text:
\section{Section title}

\abstract
We give a partial answer to the question whether the Schr\"{o}dinger equation can be derived from the Newtonian mechanics of a particle in a potential subject to a random force. We show that the fluctuations around the classical motion of a one dimensional harmonic oscillator subject to a random force can be described by the Schr\"{o}dinger equation for a period of time depending on the frequency and the energy of the oscillator. We achieve this by deriving the postulates of Nelson's stochastic formulation of quantum mechanics for a random force depending on a small parameter. We show that the same result applies to small potential perturbations around the harmonic oscillator as long as the total potential preserves the periodicity of motion with a small shift in frequency. We also show that the noise spectrum can be chosen to obtain the result for all oscillator frequencies for fixed mass. We discuss heuristics to generalize the result for a particle in one dimension in a potential where the motion can be described using action-angle variables.

\section{Introduction}

\indent

Despite the successes of quantum theory there remains the solution of the measurement problem and its unification with general relativity. Much effort has been spent assuming quantum mechanics is fundamental and applies to smallest and largest possible scales. At the smallest scales where quantum effects in gravity should take place, although we can form mathematically consistent quantum gravity theories, we have no experimental guidance yet and have extreme conceptual difficulties making sense of a quantum description of space-time. At large scales we observe that nature behaves classically which is not possible to understand within the standard postulates of quantum mechanics since those do no pretense to explain the measurement processes and the quantum to classical transition in a fundamental fashion. Perhaps the best way we know today to solve the measurement problem is to introduce ad hoc spontaneous collapse theories.

Regardless of the successes and failures of quantum mechanics, in this paper we would like to retain the Newtonian-Einsteinian notions. We try to answer the simplest possible question: Can a single non-relativistic quantum particle in a potential in one dimension can be described by Newtonian mechanics? A lot of effort has been put in deriving hidden variable theories but the answer to this question is still missing: there is no proof that it is impossible and there is no proof that all quantum effects described by a general solution of the Schr\"{o}dinger equation can be accounted classically. Indeed the latter seems almost impossible since it is very hard to imagine how a classical particle would exhibit quantum interference. We do not dogmatically believe that nature is fundamentally classical or quantum but find that it is important to explore the boundaries of existing theories. The least we can expect by answering such a question is a reformulation of quantum theory irrespective of whether the Newtonian description is physically fundamental or not.

Here we set aside the question of many particles which would involve Bell's theorem. It is widely believed that it is impossible to have a local Newtonian-Ensteinian explanation of entangled states. Although the mostly forgotten rigorous analysis of Bell inequalities by Nelson\cite{Nelson86, Nelson88, Schulz} distinguishes between passive and active locality and makes it possible for stochastic field theories to be in principle able to explain entangled states. We believe that even the explanation of superposition states for a single particle presents an enormous challenge and we do not yet worry about many particle states.

In this paper we are not able to answer the general question nor we can explain superposition states but we give a partial answer. We show that when a classical harmonic oscillator is subject to a specific random force, its fluctuating motion around the classical trajectory can be described by the Schr\"{o}dinger equation for a range of values of frequency, energy and time. We show that this generalizes to potential perturbations around the harmonic oscillator as long as the periodic motion is preserved with a small shift in frequency. We also show that the noise spectrum can be chosen to obtain the result for all oscillator frequencies for fixed mass. We further discuss a way to generalize the result to potentials admitting action-angle variables. The mathematical tools that we use to achieve this are the method of stochastic averaging and Nelson's formulation of Schr\"{o}dinger equation in terms of stochastic particle trajectories.

Before the significant discovery of Nelson\cite{Nelson66} that it is possible to give a stochastic account of Schr\"{o}dinger equation, it had been widely believed this was impossible because diffusions are dissipative but there is a notion of conserved energy in the quantum mechanical evolution. Nelson showed that it is possible to construct conservative diffusions which are equivalent to the Schr\"{o}dinger equation. However Nelson's formulation is not Newtonian: the particle is subject to random motion in its position space contrary to that the random effects should appear as forces in a Newtonian theory. Here we attempt to answer whether Nelson's formulation can be derived from a phase space stochastic process where the random term appears as a force. Indeed this is the first and perhaps the most important of the open problems stated in his book\cite{Nelson2}. This question was most openly investigated by Smolin\cite{Smolin} who gave sufficient conditions for a cosmological theory to reduce to Nelson's theory.

The same type of questions have been asked and were tried to be answered mostly by the stochastic electrodynamics community\cite{Pena}. There one assumes that an electrically charged particle is coupled to a background stochastic electric field with a specific spectrum and is also subject to electromagnetic radiation reaction. One is able to show that in equilibrium one can choose the spectrum to match with all the energy eigenstates of a harmonic oscillator. However there lacks a universal spectrum working for all energy eigenstates and the superposition states seem to be elusive. There are two main lines of attempted derivations of Schr\"{o}dinger equation both running into difficulties. In the first approach by integrating out the velocity evolution one tries to reduce to a position space process. Schr\"{o}dinger equation holds if one can neglect certain radiative terms in the equations but there is no justification for how the system reaches a state such that those terms can be neglected and how long the system stays in that state such that the approximation is valid. In the second approach it is shown that if one assumes that there are multiple ergodic energy states then stochastic variables can be described by matrix variables and one obtains Heisenberg's theory. However it seems very difficult to construct a stochastic system exhibiting classical multiple ergodic energy states which matches with the quantum energy eigenvalues and to describe the transition between energy eigenstates in such a framework. Perhaps the most important objection against stochastic electrodynamics is that it only applies to charged particles and lacks universality. However we think the questions asked and attempted to be answered in this model are valuable and give insights for further developments.

The paper is organized as follows. In Section 2 we briefly review Ito calculus and stochastic differential equations-the mathematical framework that we use in the rest of the paper. In Section 3 we give an account of Nelson's stochastic formulation of the Schr\"{o}dinger equation for a non-relativistic particle in one dimension. We introduce the two postulates of Nelson which are equivalent to the Schr\"{o}dinger equation in Madelung form.  In Section 4 we show that the Newton-Nelson law is satisfied by a particle subject to a random force proportional to white noise. In Section 5 we introduce the method of stochastic averaging to be used in the following section to derive Nelson's first postulate. In Section 6 we show that Nelson's first postulate is satisfied for a time interval depending on the energy and the frequency of the oscillator by choosing a suitable spectrum for random force. We further show how a colored spectrum yields Nelson's two postulates for oscillators of all frequencies with fixed mass. We discuss how this result generalizes to small potential perturbations around the harmonic oscillator as long as the periodic motion is preserved with a small shift in frequency. In Section 7 we give heuristics to generalize the result to arbitrary potentials admitting action angle variables. In Section 8 we discuss the results.

\section{Review of stochastic differential equations }

\indent

We give a brief review of Ito stochastic calculus and stochastic differential equations. We will only state results formally which are relevant for our purposes and refer the reader to standard textbooks on the subject(e.g. \cite{Friedman, Arnolds}). Let $\xi(t)$ be the Gaussian process with zero mean and unit variance (also known as white noise), i.e.

\begin{equation}
\langle \xi(t) \rangle = 0, \; \;  \;\forall t
\end{equation}
and for times $(t_1, t_2, ..., t_n)$, $(\xi(t_1), \xi(t_2), ..., \xi(t_n))$ are Gaussian correlated random variables with co-variance

\begin{equation}
\langle \xi(t_1) \xi(t_2) \rangle = \delta(t_1 - t_2).
\end{equation}
Note that for $t_1 \neq t_2$, $\xi(t_1)$ and $\xi(t_2)$ are independent. We define the Wiener process $W(t)$ as the formal time integral of $\xi(t)$:

\begin{equation}
W(t) = \int_0^t \xi(s) ds
\end{equation}
where we set the initial time to $t=0$ without loss of generality. We can also write this as $dW(t) = \xi(t) dt$. The Wiener process is again Gaussian since it is a linear combination of independent Gaussian random variables. Its mean is zero as can be directly seen from the definition. Its co-variance is calculated as

\begin{equation}
\langle dW(t_1) dW(t_2) \rangle = \int_0^{t_1} \int_0^{t_2} \langle \xi(s_1) \xi(s_2)\rangle ds_1 ds_2 = \text{min} ( t_1, t_2 ).
\end{equation}
From this we see that formally $dW(t)$ is of order $\sqrt{dt}$. We will be dealing with stochastic differential equations in the rest of the paper. Suppose we would like to make sense of the following initial value problem for the scalar variable $x(t)$:

\begin{equation}
\frac{dx(t)}{dt} = f(x(t)) + g(x(t)) \xi(t)
\end{equation}
with $p(x, t=0))=p_0(x)$ for some initial probability distribution $p_0(x)$. An ambiguity arises when we would like to make sense of the product $g(x(t)) \xi(t)$. We know that since $\xi(t)$ is independent of $\xi(s)$ for $s < t$,  it is independent of $g(x(s))$ for $s < t$. But the product concerns the same times. In order to remedy this difficulty we will write the equation in differential form:

\begin{equation}
dx(t) = f(x(t))dt + g(x(t)) dW(t)
\end{equation}
which is a formal way to write the integral equation:

\begin{equation}
x(t) = x_0 + \int_0^t f(x(s)) ds + \int_0^t g(x(s)) dW(s).
\end{equation}
Now if we can make sense of the integral that includes $dW(s)$ term we can define the stochastic differential equation in terms of the integral equation. There are more than one ways to define a stochastic integral. In this paper we will operate with the Ito definition. For the other famous (Stratonovich) definition see \cite{Friedman, Arnolds}. We adopt the following definition:

\begin{equation}
\int_0^t g(x(s)) dW(s) = \lim_{\Delta s \to 0} \sum_i g(x(s_i)) ( dW(s_{i+1}) - dW(s_{i}))
\end{equation}
where $\Delta s = s_{i+1}-s_i, \; \; \forall i$. Therefore the increment $dW(s_{i+1}) - dW(s_{i})$ is independent of $g(x(s_i))$. However with this definition we need to update the chain rule of calculus. Suppose that we would like to calculate the equation that is obeyed by a function of $x$, say $y=f(x)$. Remember that $dW(t)$ is of order $\sqrt{dt}$. Thus in order to correctly calculate $dy$ we should expand it up to second order. Without proof we state the Ito's lemma:

\begin{equation}
dy = \frac{df}{dx} dx + \frac{1}{2}\frac{d^2 f}{dx^2} (dx)^2 =  \frac{df}{dx} dx +  \frac{1}{2}\frac{d^2 f}{dx^2} g^2(x) dt.
\end{equation}
Note that in the expansion of $(dx)^2$ we omitted terms of order $dt^{3/2}$ and only kept those of order  $dt$ and $\sqrt{dt}$. We will also need the two dimensional version of this. Suppose we have two processes $x_1$ and $x_2$ with independent Wiener processes $dW_1(t)$ and $dW_2(t)$:

\begin{equation}
dx_i(t) = f_i(x_1, x_2)dt + g_i(x_1, x_2) dW_i(t).
\end{equation}
If $y=f(x_1, x_2)$ then we can write the differential $dy$ as

\begin{equation}
dy = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2 +  \frac{1}{2}(\frac{\partial^2 f}{\partial x_1^2} g_1^2 + \frac{\partial^2 f}{\partial x_2^2} g_2^2 )dt.
\end{equation}
We will frequently invoke these results in the following sections.

\section{Review of Nelson's stochastic mechanics}

\indent

We give a review of Nelson's stochastic formulation of non-relativistic quantum mechanics in one dimension. For more details see Nelson's original paper\cite{Nelson66}, his two books\cite{Nelson1, Nelson2} and Guerra's review\cite{Guerra}. Consider the Schr\"{o}dinger equation:

\begin{equation}
i \hbar \frac{\partial \psi(x,t)}{\partial t} = (-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + U(x)) \psi(x,t).
\end{equation}
Putting $\psi(x,t) = \sqrt{\rho(x,t)} e^{\frac{i}{\hbar} S(x,t)}$ we get the Madelung equations:

\begin{equation}
\frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial x} ( \rho \frac{1}{m} \frac{\partial S}{\partial x} )
\end{equation}

\begin{equation}
\label{Madelung2}
\frac{\partial S}{\partial t} = - \frac{1}{2m} ( \frac{\partial S}{\partial x} )^2 - U(x) + \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}} \frac{\partial^2}{\partial x^2} \sqrt{\rho}
\end{equation}
where $\rho(x,t)$ is the probability of finding the particle at $(x,t)$ and $S(x,t)$ is the phase of the wave function. We recognize first of the equations as the continuity equation with velocity $\frac{1}{m} \frac{\partial S}{\partial x}$. The second of the equations apart from the last term (quantum potential) on the right hand side is the Hamilton-Jacobi equation. Thus if $\hbar = 0$, we have the classical ensemble of particles. The Newton's equations of motion are then the equations that characteristic curves obey corresponding to this set of Madelung partial differential equations. Since the quantum potential term depends on the probability $\rho(x,t)$, giving deterministic characteristics seems not possible. However as Nelson proved\cite{Nelson66, Nelson1, Nelson2}, it is possible to give a Markovian stochastic process associated to the solution of Madelung equations in position space. We start by assuming that a particle obeys the following stochastic differential equation:

\begin{equation} \label{Nelsonfirst}
 dx(t) = b(x(t), t) dt + \sqrt{\frac{\hbar}{m}} dW(t)
\end{equation}
where $b(x(t),t)$ is a general function and $dW$ is the Wiener process. We call this as Nelson's first postulate. The diffusion equation associated to this is\cite{Friedman, Arnolds}

\begin{equation}
\frac{\partial \rho(x,t)}{\partial t} = - \frac{\partial}{\partial x} ( b(x,t) \rho(x,t) ) +  \frac{\hbar}{2m} \frac{\partial^2}{\partial x^2}\rho(x,t)
\end{equation}
where $\rho(x,t)$ is the probability of finding the particle at $x$ at time $t$. In order to match with the continuity equation we define

\begin{equation}
\frac{\partial}{\partial x}  S(x,t)= m ( b(x,t) - \frac{\hbar}{2m} \frac{\partial}{\partial x } \log \rho(x,t) )
\end{equation}
where we assumed that $\rho(x,t)$ is nowhere zero. For a discussion of what happens at zeros see \cite{Nelson2}. We want $S(x,t)$ just defined in this way to satisfy the quantum Hamilton-Jacobi equation. We could postulate it as a partial differential equation but Nelson found a way to write this solely in terms of the stochastic particle trajectory. The quantum Hamilton-Jacobi equation can be shown to be equivalent to the following equation:

\begin{equation}
\frac{1}{2}(D_+D_- + D_- D_+ ) x(t) = -\frac{1}{m} \frac{\partial U(x)}{\partial x} |_{x(t)}
\end{equation}
where $D_+$ and $D_-$ are forward and backward derivatives which will be defined below, the right hand side is the classical acceleration of the particle  evaluated on the stochastic trajectory and the left hand side is the time-symmetric stochastic acceleration. This is the stochastic analogue of Newton's second law. Thus we call this as Newton-Nelson law or Nelson's second postulate. The forward and backward derivatives are defined to be

\begin{equation}
D_+ x(t) = \lim_{\Delta t \to 0^+}  E [\frac{x(t+ \Delta t) - x(t)}{\Delta t} | x(t)]
\end{equation}

\begin{equation}
D_- x(t) =\lim_{\Delta t \to 0^+}  E [\frac{x(t) - x(t - \Delta t)}{\Delta t} | x(t)]
\end{equation}
where $E[f| x(t) ]$ denotes the expectation of $f$ conditioned on $x(t)$. For any function $F(x,t)$ we can write its forward and backward derivatives explicitly as follows

\begin{equation} \label{D+}
(D_+F)(x,t) = \frac{\partial}{\partial t} F(x,t) + b(x,t) \frac{\partial}{\partial x} F(x,t) + \frac{\hbar}{2m} \frac{\partial^2}{\partial x^2} F(x,t)
\end{equation}

\begin{equation} \label{D-}
(D_-F)(x,t) = \frac{\partial}{\partial t} F(x,t) + (b(x,t)-\frac{\hbar}{m} \frac{\partial}{\partial x} \log \rho(x,t)) \frac{\partial}{\partial x} F(x,t) - \frac{\hbar}{2m} \frac{\partial^2}{\partial x^2} F(x,t).
\end{equation}
The derivation of the formula for $D_+$ is straightforward but the calculation of $D_-$ is subtler\cite{Nelson1, Nelson2, Guerra}. Using these formulas it is straightforward to show that the Newton-Nelson law is equivalent to the $x$ derivative of the second Madelung equation (equation $\ref{Madelung2}$). It has been shown that for each solution of the Schr\"{o}dinger equation there is an associated stochastic process satisfying Nelson's postulates and if Nelson's postulates are satisfied that one can construct a wave function which satisfies the Schr\"{o}dinger equation with its absolute square the probability density of the position of particle. The stochastic formulation can be generalized to particles propagating in higher dimensions, multiple particles, fields and particles with spin\cite{Nelson2, Guerra}.

\section{Newton-Nelson law}

\indent

In this section we will show that the Newton-Nelson law is satisfied by a particle in a potential in one dimension subject to a random force. Consider a particle of mass $m$ in a potential $U(x)$ subject to a random force:

\begin{align} \label{process}
& dx(t)= v(t) dt \\ \nonumber
& dv(t)= a(x(t))dt+\sigma dW(t)
\end{align}
where $(x, v)$ denotes the position and velocity variables, $dW$ is the Wiener process, $\sigma$ is a positive constant and

\begin{equation}
a(x)= -\frac{U'(x)}{m}=-\frac{1}{m}\frac{dU(x)}{dx}.
\end{equation}
 We will make use of the following formulas for forward and backward derivatives conditioned on fixed $(x(t), v(t) )$ of a function $G(x,v,t)$ which can be found in section 5 of Guerra's review\cite{Guerra}:

\begin{align} \label{D+phase}
(D_+ G)(x,v,t) |_{(x(t), v(t))} &= \lim_{\Delta t \to 0^+} E[ \frac{G(x(t+\Delta t), v(t + \Delta t), t+\Delta t) - G(x(t), v(t), t)}{\Delta t} | x(t), v(t)]
\\ \nonumber &= \frac{\partial G}{\partial t} + v \frac{\partial G}{\partial x} + a(x) \frac{\partial G}{\partial v} + \frac{\sigma^2}{2} \frac{\partial^2G}{\partial v^2}
\end{align}

\begin{align} \label{D-phase}
 (D_- G)(x,v,t) |_{(x(t), v(t))}  &= \lim_{\Delta t \to 0^+} E[ \frac{G(x(t), v(t), t) - G(x(t-\Delta t), v(t - \Delta t), t-\Delta t) }{\Delta t} | x(t), v(t)] \\ \nonumber &= \frac{\partial G}{\partial t} + v \frac{\partial G}{\partial x} + ( a(x) - \sigma^2 \frac{\partial}{\partial v} \log \rho(x,v,t) )  \frac{\partial G}{\partial v} - \frac{\sigma^2}{2} \frac{\partial^2G}{\partial v^2}
\end{align}
where $\rho(x,v,t)$ is the probability of finding the particle at $x$ with velocity $v$ at time $t$. We also need the following result on conditional expectations for a set of random variables $(x,y,z)$:

\begin{equation} \label{expectation}
E [ F(z) | x] = \int E [F(z) | x, v] p (v| x) dv
\end{equation}
for any function $F(z)$. To derive Newton-Nelson law we will calculate the stochastic acceleration $\frac{1}{2}(D_+D_- + D_- D_+ ) x$. From equations $\ref{D+}$ and $\ref{D-}$ we see that

\begin{equation}
D_+ x(t) = D_- x(t) = v(t)
\end{equation}
where conditioning on $v(t)$ does not matter. Next we calculate $D_+ D_- x(t)$ and $D_- D_+ x(t)$ conditioned on $(x(t), v(t))$ using equations $\ref{D+phase}$ and $\ref{D-phase}$:

\begin{equation}
D_+ D_- x(t) = D_+ |_{(x(t), v(t))} v(t) = a(x(t))
\end{equation}

\begin{equation}
D_- D_+ x(t) = D_- |_{(x(t), v(t))} v(t) = a(x(t)) - \sigma^2 \frac{\partial}{\partial v} \log \rho(x,v,t).
\end{equation}
Hence

\begin{equation}
\frac{1}{2} ( D_+ D_- + D_- D_+) x(t)  |_{(x(t), v(t))}  = a(x(t)) - \frac{\sigma^2}{2} \frac{\partial}{\partial v} \log \rho(x,v,t).
\end{equation}
In order the calculate the stochastic acceleration, which is conditioned only on $x(t)$, we use equation$\ref{expectation}$:

\begin{align}
\frac{1}{2} ( D_+ D_- + D_- D_+) x(t)  |_{x(t)} &= \int \frac{1}{2} ( D_+ D_- + D_- D_+) x(t)  |_{(x(t), v(t))} p_t(v|x) dv \\ \nonumber &= a(x(t)) -  \frac{\sigma^2}{2} \int \frac{\partial p_t(v | x)}{\partial v} dv = a(x(t)).
\end{align}
Thus we have shown that the Newton-Nelson law is satisfied by the process given by equation $\ref{process}$. This result was stated without proof in \cite{Nelson1} for the particle in a potential subject to linear friction in equilibrium.

\section{Method of stochastic averaging}

\indent

In this section we introduce the method of averaging of stochastic differential equations. There are several formulations of stochastic averaging though we will only consider the theorem due to Khas'minskii\cite{Khas, Strat, Freidlin, Spanos, Pavliotis} applied to two dimensional systems in Ito form. Consider the process $(x,y)$:

\begin{equation}
dx(t) = \epsilon^2 f_1 (x(t), y(t), t) dt + \epsilon g_1(x(t),y(t),t) dW(t)
\end{equation}
\begin{equation}
dy(t) = \epsilon^2 f_2 (x(t), y(t), t) dt + \epsilon g_2(x(t),y(t),t) dW(t)
\end{equation}
where $dW$ is the Wiener process and $0 <\epsilon \ll 1$ which means that $(x,y)$ are slowly varying in time as compared to $f_i$ and $g_i$. We assume that $f_i$ and $g_i$ are sufficiently continuously differentiable and bounded.  Then for times of order $O(1 / \epsilon)$ the dynamics can be uniformly approximated the following averaged system \footnote{More precisely the original process converges weakly to the averaged process as $\epsilon \rightarrow 0$.}:

\begin{equation}
dx(t) = \epsilon^2 \bar{f}_1(x(t),y(t)) dt + \epsilon \sigma_{11}(x(t),y(t)) dW_1(t) + \epsilon \sigma_{12}(x(t),y(t)) dW_2(t)
\end{equation}

\begin{equation}
dy (t)= \epsilon^2 \bar{f}_2(x(t),y(t)) dt + \epsilon \sigma_{21}(x(t),y(t)) dW_1(t) + \epsilon \sigma_{22}(x(t),y(t)) dW_2(t)
\end{equation}
where $dW_1$ and $dW_2$ are independent Wiener processes and the averaged functions are given by

\begin{equation}
\bar{f}_i(x,y) = \lim_{T \to \infty} \frac{1}{T} \int_0^T  f_i(x,y,t) dt
\end{equation}

\begin{equation}
(\sigma \sigma^T)_{ij} (x,y) = \lim_{T \to \infty} \frac{1}{T} \int_0^T g_i(x,y,t) g_j(x,y,t) dt
\end{equation}
where $\sigma^T$ denotes the matrix transpose of $\sigma$. Note that $\sigma$ is unique up to a time dependent orthogonal transformation $R(t)$ as $\sigma R(t) (\sigma R(t))^T=\sigma \sigma^T$. It can be shown that $R(t) [dW_1(t) \: dW_2(t)]^T$ is again a Wiener process therefore replacing $\sigma $ with $\sigma R(t)$ does not change the diffusion process $(x,y)$. For applications below we need the periodic version of averaging. For periodic systems we can write

\begin{equation}
\bar{f}_i(x,y) =  \frac{1}{T} \int_0^T  f_i(x,y,t) dt
\end{equation}

\begin{equation} \label{ave}
(\sigma \sigma^T)_{ij} (x,y)  = \frac{1}{T} \int_0^T g_i(x,y,t) g_j(x,y,t)dt
\end{equation}
where $T$ is the period of oscillations which correspond to $dx=dy=0$. The way that $dx = dy =0$ corresponds to a periodic deterministic solution will be clarified in the examples in the following sections. The stochastic averaging principle is a generalization of its deterministic version which can found in \cite{Sanders}. For deterministic averaging of a one dimensional system in action-angle variables see \cite{Arnold}. For more on stochastic averaging see the review\cite{Spanos} and the books \cite{Freidlin, Pavliotis}.

\section{Nelson's first postulate for a harmonic oscillator}

\indent

Consider the harmonic oscillator with frequency $\omega$ and mass $m$ subject to a random force with position and velocity variables $(x,v)$:

\begin{align} \label{osc}
& dx(t)=v(t)dt \\ \nonumber
& dv(t)=-\omega^2x(t)dt+\epsilon \omega dW(t).
\end{align}
Assume that the initial energy of the oscillator is $E_0$ is probability 1. We will show that we can make the choice $\epsilon=\sqrt{\frac{2\hbar}{m}}$ such that the position process $x(t)$ is approximately Markovian and satisfies Nelson's first postulate for a time interval depending on $\hbar$, $E_0$ and $\omega$. We show this by approximating the dynamics given by equation $\ref{osc}$ by an averaged process using the method of stochastic averaging. We will see that this will induce a noise term for the $x$ variable which is necessary to satisfy Nelson's first postulate. Note that the noise term appearing in equation $\ref{osc}$ is not necessarily small as it is proportional to $\epsilon \omega$ as $\omega$ can be large.

To proceed note that the dynamics in phase space is not in standard form for averaging. Therefore apply the coordinate transformation

\begin{align} \label{defx}
& x=r\cos(\omega t + \phi) \\ \nonumber
& v=-\omega r \sin(\omega t+ \phi)
\end{align}
or

\begin{align}
& r=\sqrt{x^2+\frac{v^2}{\omega^2}} \\ \nonumber
& \phi=-\arctan(\frac{v}{\omega x})-\omega t.
\end{align}
To calculate the differential of $r$ and $\phi$ we use Ito's lemma and obtain

\begin{align}
& dr=\frac{x}{r}dx+\frac{v}{\omega^2 r}dv+\frac{1}{2}\frac{x^2}{ \omega^2 r^3} (dv)^2=\frac{(\epsilon\omega)^2}{2}\frac{x^2}{ \omega^2 r^3}dt+\epsilon \omega \frac{v}{\omega^2 r} dW \\ \nonumber
& d\phi=\frac{v}{\omega r^2}dx-\frac{x}{\omega r^2}dv + \frac{1}{2} \frac{2xv}{\omega^3r^4}(dv)^2-\omega dt = (\epsilon \omega)^2 \frac{xv}{\omega^3r^4}dt - \epsilon \omega \frac{x}{\omega r^2 }dW.
\end{align}
We see that both $r$ and $\phi$ are slowly varying. Therefore we apply the method of averaging over one period $T=\frac{2\pi}{\omega}$ of the harmonic oscillator. This amounts to fixing $r$ and averaging over the angle variable. Denote the time average of a function $f(x,v)$ by

\begin{equation}
 \langle f(x,v) \rangle_T=\frac{1}{T} \int^T_0 f(x(t),v(t))dt.
 \end{equation}
The evolution equations can be approximated by the following averaged equations over time intervals of order $O(1/\epsilon)$ noting that the noise terms are of order $\epsilon$:

\begin{align}
& dr=\frac{(\epsilon\omega)^2}{2}\frac{1}{ \omega^2 r^3} \langle x^2 \rangle_T dt+\epsilon \omega \frac{1}{\omega^2 r} \sqrt{\langle v^2 \rangle_T} dW_1  \\ \nonumber
& d\phi= (\epsilon \omega)^2 \frac{1}{\omega^3r^4} \langle xv\rangle_Tdt + \epsilon \omega \frac{1}{\omega r^2 } \sqrt{\langle x^2 \rangle_T} dW_2
\end{align}
where $dW_1$ and $dW_2$ are independent Wiener processes. The averaged quantities are calculated to be

\begin{align}
&\langle x^2 \rangle_T= \frac{1}{2 \pi} \int_0^{2 \pi} r^2 \cos^2\theta d\theta = \frac{r^2}{2} \\ \nonumber
&\langle v^2 \rangle_T = \frac{\omega^2}{2 \pi} \int_0^{2 \pi} r^2 \sin^2\theta d\theta = \frac{\omega^2 r^2}{2} \\ \nonumber
&\langle xv \rangle_T = -\frac{\omega}{2 \pi} \int_0^{2 \pi} r^2 \sin\theta \cos \theta d\theta = 0
\end{align}
where we have chosen $\sigma$ in equation $\ref{ave}$ as diagonal as $\langle xv \rangle_T=0 $. Substitute the averaged quantities in the averaged equations to get:

\begin{align} \label{avr}
& dr=\frac{\epsilon^2}{4}\frac{1}{  r} dt+\epsilon \frac{1}{\sqrt{2}} dW_1  \\ \nonumber
& d\phi=  \epsilon \frac{1}{\sqrt{2} r }  dW_2.
\end{align}
The averaged evolution of the amplitude of oscillations $r$ is independent of $\phi$ and the evolution of $\phi$ is determined by the evolution of $r$. Using averaged equations we can derive the averaged evolution of the position variable $x$ using equations $\ref{defx}$ and $\ref{avr}$:

\begin{align}
dx & =dr \cos(\omega t + \phi ) - r \sin(\omega t +\phi) (\omega dt + d\phi)-\frac{1}{2} r \cos(\omega t + \phi) (d\phi)^2 \\ \nonumber
& = v dt + \epsilon \frac{1}{\sqrt{2} r }(xdW_1+\frac{v}{\omega}dW_2).
\end{align}
Note that the method of stochastic averaging induced a stochastic term for the $x$ variable where in the original dynamics defined by equation $\ref{osc}$ the stochastic term only appears in the dynamics of the velocity variable. We can simplify the stochastic term noting that given $x(t)$ and $v(t)$, $dW_1(t)$ and $dW_2(t)$ are independent Gaussian processes. A linear combination

\begin{equation}
a(x(t),v(t))dW_1(t)+b(x(t),v(t))dW_2(t)
\end{equation}
of independent zero mean Gaussian processes is again a zero mean Gaussian process with variance $a^2(x(t),v(t))+b^2(x(t),v(t))$. Therefore the equation for the position variable can be written as:

\begin{equation}
dx = v dt + \frac{\epsilon }{\sqrt{2}}dW
\end{equation}
where $dW$ is the Wiener process. In general this is not a Markov process since $v$ itself is fluctuating and is dependent on $x$. However if somehow we can assume that the amplitude $r$ is constant then we can express $v$ in terms of $x$ as

\begin{equation}
v=\pm \omega \sqrt{r^2-x^2}
\end{equation}
obtaining the Markov process

\begin{equation}
dx = \pm \omega \sqrt{r^2-x^2} dt+ \frac{ \epsilon}{\sqrt{2}}dW.
\end{equation}
Now to match with Nelson's first postulate (equation $\ref{Nelsonfirst}$) we must choose

\begin{equation}
\epsilon = \sqrt{\frac{2\hbar}{m}}.
\end{equation}
With this choice of $\epsilon$ we can justify the assumption that $r$ remains approximately constant as follows. Assume that initially $r=r_0$ with probability 1. For sufficiently small times we can assume that $r$ is well approximated by $r_0$. To see this introduce the energy variable

\begin{equation}
E=\frac{1}{2}m\omega^2r^2.
\end{equation}
Using Ito's lemma its dynamics is calculated to be:

\begin{equation}
dE = (\epsilon \omega)^2 \frac{m}{2} dt + \epsilon \omega\sqrt{m E} dW_1
\end{equation}
with $E_0=\frac{1}{2} m \omega^2 r_0^2$. Set  $E(t)=E_0 + \delta E(t)$. Roughly $\delta E(t)$ grows as

\begin{equation}
\text{max}(\epsilon^2 \omega^2 m t, \epsilon \omega \sqrt{m E_0 t} )
\end{equation}
where $\sqrt{t}$ dependence arises from the Wiener term. The method of stochastic averaging is accurate for times $O(1 / \epsilon)$. Therefore if $1 / \epsilon \ll  1 / \hbar \omega^2$ and $1 / \epsilon \ll 1 / \hbar \omega^2  E_0$ which are satisfied for sufficiently small $\omega$ and $E_0$, we can assume that $\delta E(t)$ is small therefore $E(t) \approx E_0$ and $r(t) \approx r_0$ over the time interval $1 / \epsilon$ in which the method of stochastic averaging is accurate. We can write the dynamics in the following suggestive form

\begin{equation}
dx = v(x, E_0) dt + \sqrt{\frac{\hbar}{m}} dW
\end{equation}
where $v(x, E_0)=\pm \sqrt{\frac{2}{m} (E_0-\frac{1}{2}m\omega^2x^2)}$ is the classical velocity of the particle with energy $E_0$. If the stochastic term is absent then this equation would be the classical equation of motion for the particle. Hence the phase space diffusion process gives rise to a position space Markov process as a small random fluctuation around the classical trajectory.

We initially made the assumption that the random force depends on the frequency of the oscillator. Then it is natural to ask whether we can choose a universal random force term which would give the same result for an arbitrary frequency. Such a choice is indeed possible. So instead of the Markovian model start from

\begin{align} \label{osc2}
&\dot{x} (t) = v (t)\\ \nonumber
&\dot{v} (t) = -\omega^2 x(t) + \xi(t).
\end{align}
Let $\xi(t)$ be a zero mean Gaussian process with covariance

\begin{equation}
\langle \xi(t) \xi(t+\tau) \rangle = c(\tau)
\end{equation}
with its Fourier transform, the power spectrum

\begin{equation}
S(\Omega)=\int_{-\infty}^{\infty} c(\tau) e^{-i\Omega \tau} d\tau.
\end{equation}
It can be shown that upon averaging the $(r, \phi)$ evolution over one period of the oscillator, only the resonant term corresponding to $\Omega=\omega$ contributes to the averaged equations \cite{Spanos}:

\begin{align}
& dr=\frac{ S(\omega)}{8 \pi}\frac{1}{ \omega^2 r} dt+  \sqrt{ \frac{S(\omega)}{4 \pi}} \frac{1}{\omega}  dW_1  \\ \nonumber
& d\phi=  \sqrt{ \frac{S(\omega)}{4 \pi}}  \frac{1}{\omega r }  dW_2.
\end{align}
Thus if we choose $S(\Omega)=\frac{4 \pi \hbar}{m}\Omega^2$, we recover the previous results. Note that the averaged equations obtained for the colored noise are the same as the ones obtained before for white noise with spectrum dependent on frequency. Hence the processes defined by equations $\ref{osc}$ and $\ref{osc2}$ are good approximations to each other. Therefore Newton-Nelson law is approximately satisfied for the colored noise case as it is satisfied for equation  $\ref{osc}$.

Now we consider small potential perturbations around the harmonic oscillator:

\begin{align}
& dx(t)=v(t)dt \\ \nonumber
& dv(t)=-\omega^2x(t)dt -  \eta \frac{1}{m}\frac{d U(x)}{dx}|_{x=x(t)}+\epsilon \omega dW(t)
\end{align}
where $\frac{d U(x)}{dx}$ is $O(1)$ and $\eta \ll 1$. We assume that $U(x)$ preserves the periodic structure where the new frequency $\tilde{\omega}(E)$ is a small perturbation of the frequency of the harmonic oscillator:

\begin{equation}
\tilde{\omega} (E) = \omega + \eta \delta \omega.
\end{equation}
Since $\eta \delta \omega$ induces a $O(\epsilon \eta)$ correction in the stochastic term, after repeating the steps for the pure harmonic oscillator, we can approximately write

\begin{equation}
dq = v(x, E_0) dt + \frac{\epsilon}{\sqrt{2}} dW
\end{equation}
where this time $v(q, E_0)$ is the velocity associated to the perturbed potential:

\begin{equation}
v(x, E_0) = \pm \sqrt{ \frac{2}{m}( E_0 - \frac{1}{2}m\omega^2 x^2 - \eta U(x)) }.
\end{equation}

\section{Nelson's first postulate for a general potential}

\indent

In this section we would like to generalize the results for the harmonic oscillator to a general potential. However we will be able to show less. Due to the difficulty in calculating averages explicitly for general potentials we will be only able to show that we can choose the random force dependent on energy (unlike the harmonic oscillator case where the random force is independent of coordinates) such that Nelson's first law is satisfied using a heuristic averaging procedure. We will restrict to potentials for which the motion can be described using action-angle variables. Therefore consider a particle of mass $m$ in one dimension in a potential $U(x)$ subject to a random force:

\begin{align}
& dx(t)=\frac{p(t)}{m} dt \\ \nonumber
& dp(t)=-U'(x(t)) dt+\epsilon dW(t)
\end{align}
where $(x,p=mv)$ denotes the position and momentum variables and $U'(x)=\frac{dU(x)}{dx}$. We first perform the coordinate transformation from $(x, p)$ to $(x, E)$ where $E$ is the energy of the particle defined by

\begin{equation}
E (x,p) = \frac{p^2}{2m}+U(x).
\end{equation}
Using Ito's lemma we calculate $dE$ as

\begin{equation}
dE = U'(x)dx+\frac{p}{m}dp+\frac{1}{2m}(dp)^2=\frac{\epsilon^2}{2m}dt + \epsilon \frac{p(x,E)}{m} dW.
\end{equation}
Note that if the stochastic term is absent then the energy would be conserved. We can also express $dx$ in terms of $(x,E)$ by solving for $p$ in terms of $(x,E)$ in the definition of energy:

\begin{equation}
dx = \pm \frac{1}{m}\sqrt{2m(E - U(x))}dt.
\end{equation}
We now assume that the classical motion can be described by action-angle variables $(\phi, I)$\cite{Arnold}. In terms of the action-angle variables the classical deterministic equations of motion can be written as

\begin{align}
& d\phi = \omega(I) dt \\ \nonumber
& dI = 0
\end{align}
where the frequency is

\begin{equation}
w(I)=\frac{dE(I)}{dI}
\end{equation}
and the energy is a function of the action variable alone. Instead of the action variable we will use the energy variable since the energy is a function of the action but not the angle variable. We further assume that this mapping is one-to-one. Define the action function (not the action variable) as

\begin{equation}
S(I, x) = \int_{x_0}^x p(x', E) dx'
\end{equation}
for an arbitrary initial point $x_0$. Then the action variable is defined to be proportional to the action function $S$ over one period of motion:

\begin{equation}
I = \frac{1}{2 \pi} \oint p dx'
\end{equation}
and the angle variable is
\begin{equation}
\phi = \frac{\partial S(I, x)}{\partial I} = \int_{x_0}^x \frac{\partial p(x',E)}{\partial I} dx'.
\end{equation}
Taking the derivative inside the integral we have

\begin{equation}
\label{defphi}
\phi= \omega(I)\int_{x_0}^x \frac{\partial p(x',E)}{\partial E} dx' = m \omega(I) \int_{x_0}^x \frac{1}{p}dx'.
\end{equation}
We define

\begin{equation}
f(x,E)=\int_{x_0}^x \frac{1}{p}dx'
\end{equation}
so that

\begin{equation}
\phi(x, E)=m \omega(I(E)) f(x,E).
\end{equation}
Having defined the angle variable we are ready to perform the change of coordinates from $(x, E)$ to $(\phi, E)$. Using Ito's lemma we calculate $d\phi$ as

\begin{align}
d\phi &= m \frac{\partial}{\partial E} (\omega f)dE + \frac{m}{2} \frac{\partial^2}{\partial E^2} (\omega f) (dE)^2 + m\omega \frac{\partial f}{\partial x } dx \\ \nonumber
&= \omega dt + \frac{\epsilon^2}{2}(\frac{\partial}{\partial E}(\omega f) + \frac{p^2}{m}\frac{\partial^2}{\partial E^2} (\omega f))dt+\epsilon\frac{\partial}{\partial E}(\omega f) p dW.
\end{align}
We see that $\phi$ is slowly varying except the $\omega dt$ term. In order to have all the right hand side terms small we further introduce the new angle variable $\theta$ as

\begin{equation}
\theta = \phi - \omega t
\end{equation}
and compute its differential as

\begin{align}
d \theta &= d\phi-\frac{d \omega}{dE} t dE - \frac{1}{2} \frac{d^2 \omega}{dE^2} t (dE)^2-\omega dt \\ \nonumber
&= \frac{\epsilon^2}{2}(\frac{\partial}{\partial E}(\omega f) + \frac{p^2}{m}\frac{\partial^2}{\partial E^2} (\omega f)-t \frac{d^2 \omega}{dE^2} \frac{p^2}{m^2}) dt - \frac{\epsilon^2}{2m}\frac{d \omega}{dE}t dt  \\ \nonumber
& \; \; \; \; + \epsilon\frac{p}{m}(m\frac{\partial}{\partial E}(\omega f)-t\frac{d\omega}{dE})dW.
\end{align}
We have finalized the set of coordinate transformations which yielded slowly varying $(E, \theta)$ coordinates. Next we average the dynamics over a period $T=\frac{2\pi}{\omega(E)}$ fixing $E$ in $(x(t),p(t))$ to obtain the approximate averaged equations. The averaged equations for $(E, \theta)$ are

\begin{equation} \nonumber
dE = \frac{\epsilon^2}{2m} dt + \epsilon \sigma_{11}(E)dW_1 + \epsilon \sigma_{12}(E) dW_2
\end{equation}

\begin{equation}
d \theta = \frac{\epsilon^2}{2}F(E) dt+ \epsilon \sigma_{21}(E)dW_1 + \epsilon \sigma_{22}(E) dW_2
\end{equation}
where $dW_1$ and $dW_2$ are independent Wiener processes, $F(E)$ is given by

\begin{equation}
F(E) = \langle \omega \frac{\partial f}{\partial E} + \frac{p^2}{m} (2 \frac{d \omega}{dE} \frac{\partial f}{\partial E} + \omega \frac{\partial^2 f}{\partial E^2} ) \rangle_T
\end{equation}
and the matrix $\sigma$ is determined from

\begin{equation}
(\sigma \sigma^T)_{ij}= D_{ij}
\end{equation}
The components of $D$ are the averages

\begin{equation} \nonumber
D_{11}=\langle \frac{p^2}{m^2} \rangle_T
\end{equation}

\begin{equation} \nonumber
D_{12}=D_{21}= \langle \frac{p^2}{m} \omega \frac{\partial f}{\partial E} \rangle_T
\end{equation}

\begin{equation}
D_{22}=\langle p^2 \omega^2 (\frac{\partial f}{\partial E})^2 \rangle_T
\end{equation}
where we have simplified $F(E)$ and $D_{ij}$ noting that since when we take averages over the classical trajectory we can set $dx = \frac{p}{m}dt$ so that $t=mf$. Now we will go back to the dynamics of $x$ to check if Nelson's first postulate is satisfied. First calculating the dynamics of $(E, \phi)$ then
calculating the dynamics of $x$ we obtain the equation of the form

\begin{equation}
 dx = \frac{p}{m} dt + \epsilon^2 K(x,p) + \epsilon (G_1(x,p)dW_1 + G_2(x,p)dW_2 )
 \end{equation}
for some functions $K(x,p)$, $G_1(x,p)$ and $G_2(x,p)$. The stochastic term is dependent on coordinates. We invoke without rigorous justification a heuristic averaging procedure although $dx$ is not in the standard form. We fix $E$ and average the small terms over the angles.

\begin{equation}
dx = \frac{p}{m} dt + \epsilon^2 \bar{K}(E) + \epsilon (\bar{G}_1(E) dW_1 + \bar{G}_2(E) dW_2)
\end{equation}
where $\bar{K}(E)$, $\bar{G}_1(E) $ and $\bar{G}_2(E)$ are the averages of $K(x,p)$, $G_1(x,p)$ and $G_2(x,p)$. Since $dW_1(t)$ and $dW_2(t)$ are independent zero mean Gaussian processes we can write

\begin{equation}
dx = \frac{p}{m} dt + \epsilon^2 \bar{K}(E) + \epsilon G(E) dW
\end{equation}
where $dW$ is the Wiener process and $G^2(E)=\bar{G}_1(E)^2+\bar{G}_2(E)^2 $. Now as in the harmonic oscillator case for sufficiently small times $E$ is almost constant.  We are not able to calculate $G(E)$ explicitly in terms of $E$(or the action variable $I$) and $\omega(E)$ and its derivatives since the averages appearing in $D_{ij}$ seem to be difficult to evaluate analytically. Therefore unlike the harmonic oscillator case we have the weaker result: one can choose

\begin{equation}
\epsilon G(E) = \sqrt{\frac{\hbar}{m}}
\end{equation}
to satisfy Nelson's first postulate. However in this case we need to choose $\epsilon$ dependent on $E$ and this is unfavorable regarding universality: for every value of energy we need to choose a different correlation coefficient for the noise. We saw in the harmonic oscillator case that choosing the colored random force with spectrum proportional to $\omega^2$ we can satisfy Nelson's first postulate for all oscillator frequencies. If one could evaluate $G(E)$ explicitly one can check whether the same random force spectrum gives rise to Nelson's first law for arbitrary potentials admitting action-angle description.

\section{Scholia}

\indent

We have shown that it is possible to choose a random force such that fluctuations around the classical trajectory can be described by the Schr\"{o}dinger equation for the harmonic oscillator. We achieved this by showing that the Newton-Nelson law is satisfied for general potentials and Nelson's first postulate is approximately satisfied for states described by constant energy upon averaging the dynamics. We further generalized this result to the case of small perturbations around the harmonic oscillator. We also showed that there exists a colored Gaussian noise such that Nelson's two postulates are approximately satisfied for an oscillator with any frequency with fixed mass. We attempted to generalize the results for the harmonic oscillator to a general potential admitting action-angle variables. However due to the difficulty in evaluating averages and the need for an heuristic averaging principle we are only able to show that there is a random force depending on energy, mass and frequency of the system such that we can obtain Nelson's first postulate.

In this paper we did not touch the issue of superposition states. Although Nelson's formulation is equivalent to the Schr\"{o}dinger equation and describes superposition states there is no guarantee that the Nelson's process associated to a superposition state could be derived from a phase space process. The simplest idea is to let the particle assume two energies with their respective probabilities. Then the position space process should approximate the superposition state of the respective energies. However this does not work since we are not superposing quantum states but rather considering an ensemble of states and we end up in a mixed state. Thus we do not yet know whether superpositions and the most general quantum states can be described within the framework we presented here.

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


text:
ESPN will follow 'The Last Dance' with more documentaries. So look for Sammy Sosa and 'Long Gone Summer' on June 14.
By Phil Rosenthal
When "The Last Dance" with Michael Jordan and the Chicago Bulls ends its ESPN run, get ready for Sammy Sosa and Mark McGwire.
Lance Armstrong and Bruce Lee too.
ESPN on Tuesday announced it will run "30 for 30" documentaries on four successive Sundays once the "The Last Dance" concludes its five-week, 10-episode run on May 17.
Succeeding Jordan and company on May 24 and 31 will be "Lance," a two-part look at once-inspirational, then infamous, blood-doping cyclist Armstrong from director Marina Zenovich.
Bao Nguyen's "Be Water" on martial artist and actor Lee follows on June 7.
But Chicagoans are likely most interested in the June 14 telecast of "Long Gone Summer," director A.J. Schnack's examination of the Cubs' Sosa, the Cardinals' McGwire and the steroid-tainted 1998 chase of Roger Maris' home run record.
ESPN Films is moving three high-profile @30for30 to Sundays following the final episodes of #TheLastDance
Documentary subjects include @lancearmstrong, @brucelee, Mark McGwire & Sammy Sosa
Films available on ESPN+ immediately after their debuts
More: https://t.co/oLAJn7zJ1Q pic.twitter.com/jZdXs53i6m
— ESPN PR (@ESPNPR) May 5, 2020
Both McGwire and Sosa submitted to lengthy interviews for the project, according to ESPN, which said the musical score was provided by Wilco's Jeff Tweedy, who's from the St. Louis area and now lives in Chicago.
"It's a great feeling to bring three more epic documentary projects to sports fans who so need it right now," Libby Geist, vice president and executive producer of ESPN Films and Original Content, said in a statement.
Both "Lance" and "Be Water" had premieres at the Sundance Film Festival. Plans to present "Long Gone Summer" at the Tribeca Film Festival were scuttled by the new coronavirus pandemic.
The three documentaries will be made available on ESPN Plus immediately after their premieres, joining all the other documentaries in the "30 for 30" series.
The first two episodes of "The Last Dance" on April 19 set a record for the most-viewed ESPN documentary content, averaging a combined 6.1 million viewers nationally on ESPN and ESPN2.
New shelter for migrants in Woodlawn could open in the coming week
Candace Parker is signing with the Las Vegas Aces, ending a memorable 2-year run with her hometown Chicago Sky
A look at how gas prices in Illinois compare to other states

redpajama_set_name: RedPajamaCommonCrawl


It was named for the Battle of Magenta, which was fought in 1859 near the town of Magenta in northern Italy during the Franco-Piedmontese war against Austria. The deep purplish color – on of the first dyes – was discovered soon after the battle and was named to commemorate it.
Source: The 365 amazing trivia facts calendar 2007.
Pls be patient with me in my writing. In speaking, while I do not have ADHD, I nevertheless begin to converse about tangential topics to the conversation. And to bore my friends with my limitless utterances So be kind. I think I may indeed have adult adhd. I'll see a doc. But know that my writing is also "all over" the place with its many digrressions.
I'm not aware of your exact, actual question's answer 🙁 but I am aware, from Social Studies — this is one million years back (in the day) — that there was a battle, a crash of flesh colliding like freight trains with men and familiars (dogs, horses) in the fight too. If you saw the film "Lion in Winter" with Captain Picard as King you know the images and warring of which I speak … So I, aware of, apprised of, told of, explained to, … your Q reminded me of a battle I saw (in a film set during renaisance times: yes, it was the film I mentioned above) Hacking was not the provinance of kids with too much money and time and deadly malicious intent.
Meantime, the warring factions were the French Fries and ???? Perhaps it was the English. At battle's end, the french king rides in fon a beauifully proportioned and well-muscled white horse. The King trots right into the writhing carpet of dead and dying men. The King is empathetic, one can tell, but he keeps his back straight and remains regal-looking. At one point, the King removes the heavy, hot, head gear and, with helmet off, we see the King is really a queen; far too pretty for the time (no dentistry, for example). So the french Queen is seen by some stragglers who are left alone; secret out, the french had no King for the moment.
To get to the point, or a point, any point, I continue. When someone hands the queen a bloody piece of white cloth for some reason, she takes it and looks it over and says,or thinks, or one just deduces her empathy for he enemy vanquished — women, children and dogs among the dead. So, being a good actress, this queen says something like "I've never seen this color before. Let's call it purple.


Search Results: 27 results in 0.0139 seconds.
Mass Effect 3 has gone gold!
Another treat has been gifted to RPG fans at the Video Game Awards, this time in the shape of a new Mass Effect 3 gameplay trailer. In it, Shepard mounts an attack against the odds as a spectacular sequence of events unfold. It's excitement++ all-round. Also: giant sand worm things.
BioWare has revealed not just a single player demo, scheduled for January 2012, but an accompanying multiplayer beta! For those that aren't aware, Mass Effect 3 ships with an online co-op mode titled Galaxy at War. No specific platforms for either have been confirmed apart from PC, but what we do know is that you'll find early access in new copies of Battlefield 3.
It's official, we have a Star Wars: The Old Republic release date! Detailed on the SWTOR pre-order page are the editions available, related bonus items, system specification requirements and of course the dates themselves; December 20th in North America and December 22nd in Europe.
From the recently run E3 2011 expo is this STAR WARS: The Old Republic highlight wheel, featuring a conflict on the war-torn planet of Alderaan. "A militaristic Republic war hero has seized the throne in a misguided attempt to protect the people of Alderaan. Watch as Republic players confront the powerful and well-protected 'King Ulgo', in this Highlights Video."
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text:
Q: (Java) Initializing variable with lambda expression I was playing around in Java
and I tried to initialize a variable with a lambda expression
however, when trying to use the variable I get the error
java: cannot infer type for local variable test (lambda expression needs an explicit target-type)
public static void main(String[] args){
    var test = (e -> {
        return 10;
    });

    Arrays.stream(new int[]{3})
            .map(test)
            .forEach(System.out::println);

}

Honestly, I was just trying to initialize a variable using a method
and it just sort of became lambda so I just played around with lambda

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text:
Letter to Joan Wareing from Wing Commander P H Cribb
From officer commanding 582 Squadron, informing her that her husband has failed to return from an operation over enemy territory on the night of 7/8 August 1944. Mentions that he was captain of experienced crew and comments on character. No further information available and explains warning of not making any announcement which might endanger the possibility of escape or evasion. Mentions what will happen to his personal effects.
P H Cribb
England--Huntingdonshire
England--Scunthorpe
Text. Correspondence
Two page typewritten letter
Joy Reynard
ECribbPHWareingJ440809
No. 582 Squadron,
R.A.F. Station,
Little Staughton,
St Neots, Hunts.
9th August 1944.
Dear Mrs Wareing,
I deeply regret to inform you that your husband, Squadron Leader Robert Wareing D.F.C., failed to return from operations carried out over enemy territory on the night of 7th/8th August, 1944.
There is, at the time of writing, no news to give you beyond this brief statement.
Your husband was the Captain of a very experienced and capable crew, with a keen sense of loyalty to and co-operation with myself as their Squadron Commander, and with their fellows. They set an example of courage, skill and coolness of a very high order, an example which will be greatly missed. Your husband was a young man of exceptional charm of manner with a keen sense of humour, and was one of the most popular Officers in the Squadron.
It is desired to explain that the request in the telegram notifying you of the casualty to your husband was included with the object of avoiding his chance of escape being prejudiced by undue publicity in case he was still at large. This is not to say that any information about him is available, but is a precaution adopted in the case of all personnel reported "Missing".
Your husband's personal effects are being carefully collected, and will be sent to the Central Depository at Colnbrook whence they will reach you in due course.
You may rest assured that I shall communicate with you immediately any news concerning your husband comes to hand. Meanwhile, please accept the profound sympathy of my Squadron and of myself, with our hope that further information may soon be available.
P.H.CRIBB.
[underlined] Wing Commander, Commanding, 582 Squadron, RAF. [/underlined]
[page break]
Mrs Joan Wareing,
56 West Common Gardens,
[underlined] Scunthorpe, LINCS. [/underlined]
End of transcription
Wareing, Robert
582 Squadron; missing in action; RAF Little Staughton; tactical support for Normandy troops
P H Cribb, "Letter to Joan Wareing from Wing Commander P H Cribb," IBCC Digital Archive, accessed January 19, 2022, https://ibccdigitalarchive.lincoln.ac.uk/omeka/collections/document/27838.


text:
Truck Day signals Spring Training is near – What's left to do
Arlington, TX – Texas Rangers Truck Day 2/7/2014 Departs for Surprise, Arizona – Photo by David Cash
Today the equipment truck was loaded up and headed out towards Surprise, Arizona. The Texas Rangers pitchers and catchers will report to Spring Training on February 16 and start their first workout on the 17th. The position players will arrive on February 19, and start the their full workout the next day.
The Diamondbacks have already started their spring training and the Los Angeles Dodgers will have their pitchers and catchers report tomorrow.
As spring training gets closer, the build up of the 2014 team is starting to take shape. There are only a few questions left for the Texas Rangers.
Currently, Mitch Moreland is the last player the Texas Rangers are in arbitration with. Both sides must come up with an agreement or it will go to an arbitrator between now and February 21. The Rangers are offering about $2 million, but Moreland is requesting $3.35 million for a one-year deal. Neither side so far has made any changes to the offer and it could go to an arbitration hearing.
Also, as we know, Wrigley the dog has caused some drama this offseason. Derek Holland will miss at least the first third of the season. The Texas Rangers are still in need of another starting pitcher. They are reportedly talking with Tommy Hanson, a former starter for the Los Angeles Angels and Atlanta Braves. Hanson was pretty solid in his early years with the Braves, but since a rotator injury in 2011 he hasn't been the same. He had a record of 7-4 and an ERA of 5.42 last season with the Angels.
The Rangers also took a look at the top Korean pitcher Suk-min Yoon this week at their Surprise, Arizona facility. Yoon threw 30 bullpen pitches in front of the Texas Rangers staff and also the Chicago Cubs staff. To find out more about Yoon, read here.
Also, there is still a chance the Texas Rangers could bring back Nelson Cruz. Currently, it is being reported that the Seattle Mariners are talking to him and are open to a multi-year deal. But they could also pick up Kendrys Morales. If Nelson Cruz doesn't return, in my opinion, definitely make Mitch Moreland the DH and almost an everyday player.
Arlington, TX – Texas Rangers Truck Day 2/7/2014 – Photo by David Cash
Matt Harrison is also feeling very well. He's been working out and preparing for the season. He's lost some weight and his back is feeling much better. He is one piece the Texas Rangers could use in their starting rotation. Currently, it appears that it is likely Harrison will return and could be his former self. When Harrison was healthy, he was probably one of the best in the starting rotation.
Just 52 more days until Opening day. It's time for some Spring Training and baseball again.


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EU Trust Fund for Africa adopts €90 million programme on protection of migrants and improved migration management in Libya
Following up on the Joint Communication on the Central Mediterranean Route and the Malta Declaration, the EU Trust Fund for Africa upon proposal from the European Commission, adopted today a €90 million programme to step up the protection of migrants and reinforce migration management in Libya.
Today, the EU Emergency Trust Fund for stability and addressing root causes of irregular migration and displaced persons in Africa has adopted a comprehensive €90 million programme on the protection of migrants and improved migration management in Libya.
The programme will focus on the following:
Stepping up the protection of migrants and refugees (€48 million), including - inter alia - assistance to and protection of migrants and refugees at disembarkation points, in detention centres and urban settings; voluntary humanitarian returns and reintegration of migrants to their countries of origin ; creation of 'Safe Spaces' as alternatives to detention ; assistance to migrants on the move in the form of information on viable options and risks of irregular migration as well as food and non-food items.
Improve socio-economic development at municipal level and local governance (€42 million), through strengthening capacities of local authorities to provide services and foster local development and stability, through provision and access to quality services for Libyans and migrants (e.g. health facilities) and through local economic development and access to job opportunities.
The programme will be implemented through five partners: IOM, UNDP, UNHCR, UNICEF, and GIZ.
For further information, see the website of the European Commission.
Publication Date: Wed 12 Apr 2017
EU Trust Fund, Libya
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Sarin – You Can't Go Back (2021)
Sarin – You Can't Go Back (Prosthetic Records, 05 February 2021)
Sort of like a response to this article from earlier in the week where we covered an album entitled A Way Back, today we're featuring You Can't Go Back — the latest from Canadian post-metal group Sarin, named for the chemical compound that has historically been used (and sometimes rumored to have been used) as a nerve gas weapon due to its extreme toxicity.
The record immediately kicks off at full throttle: first track "Cold Open" is aptly titled in the sense that there is no time wasted on an introduction or preamble; the listener is just dropped directly into the deep end, instantly smothered with the pressure of waves upon waves of bass-heavy guitar chords crashing down amidst roared vocals.
The track seems slightly confounding at first: something about the rhythm feels like the band is employing some exotic time signature, like they're playing in 5 or 15 or something. But a closer listen reveals that the music is less complex than it seems, as each phrase is contained within a fairly standard grouping of 16 beats. But the emphasis keeps shifting back and forth, subdividing those beats into either 6+6+2+2 or 3+3+3+3+2+2, and then occasionally slipping in a few bars of "common time" 4+4+4+4 just to really mess with you. Fun times.
From there, Sarin might decide to throw together a sludge-metal rendition of the "Bela Lugosi's Dead" rhythm section with major-key triumphant-sounding guitar leads. Over the course of these six songs they may do something murky, wistful, spacey, bluesy; and every once in a while engage in a psychedelic freak-out — don't even be surprised if you find any or all of these moods within the same song, all wrapped up with a grimy post-hardcore bow on top.
Check out You Can't Go Back right now: stream/download/CD/vinyl
http://sarin.bandcamp.com
http://twitter.com/sarintheband
http://instagram.com/sarintheband
http://www.facebook.com/sarinofficial
http://www.prostheticrecords.com
http://shop.prostheticrecords.com
http://prostheticrecords.bandcamp.com
http://twitter.com/ProstheticRcds
http://www.youtube.com/user/ProstheticRecords
http://facebook.com/prostheticrecords
This entry was posted in New Releases, Reviews, Videos and tagged Canada, Ontario, Prosthetic Records, Sarin, Toronto. Bookmark the permalink.
Comatose – A Way Back (2021)
Soothsayer – At This Great Depth (2016); Echoes of the Earth (2021)


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Posts Tagged 'block
A Tweet Or Two
Tags: #MikeBrown, Big Block Of Cheese Day, block, cheese, cops, Martin Luther King Jr., mike brown, mlk, MLK Jr., New Yorker, Obama, police brutality, Political And Funny Tweets, racism, selma, SOTU, tweets, West Wing
The cast of the West Wing ✓ @PressSec ✓ Cheese puns ✓ Big Block of Cheese Day is coming back. Watch → youtu.be/ex_jJlv5HKk #AskTheWH
The White House (@WhiteHouse) January 16, 2015
Pew poll finds broad support for Obama's #Cuba changes: pewrsr.ch/1KQQaIa
lesley clark (@lesleyclark) January 16, 2015
An early look at next week's cover, by Barry Blitt: nyr.kr/1ylvsww http://t.co/jpZw3K34hB
The New Yorker (@NewYorker) January 16, 2015
Leaving K's class. Her journal entry from yesterday. Teacher called me at 8 last night to tell me how powerful it was http://t.co/yySmMkwN7L
brittany. (@bdoulaoblongata) January 16, 2015
When people of color talk oppression we are not bluffing. It's never been a game. There is no race card.
Talib Kweli Greene (@TalibKweli) January 16, 2015
Stevie Wonder is developing a musical about the Underground Railroad ti.me/1yswALU
TIME.com (@TIME) January 16, 2015
Saw Selma yesterday at Loews on Broadway, the gaggle of schoolchildren were brought to respectful silence and awe as the story unfolded.
Susan Fales-Hill (@susanfaleshill) January 16, 2015
I keep thinking about how casual the officers were around Tamir's still-living self after Tamir was shot. No assistance, no urgency.
deray mckesson (@deray) January 16, 2015
Happy to see that tickets are sold out in Philadelphia...still available in NJ selmastudenttickets.com http://t.co/QA9cXg8qOp
Pretty Foot (@PrettyFootWoman) January 15, 2015
Obama to Host 'Selma' Screening at White House on Friday bit.ly/1AW7ndM
(@Variety) January 16, 2015
UPDATE: US Supreme Court to decide whether same-sex couples nationwide have right to marry under Constitution: abcn.ws/1sHrmwe
ABC News (@ABC) January 16, 2015
Medicare Chief Marilyn Tavenner steps down nbcnews.to/1ysemf2 http://t.co/ypZmPb734o
NBC News (@NBCNews) January 16, 2015
Continue reading 'A Tweet Or Two'
White House Presents: Big Block Of Cheese Day
Tags: big, block, carney, cheese, jay, SOTU, State Of The Union, white house
The Obama White House is pulling out all the stops for the State of the Union. Join in the excitement and fun of a participatory government.
evening all
Tags: achievements, all, allegations, america's, and, approval, are, arlington, arts, auto, awards, Barack, benen, better, block, Boehner, bridge, briefing, cain, campus, car, carney, cartoon, Challenge, champions, change, coming, committee, drunk, gallup, harassment, harry, Herman, home, house, Humanities, hypocrisy, industry, infrastructure, iraq, jay, john, key, kidding, LaHood, lawyers, Mark, me, men, Michelle, monopoly, national, now, Obama, of, off, on, perry, poll, President, President's, press, program, quinnipiac, racism, ray, reid, republicans, rick, sales, senate, sexual, speech, steve, the, toon, troops, up, video, washingtonian, white, withdrawal, women, you, youth
Steve Benen: ….. "It's safe to say Speaker John Boehner does not agree with President Obama's suggestion on Tuesday that Americans are better off now than they were when he took office. "Are you kidding me?!" Boehner said loudly in response to a reporter's question on the comment."
…. Obviously, national conditions aren't close to where they need to be. …. Maybe, if guys like Boehner would start passing jobs bills and stop holding the economy back on purpose, the public would start to feel like the country is on the right track again.
But for those who take reality seriously, there's no real question as to whether the country is better off now than in January 2009:
Then the nation was hemorrhaging jobs; now it's gaining jobs.
Then the economy was shrinking; now the economy is growing.
Then the American automotive industry was on the verge of collapse; now it's starting to thrive.
Then taxpayers were sending money to Wall Street; now taxpayers are being paid back.
Then Osama bin Laden was targeting Americans and our allies; now he's dead and al Qaeda's leadership has been decimated.
Then U.S. troops were headed into the Middle East in greater numbers; now they're headed home with their heads held high.
Republicans, including John Boehner, drove the United States into a pretty deep ditch during the Bush/Cheney era, and conditions are still pretty ugly. That doesn't change the simple fact that the nation is much stronger now than the day the president was inaugurated ….
Full post here
The GOP's racial confusion:
AP: A third former employee considered filing a workplace complaint against Herman Cain over what she deemed aggressive and unwanted behavior when she and Cain, now a Republican presidential candidate, worked together during the late 1990s, the woman told The Associated Press on Wednesday. She said the behavior included a private invitation to his corporate apartment.
The woman said he made sexually suggestive remarks or gestures about the same time that two co-workers had settled separate harassment complaints against Cain, who was then the head of the National Restaurant Association.
Washington Post: Car buyers were out in force in October, snapping up trucks and SUVs and taking advantage of deals on Japanese cars.
U.S. car and truck sales were expected to top 1 million in October, a surprising number for a month when sales are usually slow. When adjusted for seasonal factors, that would be the best pace since the Cash for Clunkers program in August 2009.
Mediaite: In a Quinnipiac poll taken Oct. 25-31, embattled GOP juggernaut Herman Cain continued to show amazing staying power …. (but) The real news from this poll, taken before most of the fallout from Politico's alleged sexual harassment expose, might be that some rays of sunshine are finally hitting President Obama.
….. President Obama gained six points on his approval rating … and is now beating all Republican challengers by margins of 5-16 points. The President has seen steady progress in the polls since rolling out his American Jobs Act in September, taking it on the road, and taking the fight to the Republicans who oppose it.
Elsewhere in the poll, Democrats opened up a lead on the generic House ballot, beating Republicans 42-36, after tying them at 39 in October.
Also, the President is up to 45% approval with Gallup today – that's a rise of seven points in two weeks 😉
Wait – Suskind said the Obama administration didn't like women much:
AP: Michelle Obama has presented national arts and humanities awards to 12 community-based, after-school programs, including for at-risk kids.
The programs use dance, theater, writing, music, history and other art mediums to inspire teen moms and other young people and help them reach their potential. The first lady said at a White House ceremony Wednesday that the programs show that the arts are a lifeline – not a luxury – for many of these kids.
Most of the participants graduate from high school or earn a GED and go on to college.
The 2011 National Arts and Humanities Youth Program Awards are presented on behalf of the President's Committee on the Arts and Humanities in partnership with three national cultural agencies.
President Obama leaves Andrews Air Force Base en route to France
a ghoulishly good afternoon
Categories: All Photos and Articles
Tags: anderson, Barack, benen, block, broke, cain, campaign, drug, economy, executive, Federal, foreign, funny, gop, harassment, Herman, horse, illegal, jobs, law, Mark, nick, Obama, order, perry, plan, policy, President, republicans, rick, sexual, shortages, steve, tax, toon, toons, transactions, women
Nick Anderson
Thanks Loriah 😉
MSNBC: NBC News has confirmed that one woman received a settlement from the National Restaurant Association after complaining about inappropriate sexual conduct by Herman Cain.
NBC News is not disclosing the name of the woman nor characterizing who she is. Cain denied the allegations, saying on FOX this morning he was "falsely accused." "I have never sexually harassed anyone, anyone," he said, "and absolutely, these are false accusations."
Despite being the chief executive officer of the National Restaurant Association, he said he was unaware of any settlement with the accusers, though he didn't deny it. "If the restaurant association did a settlement, I wasn't even aware of it," he claimed, "and I hope it wasn't for much. If there was a settlement, it was handled by some of the other officers at the restaurant association."
TPM: On Fox News this afternoon, Herman Cain dismissed the Politico story dogging his campaign this week in the strongest possible terms. There's only one problem with his story of how the Politico piece came to be: his chief of staff blew a large hole in it already.
On Fox, Cain said that Politico came to him with two anonymous tales of sexual harassment settlements reached while he was the head of the National Restaurant Association. And that's why he didn't respond to Politico's request for comment.
….. That line of reasoning is fine, as far as it goes, except Cain neglected to mention one thing …. his Chief of Staff and famous smoker Mark Block confirmed to TPM that Politico's reporting of the events was accurate.
…. So Cain's strong words to Fox are already being found to be less-than-true. That's not a good sign for a campaign trying to put a tough story behind it.
Greg Sargent: 'Calling BS on GOP ideas for job creation"
As you regulars know, I've been stamping my feet for some time now about big news orgs that amplify the GOP's claim to having a "jobs plan" without asking independent experts whether Republican fiscal prescriptions would actually … create jobs.
So kudos to the Associated Press for weighing in with a bracing analysis that calls BS on the GOP's ideas for job creation, though this one is focused on the 2012 GOP presidential candidates:
Key proposals from the Republican presidential candidates might make for good campaign fodder. But independent analyses raise serious questions about those plans and their ability to cure the nation's ills in two vital areas, the economy and housing.….
President Barack Obama signs an executive order, to head off drug shortages and to prevent price gouging, in the Oval Office, October 31. In attendance was Pharmacy Manager for perioperative services Bonnie Frawley (L), Health and Human Services Secretary Kathleen Sebelius (2nd L), Jay Cuetara (2nd R), a 49-year-old cancer patient who lives in San Francisco, CA, and FDA Commissioner Peggy Hamburg (R).
See the NYT and the White House site
Daniel Bice (JS Online): Herman Cain's two top campaign aides ran a private Wisconsin-based corporation that helped the GOP presidential candidate get his fledgling campaign off the ground by originally footing the bill for tens of thousands of dollars in expenses ….. something that might breach federal tax and campaign law, according to sources and documents.
….. Election law experts say the transactions raise a host of questions for the private organization, which billed itself as a tax-exempt nonprofit, and the Cain team. "If the records accurately reflect what occurred, this is way out of bounds," said a Washington, D.C.-based election lawyer….
Michael Maistelman, a Wisconsin campaign attorney, agreed. "The number of questionable and possibly illegal transactions conducted on behalf of Herman Cain is staggering" ….
Steve Benen: This looks awfully bad for Team Cain. The report points to evidence that two of Cain's top aides set up a now-defunct non-profit entity that illegally financed the campaign, effectively helping it get off the ground. There's nothing in FEC filings to suggest the campaign ever paid Prosperity USA back for the substantial funds it spent on Cain's behalf.
rise and shine – it's friday
Categories: All Photos, All Videos, Articles and West Wing Week
Tags: act, ad, al, american, are, Barack, block, Boehner, cain, Congress, cure, driving, drunk, drunken, economy, ed, eugene, fannie, freddie, friday, gop, growth, Herman, house, jobs, john, mac, mae, Mark, Michelle, mitt, Obama, party, President, republicans, robinson, romney, Schultz, sharpton, smoking, suppression, tea, the, toon, video, voter, week, west, Where, wing
AJC: Pushing a campaign to act without Congress, President Barack Obama will announce on Friday two more executive actions on the economy, both of them small steps intended to give a boost to businesses.
The moves cap a week in which Obama has sought to employ the power of his office as he struggles to make headway on his jobs bill on Capitol Hill.
Obama is directing government agencies to shorten the time it takes for federal research to turn into commercial products in the marketplace. The goal is to help startup companies and small businesses create jobs and expand their operations more quickly.
On the other front, Obama is calling for creation of a centralized online site, to be known as BusinessUSA, for companies to easily find information on federal services. The site is to be up and running within 90 days and will be designed with input from U.S. businesses.
Obama is announcing both steps in presidential memos to be released Friday morning, according to administration officials.
Washington Post: The U.S. economy grew at its fastest clip in a year during late summer as consumers and businesses shrugged off fears of a new recession, according to government data released Thursday that helped drive the stock market to its best day since August.
Investors were also cheered by overnight news that European leaders have reached an agreement on how to address their continent's debt crisis, and the Standard & Poor's 500-stock index ended the day up 3.4 percent. European markets were up even more sharply, with the German Dax index up 5.3 percent.
The agreement in Europe still has many details to be filled in, and the 2.5 percent pace of U.S. economic expansion in the third quarter isn't enough to bring unemployment down quickly, even if it is sustained. But on both sides of the Atlantic, the news on Thursday offered a sense of relief: Maybe the world isn't falling apart after all.
National Journal: On the surface, it seems surprising that the U.S. economy nearly doubled its growth in the third quarter.
July brought tense debt-ceiling negotiations and threat of a government default. In August, a top credit ratings agency cut the United States' top AAA rating. Concerns over European sovereign debt seeped into the U.S. and grew in September as the situation across the Atlantic appeared to be rapidly unraveling. Congress announced no major steps to fix the economy and the weak housing and labor markets treaded water.
But on Thursday, the Commerce Department announced that gross domestic product grew by 2.5 percent during that time, up from just 0.9 percent in the first half of the year.
While 2.5 percent GDP growth hardly signals boom times, it seems to fly in the face of those major headwinds. But economists say it reflects a gradual climb out of recession rather than a rapid shift in the country's fortune.
…. The good news is that the third quarter shared many of the same uncertainties that are likely to shape the fourth: uncertainty over the budget and deficit, threat of a downgrade, and the European debt crisis. The economy, while hardly in fighting shape, pulled through the third quarter to eke out 2.5 percent GDP growth — which is not too shabby.
AP: He is the man with the mustache who takes a rebellious drag on a cigarette in the Herman Cain Internet ad gone viral …. Meet Mark Block, Cain's unorthodox campaign manager …. a Republican strategist and tea party leader who's left a trail of questionable campaign work behind him.
Block has been accused of voter suppression and was banned from running Wisconsin political campaigns for three years to settle accusations he coordinated a judge's re-election campaign with a special interest group.
Records show Block has faced foreclosure on his home, a tax warrant by the Internal Revenue Service and a lawsuit for an unpaid bill. He also acknowledges he was arrested twice for drunken driving…..
Eugene Robinson: The hard-right conservatives who dominate the Republican Party claim to despise the redistribution of wealth, but secretly they love it – as long as the process involves depriving the poor and middle class to benefit the rich, not the other way around.
That is precisely what has been happening, as a jaw-dropping new report by the nonpartisan Congressional Budget Office demonstrates. Three decades of trickle-down economic theory, see-no-evil deregulation and tax-cutting fervor have led to massive redistribution. Another word for what's been happening might be theft.
ThinkProgress: House Republicans took the government to the brink of shutdown last spring by demanding across-the-board budget cuts to many vital programs. Instead of focusing on job creation, as Americans wanted them to, the GOP turned its attention to slashing funds for programs that funded assistance for women and children, local law enforcement, the social safety net, environmental protections, and many other programs they deemed as either too expensive or unnecessary. Worse, when challenged on why they hadn't made the effort to tackle high unemployment, Republicans insisted that their slash-and-burn budget cuts were meant to create jobs.
….. According to a new report from the Center for American Progress' Scott Lilly, those cuts didn't result in the job creating boon Republicans insisted would follow. Instead, it has done just the opposite, as those cuts will result in the destruction of roughly 370,000 jobs……
First lady Michelle Obama visited three Florida cities on Thursday for campaign fundraisers…. "This has been a great day," the first lady told supporters in Ft. Lauderdale. "This is my third city in one day. And I go home tonight and Barack and I get up and go to parent-teacher conference tomorrow morning."
"Just so you know," she said. "Just handling our business."
The Hill: The government's bailout of Fannie Mae and Freddie Mac might end up costing taxpayers billions less than originally estimated, the mortgage giants' regulator said Thursday.
In a new report, the Federal Housing Finance Agency (FHFA) said the actual performance of the government-sponsored enterprises (GSEs) had been "substantially better" than originally projected, lowering the overall bill for the bailout, a longtime sore spot with GOP lawmakers.
…. So far, Fannie and Freddie are beating bailout expectations. In the first year of projections, the GSEs actually tapped the Treasury Department for $19 billion to $73 billion less than anticipated, depending on what scenario was applied.
picture perfect: no 5
By Chipsticks Closed
Categories: All Photos and Picture Perfect
Tags: Barack, Basketball, block, blocking, blocks, funny, love, Obama, pete, President, reggie, souza
President Barack Obama blocks a shot while playing basketball with personal aide Reggie Love at St. Bartholomew's Church in New York, Sept. 23, 2009. (Official White House photo by Pete Souza)
Souza showed Obama a picture on the back of his camera from a friendly one-on-one game Obama played with presidential aide (and former Duke basketball player) Reggie Love. Obama, who was talking trash during the game, is seen leaping up to block a shot, looking great. Souza said when the president saw the photo, he told him, "I order you to make this a jumbo." Before it was posted, the president had Love sign the print with a note complimenting the president on his block.
See all the Picture Perfect photos here


Ms. Lopez's Class is a fun, respectable and safe learning environment!
Love to dance? We Go Noodle Everyday!!!!!! Make sure you open your Go Noodle student account so you can dance at home too!
Class Dojo is an important communication tool. It is the best way to communicate with me in a daily basis. Parents and students can open accounts and track behavior, classroom stories and keep up with weekly announcements. It is easy to sign up and there is also a mobile app available.
My name is Ms. Lopez and I am very excited to be a part of your educational journey this year. I graduated from ASU with a Bachelor's in Education and GCU with a Master's in Educational Leadership. This will be my seventh year teaching. I promote a fun, safe and respectable learning environment. My philosophy is that students learn when they feel safe, loved and enjoy being in the classroom, so I am consistently thinking of fun ways to teach and always promote hands on and engaging lessons. I enjoy watching and playing sports. I also like to have fun! I am so passionate about this profession and I know we will have a Pawsome year!!!
Welcome 6th Grade ELA/SS Cheetahs!!


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Share this Story: B.C. Housing defends Oppenheimer strategy, is 'confident' most can be housed
B.C. Housing defends Oppenheimer strategy, is 'confident' most can be housed
The tent city was given notice of eviction Monday morning, ordering them to clear their tents and leave by Wednesday evening.
Zak Vescera
The tent city at Oppenheimer Park was given notice of eviction Monday morning, ordering them to remove their tents and leave by Wednesday evening. Photo by Arlen Redekop /PNG
B.C. Housing is defending its plan to house homeless residents of Oppenheimer Park, though it acknowledges it may not be able to find permanent homes for them all.
The tent city was given notice of eviction Monday morning, ordering them to remove their tents and leave by Wednesday evening.
B.C. Housing defends Oppenheimer strategy, is 'confident' most can be housed Back to video
A Vancouver city release says B.C. Housing has identified "approximately 140" units of supportive housing in recent months to accommodate residents, some of whom have been in the park since winter. They include addictions services and 24/7 support from staff, though B.C. Housing regional director Brenda Prosken acknowledged there was a need for more units for mobility-restricted tenants.
The camp's population is ever-changing, but on Monday afternoon there were roughly 110 tents, which could hold as many as 200 people. As of Tuesday afternoon, the city said roughly 50 people had accepted housing offers.
Prosken acknowledged it's possible not everyone who wants a permanent home will get one, especially after a "surge" of recent arrivals.But she's confident everyone who wants to move indoors can do so.
"If those numbers continue to escalate in terms of those people who are at the camp demonstrating continuous need, we may have some shelter options available to those individuals," she said.
B.C. Housing's strategy has come under attack from activists who say the strategy of "holding" rooms for residents unintentionally leaves people without shelter.
"It's just moving the problem around and not offering any new housing," Fiona York, Carnegie Community Action Project coordinator and administrator, said Monday. "It's dispersing people for optics."
Prosken says housing is assigned based on need and not a first-come-first-serve basis, noting 35 Oppenheimer residents were housed before Monday.
"Because they were in immediate need of accommodation, they would and could fall under that prioritization category," said Prosken.
Some residents are packing up to leave Oppenheimer park in Vancouver. The Vancouver Park Board issued notice Monday morning instructing campers living in Oppenheimer Park to remove their tents and structures by Wednesday evening. Photo by Arlen Redekop /PNG
She says the agency co-ordinated "natural vacancies," and expedited repairs on units that were uninhabitable.
"Obviously, we never like rooms to stay empty for any long period of time," said Tanya Fader, Portland Hotel Society interim executive director. "But … a lot of them weren't ready before now repair-wise."
Fader says a "housing freeze" was in effect for roughly two months, but PHS still housed new tenants in "extraordinary" circumstances.
PHS has 35 units for Oppenheimer residents, 17 of which were filled by Tuesday morning and some of which were still undergoing repairs.Most other units are in buildings operated by the Atira Women's Resource Society, Lookout Housing and Rain City Housing, most of whom say they have more vacancies than is typical for this time of year.
Fader says that despite speculation most Oppenheimer residents are already on B.C. Housing waiting lists and aren't "jumping the queue" to get a home.
Residents pack up at Oppenheimer park in Vancouver. Photo by Arlen Redekop /PNG
"Some of them have been homeless for five years," she said.
Activists and some Oppenheimer residents say they intend to disobey the eviction time of 6 p.m. on Wednesday.
Asked Monday, Vancouver Mayor Kennedy Stewart didn't say whether the city would pursue an injunction to clear residents who remain. A city release says police will keep a presence in the park but will not immediately remove people after the deadline expires.
Michael Edward, a downtown resident who helped raise the first tent at Oppenheimer in 2014, said he doubts any clearance will be permanent.
"So long as you have people living in those SROs, there will be people willing to stay in the park," he said Monday.
Deputy city manager Paul Mochrie and York said it comes down to the larger issue of a housing shortage.
"It's way beyond the capacity of local governments … The housing mandate really sits with the province and the feds," said Mochrie on Monday. "Obviously, finger-pointing between governments doesn't solve any problems here, but the city is definitely looking to play a role, but it has to be a much broader solution."
-With files from Dan Fumano
zvescera@postmedia.com
twitter.com/zakvescera


Jackie, Healthy Witness: Thankful Thursday!
Are you taking care of any issues that you have now? Maybe your knees, feet or back ache or maybe your BP is borderline or you have swelling or arthritis somewhere in your body or a symptom that's just not quite right. Have you ever ask your doctor about them or sought out a specialist?
Take care of your body now. It's the only one you have. The more proactive you are early on, the less maintenance or issues you will have later. It is obviously no guarantee, but it is worth taking care of things now when they might be a small problem rather than wait until they get worse. I will tell you it is worth it because you are worth it.
It's no different than a car or a house or anything that you own that requires maintenance and parts. It is best to fix or repair them as they start to weaken or not function 100% than it is to wait until it's too late and more difficult to fix/repair.
I am so THANKFUL that I was able to take care of my feet problems now. I was born with bunions that progressively got worse until adulthood. When I was 10 years old, a podiatrist told me I would have surgery eventually. I had successfully lost the weight I wanted to lose (2011-2013) and after two years of Maintenance (2013-2015), I wanted to add Fitness. During the time of adding Fitness (2015), my feet revealed more pain and problems then before. It was clear that now was the time to fix them.
It is your personal responsibility for yourself and loved ones to keep yourself a priority. It is not selfish it is wise. You are loved and appreciated. Live your days as healthy as you can within your resources and do it with support. I am here to encourage you, share resources, and remind you that you are loved and you are worth the time.
I am off to rest now, something I don't do very well. But it is because of others who encourage me and support me that help me make these decisions to be wise and properly care for myself so I can serve and care for others. It is my passion to serve you and encourage you because I know support equals long-term success.
Don't put off something you KNOW needs your attention... Act now, while you're in control! Message me any personal or unique need that you may have. I enjoy consulting others in nutrition and fitness as well as healthy habits to support your personal goals.


The California Data Breach Report "provides an analysis of the data breaches reported to the California attorney general from 2012-2015." In nearly all cases, the breaches exploited vulnerabilities for which fixes had been available for more than a year. California state law states, "A business that owns, licenses, or maintains personal information about a California resident shall implement and maintain reasonable security procedures and practices appropriate to the nature if the information." The report goes on to say that organizations that do not implement the Center for Internet Security's (CIS) 20 Critical Security Controls would be found to demonstrate "a lack or reasonable security."
Two US Navy admirals have asked Defense Secretary Ash Carter to add industrial control systems (ICS) to the monthly scorecard that defense agencies submit to Carter's office. If left unaddressed, the issue could affect the "ability to execute assigned missions."
A US federal judge has found that the US Department of Defense (DoD) funded research at Carnegie Mellon University's Software Engineering Institute (SEI) on the Tor network. That research was subpoenaed by the FBI and used to determine the identity of a suspect in the Silk Road 2.0 case.
In a pre-release advisory, OpenSSL developers say they will release patches for several security issues, some high severity, on Tuesday, March 1. The versions of the cryptographic library scheduled for release next week are 1.0.2g and 1.0.1s. The advisory also reminds users that support for OpenSSL version 1.0.1 will end on December 31, 2016.
Police in Germany now have the authority to use spyware to infiltrate suspects' computers and mobile devices under certain circumstances. Police must obtain a court order to use the spyware, and may use it only in cases where life is at risk or a threat to the state. Some are skeptical about the malware's ability to be used in a targeted way that does not infringe on users' privacy.
According to PwC's Global Economic Crime Survey 2016, nearly one-third of organizations surveyed said they had experienced cybercrime. The report explains the surprisingly low percentage by noting, "the insidious nature of this threat is such that of the 56 percent who say they are not victims, many have likely been compromised without knowing it." The report also found that just 37 percent of organizations have established a cyber incident response plan. "Many boards are not sufficiently proactive regarding cyber threats." The report draws its statistics from responses from more than 6,000 organizations in 115 countries.
Mozilla has granted Symantec an exception from the SHA-1 certificate ban, allowing the company to issue nine SSL/TLS certificates signed with the weak hashing algorithm to payment processor Worldpay for 10,000 payment terminals. Worldpay did not migrate all of its SSL/TLS servers to SHA-2 certificates by the January 1, 2016, deadline, and did not obtain new SHA-1 certificates prior to the end of last year.
Nissan has pulled a mobile app for its Leaf vehicles due to unsecure APIs that could be exploited to take remote control of certain functions. The app required no authentication. Attackers would need to know only the vehicle identification number (VIN) assigned to the car to access the vehicle's climate control and battery charge management systems. Nissan plans to relaunch the app, NissanConnectEV, when it is fixed.
Google is offering its Project Shield distributed denial-of-service (DDoS) attack fighting service free to organizations that publish news. Project Shield debuted in 2013 as part of Google's Page Speed service. Google views DDoS attacks as a form of censorship, so protecting news sites from those attacks maintains the flow of information. Human rights websites and election-monitoring websites are also eligible for the free service.
A US district judge in North Carolina has sentenced Nikhil Nilesh Shah to 30 months in prison for sabotaging his former employer's servers. Shah was an IT manager at SmartOnline. He left that company in March 2012, and in June of that same year, he sent malicious code to his previous employer's servers, deleting much of the company's intellectual property.
Asus has agreed to the terms of a settlement with the US Federal Trade Commission (FTC) regarding vulnerabilities in its home routers and cloud services. The FTC noted that Asus frequently "did not address security flaws in a timely manner and did not notify customers about the risks posed by the vulnerable routers." The settlement calls for Asus to establish and maintain a comprehensive security program and to undergo audits every two years for the next 20 years.


text:
package org.eclipse.bpel4chor.model.pbd.impl;

import java.util.Collection;

import org.eclipse.bpel4chor.model.pbd.MessageExchange;
import org.eclipse.bpel4chor.model.pbd.MessageExchanges;
import org.eclipse.bpel4chor.model.pbd.PbdPackage;

import org.eclipse.emf.common.notify.NotificationChain;

import org.eclipse.emf.common.util.EList;

import org.eclipse.emf.ecore.EClass;
import org.eclipse.emf.ecore.InternalEObject;

import org.eclipse.emf.ecore.util.EObjectContainmentEList;
import org.eclipse.emf.ecore.util.InternalEList;

/**
 *
 * An implementation of the model object 'Message Exchanges'.
 *
 *

 * The following features are implemented:
 *

 *
{@link org.eclipse.bpel4chor.model.pbd.impl.MessageExchangesImpl#getMessageExchange Message Exchange}

 *

 *

 *
 * @generated
 */
public class MessageExchangesImpl extends ExtensibleElementsImpl implements MessageExchanges {
	/**
	 * The cached value of the '{@link #getMessageExchange() Message Exchange}' containment reference list.
	 *
	 *
	 * @see #getMessageExchange()
	 * @generated
	 * @ordered
	 */
	protected EList<MessageExchange> messageExchange;

	/**
	 *
	 *
	 * @generated
	 */
	protected MessageExchangesImpl() {
		super();
	}

	/**
	 *
	 *
	 * @generated
	 */
	@Override
	protected EClass eStaticClass() {
		return PbdPackage.Literals.MESSAGE_EXCHANGES;
	}

	/**
	 *
	 *
	 * @generated
	 */
	public EList<MessageExchange> getMessageExchange() {
		if (messageExchange == null) {
			messageExchange = new EObjectContainmentEList<MessageExchange>(MessageExchange.class, this, PbdPackage.MESSAGE_EXCHANGES__MESSAGE_EXCHANGE);
		}
		return messageExchange;
	}

	/**
	 *
	 *
	 * @generated
	 */
	@Override
	public NotificationChain eInverseRemove(InternalEObject otherEnd, int featureID, NotificationChain msgs) {
		switch (featureID) {
			case PbdPackage.MESSAGE_EXCHANGES__MESSAGE_EXCHANGE:
				return ((InternalEList<?>)getMessageExchange()).basicRemove(otherEnd, msgs);
		}
		return super.eInverseRemove(otherEnd, featureID, msgs);
	}

	/**
	 *
	 *
	 * @generated
	 */
	@Override
	public Object eGet(int featureID, boolean resolve, boolean coreType) {
		switch (featureID) {
			case PbdPackage.MESSAGE_EXCHANGES__MESSAGE_EXCHANGE:
				return getMessageExchange();
		}
		return super.eGet(featureID, resolve, coreType);
	}

	/**
	 *
	 *
	 * @generated
	 */
	@SuppressWarnings("unchecked")
	@Override
	public void eSet(int featureID, Object newValue) {
		switch (featureID) {
			case PbdPackage.MESSAGE_EXCHANGES__MESSAGE_EXCHANGE:
				getMessageExchange().clear();
				getMessageExchange().addAll((Collection<? extends MessageExchange>)newValue);
				return;
		}
		super.eSet(featureID, newValue);
	}

	/**
	 *
	 *
	 * @generated
	 */
	@Override
	public void eUnset(int featureID) {
		switch (featureID) {
			case PbdPackage.MESSAGE_EXCHANGES__MESSAGE_EXCHANGE:
				getMessageExchange().clear();
				return;
		}
		super.eUnset(featureID);
	}

	/**
	 *
	 *
	 * @generated
	 */
	@Override
	public boolean eIsSet(int featureID) {
		switch (featureID) {
			case PbdPackage.MESSAGE_EXCHANGES__MESSAGE_EXCHANGE:
				return messageExchange != null && !messageExchange.isEmpty();
		}
		return super.eIsSet(featureID);
	}

} //MessageExchangesImpl

redpajama_set_name: RedPajamaGithub


text:
EXCLUSIVE: Michael Bolton In Talks To Join Dancing With The Stars
Aug. 24 2010, Published 7:40 a.m. ET
Singer Michael Bolton is making a comeback – on the dance floor!
RadarOnline.com can exclusively reveal that the 57-year-old Grammy Award winner is in negotiations to join the cast of Dancing With The Stars.
"Michael's management company had been in talks with the show for over a month," an inside source told RadarOnline.com. "They're all very excited about it, especially Michael."
DWTS co-host Brooke Burke made an appearance Monday on Lopez Tonight where she denied knowing who any of the contestants are, but slyly name-dropped Bolton as a good potential for the show.
Rumors are also swirling that Jersey Shore's Mike "The Situation" Sorrentino may also be joining the show which will begin its 11th season this fall.
VIDEO: Dancing With The Stars Viewers Choose Their Favorite Pros
Bolton achieved great success in the 80's and 90's with hits such as How Am I Supposed To Live Without You and When A Man Loves A Woman.
The crowd-pleasing ABC show has hosted famous contestants in the past such as Kate Gosselin, Shawn Johnson, Jake Pavelka, Pamela Anderson, Kelly Osbourne and Lance Bass.
VIDEO INTERVIEW: Dancing With The Stars Contestants Rate The Judge's Comments
VIDEO: In The Closet Of Dancing With The Stars
VIDEO: Ashley Tisdale Gives "Fake Obama" Lap Dance