text:
Q: Make figure active without making it visible I'm looping through a large number of figures while adding to axes in each of them.  Because there are many, I don't want them to be visible.  However, when I activate a figure, it becomes visible, as illustrated with the code below:
>> f1 = figure('visible', 'off');
>> f2 = figure('visible', 'off');
>> figure(f1);

Of course, I could have it followed immediately by set(f1, 'visible', 'off');, but is there some direct way of keeping visible set to off without resetting it "manually" after each figure call?

A: Add the axes when you create the figure, and use the axes handle to manipulate it later without activating explicitly the figure. Example
f1 = figure('visible', 'off');
a1=axes();
f2 = figure('visible', 'off');
a2=axes();
...
x=-1:0.01:1; y1=sin(2*pi*x); y2=cos(2*pi*x);
plot(a1, x, y1);
plot(a2, x, y2);

you can now make them visible to see the result;
set(f1, 'visible', 'on');
set(f2, 'visible', 'on');

Hope it helps.


text:
Anthony "Tony" Harris (* 20. April 1949 in Cleveland, Ohio) ist ein ehemaliger US-amerikanischer American-Football-Spieler. Er spielte auf der Position des Runningbacks, Wide Receivers und Defensive Backs in der National Football League (NFL).

Karriere
Harris spielte zwei Jahre auf der Position des Runningbacks für die Toledo Rockets College Football. Zuvor hatte er zwei Jahre wie zuvor in der Highschool als Quarterback gespielt. Im NFL Draft 1971 wurde er als 101. Spieler von den San Francisco 49ers ausgewählt. Er war der letzte gedraftete Runningback Toledos bis 2002 Chester Taylor ausgewählt wurde. Die 49ers planten dabei Harris als Runningback, Defensive Back oder Wide Receiver einzusetzen. Er spielte vier Spiele für die 49ers. 1972 wechselte er zu den Denver Broncos, spielte jedoch nie ein Spiel für diese.

Einzelnachweise

American-Football-Spieler (Toledo Rockets)
American-Football-Spieler (San Francisco 49ers)
American-Football-Spieler (Denver Broncos)
Geboren 1949
US-Amerikaner
Mann

redpajama_set_name: RedPajamaWikipedia


The literal definition of "egality" is equality. But what does this practically mean?
Equality is not something that is the default of nature, in that we all have individual strengths, challenges, and distinct starting points. This individuality is one of society's great strengths, and not something to be overlooked.
But there is a recognition that - independent of who we are - there are certain basic rights, human rights among them, that deserve to be extended to all of us. In America, we speak of "life, liberty and the pursuit of happiness". In France, we speak of "liberté, égalité, fraternité". Similar sentiments are echoed the world over: we each are equally deserving of consideration when it comes to our rights, to justice and to an opportunity to pursue our dreams.
In practice, as we all have witnessed in one form or another, those rights are not equally available to everyone. It is necessary, as many continually do, to find and address these imbalances where we can.
A goal of Egality as an enterprise is to try to address these imbalances in a positive way. And critical to positively addressing sustainable development and inequality is to realize the importance of building, not tearing down. Rights do not have to be taken from one person or group to extend them to another. The beauty of sustainable development is that evolving a positive world system benefits everyone throughout it.
Inseparable from having rights is to have responsibilities. And a genuine responsibility is to use our power to equally extend those rights to all of us. That is root of our own rights.


According to Whois record of Nicegap.ir, it is owned by Mohammad Ahmadi since 2018. Nicegap was registered with NIC-IR.
The current Nicegap.ir owner and other personalities/entities that used to own this domain in the past are listed below.
If you would like to share more "whois" details on Nicegap with us, please contact us!
If you are Nicegap owner and would like to increase privacy protection level for your data - please, deal with NIC-IR which is your site's registrar.
Whois history of Nicegap.ir is provided using publicly open domain data.


text:
7 Alzheimers Dementia Memoirs Worth Reading
By Susan Macaulay on	 March 31, 2014 Books, Care Partnering, Memoirs, Resources
There's a growing tidal wave of information and perspectives on Alzheimer's and dementia caregiving – some of it useful and interesting, some not so much!
Former caregiver and author Martha Stettinius and I share some of our "top reads" in the memoir category.
1. Passages in Caregiving: Turning Chaos into Confidence
Gail Sheehy, 2010
Me: I inhaled Sheehy's earlier Passages and Pathfinders and am in the process of doing the same with this part memoir, part how-to-make-the-experience-more-joyful book on caregiving. Sheehy lost her husband to cancer; this book is about the journey through their last chapter together. Lots of stories, loads of tips, easy to read. My copy is already full of underlines, tick marks and notes in the margins…
2. Love in the Land of Dementia
Deborah Shouse, 2013
Stettinius: A short, sweet read that left me feeling both hopeful and inspired. Not only is Shouse a talented writer, but she shows us in simple but beautifully-wrought scenes how she came to discover small ways each day to enjoy her mother's company despite her mother's advancing dementia. To read my full review of this book, click here.
3. Inside the Dementia Epidemic: A Daughter's Memoir
Martha Stettinius, 2013
Me: One of the best Alzheimers dementia memoirs I have yet to read. While other best sellers left me uninspired, I looked forward to sitting down with Stettinius and her Mom every night and learning more about their personal journeys. Many of the vignettes resonated so strongly with me that I found myself nodding in agreement, laughing and crying all the way through. (More with Stettinius video interview here)
4. The Long Hello: The Other Side of Alzheimer's
Cathie Borrie, 2010.
Stettinius: A lyrical, poignant story of how Borrie cares for her mother for seven years. Though her mother has Parkinson's and Alzheimer's, Borrie describes her as "one hundred percent of who she is." Borrie writes down or tape records their conversations over these years. I read this beautiful book in one sitting. It reminded me so much of my own conversations with my mother as her language has grown more elliptical, and of the mixed emotions we often feel as caregivers.
5. Dancing with Rose: Finding Life in the Land of Alzheimer
Lauren Kessler, 2007
Me: This was the first book I read on Alzheimer's and caregiving. To better understand Alzheimer's (from which her mother had died), Kessler takes a job as a bottom-of-the-rung caregiver at an "memory care" facility where she learns lessons that challenge what we think we know about the disease and what really happens in such facilities. I'm reading it again.
6. A Funny Thing Happened on My Way to the Dementia Ward: Memoir of a Male CNA
Charles Schoenfeld, 2012
Stettinius: Schoenfeld, a retired FedEx driver, has written a touching and respectful memoir about caring over seven years for the residents of a nursing home. It's refreshing to read the perspective of a CNA, and he writes beautifully about his fondness for each resident.
7. Still Alice
Lisa Genova, 2007
Me: a fictional memoir narrated by Alice Howland, a celebrated Harvard professor (and mother of three grown children) who at the height of her career notices forgetfulness creeping into her life. The author, who is neuroscientist, imagines what it must be like for the dementia sufferer as she quite literally loses her mind. Fascinating and sometimes a little frightening, the story is full of insights into the disease from a "patient's" perspective.
We hope you are as inspired by these books as we have been!
Susan Macaulay is the creator and curator of My Alzheimer's Story. Susan also created and curates AmazingWomenRock.com and SheQuotes.com; she blogs about things other than Alzheimer's disease and dementia at AmazingSusan.com
Martha Stettinius is the author of the acclaimed "Inside the Dementia Epidemic: A Daughter's Memoir." She was a "sandwich generation" caregiver for her mom, Judy, who had vascular dementia and probable Alzheimer's disease, and who lived with Stettinius, then in assisted living, a rehab center, a "memory care" facility, and a nursing home. Stettinius blogs and consults for www.caregivers.com and www.eCareDiary.com; her blog and book website are at www.insidementia.com.
Will Gibson on July 24, 2014 6:51 pm
There is a widely held misconception in the African American community that the Alzheimer's disease is mostly a white disease. This is not only a false notion, it is a view fraught with danger for black people. There is a terrific book by author Will Gibson that dispels this falsehood. ESTELLA & SYLVIA, while a novel that tells a warm, loving family story, deals with many of the issues and challenges regarding Alzheimer's.
As Frances Blake, a book lover who enjoys a good story while caring for her sick mother, says about ESTELLA & SYLVIA: "This is one of the most unique family stories I've ever read. I found myself rooting for the whole family. It also helped me as a caregiver and helped my friends understand what I was going through. Most important, my friends enjoyed it immensely as good fiction."—
Estella & Sylvia

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{\mbox{}\hspace{-.6cm}.\hspace{.4cm}#1}}
\newcommand{\Subsection}[1]{\subsection{\mbox{}\hspace{-.6cm}.\hspace{.4cm}
\em #1}}

\vspace{.8cm}
\noindent
{\bf Abstract}
The theory of phenomenological Non-equilibrium Thermodynamics
is extended by includimg stochastic processes in order to account for recently derived
thermodynamical relations such as the Jarzynski equality.
Four phenomenological axioms are postulated resulting in a phenomenological interpretation of Jarzynski's equality.  Especially, considering the class of Jarzynski processes Jarzynski's equality follows from the axiom that the statistical
average of the exponential work is protocol independent.
\vspace{.5cm}\newline
{\bf Keywords} Non-equilibrium Thermodynamics$\cdot$Jarzynski Equality$\cdot$
Stochasticity

\section{Introduction}

Among many branches of thermodynamics {\bf --}such as Thermodynamics of
Irreversible
Processes, Rational Thermodynamics, Extented Thermodynamics, Endoreversible
Thermodynamics, Finite Time Thermodynamics, Quantum Thermodynamics, Mesoscopic
Theory, GENERIC \C{MU1}{\bf --}
Stochastic Thermodynamics is another branch which introduces probabilities into
the thermodynamical description \C{SEIF}. But in contrast to the above
mentioned branches,
Stochastic Thermodynamics allows processes of negative process entropy,
some times called "violations" of the Second Law.
This item does not have any consequences on the phenomenological level because
the negative process entropies vanish by establishing mean values using the
introduced probabilities. Consequently it is obvious, how to obtain the
phenomenological level
in Stochastic Thermodynamics, but how vice-versa to incorporate stochastic
processes with their
probabilities and process entropies into phenomenological Non-equilibrium
Thermodynamics is an open question which is investigated in this paper.
This is not done in full generality, but only for a special process class
{\bf--}called the Jarzynski process class{\bf--} because we want to clarify the
status of the integral fluctuation theorem
{\bf--}the Jarzynski equality \cite{1}{\bf--} in the framework of
Non-equilibrium Thermodynamics.
\vspace{.3cm}\newlin
The paper is organized as follows: After having introduced the Jarzynski
process, a sketch of Non-equilibrium Thermodynamics of
discrete systems is given for introducing the items which later on are
needed. The Jarzynski process of phenomenological Non-equilibrium
Thermodynamics is replaced by a set of stochastic processes which generate probability
densities defined on the process work as a stochastic variable. These stochastic
processes are decomposed into regular and non-regular processes distinguished by their
process entropy production: regular ones have positive process entropy
production, whereas that of the non-regular processes is negative.
The mean values of process dissipation and work are considered with respect
to the phenomenological Second Law. Two phenomenological axioms establish
Jarzynski's equality. The reversible case and equilibrium are shortly
discussed. A summary and a discussion finish the paper.

\section{The Jarzynski Process\label{JP}}

We consider a discrete system\footnote{A so-called Schottky
system \cite{SS}: a ``box'' interacting with its environment which is not to be confused with discrete
systems in Stochastic Thermodynamics which evolve on a discrete state space \cite{SEIF}.}
which interacts with its environment by power and
heat exchange performing a Jarzynski process. Such a process
begins at time $t=0$ in a fixed
equilibrium state $A^{eq}$ whose work variable (Jarzynski's switching
parameter) is $\lambda = 0$ and arrives at time $t=\tau$ in a
non-equilibrium state\footnote{A so-called {\em non-thermal state}.} $C^{neq}$
with the work variable $\lambda = 1$. During the time
$0\leq t\leq\tau$, the system exchanges irreversibly work and
heat with its controlling equilibrium environment\footnote{A heat reservoir.}
of constant thermostatic temperature $T^*$. For times
$\tau< t\leq d$ the thermal contact with the controlling
heat reservoir maintains, but the switching parameter is fixed
at $\lambda = 1$, that means, no additional work, but heat is
exchanged between the system and the reservoir during this time
interval. Consequently, the final equilibrium state $B^{eq}$
has the same work variable $\lambda = 1$ as $C^{neq}$,
and according to the process control by the heat reservoir
its temperature is $T^*$. Because the state space of
the system in consideration is given by the work variable $\lambda$
and by the contact temperature $\Theta$\ \footnote{The contact
temperature is a non-equilibrium analogue of the
thermostatic temperature. More details in sect.\ref{CT} \cite{5,6,9}.},
we can represent the Jarzynski process as follows
\begin{eqnarray}\label{J1}
{\cal J}_\lambda:\quad &t=0:&\ A^{eq}(\lambda=0,T^*)
\quad\longrightarrow\quad [\st{\td}{\lambda}(t),\st{\td}{Q}(t)]
\quad\longrightarrow
\\ \label{J2}\longrightarrow\quad
&t=\tau:&\ \ C^{neq}(\lambda=1,\Theta).\quad\longrightarrow\quad
[\st{\td}{\lambda}=0,\st{\td}{Q}(t)]\quad\longrightarrow
\\ \label{J3}\longrightarrow\quad
&t=d:&\ \ B^{eq}(\lambda=1,T^*).
\end{eqnarray}
Here, the time-dependent work variable $\lambda(t)$ {\bf--}the protocol{\bf--}
is arbitrary, but controlled, whereas the heat exchange
$\st{\td}{Q}(t)$ is uncontrolled and depends on the given protocol.
\vspace{.3cm}\newlin
The interpretation of the work which is done during the transfer
from $\lambda =0$ to $\lambda =1$ {\bf--}during the protocol{\bf--} is different in
Non-equilibrium and Stochastic Thermodynamics: in Non-equilibrium
Thermodynamics the work is a phenomenological variable, whereas in Stochastic
Thermodynamics the work is introduced as a stochastic variable $w$ generating a
propability distribution $p_\lambda(w)$, if identical protocols $\lambda(t)$ are
performed. Considering a Jarzynski process \R{J1} to \R{J3}
with a heat reservoir at inverse (thermostatic) temperature $\beta$,
the probability distribution
is not arbitrary, but constrained by the relation which is known as Jarzynski equality
\begin{equation}\label{eq Jarzynski equality}
 \int_{-\infty}^\infty p_\lambda(w) \exp(-\beta w)dw = \exp(-\beta\Delta^{AB} F),
\end{equation}
where $\Delta^{AB} F$ denotes the difference of the
\emph{equilibrium} free energy of the system between the
states $A^{eq}$ and $B^{eq}$. Note
that ~(\ref{eq Jarzynski equality}) remains valid even if we only look at
changes of the system from $A^{eq}$ to $C^{neq}$ because there is no work
performed on the system from $C^{neq}$ to
$B^{eq}$ and $p_\lambda(w)$ remains unchanged during that part. Hence,
~(\ref{eq Jarzynski equality}) teaches that non-equilibrium stochastic
fluctuations contain valuable information about equilibrium quantities.
\vspace{.3cm}\newlin
Jarzynski's equality and was originally discovered in
1997~\cite{1, JarzynskiPRE1997}. Since then it was derived under many circumstances, for instance, it follows from
the detailed (Crooks) fluctuation theorem~\cite{CrooksJSP1998, CrooksPRE1999, CrooksPRE2000}, it is also valid in the
strong coupling regime~\cite{JarzynskiJSM2004} or for quantum systems~\cite{TasakiArXiv2000, MukamelPRL2003,
CampisiTalknerHaenggiPRL2009}. Furthermore, early experimental confirmations can be found in~\cite{TrepagnierEtAlPNAS2004, CollinEtAlNature2005, BustamanteLiphardtRitortPhysToday2005} and a recent review
is given in~\cite{JarzynskiReview2011}. In the next section we will look at the Jarzynski process from a purely
phenomenological perspective (without probabilities and stochasticity) before we then ask how to incorporate the Jarzynski equality on a phenomenological level ?

\section{A Thermodynamical Sketch}
\subsection{Basic phenomenological variables}

The state of a discrete thermodynamical system is described by
basic variables. Kind and number of these variables depend on the nature
of the system under consideration and on the process going on in that system.
The number of variables in non-equilibrium is clearly greater than in
equilibrium. Therefore, equilibrium needs a minimal number of basic variables
spanning the so-called equilibrium sub-space. According to a special
formulation of the Zeroth Law \cite{2}, the equilibrium variables of a thermally
homogeneous\footnote{A system without adiabatic partitions.}
system are
\begin{equation}\label{1}
\mvec{z}_{eq}\ =\ (U,\mvec{a},\mvec{n})\qquad\mbox{or}\qquad
\mvec{z}_{eq}\ =\ (T,\mvec{a},\mvec{n})
\end{equation}
(internal energy U, work variables \mvec{a}, mol numbers \mvec{n},
thermostatic temperature T). In equilibrium,
there exists an one-to-one mapping between the
internal energy of the system and its thermostatic temperature
$U\leftrightarrow T$.
\vspace{.3cm}\newlin
More basic variables than in equilibrium are needed in non-equilibrium
\begin{equation}\label{2}
\mvec{z}\ =\ (U,\mvec{a},\mvec{n},\mvec{z}_{neq}).
\end{equation}
The set of the non-equilibrium variables {$\mvec{z}_{neq}$}
depends on the nature of the system in
consideration: e.g. the orientation of needle-shaped molecules may be an
example in the case of complex materials, time derivatives of the equilibrium
variables and dissipative fluxes are other examples. Here with regard to Jarzynski
processes, we introduce the contact temperature $\Theta$ of the system as one
non-equilibrium variable \cite{5,6,9}. Other basic non-equilibrium
variables {\bf--}e.g. the internal variables{\bf--} are marked by a
place-holder $\mvec{\xi}$. Consequently, the non-equilibrium variables are
\begin{equation}\label{2a}
\mvec{z}_{neq}\ =\ (\Theta, \mvec{\xi}),
\end{equation}
and the basic variables spanning the state space of the system are
\begin{equation}\label{7}
\mvec{z}\ =\ (U, \mvec{a}, \mvec{n}, \Theta, \mvec{\xi}).
\end{equation}
The non-equilibrium contact temperature $\Theta$ is independent of the other variables of
the state space, especially independent of the internal energy \cite{10}.

\subsection{Non-equilibrium entropy, First Law}

A non-equilibrium entropy is a state function on the non-equilibrium state space \R{7}.
The time rate of this non-equilibrium entropy is an analogue to Gibbs'
fundamental equation \cite{9}
\begin{equation}\label{6}
\st{\td}{S}\ :=\ \frac{1}{\Theta}\st{\td}{U} -  \frac{\mvec{A}}{\Theta}\cdot
\st{\td}{\mvec{a}} - \frac{\mvec{\mu}}{\Theta}\st{\td}{\mvec{n}} +
\alpha\st{\td}{\Theta} + \mvec{\beta}\cdot\mvec{\st{\td}{\xi}}
\end{equation}
The conjugate quantities to the state space variables are: the reciprocal contact
temperature $1/\Theta$, the generalized forces $\mvec{A}$ over the contact
temperature, the chemi\-cal potentials $\mvec{\mu}$ over the contact
temperature and  the conjugate quantities $\alpha$ and $\mvec{\beta}$ related to
the contact temperature and to the internal variables\footnote{If the contact
temperature $\Theta$ in \R{6} is replaced by the thermostatic
temperature $T^*$ of the controlling heat reservoir, the expression
\R{6} looses its property as state function because $T^*$ belogs to the controllimg
heat reservoir and is therefore not a state variable.
In equilibrium, $\Theta$ is replaced by the thermostatic temperature $T$ of the system.}.
\vspace{.3cm}\newlin
Introducing the molar enthalpy $\mvec{h}$ and the external change of mol
numbers $\st{\td}{\mvec{n}}\!{^e}$ by the material exchange between system and
environment \cite{8}, the First Law and the power $\st{\td}{W}$ are
\begin{equation}\label{8}
\st{\td}{U}\ =\ \st{\td}{Q} + \st{\td}{W} + \mvec{h}\cdot\st{\td}{\mvec{n}}{^e}
,\qquad \st{\td}{W}\ :=\ \mvec{A}\cdot\st{\td}{\mvec{a}}.
\end{equation}
Approaching Jarzynski processes, we consider here closed discrete systems
without chemical reactions
\begin{equation}\label{9}
\st{\td}{\mvec{n}}\ \doteq\ \mvec{0},\qquad\st{\td}{\mvec{n}}{^e}\ \doteq\
\mvec{0}.
\end{equation}
By taking (\ref{8}) and (\ref{9}) into account, (\ref{6}) results for closed
systems in
\begin{equation}\label{N30}
\st{\td}{S}\ =\ \frac{1}{\Theta}\st{\td}{Q} +
\alpha\st{\td}{\Theta} +\mvec{\beta}\cdot\mvec{\st{\td}{\xi}} .
\end{equation}
This is the usual expression of decomposing the entropy time rate in phenomenological
Non-equilibrium Thermodynamics~\C{GM}:
$\st{\td}{Q}/\Theta$ is the entropy flux and
$\alpha\st{\td}{\Theta} +\mvec{\beta}\cdot\mvec{\st{\td}{\xi}}$ the entropy
production. Closing the system enforces vanishing of the entropy flux.

\subsection{Contact temperature, free energy and work\label{CT}}

We consider a closed non-equilibrium system which is in contact with a heat
reservoir of thermostatic temperature $T^*$. The heat exchange between them
is $\st{\td}{Q}$. We now define the contact temperature $\Theta$ of the
non-equilibrium system \cite{5,6} by the inequality
\begin{equation}\label{N30a}
\left(\frac{1}{\Theta}-\frac{1}{T*}\right)\st{\td}{Q}\ \geq\ 0.
\end{equation}
This ``defining inequality of the contact temperature'' states that $\Theta = T^*$
 if and only if $\dot Q = 0$.
\vspace{.3cm}\newlin
Taking (\ref{N30}), (\ref{8}), (\ref{9}) and \R{N30a} into account, we obtain
\begin{equation}\label{N30b}
\st{\td}{S} -\alpha{\st{\td}\Theta}-\mvec{\beta}\cdot\mvec{\st{\td}{\xi}}\ \geq\
\frac{1}{T^*}\Big(\st{\td}{U}-\st{\td}{W}\Big),
\end{equation}
an inequality which stems from introducing the contact temperature by \R{N30a}
and which is independent of the Second Law.
Here, $T^*$ is the constant thermostatic temperature of a controlling heat
reservoir, such one {which appears} in the Jarzynski process.
\vspace{.3cm}\newlin
Next, we define the state function of
non-equilibrium free energy as\footnote{This definition is in contrast to
definitions used in Stochastic Thermodynamics~\cite{GaveauSchulmanPLA1997, CrooksPRE2007, EspositoVandenBroeckEPL2011}
where the contact temperature is not used and is replaced by the
thermostatic temperature $T^*$ of the controlling reservoir. Thus, $F$ becomes
the free energy in the equilibrium state $B^{eq}$ \R{J3}, whereas \R{10} refers to $C^{neq}$.}
\begin{equation}\label{10}
F(\mvec{a},\Theta,\mvec{\xi})\ :=\ U - \Theta S
\quad\Longrightarrow\quad
\st{\td}{F}\ =\ \st{\td}{U} - (\Theta S)^\td.
\end{equation}
Inserting (\ref{10}) into (\ref{N30b}) results in
\begin{equation}\label{N30c}
\st{\td}{S} -\alpha{\st{\td}\Theta}-\mvec{\beta}\cdot\mvec{\st{\td}{\xi}}\ \geq\
\frac{1}{T^*}\Big(\st{\td}{F} +(\Theta S)^\td  -\st{\td}{W}\Big).
\end{equation}
Integration along a Jarzynski process ${\cal J}_\lambda:
A^{eq}\longrightarrow B^{eq}$ yields
\begin{equation}\label{N30d}
S_B - S_A -{\cal J}_\lambda\!\!\!\int_A^B(\alpha{\st{\td}\Theta}+\mvec{\beta}\cdot\mvec{\st{\td}{\xi}})dt\ \geq\
\frac{1}{T^*}\Big(\Delta^{AB}F +\Theta_BS_B - \Theta_AS_A - W^{AB}\Big).
\end{equation}
According to (\ref{J1}) and (\ref{J3}),
\begin{equation}\label{N30e}
\Theta_A\ =\ \Theta_B\ =\ T^*
\end{equation}
is valid\footnote{An integration only between $A^{eq}$ and $C^{neq}$ would not
result in \R{N30e} because of $\Theta_C\neq T^*$.},
and we obtain an inequality valid along Jarzynski processes
\begin{equation}\label{N30f}
{\cal J}_\lambda\!\!\!\int_A^B(\alpha{\st{\td}\Theta}+\mvec{\beta}\cdot\mvec{\st{\td}{\xi}} )dt\ =:\
\Sigma^{AB}\ \leq\ \frac{1}{T^*}\Big(W^{AB} - \Delta^{AB}F\Big)\
=:\ {\frac{D^{AB}}{T^*}}
\end{equation}
which stems { as \R{N30b}} from the defining inequality of the contact temperature (\ref{N30a}).
According to the decomposition of the entropy rate \R{N30}, the bracket in
\R{N30f}$_1$ is the entropy production, so that
$\Sigma^{AB}$ becomes the process entropy production
between $A^{eq}$ and $B^{eq}$, and
$D^{AB}$  is the process dissipation which
is in generally greater than $T^*\Sigma^{AB}$.
Note that $D^{AB}/T^*$ is also sometimes called the entropy production because
it coincides with the entropy increase of system \emph{and}
bath~\cite{1, SEIF, JarzynskiReview2011}.
\vspace{.3cm}\newlin
The process work $W^{AB}$ along the Jarzynski process ${\cal J}_\lambda$ is
related to the given protocol $\lambda(t)$ according to \R{J1} and to the generalized
forces $ L({\lambda,\Theta})$
\begin{equation}\label{J4}
W^{AB}_\lambda\ =\ {\cal J}_\lambda\!\!\!\int_A^B L({\lambda,\Theta})
\st{\td}{\lambda}(t)dt.
\end{equation}
This quantity will become a key position when stochastic processes
are introduced below. We now take the Second Law into account.

\subsection{Second Law}

According to the Second Law, the entropy production
\begin{equation}\label{N31}
\alpha\st{\td}{\Theta} +\mvec{\beta}\cdot\mvec{\st{\td}{\xi}}\ \geq\ 0,
\end{equation}
is not negative in phenomenological Non-equilibrium Thermodynamics \C{GM}.
Thus, (\ref{N30}) and (\ref{N30f})$_1$ become with (\ref{N30a}) and
(\ref{N31})
\begin{equation}\label{N32}
\st{\td}{S}\ \geq\ \frac{1}{\Theta}\st{\td}{Q}\ \geq\
\frac{1}{T^*}\st{\td}{Q},\qquad 0\ \leq\ \Sigma^{AB}\ \leq\
\frac{D^{AB}}{T^*}.
\end{equation}
From \R{N30f}$_2$ and \R{N32}$_3$ follows the well known fact which is also
valid for Jarzynski processes
\begin{equation}\label{aJ4}
W^{AB}\ \geq\ \Delta^{AB}F.
\vspace{.3cm}\end{equation
The reversible case is defined by
\begin{equation}\label{J4a}
(\alpha\st{\td}{\Theta} + \mvec{\beta}\cdot\mvec{\st{\td}{\xi}})^{rev}
\ \equiv\ 0,\qquad \mbox{and}\quad
\Theta\ \equiv\ T^*\ =\ \mbox{const}.
\end{equation}
Consequently,  we obtain according to \R{N30f} and \R{J4a}$_1$  for the reversible
case
\begin{equation}\label{J4b}
\Sigma^{AB}_{rev}\ =\ D^{AB}_{rev}\ =\ 0\quad\longrightarrow\quad
W^{AB}_{rev}\ =\ \Delta^{AB}F.
\vspace{.3cm}\end{equation
This sketch outlines the tools which we need in the sequel.

\section{Introducing Stochasticity}

In contrast to macroscopic systems, the behaviour of small (mesoscopic) systems is inherently stochastic:
for describing them, stochastic variables have to be used generating
probability distributions. Although the average behaviour of stochastic systems
still obeys the phenomenological laws of thermodynamics, there is much more to
discover as Stochastic Thermodynamics lets suppose.
Here, for the special exampe of Jarzynki processes, we are interested in a ``top-down'' approach, i.e., we ask how
to modify phenomenological Non-equilibirum Thermodynamics in order to account for fluctuations and stochasticity.
This is in contrast to Stochastic Thermodynamics, which follows rather from a
``bottom-up'' approach by relying on microscopically derived equations of motion
(e.g., Langevin or master equations)~\cite{SEIF}.

\subsection{Process work as a stochastic variable}

We consider one of the numerous protocols $\lambda(t)$ performing
a Jarzynski process. According to \R{J4}, the work $W^{AB}_\lambda$ is
required. Whenever the same protocol is used in Non-equilibrium
Thermodynamics, the same work is required for performing the
corresponding Jarzynski process. This situation is
totally different for {\em stochastic systems}: several experiments,
all performed with the same given protocol $\lambda(t)$ require
several different works for performing the Jarzynski process
with the result, that \R{J4} cannot hold true for stochastic systems.
Consequently, we postulate that the process work is a stochastic quantity.

\subsubsection{The first basic axiom}

$\blacksquare${\bf First\,Basic\,Axiom:}
The process work along a Jarzynski process is a stochastic variable
given by a stochastic equation
\begin{equation}\label{J5}
{\cal W}^{AB}_\lambda\ =\  {\cal J}_\lambda\!\!\!\int_A^B {\cal L}({\lambda,\Theta})
\st{\td}{\lambda}(t)dt\ \in\ \mathbb{R}\ \longrightarrow\
p_\lambda({\cal W}^{AB}_\lambda)
\end{equation}
replacing \R{J4}. Performing the same protocol numerously, the values of the
process works generate a probability distribution function $p_\lambda$ on
${\cal W}^{AB}_\lambda$.
\hspace{2.4cm}$\blacksquare$
\vspace{.3cm}\newlin
The same protocol $\lambda(t)$ generates by the stochastic properties
of the material {\bf--}introduced by the stochastic mapping ${\cal L}(\lambda,\Theta)$
for the generalized forces \C{OST}{\bf--} different process works
${\cal W}^{AB}_\lambda$
which all together implement a probability distribution function
$p_\lambda({\cal W}^{AB}_\lambda)$
on the stochastic variable of the process work. Consequently,
in the framework of Non-equilibrium Thermodynamics, the probability
distribution function $p_\lambda({\cal W}^{AB}_\lambda)$ is a measurable quantity
which can be found out by performing a sufficently high number of
Jarzynski processes always using the same protocol $\lambda(t)$.
Not only the work becomes a stochastic quantity but also related quantities such
as heat, entropy and entropy production.

\subsubsection{Jarzynski process class}

We can suppose, that by replacing the phenomenological quantities of
Non-equi\-li\-brium
Thermodynamics by stochastic ones, some results of the phenomenological
theory will change: Stochastic and Non-equilibrium Thermodynamics are
different to each other, and the following question arises: What are the
phenomenological conditions under which Non-equilibrium Thermodynamics turns
out to be a special case of Stochastic Thermodynamics ?
For instance, it is easy to see that, taking Jarzynski's inequality
into account, processes of negative process dissipation appear in Stochastic
Thermodynamics,
a fact which is strictly forbidden in Non-equilibrium Thermodynamics.
\vspace{.3cm}\newlin
Up to now, we considered one arbitrary, but fixed protocol belonging to a special
Jarzynski process. According to \R{J1}$_3$ and \R{J2}$_3$, Jarzynski processes
can be performed with several different protocols.
All these protocols together form the {\em (stochastic) Jarzynski process class}
\begin{equation}\label{J5a}
\{{\cal J}_\lambda\}\ :=\ \Big\{\wedge\lambda:\
\lambda(t)\in {\cal J}_\lambda,\ p_\lambda({\cal W}^{AB}_\lambda)\Big\}.
\end{equation}
The introduction of this process class allows to derive connections
between the probability densities of different protocols in the sequel.

\subsection{Exponential mean process work\label{EMPW}}

Approaching Jarzynski's equality, we start out with Jensen's
inequality\footnote{Jensen's inequality
states that for any convex function $f$ and random variable $X$ we have
$\mathbb{E}[f(X)] \ge f[\mathbb{E}(X)]$ where $\mathbb{E}[...]$ denotes an expectation value. Since
$f(x) = e^{-\beta x}$ is convex, Eq.~(\ref{J8}) follows.}
\begin{eqnarray}\label{J8}
\int p(x)\exp(-\beta x)dx\ \geq\ \exp\Big(-\beta\int p(x)xdx\Big),
\\ \label{J8a}
\int p(x)dx\ =\ 1,\quad p(x)\ \geq\ 0,\quad\beta\ >\ 0,
\end{eqnarray}
and we identify $x$ with the stochastic process work, and the
probability function $p_\lambda(x)$ belongs to an arbitrary protocol
$\lambda(t)$
\begin{equation}\label{J9}
x\ \equiv\ {\cal W}^{AB}_\lambda,\qquad
\int p_\lambda(x)x dx\ =:\ W^{AB}_\lambda.
\end{equation}
Here, the phenomenological process work $W^{AB}_\lambda$ is introduced
as the mean value over all stochastic process works ${\cal W}^{AB}_\lambda$.
According to this setting, Jensen's inequality \R{J8} results in
\begin{equation}\label{J10}
\int p_\lambda({\cal W}^{AB}_\lambda)
\exp(-\beta {\cal W}^{AB}_\lambda)d{\cal W}^{AB}_\lambda\
\geq\ \exp\Big(-\beta W^{AB}_\lambda\Big).
\end{equation}
As already mentioned, the theoretical concept of process work is
different in Stochastic and Non-equilibrium thermodynamics: according to \R{J9}, we
have to distinguish between stochastic and phenomenological work:
${\cal W}^{AB}_\lambda\neq W^{AB}_\lambda$ \C{OST}.
\vspace{.3cm}\newlin
Applying the mean value theorem on the lhs of \R{J10}, we obtain
\begin{equation}\label{bJ10}
\int p_\lambda(x)\exp(-\beta x)dx\ =\ \exp(-\beta M_\lambda)\
\geq\ \exp(-\beta W^{AB}_\lambda)
\end{equation}
Here, $M_\lambda$ is the {\em exponential mean process work} which
is different from the phenomenological one according to \R{bJ10}
\begin{equation}\label{cJ10}
M_\lambda\ \leq\ W_\lambda^{AB}.
\end{equation}
We now consider the exponential mean process work in two special cases
of the probability distribution: the non-stochastic case and the
reversible one.

\subsubsection{The non-stochastic case}

If in every repetition of the same protocol
we measure the same work value $W^{ABnst}_\lambda$,
we refer to this case as ``non-stochastic'' and the corresponding probability density is
\begin{equation}\label{cJ10a}
p^{nst}_\lambda(x) = \delta\left(x-W^{ABnst}_\lambda\right).
\end{equation}
Then, according to \R{bJ10} we obtain
\begin{equation}\label{cJ10b}
\exp(-\beta W^{ABnst}_\lambda)\ =\ \exp(-\beta M^{nst}_\lambda)\
\geq\ \exp(-\beta W^{ABnst}_\lambda),
\end{equation}
resulting in
\begin{equation}\label{cJ10c}
M^{nst}_\lambda\ =\ W^{ABnst}_\lambda\ \geq\ \Delta^{AB}F,
\end{equation}
because the phenomenological process work obeys the Second Law \R{aJ4} also for
non-stochastic processes.

\subsubsection{The reversible case}

According to \R{J4b}$_2$, the reversible case is defined by
\begin{equation}\label{cJ10d}
W^{ABrev}_\lambda\ =\ \Delta^{AB}F.
\end{equation}
Thus, \R{bJ10} yields
\begin{equation}\label{bJ10e}
\int p^{rev}_\lambda(x)\exp(-\beta x)dx\ =\ \exp(-\beta M^{rev}_\lambda)\
\geq\ \exp(-\beta\Delta^{AB}F)
\end{equation}
resulting in $M^{rev}_\lambda\ \leq\ \Delta^{AB}F$.
In comparison with (\ref{cJ10c}) and (\ref{cJ10d}), we obtain
the chain of inequalities
\begin{equation}\label{bJ10g}
M^{rev}_\lambda\ \leq\ \Delta^{AB}F\ =\ W^{ABrev}_\lambda
\leq\ M^{nst}_\lambda\ =\ W^{ABnst}_\lambda.
\end{equation}
Consequently, the exponential mean process work $M_\lambda$ is
process-dependent $\bf-$reversible or non-stochastic$\bf-$ for the
present\footnote{This process dependence vanishes by introducing Jarzynski's
equality in sect.\ref{JE}, giving rise to the second basic axiom in sect.\ref{SBA}.}.

\subsection{Regular and non-regular processes}

The stochastic process work \R{J5} takes values which can be
greater or smaller than the free energy difference $\Delta^{AB}F$.
Consequently, the integral in \R{bJ10} can be decomposed into two parts
\begin{equation}\label{fJ10}
\int_{x\geq\Delta} p_\lambda(x)\exp(-\beta x)dx
+\int_{x <\Delta} p_\lambda(x)\exp(-\beta x)dx\
=\ \exp(-\beta M_\lambda).
\end{equation}
We now denote processes with ${\cal W}^{AB}_\lambda\geq\Delta^{AB}F$
as {\em regular processes} and such with ${\cal W}^{AB}_\lambda<\Delta^{AB}F$ as
{\em non-regular} ones. Consequently, the first integral of \R{fJ10} runs over the regular
processes, whereas the second one runs over the non-regular processes.
\vspace{.3cm}\newlin
Application of the mean value theorem to the lhs of \R{fJ10} is possible and results
by use of \R{bJ10} in
\begin{eqnarray}\label{gJ10}
\exp(-\beta M^+_\lambda)P^+_\lambda\
+\ \exp(-\beta M^-_\lambda)P^-_\lambda\
=\ \exp(-\beta M_\lambda)\ \geq\ \exp(-\beta W^{AB}_\lambda),
\\ \label{iJ10}
P^+_\lambda\ :=\ \int_{x\geq\Delta} p_\lambda(x)dx,\qquad
P^-_\lambda\ :=\ \int_{x <\Delta} p_\lambda(x)dx,\qquad
P^+_\lambda + P^-_\lambda\ =\ 1.
\end{eqnarray}
By construction, we obtain
\begin{equation}\label{hJ10}
 M^-_\lambda < \Delta^{AB}F ~{\le}~  M^+_\lambda.
\end{equation}
More specifically, the exponential mean process works $M^+_\lambda$ and $M^-_\lambda$ of the regular and the
non-regular processes depend on the precise form of the probability densities.
According to \R{gJ10}, \R{fJ10} and \R{iJ10}, we obtain
\begin{eqnarray}\label{hJ10a}
M^+_\lambda\ =\ \frac{1}{\beta}\Big[\ln\int_{x\geq\Delta} p_\lambda(x)dx-
\ln\int_{x\geq\Delta} p_\lambda(x)\exp(-\beta x)dx\Big],
\\ \label{hJ10b}
M^-_\lambda\ =\ \frac{1}{\beta}\Big[\ln\int_{x<\Delta} p_\lambda(x)dx-
\ln\int_{x<\Delta} p_\lambda(x)\exp(-\beta x)dx\Big].
\end{eqnarray}
Together with \R{hJ10} and \R{iJ10}$_{1,2}$,
this yields after a short algebraic manipulation
\begin{equation}\label{hJ10d}
\frac{P^-_\lambda}{P^+_\lambda}\ {<}\
\frac{\int_{x<\Delta} p_\lambda(x)\exp(-\beta x)dx}{\int_{x\geq\Delta} p_\lambda(x)\exp(-\beta x)dx}.
\vspace{.3cm}\end{equation
With the help of (\ref{gJ10}) and the normalization condition
{\R{iJ10}$_3$, we can solve for $P^\pm_\lambda$. This results in
\begin{eqnarray}\label{nJ10}
P^+_\lambda\ =\
\frac{\exp(-\beta M^-_\lambda)-\exp(-\beta M_\lambda)}
{\exp(-\beta M^-_\lambda)-\exp(-\beta M^+_\lambda)},
\\ \label{nJ10a}
P^-_\lambda\ =\ \frac{\exp(-\beta M_\lambda)-\exp(-\beta M^+_\lambda)}
{\exp(-\beta M^-_\lambda)-\exp(-\beta M^+_\lambda)}.
\end{eqnarray}
From the positivity of $P^\pm_\lambda$ we further obtain
\begin{equation}\label{nJ10c}
M^-_\lambda \leq M_\lambda,\quad M_\lambda\leq M^+_\lambda.
\end{equation}

\subsection{Mean process work}

Up to now, the exponential mean process work was considered in sect.\ref{EMPW}.
Let us {now} go one step back and look at the mean value of the stochastic work itself.
Starting out with \R{J9} and decomposing
its lhs according to \R{iJ10}, we obtain by use of the mean value theorem and
\R{aJ4}
\begin{equation}\label{I3a}
W^+_\lambda P^+_\lambda + W^-_\lambda P^-_\lambda\ =\ W^{AB}_\lambda\
\geq\ \Delta^{AB}F.
\end{equation
According to the decomposition \R{iJ10}, we have for the mean values of the process
works belonging to the regular (+) and non-regular (-) processes
\begin{equation}\label{I3b}
W^-_\lambda\ \leq\ \Delta^{AB}F\ \leq\ W^+_\lambda.
\end{equation}
From \R{I3a} follows
\begin{equation}\label{I3e}
(W^+_\lambda -W^{AB}_\lambda) P^+_\lambda+(W^-_\lambda -W^{AB}_\lambda) P^-_\lambda\ =\ 0,\quad\longrightarrow\quad
1\ =\ \frac{(W^{AB}_\lambda-W^-_\lambda) P^-_\lambda}{(W^+_\lambda -W^{AB}_\lambda) P^+_\lambda}.
\end{equation}

\section{Jarzynki's Equality\label{JE}}

\subsection{{Two phenomenological lemmata}}

Approaching Jarzynski's equality, an axiom is needed which
postulates suitable properties of the exponential mean process work $M_\lambda$
which is up to now protocol dependent according to \R{bJ10g}. For more physical
elucidation, we formulate this axiom in two steps by two lemmata, so to say as
auxiliary axioms.
Although the exponential mean process work $M_\lambda$ is smaller than
$W^{AB}_\lambda$ according to \R{cJ10}, we
demand that it satifies the
Second Law \R{aJ4} like the phenomenological process work:\nolinebreak
\vspace{.3cm}\newlin
$\blacksquare${\bf Lemma I:}
\begin{equation}\label{dJ10}
\Delta^{AB}F\ \st{\td}{\leq} \ M_\lambda.
\hspace{4.3cm}\blacksquare\hspace{-5cm}
\vspace{.3cm}\end{equation
Lemma I together with the inequalities \R{cJ10}, \R{hJ10}, and \R{nJ10c} can be summarized as
\begin{equation}\label{dJ10z}
 M^-_\lambda\ <\
\Delta^{AB}F\ \leq \ M_\lambda\
\left\{
\begin{array}{l}
\leq\ W^{AB}_\lambda\\
\leq\ M^+_\lambda\\
\end{array}
\right.
\end{equation}
or, equivalently,
\begin{equation}\label{dJ10y}
\exp{(-\beta M^-_\lambda)}\ >\ \exp{(-\beta\Delta^{AB}F)}\ \geq\
\exp{(-\beta M_\lambda)}\
\left\{
\begin{array}{l}
\geq\ \exp{(-\beta W^{AB}_\lambda)}\\
\geq\ \exp{(-\beta M^+_\lambda)}\\
\end{array}
\right..
\vspace{.3cm}\end{equation
Because Lemma I is demanded for all protocols -- also for reversible ones -- we obtain from \R{bJ10g}$_1$ in comparison
with \R{dJ10}
\begin{equation}\label{dJ10a}
M_\lambda^{ABrev}\ =\ \Delta^{AB}F,
\end{equation}
and \R{bJ10e} results in
\begin{equation}\label{dJ10b}
\int p^{rev}_\lambda(x)\exp(-\beta x)dx\ =\ \exp(-\beta\Delta^{AB}F).
\end{equation}
That is to say, Lemma I implies the Jarzynski equality
~(\ref{eq Jarzynski equality}) for reversible protocols.
Taking the second inequality of \R{dJ10y} into account, \R{nJ10a} results in
\begin{equation}\label{dJ10t}
P^-_\lambda\ \leq\ \frac{\exp(-\beta\Delta^{AB}F)-\exp(-\beta M^+_\lambda)}
{\exp(-\beta M^-_\lambda)-\exp(-\beta M^+_\lambda)},
\end{equation}
that means,
Lemma I gives also  a constraint on the integrated probability $P_\lambda^-$ of the
non-regular processes, an inequality which we need later on.
\vspace{.3cm}\newlin
Now, to extend the validity of the Jarzynski equality to arbitrary protocols we introduce
a second
Lemma, which states that the non-regular admixture \R{nJ10a} should have an influence as great as possible by choosing
$M_\lambda$ independently of the special protocol. Hence, we demand
\vspace{.3cm}\newlin
$\blacksquare${\bf Lemma II:}
\begin{eqnarray}\nonumber
\Big(P^-_\lambda\rightarrow\mbox{max, for all protocols}\Big)
\longrightarrow\hspace{5.5cm}
\\ \label{J11}\longrightarrow
\Big(\exp(-\beta M_\lambda)\rightarrow\mbox{max}\Big)
\longrightarrow\Big(M_\lambda\ \st{\td}{=}\ \Delta^{AB}F\Big).
\hspace{1cm}
\blacksquare\hspace{-.7cm}
\vspace{.3cm}\end{eqnarray
Note that Lemma II implies Lemma I, but we found it intuitive to start with Lemma I separately.
Furthermore, the second inequality of \R{dJ10y} and the inequality \R{dJ10t} change into equations by Lemma II.
Now, multiplication of \R{bJ10}$_1$ with $\exp(\beta M_\lambda)$ and taking
Lemma II into account results in a
phenomenological vindication of Jarzynski's equality
\begin{equation}\label{J13}
 \boxed{
  \{{\cal J}_\lambda\}:\hspace{1cm} \int p_\lambda(x)\exp\Big(-\beta (x-\Delta^{AB}F)\Big)dx\ =\ 1
 }
\end{equation}
including \R{dJ10b} which can be derived without using Lemma II.
\vspace{.3cm}\newlin
Finally, taking Jarzynski's equality into account, from \R{nJ10} and \R{nJ10a}
follows with \R{J11} for the admixtures of the regular and non-regular processes
\begin{eqnarray}\label{I2}
P^+_\lambda\ =\
\frac{\exp(-\beta M^-_\lambda)-\exp(-\beta \Delta^{AB}F)}
{\exp(-\beta M^-_\lambda)-\exp(-\beta M^+_\lambda)},
\\ \label{I3}
P^-_\lambda\ =\ \frac{\exp(-\beta \Delta^{AB}F)-\exp(-\beta M^+_\lambda)}
{\exp(-\beta M^-_\lambda)-\exp(-\beta M^+_\lambda)},
\end{eqnarray}
resulting in
\begin{equation}\label{aJ11c}
\frac{P^-_\lambda}{P^+_\lambda}\ =\
\frac{1-\exp\Big(-\beta (M^+_\lambda-\Delta^{AB}F)\Big)}
{\exp\Big(+\beta (\Delta^{AB}F-M^-_\lambda)\Big)-1}.
\end{equation}
Note that the restriction on the probability densities $p_\lambda({\cal W^{AB}_\lambda})$ by the
phenomenolo\-gi\-cal Lemmata I and II ge\-ne\-rating Jarzynski's equality can be tested by experimental investigation
according to our basic assumption that these probability densities are experimentally given in the view of Non-equilibrium Thermodynamics.
Especially, testing the restrictions on $P_\lambda^\pm$ might require
much less statistics than the validation of the Jarzysnki equality itself for which it is extremely important to
sample the very rare events where the dissipated work is much smaller than the free energy difference
\cite{JarzynskiPRE2006, HalpernJarzynskiArXiv2016}.

\subsection{The Second Basic Axiom\label{SBA}}
\label{sec the second basic axiom}

Because $\Delta^{AB}F$ is a constant belonging to all Jarzynski processes
between $A$ and $B$, we obtain from \R{J13} for two different protocols
$\lambda(t)$ and $\mu(t)$ of $\{{\cal J}_\lambda\}$
\begin{equation}\label{J11c}
\int p_\lambda(x)\exp(-\beta x)dx\ =\ \int p_\mu(x)\exp(-\beta x)dx\ =\
\exp(-\beta \Delta^{AB}F).
\end{equation}
That means, the expectation value of the exponential process work
$\exp(-\beta{\cal W}^{AB}_\lambda)$
is protocol-independent, and all protocols of the Jarzynski process class have
to satisfy Jarzynski's equality, a fact which restricts the possible probability
densities. Consequently, Jarzynski's equality is an object of experimentally testing
because in Non-equilibrium Thermodynamics we do not start out with special
given proba\-bi\-li\-ty densities $p_\lambda({\cal W^{AB}_\lambda})$.
\vspace{.3cm}\newlin
Jarzynski's equality is here established by the two phenomenological lemmata
\R{dJ10} and \R{J11}, whereby the second one includes the first. The two-step
procedure is chosen because of the more evident physical interpretation. The
main result of Jarzynski's equality is that the mean value of the exponential
process work is protocol-independent according to Lemma II \R{J11}$_3$.
This fact can be used for replacing the two phenomenological lemmata by another
basic axiom:\vspace{.3cm}\newline
$\blacksquare${\bf Second Basic Axiom:}\vspace{-.2cm}
\begin{center}
\fbox{
\parbox[t]{6.3cm}{The mean value of the exponential
process work is protocol-independent.}}
\end{center}
\hfill$\blacksquare$
\newline
Using this axiom, \R{J11} follows immediately from \R{bJ10g}, because
protocol independence of the mean value of the exponential  process work means
$M^{rev}_\lambda = M^{nst}_\lambda$. This more formal axiom allows to establish
Jarzynski's equality with out use of Lemmata I and II which can be regarded as physical
interpretation behind the Second Basic Axiom.
\vspace{.3cm}\newlin
Jarzynski's equality, derived in the framework of Stochastic Thermodynamics
\cite{1,SEIF}, is an {\em integral fluctuation relation} with regard to the
Jarzynski process class. Whatever its derivation in Stochastic Thermodynamics
may be, from the point of view of Non-equilibrium Thermodynamics, Jarzynski's
equality can be phenomenologically established by the Second Basic Axiom.
The procedure for implementing stochastic processes into Non-equilibrium Thermodynamics is totally different
from that used in Stochastic Thermodynamics because we neither make use of any underlying equation of motion nor any specific Hamiltonian in our framework.
In Non-equilibrium Thermodynamics non-regular processes
appear instead of reversed processes with the difference that non-regular
processes are measurable contributions to the non-regular admixture.
The Second Basic Axiom which allows to establish Jarzynski's equality is a
phenomenological statement on the protocol-independence of the mean values
of the exponential process works.

\section{Some Results}
\subsection{Dissipation and non-stochasticity}

Jarzynski's equation allows to express the dissipation. Starting out with
\R{J13} and \R{J10}
\begin{equation}\label{J11d}
\int p_\lambda(x)\exp(-\beta x)dx\ =\ \exp(-\beta \Delta^{AB}F)\ \geq\
\exp(-\beta W^{AB}_\lambda),
\end{equation}
we obtain
\begin{equation}\label{J11e}
\int p_\lambda(x)\exp\Big(-\beta (x-W^{AB}_\lambda)\Big)dx\ =\
\exp\Big(+\beta (W^{AB}_\lambda-\Delta^{AB}F)\Big)\ \geq\ 1,
\end{equation}
and the dissipation is
\begin{equation}\label{J11f}
D^{AB}_\lambda\ :=\ \beta (W^{AB}_\lambda-\Delta^{AB}F)\ =\
\ln\int p_\lambda(x)\exp\Big(-\beta (x-W^{AB}_\lambda)\Big)dx\ \geq\ 0.
\end{equation}
The phenomenological process work $W^{AB}_\lambda$ is given by \R{J9}$_2$.
\vspace{.3cm}\newlin
A further result due to Jarzynski's equality is obtained for non-stochastic
processes: taking \R{J11}$_3$ into account, \R{cJ10c} yields
\begin{equation}\label{I1}
\Delta^{AB}F\ =\ M^{nst}_\lambda\ =\ W^{ABnst}_\lambda\ \geq\ \Delta^{AB}F
\ \longrightarrow\ \Delta^{AB}F\ =\ W^{ABnst}_\lambda,
\end{equation}
that means, non-stochastic processes are always reversible, if Jarzynski's
equality holds:
\begin{equation}\label{J11g}
\mbox{Jarzynski's equality}\ \longrightarrow
\left\{
\begin{array}{ccc}
\mbox{non-stochastic} & \longrightarrow & \mbox{reversible} \\
\mbox{stochastic} & \longleftarrow & \mbox{irreversible} \\
\end{array}
\right.
\vspace{.3cm}\end{equation
Because reversible "processes" are defined as trajectories in the equilibrium
sub-space \C{WM1,WM2,WM3}, they are idealized objects which do not exist in nature.
Nevertheless, the reversible processes have to be included into the theoretical
framework because they belong to it as a closure of the theory.
According to \R{J11g}, all irreversible processes
create stochasticity in the sense that the work distribution is different from a delta
distribution, but why was this not recognized so far within the framework of
phenomenological Non-equilibrium Thermodynamics? This is due to the fact that most
experiments were carried out on macroscopic systems where the number of repetitions
of the experiment as well as the measurement device is not sensible enough to
discriminate between different work values for a given protocol. Hence, the
different process works appear as being equal according to \R{J4} and \R{J5}
\begin{equation}\label{I1a}
{\cal J}_\lambda\!\!\!\int_A^B \Big[{\cal L}({\lambda,\Theta})-
L({\lambda,\Theta})\Big]\st{\td}{\lambda}(t)dt\ \approx\ 0.
\end{equation}
Consequently, conventional Non-equilibrium
Thermodynamics is a special case of Stochastic Thermodynamics, if Jarzynski's equality
holds and the stochasticity of the irreversible processes is ignored. Another possibility
to ignore stochasticity is to remove the non-regular processes from the theoretical
concept, discussed in the next section.

\subsection{Stochasticity without "violations" ?}

Taking lemma II \R{J11} into account, Jarzynski's equality writes according to \R{fJ10}
\begin{equation}\label{E1}
\int_{x\geq\Delta} p_\lambda(x)\exp(-\beta x)dx
+\int_{x <\Delta} p_\lambda(x)\exp(-\beta x)dx\
=\ \exp(-\beta\Delta^{AB}F).
\end{equation}
Here, the $(x\geq\Delta)$-terms belong to the regular processes.
According to \R{iJ10}$_2$, the non-regular admixture includes all experiments
belonging to protocols $\lambda(t)$ whose process work $x$ is smaller than
the difference $\Delta^{AB}F$ of the free energy. These expe\-ri\-ments are
some times denoted as "violations" of the Second Law. This expression should
be used with care because the phenomenological Second Law is not a statement
valid for stochastic variables.
\vspace{.3cm}\newlin
We now investigate the consequences, if the non-regular processes are eliminated
and only the regular processes are considered. Thus, we set
\begin{equation}\label{E2}
x\ <\ \Delta^{AB}F:\quad p_\lambda(x)\ \doteq\ 0\quad\longrightarrow
\quad \int_{x\geq\Delta} p_\lambda(x)dx\ =\ 1,
\end{equation}
and a naive application of Jarzynski's equality \R{E1} becomes
\begin{equation}\label{E3}
\int_{x\geq\Delta} p_\lambda(x)\Big(\exp(-\beta x)-
\exp(-\beta\Delta^{AB}F)\Big)dx\ =\ 0.
\end{equation}
Because the big bracket in \R{E3} is negative, the probability density is
\begin{equation}\label{E4}
x\ \geq\ \Delta^{AB}F:\quad p_\lambda(x)\ =\ \delta(x-\Delta^{AB}F),
\end{equation}
that means: if Jarzynski's equality holds and
if the non-regular admixture va\-ni\-shes, the stochastic process
works $x={\cal W}^{AB}_{\lambda}$ have for all protocols the same value
$\Delta^{AB}F$. Consequently, the regular processes are non-stochastic
according to \R{cJ10a} and reversible according to \R{J11g}. Thus, we proved the
following statement:
\begin{center}\fbox{
\parbox{10.5cm}{If Jarzynski's equality holds and the non-regular
admixture vanishes, all regular processes are non-stochastic and reversible.}}
\end{center}
or shorter in other words: irreversible processes generate stochasticity and
no stochas\-ti\-ci\-ty without non-regular processes.

\subsection{A special family of probability densities}

In principle, there are many different probability distributions
possible which sa\-tis\-fy the Jarzynski equality. But of course, these probability distributions
are not arbitrary because they have to satify Jarzynski's equality as a constraint.
We now are going to consider a special, but characteristic family for which the
non-regular processes are much less frequent than the regular ones.
For this pupose we start out with Jarzynski's equality \R{J13} written down in the
special decomposition into regular and non-regular processes
\begin{eqnarray}\nonumber
\int_{x\geq\Delta} p_\lambda(x)\exp\Big(-\beta (x-\Delta^{AB}F)\Big)dx +
\int_{x <\Delta} p_\lambda(x)\exp\Big(-\beta (x-\Delta^{AB}F)\Big)dx\ =\
\\ \label{aI3}
=\ 1\ =\ \int_{x\geq\Delta} p_\lambda(x)dx +
\int_{x <\Delta} p_\lambda(x)dx,\hspace{.4cm}
\end{eqnarray}
resulting in
\begin{eqnarray}\nonumber
\int_{x\geq\Delta} p_\lambda(x)
\Big[\exp\Big(-\beta (x-\Delta^{AB}F)\Big)-1\Big]dx\ =\hspace{3cm}
\\ \label{bI3}
=\ \int_{x<\Delta} p_\lambda(x)
\Big[1-\exp\Big(-\beta (x-\Delta^{AB}F)\Big)\Big]dx.
\end{eqnarray}
By changing the integral limits: first integral $x=:\Delta^{AB}F+y$, second integral
\newline$x=:\Delta^{AB}F-y$, $y\geq 0$
\begin{eqnarray}\nonumber
\int_0^\infty p_\lambda\Big(\Delta^{AB}F+y\Big)\Big[\exp(-\beta y)-1\Big]dy\ =\hspace{3cm}
\\ \label{cI3}
=\ \int_0^\infty  p_\lambda\Big(\Delta^{AB}F-y\Big)\Big[1-\exp(\beta y)\Big]dy.
\vspace{.3cm}\end{eqnarray
A special probability density obeying \R{cI3} and consequently also satisfying
Jarzyns\-ki's equality is
\begin{equation}\label{eI3}
 p_\lambda\Big(\Delta^{AB}F-x\Big)\ =\
\frac{1-\exp(-\beta x)}{\exp(\beta x)-1}p_\lambda\Big(\Delta^{AB}F+x\Big),\quad
x>0.
\end{equation}
Because
\begin{equation}\label{fI3}
\frac{1-\exp(-\beta x)}{\exp(\beta x)-1}\ =\ \exp(-\beta x),\quad x>0,
\end{equation}
we obtain for the probability density \R{eI3}
\begin{equation}\label{gI3}
 p_\lambda\Big(\Delta^{AB}F-x\Big)\ =\ \exp(-\beta x)
p_\lambda\Big(\Delta^{AB}F+x\Big),\quad
x>0,
\end{equation}
showing that the probability density belonging to the non-regular
processes is exponentially smaller than that belonging to the
regular ones. This special chosen case motivates to formulate a further
axiom in the next section which holds generally for all probability
densities and all protocols of the Jarzynski process class and not only for the special
chosen case.
\vspace{.3cm}\newlin
The result \R{gI3}is here derived by a phenomenological procedure.
As Crooks found out in Stochastic Thermodynamics
~\cite{CrooksJSP1998, CrooksPRE1999, CrooksPRE2000}, the distribution
belonging to the non-regular processes can be linked to the experiment
by considering the \emph{inverse} protocol of $\lambda(t)$, defined by
$\lambda^\dagger(t) \equiv \lambda(\tau-t)$, $t\in[0,\tau]$, and the inversion of
any magnetic field and rotation. \R{gI3} is then known as a special case
of the so-called \emph{detailed fluctuation theorem}~\cite{SEIF}.

\section{The Non-regular Admixture}

In the last section,
a special class of probability densities was distinguished by an arbitrary choice
which results in the fact that the probability density of the non-regular processes
is smaller than that of the regular processes. This statement
has to be generalized for other process
quantities presupposing Jarzynski's equality.

\subsection{The third axiom}

Inspired by the last section, we postulate the third axiom:
\vspace{.3cm}\newlin
$\blacksquare${\bf Third Axiom:} For all Jarzynski processes belonging
to an arbitrary chosen protocol $\lambda (t)$, the fraction
of the non-regular processes  is not greater than that of the regular ones
\begin{equation}\label{I4}
P^-_\lambda\ \st{\td}{\leq}\ P^+_\lambda
\quad\longrightarrow\quad
P^-_\lambda \leq \frac{1}{2},\quad P^+_\lambda\geq\frac{1}{2}.
\hspace{2.4cm}\blacksquare\hspace{-2.4cm}
\vspace{.3cm}\end{equation
This axiom goes beyond the Jarzynski equality or the Crooks
fluctuation theorem because we posutlate it for every protocol of the Jarzynski process
class.
\vspace{.3cm}\newlin
A first consequence of \R{I4} is according to \R{I3e}
\begin{equation}\label{Z4}
W^{AB}_\lambda-W^-_\lambda\ \geq\ W^+_\lambda -W^{AB}_\lambda
\quad\longrightarrow\quad 2W^{AB}_\lambda\ \geq\  W^+_\lambda + W^-_\lambda.
\end{equation}
This result \R{Z4}$_1$ is pretty clear: the difference between the
phenomenological process work and the  mean value of the process works of
the non-regular admixture is not smaller than that for the regular processes.
This statement is so evident that it could serve as an axiom instead of \R{I4}.
We now go back to the exponential mean values of the process work.
\vspace{.3cm}\newlin
Introducing the abbreviations according to \R{hJ10}$_3$
\begin{equation}\label{Z4a}
a\ :=\ \beta(M^+_\lambda - \Delta^{AB}F)\ \geq\ 0,\qquad
b\ :=\ \beta(\Delta^{AB}F -M^-_\lambda)\ \geq\ 0,
\end{equation}
we obtain from \R{aJ11c} and \R{I4}$_1$
\begin{eqnarray}\nonumber
\frac{P^-}{P^+}\ =\ \frac{1-\exp(-a)}{\exp b -1}\ \leq\ 1&\longrightarrow&
2\ \leq\ \exp b + \exp(-a)
\\ \label{Z4b}
&\longrightarrow& \ln\Big(2-\exp(-a)\Big)\ \leq\ b,
\end{eqnarray}
an inequality connecting $M^+_\lambda$, $M^-_\lambda$ and $ \Delta^{AB}F$
according to the third axiom. Because of
\begin{equation}\label{1Z4a}
2-\exp(-a)\ \leq\ \exp a
\end{equation}
\R{Z4b} results in
\begin{equation}\label{2Z4a}
\ln\Big(2-\exp(-a)\Big)\ =\ b_{min}(a)\ \leq\ a.
\end{equation}
Consequently, if $a$ is given, two kinds of $b$ ($b^-$ and $b^+$)
are possible satisfying \R{Z4b} in accordance with the third axiom \R{I4}$_1$
\begin{equation}\label{3Z4a}
b_{min}(a)\ \leq\ b^-\ \leq\ a\ \leq\ b^+.
\end{equation}
Inserting the abbreviations \R{Z4a}, we obtain inequalities which have
$M^+_\lambda$, $M^-_\lambda$ and $ \Delta^{AB}F$ to satisfy so that
the third axiom \R{I4}$_1$ is true
\begin{equation}\label{4Z4a}
\ln\Big(2-\exp\Big(-\beta(M^+_\lambda-\Delta^{AB}F)\Big)\Big)\ \leq\
\left\{\begin{array}{l}
\beta(\Delta^{AB}F-M^-_\lambda)\leq\beta(M^+_\lambda-\Delta^{AB}F)
\\
\beta(M^+_\lambda-\Delta^{AB}F)\leq\beta(\Delta^{AB}F-M^-_\lambda)\\
\end{array}\right.
\end{equation}
In any case, the third axiom enforces
\begin{equation}\label{5Z4a}
\fbox{$
\ln\Big[2-\exp\Big(-\beta(M^+_\lambda-\Delta^{AB}F)\Big)\Big]\ \leq\
\beta(\Delta^{AB}F-M^-_\lambda)
$},
\end{equation}
an inequality which can be tested by experiments.
\vspace{.3cm}\newlin
The third axiom allows an estimation of the non-regular admixture. Starting
out with \R{Z4b}$_1$, a $y$ exists
\begin{equation}\label{6Z4a}
\frac{P^-}{P^+}\ =\ \frac{1-\exp(-a)}{\exp b -1}\ \leq\ \exp(-y)\ \leq\ 1, \qquad
y\ \geq\ 0,
\end{equation}
and we obtain
\begin{equation}\label{7Z4a}
\ln\frac{\exp b -1}{1-\exp(-a)}\ \geq\ y.
\end{equation}
We now apply the estimation
\begin{equation}\label{8Z4a}
\ln x\ \geq\ 2\frac{x-1}{x+1}, \qquad x\ >\ 0.
\end{equation}
Inserting
\begin{eqnarray}\label{7aZ4a}
x-1\ \longrightarrow\quad
\frac{\exp b -1}{1-\exp(-a)} -1 &=& \frac{\exp b+\exp(-a)-2}{1-\exp(-a)}
\\ \label{7bZ4a}
x+1\ \longrightarrow\quad
\frac{\exp b -1}{1-\exp(-a)}+1 &=& \frac{\exp b-\exp(-a)}{1-\exp(-a)}
\end{eqnarray}
into \R{8Z4a}, we obtain by taking \R{Z4b}$_2$ into consideration
\begin{equation}\label{9Z4a}
\ln\frac{\exp b -1}{1-\exp(-a)}\ \geq\
2\frac{\exp b+\exp(-a)-2}{\exp b-\exp(-a)}\ =:\ y\ \geq\ 0.
\end{equation}
Consequently, \R{6Z4a}$_1$ results in
\begin{eqnarray}\label{10Z4a}
\frac{P^-}{P^+} &\leq&
\exp \Big[2\frac{2-\exp b-\exp(-a)}{\exp b-\exp(-a)}\Big]\ =
\\ \label{11Z4a}
&=& \exp \Big[2\frac{2-\exp\Big(\beta(\Delta^{AB}F-M^-_\lambda)\Big)
-\exp\Big(-\beta(M^+_\lambda-\Delta^{AB}F)\Big)}
{\exp\Big(\beta(\Delta^{AB}F-M^-_\lambda)\Big) -\exp\Big(-\beta(M^+_\lambda-\Delta^{AB}F)\Big)}\Big],\hspace{1.2cm}
\end{eqnarray}
if \R{Z4a} is inserted.
\vspace{.3cm}\newlin
We now investigate the consequences of the third axiom \R{I4} for equilibrium
and in the case of reversible protocols.

\subsection{Equilibrium: the fourth axiom}

Equilibrium and reversible processes are two different concepts which should be
distinguished properly: whereas reversible "processes" as trajectories in the
equilibrium sub-space \C{WM2} are defined by \R{cJ10d},
equilibrium is defined by {\em equilibrium conditions} which by
definition are also valid for reversible processes.
In more detail: taking into account that \R{I3e} is also valid
for reversible processes, we obtain by use of \R{cJ10d}
\begin{equation}\label{Z4j}
1\ =\ \frac{(W^{AB}_{\lambda rev}-W^-_{\lambda rev}) P^-_{\lambda rev}}
{(W^+_{\lambda rev} -W^{AB}_{\lambda rev}) P^+_{\lambda rev}}\ =\
\frac{(\Delta^{AB}F-W^-_{\lambda rev}) P^-_{\lambda rev}}
{(W^+_{\lambda rev} -\Delta^{AB}F) P^+_{\lambda rev}}.
\vspace{.3cm}\end{equation
We now have to postulate the equilibrium condition in agreement with the
definition of reversible processes:
\vspace{.3cm}\newlin
$\blacksquare${\bf Fourth Axiom (Equilibrium):}
In equilibrium {\bf--}and consequently also for reversible processes{\bf--}
regular and non-regular processes are equally frequent\footnote{In Stochastic
Thermodynamics, this axiom runs: Forward and backward path probabilities are
equal in equilibrium \C{Ford}}: according to \R{Z4j},
we obtain
\begin{equation}\label{Z4l}
\frac{P^-_{\lambda eq}}{P^+_{\lambda eq}}\ \st{\td}{=}\ 1
\ \longleftrightarrow\
\frac{\Delta^{AB}F-W^-_{\lambda rev}}{W^+_{\lambda rev} -\Delta^{AB}F}\
=\ 1\ \longleftrightarrow\
2\Delta^{AB}F\ =\ W^+_{\lambda rev}+W^-_{\lambda rev}.
\hspace{.2cm}\blacksquare
\vspace{.3cm}\end{equation
The fourth axiom \R{Z4l}$_1$ points out that in equilibrium the regular
processes are equalized by the non-regular ones. This corresponds to the assumption
in Stochastic Thermodynamics that in equilibrium the global detailed balance is
satisfied and that probability fluxes are balanced and no net currents appear.
 Here, the non-regular processes come into consideration by splitting the process
integrals according to \R{fJ10} and \R{iJ10}.
\vspace{.3cm}\newlin
According to the fourth axiom \R{Z4l}$_1$ and \R{Z4b}, we obtain by taking
\R{Z4a} and \R{2Z4a} into account
\begin{eqnarray}\label{Z5}
\ln\Big[2-\exp(-a_{rev})\Big] &=& b_{rev}\ \leq\ a_{rev},
\\ \label{Z6}
\ln\Big[2-\exp\Big(-\beta(M^+_{\lambda rev}-\Delta^{AB}F)\Big)\Big] &=&
\beta(\Delta^{AB}F-M^-_{\lambda rev}),
\\ \label{Z7}
\frac{b_{rev}}{\beta}\ =\
\Delta^{AB}F -M^-_{\lambda rev}\
&\leq& M^+_{\lambda rev} - \Delta^{AB}F\ =\ \frac{a_{rev}}{\beta},
\end{eqnarray}
and from \R{Z4l}$_3$ and \R{Z7} follows
\begin{equation}\label{Z4p}
2\Delta^{AB}F\ =\ W^+_{\lambda rev}+W^-_{\lambda rev}\ \leq\
M^+_{\lambda rev} + M^-_{\lambda rev}.
\vspace{.3cm}\end{equation
The fourth axiom \R{Z4l}$_1$ enforces the equality in \R{5Z4a}
\begin{equation}\label{Z8}
\ln\Big[2-\exp\Big(-\beta(M^+_{\lambda rev}-\Delta^{AB}F)\Big)\Big]\ =\
\beta(\Delta^{AB}F-M^-_{\lambda rev})
\end{equation}
which also can be experimentally tested, if the corresponding protocol is reversible,
that means, if the protocol of the Jarzynski process \R{J1} to \R{J3} is suffiently slow $\lambda(\alpha t),\ \alpha\rightarrow 0$, for approximating reversibility.

\section{Summary}

\begin{itemize}

\item Processes of a closed discrete system between to fixed equilibrium states
controlled by a heat reservoir and divided into two parts --the first part with
power exchange caused by fixed initial and fixed final work variables, the se\-cond
one as relaxation by constant work variables to the final equilibrium state--
constitute the Jarzynski process class (\ref{J1}) to (\ref{J3}) and (\ref{J5a}).

\item Experimental fact is that the process work is different for several identical
Jarzynski processes enforcing to treat the process work as a stochastic variable
whose different values establish the domain of a probability density \R{J5}.

\item The processes of the Jarzynski process class
fall into two branches: regular processes of non-negative process dissipation
(\ref{iJ10})$_1$ and non-regular processes of  negative process dissipation
(\ref{iJ10})$_2$.

\item Although the process dissipation is negative for non-regular
processes, the mean value of all Jarzynski processes --the phenomenological
dissipation-- is not negative (\ref{N32})$_2$.

\item Two phenomenological lemmata\newline
I) the exponential mean process work obeys the Second Law and\newline
II) the non-regular processes have an extent as great as lemma I) allows,\newline
are the phenomenological back-ground of the basic axiom
\begin{center}\fbox{
\parbox[t]{7.9cm}{
The mean value of the exponential process work is independent of the Jarzynski
process class.}}
\end{center}
This basic axiom implements Jarzynski's inequality straightforward, \R{J11} and \R{J13}.

\item If Jarzynski's equality holds, non-stochastic processes are always reversible.

\item If Jarzynski's equality holds, irreversible processes generate stochasticity.

\item Beyond Jarzynski's equality: the extent of the non-regular processes
is not greater than that of the regular ones \R{I4}$_1$.

\item Beyond Jarzynski's equality: in equilibrium {\bf--}and consequently for
reversible "processes"{\bf--} the fractions of non-regular and regular
processes are equal, \R{Z4l}$_1$.

\end{itemize}

\section{Discussion}

The tools of Stochastic Thermodynamics are based on Statistical
Mechanics, whereas Non-equilibrium Thermodynamics is a purely phenomenological
theory. Both theories describe phenomenological processes, Stochastic Thermodynamics
by mean values over its statistical back-ground and Non-equilibrium Thermodynamics
by phenomenological Laws. For a special class of processes {\bf--}the Jarzynski process
class{\bf--} an equality, called Jarzynski's equality, was derived in 1997 by a statistical
procedure containing a probability density stemming from the statistical back-ground.
Now the question arises: can Jarzynski's equality be derived phenomenologically,
if Non-equilibrium Thermodynamics is equipped with stochastic processes ?
\vspace{.3cm}\newlin
Considering the Jarzynski process class, an experimental fact is that the process
work is a stochastic variable, that means, performing a Jarzynski process identically
repeated, the process work fluctuates, and a probability density is experimentally
generated. What are now the phenomenological axioms which this probability
density has to obey, so that Jarzynski's equality is valid ? The answer {\bf--}based
on two phenomenological axioms{\bf--} is easy: the mean value of the stochastic
exponential process work is the same for all Jarzynski processes.
\vspace{.3cm}\newlin
The Jarzynski processes which are identically repeated can be split into regular
and non-regular ones. By definition, the process dissipation of the non-regular
processes is smaller than the corresponding difference of the free energy.
By contrast,
the process dissipation of the regular processes is not smaller than the free energy
difference as it is valid for all non-stochastic processes.
The non-regular processes are sometimes confusingly called
``violations'' of the second law of thermodynamics.
\vspace{.3cm}\newlin
Finally statement: Non-equilibrium Thermodynamics of the Jarzynski process class
can be extended by stochastic processes satisfying Jarzynski's equality which obeys
phenomenological axioms.
\vspace{.5cm}\newline
{\bf Acknowledgement} The co-working of Dr. Philipp Strasberg, Institut f\"ur
Theoretische Physik, TU Berlin, is gratefully acknowleged. Numerous helpful and
sometimes controversal discussions with him makes my phenomenological view
concerning Jarzynski's equality more strict. Several parts of the paper are formulated
by him. All references concerning Stochastic Thermodynamics are generated by
Dr. Strasberg. The paper did not come into being without his help.

redpajama_set_name: RedPajamaArXiv


The Ghana Standards on Accessibility Design (GSAD) document provides specification on how buildings, roads and recreational centers should be designed so as to make them accessible to all, especially the PWDs.
Deputy Minister for Gender, Children and Social Protection, Della Sowah, said addressing crucial challenges that affected the people was at the core of Ghana's sustainable development.
Disability experts say though PWDs have a legal entitlement and a right to access mainstream opportunities, services, programs, public events and infrastructure, many in Ghana experience profound exclusion.


This entry was posted on 12-22-2018 at 19:00 and is filed under General Information, Just For Fun!, Reviews, Uncategorized with tags A Christmas Story, authors, Christmas, crime fiction, Dead Holiday, death is only a click away, ebooks, holiday, House of Misery, Jeffrey Martin, Mark Blankenship, serial killer, serial killers, short stories, Smashwords, thriller, writing.	You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


This and 3 other objects are a part of a set whose first object is His Royal Highness Coffee Service.
This is a Creamer. It is dated ca. 1940. Its medium is triple electro-plated nickel silver, turned ebony. It is a part of the Product Design and Decorative Arts department.


text:
require 'dm-is-localizable/backend'

module DataMapper
  module I18n
    module Backend

      class Default
        include Backend::API

        DEFAULT_LOCALE_TAG                  = 'en-US'
        DEFAULT_LOCALE_REPOSITORY_NAME      = :default
        DEFAULT_LOCALE_STORAGE_NAME         = 'locales'
        DEFAULT_TRANSLATION_MODEL_NAMESPACE = nil
        ACCEPTS_NESTED_ATTRIBUTES           = true

        # RFC 4646/47
        DEFAULT_LOCALE_TAG_FORMAT = %r{\A(?:
          ([a-z]{2,3}(?:(?:-[a-z]{3}){0,3})?|[a-z]{4}|[a-z]{5,8}) # language
          (?:-([a-z]{4}))?                                        # script
          (?:-([a-z]{2}|\d{3}))?                                  # region
          (?:-([0-9a-z]{5,8}|\d[0-9a-z]{3}))*                     # variant
          (?:-([0-9a-wyz](?:-[0-9a-z]{2,8})+))*                   # extension
          (?:-(x(?:-[0-9a-z]{1,8})+))?|                           # privateuse subtag
          (x(?:-[0-9a-z]{1,8})+)|                                 # privateuse tag
          /* ([a-z]{1,3}(?:-[0-9a-z]{2,8}){1,2}) */               # grandfathered
          )\z}xi

        attr_accessor :default_locale_tag
        attr_accessor :locale_tag_format
        attr_accessor :locale_repository_name
        attr_accessor :locale_storage_name
        attr_accessor :translation_model_namespace
        attr_writer   :accepts_nested_attributes

        def initialize
          @default_locale_tag          = DEFAULT_LOCALE_TAG
          @locale_repository_name      = DEFAULT_LOCALE_REPOSITORY_NAME
          @locale_storage_name         = DEFAULT_LOCALE_STORAGE_NAME
          @locale_tag_format           = DEFAULT_LOCALE_TAG_FORMAT
          @translation_model_namespace = DEFAULT_TRANSLATION_MODEL_NAMESPACE
          @accepts_nested_attributes   = ACCEPTS_NESTED_ATTRIBUTES
        end

        def normalized_locale_tag(tag)
          tag # noop, overwrite for specific behavior
        end

        def available_locales
          DataMapper::I18n::Locale.all
        end

        def accepts_nested_attributes?
          @accepts_nested_attributes
        end
      end # class Default

    end # module Backend
  end # module I18n
end # module DataMapper

redpajama_set_name: RedPajamaGithub


text:
Home » Successful liftoff for soil moisture mission
Successful liftoff for soil moisture mission
02/03/15 2:25 PM By Sara Wyant
WASHINGTON, Feb. 3, 2015 – After two previous failed attempts, NASA successfully launched its first Earth satellite on Jan. 31 – designed to detect soil moisture, improve climate and weather forecasts, track yields and help scientists more accurately account for how much carbon plants are removing from Earth's atmosphere each year.
The Soil Moisture Active Passive (SMAP) observatory, is on a three-year mission aiming to expand our understanding of a key component of the Earth system that links the water, energy and carbon cycles driving our living planet. SMAP's combined radar and radiometer instruments will peer into the top 2 inches of soil, through clouds and moderate vegetation cover, day and night, to produce the highest-resolution, most accurate soil moisture maps ever obtained from space. The mission will map global soil moisture with about 5.6-mile (9-kilometer) resolution.
"SMAP will improve the daily lives of people around the world," said Simon Yueh, SMAP project scientist at NASA's Jet Propulsion Laboratory (JPL) in Pasadena, California in a statement. "Soil moisture data from SMAP has the potential to significantly improve the accuracy of short-term weather forecasts and reduce the uncertainty of long-term projections of how climate change will impact Earth's water cycle."
To Mexico on a renewable fuels mission
The best USDA trade mission ever?
Iowa farmers brace for another year of belt-tightening, survey finds
The SMAP Applications Working Group is collaborating with 45 "early adopters" to test and integrate the mission's data products into different applications. For example, since 2005, USDA's Risk Management Agency's Strategic Data Acquisition and Analysis (SDAA) team, has participated in the working group established by the U.S. Office of Science and Technology Policy in the area of agricultural land imaging as part of the Landsat Data Continuity Mission. The SDAA has been a participating member of the Landsat Science Team since 2012 for the purpose advising policy and championing the needs of USDA, crop insurance, and the agricultural community for the current and all future Landsat sensors.
The first release of SMAP soil moisture data products is expected within nine months. Fully validated science data are expected to be released within 15 months. For more information, see the SMAP website.
For more news, go to www.agri-pulse.com.
Sara Wyant
Editor/Publisher


We had another great day at the WBSDS at Newbury today! It was a big show with 9 simultaneous rings running, but as it was quite local there were lots of friendly faces there to make it a fun day!
I was given lots of advice by my experienced friends on how to deal with Pepper's contacts in the ring, and having taken their advice I feel that I will be less daunted by KC shows in future.
Our first run was Combined Agility - I went and spoke to the judge while we were walking the course and asked her permission to put Pepper back on the contact if she didn't stop. She was really friendly and encouraging and said that it would be fine and she was grateful that I had asked. When it came to our turn I found that I was less stressed as I had a strategy and knew that the judge was OK with it! Pepper stopped perfectly on all of the contacts but unfortunately we had a pole down! I shouldn't be upset about the pole as my aim was to get good contacts and she delivered on that!
Second run was Graded Agility - I spoke to the judge again before we started running - he was OK but not quite as welcoming as the previous judge. We ran the course and she leapt off the A frame - I tried to get her back in a 2o2o position but she was being a little tinker! Eventually she got in position and the judge said I could carry on but that was my last chance! I really should have just left the ring, but carried on and she didn't stop on the dog walk - she didn't miss the contact but she did leap so I really should have stopped her again!
Next was Jumping - a nice course and Pepper was flying - right up to the point when she went over a jump and got distracted by the photographer sitting in front of her! She had to go over and investigate and when I got her back she took the jump the wrong way! I shouldn't get annoyed with the photographer - she needs to get used to distractions and it was my fault for not keeping her attention in the first place!
Final round was Combined Jumping - a course with a difficult start that I was trying to find an easy way to handle! In the end I decided to handle it in the way we would in training which meant having to do 12 weaves on the right hand side and a rear cross on the next jump. She never ceases to amaze me - perfect weaves and lovely rear cross - she flew round the rest of the course clear and we were placed 2nd! A great way to end the day and another rosette to add to the collection.
Ruby helping to model our rosette!
Pepper and I went to the UKA show at Broadlands today!
As usual it was a mixed day with only one clear round, our first run of the day in the Beginners Jumping. We were placed 2nd and received another blue rosette to add to our collection.
She did a beautiful run and I got my hopes up for great things for the rest of the day but sadly it was all downhill from there!
We need 2 points in Steeplechase to go up to the Senior level, and I was really hopeful that with two Steeplechase runs on the day we would be able to get those points - unfortunately not! We were E'd for the first one as Pepper got so excited she released herself before I was ready and it completely threw me. She missed the 3rd jump but when I got her back on track she flew round the rest of it.
We then had a painful few hours waiting for our next runs. We had decided to enter the pairs with our friends June and Kola and as time approached when we were due to run that, we also needed to be at the Novice Steeplechase and Beginners Agility! Typical!
It was fun doing the pairs - June and Kola ran first doing the Agility section - they did well but unfortunately had a problem with the weaves. We handed over the baton and Pepper and I headed off for the Jumping section. The opening was really the hardest part and we managed to navigate it exactly as I planned, but as we headed round the course Pepper dived off into a tunnel so a real shame, but at least we'll feel less daunted to try this again in the future.
We ran straight over to the Steeplechase and I should really have taken a bit of time to settle down and remind myself of the course, but I was distracted worrying about getting it done so that we could get to the agility. Unfortunately Pep knocked the first jump, very unlike her, but after that she did really well! All my fault for rushing!
Finally we made it to the Agility. I had really hoped this would be earlier in the day and had resolved to run it NFC so that I could train the contacts. By the time I got to the agility I was frazzled so just ran the course - Pepper leapt off every contact and took the wrong jump so a big fat E to end the day! I've really got to get my act together with the contacts as it's really holding us back! Back to the drawing board!
Ruby and Pepper don't tend to choose to sit together, they prefer their own space, but when they do find themselves in close proximity it never ceases to amaze me how often they sit in exactly the same position!


text:
Zimbabwe: Mugabe Slammed for Wasting Resources
The million dollar party celebrating Robert Mugabe's 90th birthday has been slammed as a disturbing sign of greed and waste of resources, in a country where the majority are struggling to feed their families.
The ageing ZANU PF leader was honoured with a lavish party at Rudhaka Stadium in Marondera on Sunday, attended by an estimated 10,000 guests that included musicians and athletes as well as political and religious figures.
The Sunday Mail newspaper said a record 90 beasts were slaughtered to feed the guests and dozens more live animals were donated to Mugabe by the ZANU PF provinces. A huge cake was on display during festivities, where Mugabe claimed to be in good health, saying he felt "energetic like a boy of nine".
Former Zimbabwean diplomat and commentator, Clifford Mashiri, disagreed, saying Mugabe was wheeled around the party on the back of a truck instead of walking to greet guests and supporters. This was unlike previous years when he made the rounds on his two feet.
Mashiri said the ageing leader had also exposed his frailty during an interview aired on state television Thursday, when he was seen struggling to speak and rambled on for long periods after each question.
Mashiri also blasted Mugabe and ZANU PF for spending so much on a birthday party, while the country was still dealing with the devastating effects of recent floods and many have nothing to eat. Mashiri said the event was nothing for them to brag about.
"Whether it was him who pooled together the money or it was his inner circle, it's a waste of resources at a time when ordinary people are suffering, including civil servants to whom he made empty promises of salary increases several times. It is unfortunate Mugabe doesn't show any sensitivity to that," Mashiri explained.
Ahead of the stadium celebrations, Mugabe attended a party organised by his office staff at State House Saturday. This was soon after returning from Singapore where he allegedly had an operation to remove a cataract in his left eye, although most commentators felt it was more than that.
The highlight of the office party was reportedly a gigantic birthday gift presented to him by the staff, in the form of a 1.2 ton chair carved in stone and adorned with gold, diamonds, ivory and several animal skins.
Dubbed the "Simbahwe chair", this throne took two years to make and was reportedly a labour of love to honour Mugabe, by three artists who demanded no fee. Mashiri questioned where the gold, diamonds and ivory came from and how his staff could have afforded such an extravagant gift.
"It's most unfortunate there is a cult of personality in Mugabe and as such some people are worshipping him. This is also a serious contradiction in the fight to stop the poaching and abusive killing of animals in the country," Mashiri said.
The party did not disappoint in drama either. In what Mashiri described as "sucking up", Vice President Joice Mujuru reportedly praised Mugabe as an icon then turned to his daughter Bona and said: "Bona, you are not the first daughter, I am the first daughter."
Statement by Minister Donald Grant and Penny Vinjevold ? Head of Education: WCED reaction to court judgment
Namibia: Passion Killings Claimed 36 Lives


text:
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pk.contender.earmouse

Class ModuleStats

java.lang.Object

pk.contender.earmouse.ModuleStats

public class ModuleStats
extends java.lang.Object

Keeps track of all the answers a user has given to exercises of a specific Module.

Provides methods for entering answers and obtaining useful information about this data and
 for reading and writing recorded statistical data to local storage.

Author:

Paul Klinkenberg <pklinken.development@gmail.com>

Field Summary

Fields

Modifier and Type
Field and Description

private java.util.List<ModuleAnswer>
moduleAnswerList

All the recorded answers

private java.io.File
statsFile

Reference to local storage

Constructor Summary

Constructors

Constructor and Description

ModuleStats(Context context,
           int id)

Construct an instance to be associated with the given Module Module.id, attempts to load from local storage if the file
 exists, otherwise creates an fresh instance.

Method Summary

Methods

Modifier and Type
Method and Description

void
addAnswer(int exerciseIndex,
         boolean result)

Add an answer to moduleAnswerList

int
calculateSuccessRate()

Calculates the success rate of the registered answers.

int
exerciseCount(int exerciseIndex)

Returns the number of times exerciseIndex is registered.

int
exercisesCompleted()

Returns the total amount of exercises completed

int
exerciseSuccessRate(int exerciseIndex)

Returns the success rate in % of a particular exerciseIndex, returns 0 if no exercises were found

private void
initModuleStatsFromJson(java.io.FileReader fr)

Reads an existing ModuleStats instance from a JSON file

boolean
purgeStats()

Delete all statistical data for this instance.

void
saveModuleStats()

Save this instance to local storage, if the file doesn't exist, creates it, otherwise overwrites existing data.

private void
saveModuleStatsToJson(java.io.FileWriter fw)

Save moduleAnswerList to the given FileWriter as a JSON stream

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

moduleAnswerList

private java.util.List<ModuleAnswer> moduleAnswerList

All the recorded answers

statsFile

private final java.io.File statsFile

Reference to local storage

Constructor Detail

ModuleStats

public ModuleStats(Context context,
           int id)

Construct an instance to be associated with the given Module Module.id, attempts to load from local storage if the file
 exists, otherwise creates an fresh instance.

Parameters:

context - The application context, used for file operations

id - the ID of the Module this instance will be associated with

Method Detail

initModuleStatsFromJson

private void initModuleStatsFromJson(java.io.FileReader fr)
                              throws java.io.IOException

Reads an existing ModuleStats instance from a JSON file

Parameters:

fr - The FileReader to read from

Throws:

java.io.IOException

saveModuleStatsToJson

private void saveModuleStatsToJson(java.io.FileWriter fw)
                            throws java.io.IOException

Save moduleAnswerList to the given FileWriter as a JSON stream

Parameters:

fw - The FileWriter to save to

Throws:

java.io.IOException

saveModuleStats

public void saveModuleStats()

Save this instance to local storage, if the file doesn't exist, creates it, otherwise overwrites existing data.

addAnswer

public void addAnswer(int exerciseIndex,
             boolean result)

Add an answer to moduleAnswerList

Parameters:

exerciseIndex - The index of the exercise the answer refers to

result - The correctness of the answer

calculateSuccessRate

public int calculateSuccessRate()

Calculates the success rate of the registered answers.

Returns:

The percentage of registered answers that is correct or -1 if there is no available data.

exercisesCompleted

public int exercisesCompleted()

Returns the total amount of exercises completed

An exercise is completed when the user
 gave the correct answer, even if there were wrong answers before that.
 Since one moves on to the next exercise after giving a correct answer, the number of exercises completed
 is the same as the number of correct answers given.

Returns:

the number of exercises in this module that were answered correctly.

exerciseSuccessRate

public int exerciseSuccessRate(int exerciseIndex)

Returns the success rate in % of a particular exerciseIndex, returns 0 if no exercises were found

Parameters:

exerciseIndex - the individual exercise whose success rate to return

Returns:

the success rate in % of the exercise with index exerciseIndex, or 0 if no records were found.

exerciseCount

public int exerciseCount(int exerciseIndex)

Returns the number of times exerciseIndex is registered.

Parameters:

exerciseIndex -

Returns:

the number of times exerciseIndex is registered in moduleAnswerList

purgeStats

public boolean purgeStats()

Delete all statistical data for this instance.

Returns:

True on success, false otherwise.

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and Chapter 3 in the actual book.
to my life. Let's go cherry picking…shall we?
I'll give you one more then call it a wrap.
"God quickens you. He makes you alive. He plants creative ideas in your mind. He creates desires in your heart – not worldly ones, but godly ones.
Marinate on that for awhile. Let it soak all the way in.
Try to fully understand it.
You have two choices…ignore it or find a way to apply it.
I'll apply it… thank you very much.
My life is my testimony. Great things are on the horizon.
I'm just getting started. Watch me now….


The dreaded green line can appear vertically anywhere on the display. Affected users see lines of different brightness levels along the edge or middle of the screen that persist after reboot or using different apps. This is a hardware issue, and Apple is replacing the few iPhone X devices exhibiting this problem. Take advantage of Apple's legendary customer service and contact the company or visit the Apple Store for a warranty claim.
Another rare problem involves a static or buzzing noise coming from the front earpiece speaker when playing audio at high volume. The crackle happens with various types of audio. While a similar problem on the iPhone 8 was fixed via iOS software update, Apple has been replacing iPhone X units affected by this speaker issue. Contact Apple or visit the Apple Store for a warranty claim.
Apple has promised an upcoming software fix for an issue where the iPhone X becomes unresponsive when exposed to cold weather. Some users noticed the touch screen stops responding for a couple of seconds after suddenly exposing the device to extreme cold. Apple issued a statement acknowledging the quirk and explained that a future software update will prevent this behavior.
Why are colors on the iPhone X screen so warm? iPhone X users might be caught off guard by a yellow or orange tint, however this is actually by design. The True Tone display automatically adjusts display colors based on the surrounding environment. For those who prefer a bright blue screen, this feature can be turned off. Click here for instructions on how to disable True Tone on the iPhone X.
Most iPhone X users are rightfully concerned about drop damage, as the device has an all-glass front and back. Add the components for the front facing TrueDepth camera to the mix, and any impact could result in problems. Reports of iPhone X devices being dropped and surviving without visible cracks in the glass are reassuring, until its discovered that Face ID is broken. These delicate parts are sensitive to impact, which is just one reason why AppleCare+ for iPhone X is a good idea. Interested owners can purchase an AppleCare+ plan to protect against accidental damage within 60 days of buying an iPhone X.
This biggest confusion over people is Face ID.


Can Corporations Receive a Dollor-for-Dollar Arizona Tax Credit?
Yes, any eligible business with an Arizona tax liability can donate to the Institute for Better Education to benefit either the Low-Income Scholarship Program and/or the Disabled/Displaced Scholarship Program. These programs help children who are low-income, disabled, or have been a ward of the state (displaced) to attend the private school that best fits their needs, all while giving the donating corporation a dollar-for-dollar tax credit! Your corporation can even recommend a school to benefit from these scholarships!
Eligible businesses are C-Corporations, S-Corporations, Insurance companies who pay taxes on their premiums collected in Arizona, and LLCs that file taxes as an S-Corporation.
The donation must be made to an STO (School Tuition Organization). Institute for Better Education (IBE) is a certified STO by the Arizona Department of Revenue and a federal non-profit organization (501(c)(3)).
The corporation can designate their donation for either the Low-Income Corporate Scholarship Program and/or the Disabled/Displaced Corporate Scholarship Program.
The minimum contribution amount is $5,000 for S-corporations or LLCs that file as an S-corp. There is no minimum for C-corporations or insurance companies. There is no maximum contribution, but the state sets a donation cap for all contributions received for both programs each year. The Low-Income Corporate Tax Credit donation limit increases 20% each year. For fiscal year 2018/19, the limit is $89,161,006. The Disabled/Displaced Corporate Tax Credit donation limit is constant each year at $5 million dollars.
The donation approval process (see below) starts July 2, 2018 and is on a first come/first serve basis until the tax credit limit is reached. In past years, both tax credit programs have reached their limit quickly.
The corporation may carry forward any unused tax credits for 5 years.
Determine your corporation's donation amount based on your state income tax liability.
Complete the Corporate Donation Application and email to the Corporate Program Coordinator, Wendi Hauser, at wendi@ibescholarships.org.
IBE will submit your donation request to the Arizona Department of Revenue (ADOR) for pre-approval.
The ADOR will respond with an approval/denial within a few business days. IBE will then notify the corporation and the corporation has twenty days from the date of approval to provide IBE with the donation amount approved. IBE then notifies the ADOR that the donation is complete.
The Corporate Tax Credit donation amount can then be claimed when filing your company's taxes.
The Corporate Tax Credit is available to an S-Corporation as defined in Section 1361 of the Internal Revenue Code.
LLCs taxed as S-Corporations for federal and state purposes are eligible to donate to IBE for a Corporate Tax Credit.
The S-Corporation may make an irrevocable election to pass the credit through to its individual shareholders instead of claiming the credit against income taxed at the corporate level (on the 120S).
The S-Corporation does not need to have a liability in order to pass the credit through to its shareholders. The shareholders will apply their share of the tax credit against their individual income tax liability.
Any pro rata shares that would be distributed to non-eligible partners (individual partners in a partnership) or estates or trusts (shareholders in an S-Corporation) are lost.
The donation to IBE must be made by the S-Corporation in order to be eligible for the tax credit, not the individual taxpayer.
For Low-Income Corporate Tax Credit donations - Form 335 and Form 335-S will be completed by the S-Corporations and submitted with their 120S. A copy of Form 335-S is also to be forwarded to the shareholders.
For Disabled/Displaced Corporate Tax Credit donations – Form 341 and Form 341-S will be completed by the S-Corporation and submitted with their 120S. A copy of Form 341-S is also to be forwarded to the shareholders.
There is no limit on the Corporate Tax Credit claimed by the individual taxpayer. If the taxpayer's share of the S-corporation credit is $3,000 then their credit will be $3,000.
Form 335-S or 341-S will be completed by the S-Corporation and forwarded to the shareholder. The individual taxpayer will claim their share of the credit on Form 341-I.
The Corporate Tax Credit does not impact the individual taxpayer's ability to claim the Original or PLUS tax credit.
The individual taxpayer may carry forward any unused credit for 5 years.


Amazing Projects is a Sydney building company that focuses on giving our customers the perfection they want.
Building a new home usually represents the single biggest investment of your life. It can be that once-in-a-lifetime project—to build a special place for you and your family to live, to entertain and to relax. It turns your plans into your dream home.
To build the home of your dreams you need to be able to trust your builder to take your ideas and make them work. Amazing Projects will build your home to match your needs. Being amazing is about giving our customers the perfection they want.
Amazing Projects is a dedicated and professional team who can help and guide you through design, planning applications, and construction. We will also make sure that your home is painted, cleaned and ready for you to move in.
We will design your home and assist you or manage the development application and planning processes.
We do work in all parts of Sydney. Contact us now to discuss your project – it is a great way to get started. Our consultations and quotes are free with no obligations. If you want an Amazing Project call us or contact us by email.


We offer outstanding levels of care and support for all residents at our magnificent care homes in Norfolk, where services range from residential and nursing care for older people through to care of people with learning disabilities or first class dementia care in one of our superb 'Memory Lane' communities'. All of our Norfolk nursing homes provide exceptional facilities combined with an excellent variety of daily activities to choose from, including days out to the local zoo and unique music therapy classes. Programmes are planned to ensure that residents are engaged, individually and in groups, so that they continue to get the most from life.
In addition to first class nursing and residential care, support staff in our residential homes provide the highest quality cooking and hospitality, helping to guarantee an enjoyable stay with us.


My Water is Hard?My Water is Hard?
Fresh Water Ind. has been serving Newmarket, Aurora and surrounding communities for over 25 years. We are family owned and operated and are the oldest GTA full service water company.


text:
Q: Making a Laser which stops at objects in spritekit Is it possible to make a laserbeam in ios spritekit that will change length depending on the angle and if there is  any objects in the way? its hard to explain but check the image.

A: This is possible. 'Line Of Sight' type of techniques are very well explained in Sprite Kit Programming Guide: Simulating Physics chapter.
Listing 8-10  Casting a ray from the center of the scene is a code snippet that is on the right track.
Provided the objects that the laser will hit all have physics bodies then you should be able to use enumerateBodiesAlongRayStart:end:usingBlock: from SKPhysicsWorld class. Think of this method as firing an invisible ray first, so you can get the end point for the laser beam you wish to draw.

redpajama_set_name: RedPajamaStackExchange


A ruling by The European Court of Human Rights means that companies NOW have the right to monitor the private online messages of their staff.
Will this ruling change how you use private messaging? Click here to take our poll.


This entry was posted in Religion and tagged Catholics, Christianity, Religion, Religious Doctrine, Stupidity by Les. Bookmark the permalink.
Standards on the host & wine are possibly for the best, twinkles & bourbon would be entirely too weird.
I dunno. Twinkies and bourbon could possibly lure me back into church attendance.
Pity Heinlein didn't think of that for the Fosterites in "Stranger in a Strange Land".
The methodists, and I believe most other Protestant denominations, don't believe in actual transubstantiation. For them its figurative. Its only the catholics who think they are getting actual bits of Jesus.


text:
Q: Ado.net, restaurar la base de datos maestra estoy haciendo un programa para restaurar backups de sql server desde visual basic , pero al momento de restaurar me sale un error , gracias por la ayuda

Este es el código para restaurar , la conexion la hago en modulo aparte

A: para restaurar una base de datos en modo de usuario único, lo puedes hacer de dos formas:
- manual (a través del 'Sql Management Studio'): click en el botón derecho sobre la Base de Datos en cuestión, eliges 'Propiedades' y en la columna de la izquierda, 'Opciones'. Verás en la ventana de la derecha la línea de 'Restringir Acceso'. Ahí debes elegir 'SINGLE_USER'
- mediante programación : en el string para instalar el BackUp debes incluir la condición 'SET SINGLE_USER'
Saludos