödinger equation
as a simple consequence.

\section{Authors Declaration Interests}
The authors declare that they have no know competing financial interests or personal relationship that could have appeared to influence the work reported in this paper.

redpajama_set_name: RedPajamaArXiv


text:
Q: php artisan list - PHP Fatal error: Undefined class constant 'HEADER_FORWARDED' Running php artisan list on my Laravel app is giving me an odd error. E.g.
php artisan list
PHP Fatal error:  Undefined class constant 'HEADER_FORWARDED' in /path/to/app/vendor/fideloper/proxy/config/trustedproxy.php on line 56

Any suggestions?

redpajama_set_name: RedPajamaStackExchange


The designer's second presentation for the French megabrand was a carefully considered, Michael Jackson-inspired powerhouse, built layer upon layer with the kind of shameless manipulation that characterizes the greatest showmen.
Ah yes, MJ. Abloh insisted he had The Wiz in mind before he even got to Michael. Nevertheless, in his Vuitton voyage through the Seven Ages of Man, what more provocative focus could he choose than the manchild who changed the world? "The ultimate muse," Abloh called him. "We watched him grow, but he stayed a boy the whole time." In his elaborate show notes (a very good read, I must say) he talked about Jackson's extraordinary power, "every jacket, glove, sock and hat fashioned into an instrument of awe." What fashion designer wouldn't sell their soul to Satan for that power?
Alboh dreams too. The extravagant craftsmanship of the Vuitton workshops allowed him to elevate the ordinary, just like MJ with his gloves and socks. A plaid shirt sprinkled with crystals? "Raindrops," Abloh called them. More crystals, a Milky Way of them, extravagantly crusted a showpiece blouson. The Vuitton monogram was artfully debossed on a cashmere twill puffa. Maybe it was the Ancient Egyptian pleated shendyt worn by MJ in the "Remember the Time" video that inspired the long, flowing pleated skirts. It was definitely "We Are The World" that inspired the flag collages in intarsia-ed leather and the houndstooth pattern that was actually a million little continents of Africa (the song's lyrics were printed on a shirt).
The enthusiastic creativity of these touches truly illuminated the collection, though you could easily get the feeling that Abloh's heart was elsewhere. "I may be the streetwear guy, quote-unquote, but tailoring is the story I want to tell," he claimed. "I want to get my young audience out of sneaker world." Not at Vuitton's prices, baby. Still, there are many less noble aspirations in the world. And the yellow brick road has always led to the most aspirational destination of all in American mythology. MJ danced and sang his way to Oz. VA is taking the fashion route.


text:
Colorado > Denver > Schools > School Profile
Wyatt Academy Middle School
3620 Franklin Street, Denver, CO 80205
Denver County 1 School District
Type Public charter
Public charter
School Board Finder
Learn more about how to get involved
Powered by XQ
Equity overview
Low-income students
This rating measures how much students at this school improved from one year to the next, compared to students with similar proficiency levels at other schools in the state.
This section shows how much academic progress (also called "growth") students at this school are making given where they were last year, compared to similar students in the state.
Currently, this information is unavailable. Update me when new information is published on this school's profile.
Find out what state standardized test scores really tell you about a school – and what they don't.
% Proficient
State avg: 34%
Notice something missing or confusing?
Studies show that diversity in school leads to long-term benefits for students. Discover why!
In this section, we publish a rating that reflects how well this school is serving disadvantaged students, compared to other schools in the state, based on college readiness, learning progress, and test score data provided from the state's Department of Education.
The state does not provide enough information for us to calculate an Equity Rating for this school.
Would you find it helpful to have this kind of information for the school? YesNo
What makes a great teacher? The truth may surprise you.
This section includes information about school staff, including the ratio of students to teachers, and teacher tenure and certification.

redpajama_set_name: RedPajamaCommonCrawl


I checked our Mobile App roadmaps and unfortunately, this functionality will not make it into the next build (the next release is scheduled for end of Jan), but will be considered for a future build.
I apologize for our not-so-speedy reply on your request, Mark! I've alerted the mobile project manager. You should get a reply about wether this is planned in our upcoming releases.
You can change it online in a browser... which we did. Add this feature to the mobile app and outlook plugins.


text:
Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis.
P. J. Block, M. Skinner, Merrill Benson, A. S. Cohen
A monoclonal immunoglobulin has been isolated from the peritoneal fluid of a patient with primary amyloidosis. The immunoglobulin was reduced and alkylated and the light and heavy chains were compared to the major protein constituent of that patient's hepatic amyloid fibrils. N-terminal amino acid sequences of the light chain and amyloid fibril were identical when carried to 20 residues and were typical of a kappa I light chain. Molecular weight studies suggested that the fibril protein was composed of an intact light chain with a molecular weight of 23,000. The hypothesis that amyloid fibril protein in primary amyloid is derived from circulating monoclonal immunoglobulin is discussed.
Arthritis and Rheumatism
Immunoglobulin Light Chains
Ascitic Fluid
Amyloid
Amyloidogenic Proteins
Primary amyloidosis
Block, P. J., Skinner, M., Benson, M., & Cohen, A. S. (1976). Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis. Arthritis and Rheumatism, 19(4), 755-759.
Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis. / Block, P. J.; Skinner, M.; Benson, Merrill; Cohen, A. S.
In: Arthritis and Rheumatism, Vol. 19, No. 4, 07.1976, p. 755-759.
Block, PJ, Skinner, M, Benson, M & Cohen, AS 1976, 'Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis.', Arthritis and Rheumatism, vol. 19, no. 4, pp. 755-759.
Block PJ, Skinner M, Benson M, Cohen AS. Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis. Arthritis and Rheumatism. 1976 Jul;19(4):755-759.
Block, P. J. ; Skinner, M. ; Benson, Merrill ; Cohen, A. S. / Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis. In: Arthritis and Rheumatism. 1976 ; Vol. 19, No. 4. pp. 755-759.
@article{6b4f99a9b090437191ecd77dcf693c08,
title = "Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis.",
abstract = "A monoclonal immunoglobulin has been isolated from the peritoneal fluid of a patient with primary amyloidosis. The immunoglobulin was reduced and alkylated and the light and heavy chains were compared to the major protein constituent of that patient's hepatic amyloid fibrils. N-terminal amino acid sequences of the light chain and amyloid fibril were identical when carried to 20 residues and were typical of a kappa I light chain. Molecular weight studies suggested that the fibril protein was composed of an intact light chain with a molecular weight of 23,000. The hypothesis that amyloid fibril protein in primary amyloid is derived from circulating monoclonal immunoglobulin is discussed.",
author = "Block, {P. J.} and M. Skinner and Merrill Benson and Cohen, {A. S.}",
journal = "Arthritis and Rheumatology",
T1 - Identity of a peritoneal fluid immunoglobulin light chain and the amyloid fibril in primary amyloidosis.
AU - Block, P. J.
AU - Skinner, M.
AU - Benson, Merrill
AU - Cohen, A. S.
N2 - A monoclonal immunoglobulin has been isolated from the peritoneal fluid of a patient with primary amyloidosis. The immunoglobulin was reduced and alkylated and the light and heavy chains were compared to the major protein constituent of that patient's hepatic amyloid fibrils. N-terminal amino acid sequences of the light chain and amyloid fibril were identical when carried to 20 residues and were typical of a kappa I light chain. Molecular weight studies suggested that the fibril protein was composed of an intact light chain with a molecular weight of 23,000. The hypothesis that amyloid fibril protein in primary amyloid is derived from circulating monoclonal immunoglobulin is discussed.
AB - A monoclonal immunoglobulin has been isolated from the peritoneal fluid of a patient with primary amyloidosis. The immunoglobulin was reduced and alkylated and the light and heavy chains were compared to the major protein constituent of that patient's hepatic amyloid fibrils. N-terminal amino acid sequences of the light chain and amyloid fibril were identical when carried to 20 residues and were typical of a kappa I light chain. Molecular weight studies suggested that the fibril protein was composed of an intact light chain with a molecular weight of 23,000. The hypothesis that amyloid fibril protein in primary amyloid is derived from circulating monoclonal immunoglobulin is discussed.
JO - Arthritis and Rheumatology
JF - Arthritis and Rheumatology


From decaf darlings to the best caffeinated green and black teas, we offer a tea to suit every caffeine preference. For a stand-out breakfast tea, try Republic Chai or one of our premium Matcha blends which contains the most caffeine of our teas. Try one of our green teas for pure taste and light caffeine content. Our 100% white tea collection steeps a subtle, mellow cup and contains the least caffeine of our teas. Flavored or unflavored, tea bags or loose leaf, the choice is yours!


Imagine a home where the fine art of living has been perfected. Welcome to The Orsini, home to the best apartments in downtown Los Angeles. From the moment you enter the elegant Mediterranean inspired 2-story foyer at The Orsini, you will realize that you have come home to a new experience in luxury apartment living. We are in the heart of Los Angeles, California and the pulse of extraordinary living. Enjoy a short commute to the LA Fashion District, Staples Center, Disney Concert Hall and all the amazing dining and glamorous shopping that living in Los Angeles has to offer. All listed pricing is subject to immediate change without prior notice. Lease now to lock in this great low rate!


''You manage what you measure,'' as they say, and the fact that no one had to measure options values helped lead to an explosion in their issuance after 1994. Employees knew they could be valuable, and so did bosses. Most shareholders either did not care or accepted the argument that reporting the expenses would depress share prices and thus be bad for them.
The options boom made many executives amazingly rich, and led to a lot of envy from those who were missing out. Did that help to reduce ethical standards in auditing firms, or among lawyers and investment bankers working on corporate deals? That cannot be proved.
But the options wealth for bosses of traded companies provided an excuse for Mr. Grasso to collect extra bonuses at the exchange, to compensate him for not receiving options. Then he got pension benefits on those bonuses, which provided him with the bulk of the $139.5 million payout that led to his ouster.
The companies that issued options worried about the one place all those options did show up in their financial statements: as dilution on the balance sheet. So, many companies borrowed money to repurchase shares.
The government was another beneficiary. Executives paid ordinary income tax when they cashed in the bonuses, and that led to a surge in government tax receipts that was little understood at the time. The budget bean counters assumed that those taxes were offset by reduced corporate taxes, because companies got deductions for the option profits, and so did not affect the deficit. But a lot of the profits came on options from unprofitable companies.
That helped to produce the Clinton budget surpluses, and the bursting of the bubble meant that most of the forecasted surpluses were going to vanish anyway, even before the Bush administration cut taxes.
Now the accounting standards board is trying again, and this time it will probably succeed, although there is no guarantee. Some companies are pushing a ''compromise'' that would deduct the expense -- at ridiculously low values -- of only those options given to top executives. There is no logical reason options given to one employee would be an expense while those given to another would not. But the hope is that politicians will be able to claim they are voting for little-guy recipients, not greedy corporate bosses.
Most shareholders understand that option compensation is an expense just like any other. Over the anguished opposition of managements, they are voting this spring at annual meetings for the expensing of options. That should sway votes in Washington.
Some lessons are hard to learn, however. Some European banks are petrified over having to disclose how much money they are actually making, or losing, on their derivatives portfolios. They have tried, with some success in the media, to turn a fight over rules of the International Accounting Standards Board into an argument about Anglo-American cultural imperialism, rather than about good accounting. The European Commission may block the rules.
It has been said that Europe is 10 years behind America. That would, perhaps, be one way to prove it.


WallArt offers a collection of modern and contemporary textured wall designs. 3D Wall Panels with 24 different designs, varying from basic or industrial to modern or romantic: Each design has its own look and creates an unique atmosphere. Together, the tiles from WallArt form a repeating pattern and after assembling the decorative wall panels you can paint or spray them in any desired colour.
As the original brand, since 2010, WallArt 3D Wall Panels give an extra dimension to your walls and compliment just about any room in the house. Our wall panels are used for Residential & Commercial projects: Hotels, Office lobbies, Retail outlets, Restaurants, Trade show exhibits and more. They make for beautiful and extravagant looking accent walls. No doubt about that!
Have a look at some of our Mywallart 3D Wall Panels.
WallArt is a European Design Company with a sales office in Hong Kong and a professional production facility in China. At this moment WallArt 3D Wall Panels is expanding their sales network and currently we are looking for E-Commerce partners, new Resellers and Distributors. Contact us now and start business today!
Big Brother is a television reality game show. The show follows a group of houseguests living together 24 hours a day in the "Big Brother" house, isolated from the outside world. The 3D decorative wall panels from WallArt are an eyecatcher in the most popular TV Show in USA Big Brother. Big Brother exceeded even our wildest expectations with these magnificent walls made of our Gaps design. Simply breathtaking!
The Tomorrow People is an American science fiction television series developed by Greg Berlanti that airs on The CW American television. The series follows a group of young people who possess psionic powers as the result of human evolution. This TV Series on CW Television Networks they use in several scenes our 3D Wall Panels Bricks design !
In a commercial being aired on Italian TV, the former Inter Milan manager provides the voice over for an imaginary press conference. The Portuguese manager doesn't actually appear in the advert though - that's because he's 'the Transparent One'.
These lightweight 3D Wall Panels are handy and easy to install by means of the supplied tile spacers. You can easily fix them with 'WallArt Hybrid Adhesive' or a similar adhesive to almost any surface. We will demonstrate it in our installation film.
Welcome to the world of eco friendly 3d wall decoration! WallArt – The Original Brand – introduced this eco-friendly interior product in 2010. With the 3D Wall Panels and 3d wall tiles you will give an extra dimension to your walls! These 3d wall coverings are made of an eco friendly material and are easy to install, both DIY and professional use of this 3d wall covering has proven to create the looks of a beautiful 3d wall.
WallArt wall decoration is original, contemporary and versatile and you turn your interior in a spectacular metamorphose, in less than no time. WallArt 3d wall panels are 50 x 50 cm, allow you to decorate an entire room from head to foot. Besides, you can add accents to a part of the wall. Whatever you choose: the result is stunning! Take a look at the variation of all the 3D Wall Panels and our range of 3d wallpaper, you will be surprised to see so many stunning decorative wall panels !
3D Plant Fibre Wall Panels are Highly durable and scratch resistant. These WallArt 3D wall tiles offer great visual appeal and a beautiful, lustrous finish to any wall. There is no better design space than the effects produced by a rich and well-designed 3D wall panel lit by the best lighting. No matter if the wall panels or the lighting is of light or bold colors, the combination will always produce the atmosphere and the mood you would like to create.
Some Highlights of WallArt 3d Wall Decor: * Suitable for both new construction and renovations * Markets include hospitality, healthcare, corporate, commercial, education etc. * Ideal to use for focal point, background wall cover or visual transition spaces * Modular design with ofcourse an extensive array of carving expressions * Precision fit fiber panels, moldings, reveals & trimming * Bespoke 3D plant fiber wall panel system.


text:
business|02 February, 2020
150 HINO trucks join Dubai Refreshment PJSC - Local PepsiCo Bottler
The Al-Futtaim company extends its association with the global food and beverage leader
Dubai, United Arab Emirates: Al-Futtaim's Commercial Vehicle Division, exclusive distributor of HINO trucks in the UAE, has announced that is has delivered a major order of 150 Light Duty HINO 300 series trucks to Dubai Refreshment PJSC, local bottler and distributor of PepsiCo Beverages in Dubai and the Northern Emirates.
This new agreement further extends HINO's association with PepsiCo bottler in Dubai which will use the new trucks to boost its fleet of refreshment delivery trucks in the UAE.
Following the handover, Ramez Hamdan, General Manager, HINO, Al-Futtaim's Commercial Vehicle Division said, "Our presence in the F&B industry has been increasing year-on-year, thanks to the trust our partners put in the quality of our trucks and our 'HINO Total Support' aftersales concept. The 150 HINO trucks are tailor-made to enhance PepsiCo beverages' delivery operations in the UAE, and we have coupled the delivery with eco-driving training sessions to help the fleet users drive more efficiently and improve their road safety".
"We are extremely pleased to continue our association with HINO. We chose HINO trucks for their quality, performance and fuel efficiency, as they adhere to Euro-4 emission standards, helping our operations to have a positive impact on the environment, in line with our commitment to support the UAE government's 2021 vision," said Mr. Mr. Tarek ElSakka, Chief Executive Officer of Dubai Refreshment PJSC.
The eco-driving trainings come as part of the 'HINO Total Support' and in addition to helping reduce the risk of major road accidents, the trainings also help drivers optimize total cost of truck ownership, contributing to the bottom line, improve fuel efficiency, minimise the cost of wear and tear, reduce service and maintenance costs as well as insurance premiums.
Conforming to UAE safety standards, HINO has partnered with GORICA for the customization of the body of the trucks to suit the specialized needs of Dubai Refreshment PJSC – local PepsiCo bottler.
For more information on HINO products and solutions, please call 800 HINO SALES (800 4466 72537) or visit hino.ae.


Do you know how to form superlative adjectives?
Today's program is going to be about such words as 'some' and 'any'.
How to express an action which will happen at a particular time in the future?
In this program you are going to learn the Past Continuous tense.
The special words of today's program are 'have to' and 'has to'.
Today you're gonna learn how to compare things and people.
Learn everything about the Present Perfect tense.
In this program you are going to talk about articles.
Our host will explain how to make polite requests.
Learn how to give advice to people grammatically correct.
Let's talk about things which were true in the past, but they are not true anymore.
The special construction of the program is «going to».


As a small nonprofit ANPR relies on member donations and other fundraising efforts to cover our expenses as an organization. In addition to membership dues, each year we hold an annual fund drive. Since 2008, the year-end appeal has been called the Fall Fund Campaign. You may give a tax-deductible contribution to ANPR any time of the year by visiting our donation page.
Become a member of these programs and earn money for ANPR just by shopping at your favorite retailers.
© 2015. All rights reserved. ANPR is a registered 501(c)(3) nonprofit organization.


In 2014, Japan saw the release of big Pocket Monster games Pokémon Omega Ruby and Alpha Sapphire. The games sold a ton in Japan, but not enough to reign supreme. A new challenger has taken the crown: Yokai Watch.
That's right, Famitsu is reporting Yokai Watch 2 outsold Pokémon ORAS in 2014.
What's interesting about this comparison is that both games were for the Nintendo 3DS and that both had two editions (Yokai Watch 2 was available in Yokai Watch 2 Ganso or Honke, while Pokémon came in Omega Ruby and Alpha Sapphire).
Another thing to keep in mind is that, of course, ORAS is a remake, while Yokai Watch 2 Ganso/Honke are new games. But, if you are doubting Yokai Watch's popularity in 2014, the fifth* biggest selling game was Yokai Watch 2 Shinuchi, a new version of Yokai Watch 2. It went on sale December 13 and sold 2,007,327. What's even more impressive was that the sixth biggest selling game in Japan in 2014 was the first Yokai Watch game, which originally launched on July 11, 2013. This past year, it sold 994,346 copies in Japan.
Of course, that was 2014. Pokémon has an incredibly impressive track record and has been popular for a very, very long time. That doesn't look like it will be changing anytime soon. The question now is whether or not Yokai Watch has the same kind of staying power. We shall see.
*The third biggest seller last year in Japan was Monster Hunter 4G at 2.3 million and the fourth Super Smash Bros. for Nintendo 3DS at 2.06 million.


text:
«Папильон» (или «бабочка из Меца»; ) — в филателии узкая полоска тонкой шёлковой бумаги с адресом и небольшим письменным сообщением, отправленная в сентябре-октябре 1870 года в период франко-прусской войны 1870—1871 годов из окружённой прусской армией крепости Мец на неоккупированные территории Франции с помощью воздушного шара (аэростата).

Название
Название «бабочки из Меца» связано с фамилией изготовлявшего воздушные шары Папильона (Papillon), поэтому порой встречающийся перевод названия этих листков как «письмо-бабочка» ( — «бабочка») неверен.

Описание
После того, как 19 июля 1870 года Франция объявила войну Пруссии, к 19 августа наступающие прусские войска осадили почти половину французской Рейнской армии в крепости Мец на северо-востоке Франции. Чтобы поддерживать сообщение с неоккупированными районами Франции, был запущен ряд воздушных шаров без экипажа с письмами осаждённого гарнизона. 27 октября 1870 года осаждённый в Меце гарнизон прекратил сопротивление.

Вначале задумали и стали осуществлять идею переправки корреспонденции с помощью воздушных шаров два армейских фармацевта по фамилии Жаннель (Jeannel) и Папильон (Papillon). В период с 5 по 15 сентября 1870 года ими были отправлены 14 воздушных шаров, на каждом из которых было от 40 до 140 «папильонов». Из них восемь шаров были перехвачены прусскими войсками. Из примерно полутора тысяч таких отправленных ими листков сегодня известны только 19 экземпляров.

С 16 сентября по 3 октября отправкой воздушных шаров с папильонами занимались французские военные инженеры и британский военный корреспондент Джордж Робинсон (George Thomas Robinson). При этом использовались более крупные воздушные шары, которые, впрочем, тоже были без экипажа. Было отправлено одиннадцать воздушных шаров, на каждом из которых насчитывалось от 5 тысяч до 32 тысяч папильонов. Шесть из них были захвачены прусской армией. На сегодняшний день известно около 180 таких папильонов.

3 октября 1870 года отправка почтовых отправлений воздушными шарами была прекращена из соображений безопасности.

Папильоны с успешно достигших неоккупированных районов воздушных шаров сортировались и паковались в пакеты в соответствии с местом назначения и доставлялись в соответствующие региональные почтовые отделения. Перехваченные папильоны, как правило, так и не попадали на почту. По этой причине на многих сохранившихся папильонах нет никаких почтовых отметок (почтовых штемпелей). Небольшое число папильонов пересылались в конвертах, так называемых , предоставляемых французской почтовой службой.

Папильоны, адресованные за границу, встречаются редко. Всего известны 17 таких отправлений, десять из которых адресованы в Бельгию или Англию.

См. также
 Воздушная почта
 История почты и почтовых марок Франции

Примечания

Ссылки

redpajama_set_name: RedPajamaWikipedia