text:
\section{Introduction}
\label{intro}
In \cite{2017,compa}, we have proposed a model mimicking quantum mechanics (QM) with local, realistic, and stochastic features.

In the proposed model, the results of nonrelativistic spinless QM systems have been interpreted as probability distributions of similarly-prepared ensembles of particles that are emitted by one or multiple sources. At a given time, individual particles have definite values for position and momentum, among other observables, thus fulfilling \textit{realism}.

The \textit{stochastic} behavior that is manifested by the empirical evidence of QM is explained by assuming a fundamental randomness both in preparation and in particles trajectories. The emergence of QM behavior is a consequence of the particular rules of motion chosen. The motion of individual particles and their interaction with external forces take place on a discrete space–time under the form of a lattice. Particle trajectories are asymmetric random walks, with transition probabilities being simple functions of a few quantities that are either randomly attributed to the particles during their preparation, or stored in the lattice nodes that the particle visits during the walk.

The lattice-stored information is progressively built as the nodes are visited by successive emissions. This process, where particles leave a ``footprint" in the lattice that is used by subsequent particles, is ultimately responsible for the QM behavior. Therefore the interactions between subsequent emissions fulfill \textit{localism}, albeit through the mediation of the lattice. Interference and the Born rule emerge as a consequence of this mechanism.

In its nature, the proposed model could be regarded as the first full embodiment of the non-ergodic interpretation (NEI) proposed since the 1980s by V. Buonomano \cite{buonomano,selleri}. The NEI idea was that, although separated by large time intervals, subsequent particles may interact with one another by means of memory effects in a medium (in our case, the lattice). That is, the usual assumption that time average coincide with ensemble averages is not true \cite{khrennikov2}.

In this paper, we aim at extending the model to cover spin.

The main characteristics that distinguishes quantum spin from classical magnetic moment behavior is probably the quantization of the former after a measurement, e.g., by a Stern-Gerlach (SG) apparatus, is performed.
This behavior is described in standard QM using matrices and eigenvectors.
In alternative theories, spin has been derived from path integrals \cite{nielsen} and
Nelson's stochastic mechanics \cite{dankel,garbaczewski}. %
The local-realistic mechanism proposed here only involves a few additional quantities carried on by the particles.
These variables are subject to stochastic preparation at sources and time evolution, including interaction with the lattice nodes storing the information about the magnetic field.

The proposed model is also used to demonstrate spin entanglement in addition to momentum entanglement already incorporated in \cite{2017}.
Bell theorem is regularly used to dismiss any possibility that a local realist theory could explain spin entanglement results.
This is because its main assumptions are the existence of hidden variables (``realism") and locality.
Rejecting either of the two leads to the standard approach of indeterminism (Copenhagen interpretation) or the non-standard approach of nonlocality (e.g., Bohm interpretation).
However, Bell's theorem is also based on a third assumption, that of ``measurement independence" (MI) that is rather strong.
The fact that MI is not satisfied have been often explained by some kind of (super)determinism or ``conspiracy".
However, other, less unpleasant reasons to renounce to MI exist.

Although non-ergodic solutions would in principle belong to this category, no proposal has been issued in this direction as per the best knowledge of the author \cite{buonomanoEPR}.
In \cite{vervoort} a toy contextual model for which MI is not satisfied is presented.
Other attempts include the use of negative probabilities \cite{mueckenheim}, or heuristic models that correctly reproduce the QM correlations by renouncing to MI in an abstract and physically unexplained fashion \cite{dilorenzo}.
A different approach is to renounce to the ``fair detection" assumption and assume instead that the probability of joint detection (detector efficiency) depends on the settings \cite{thompson,selleri}.

However, the most interesting attempt to reproduce Malus' law and QM predictions in a local-realistic context seems the event-based class of models proposed by \cite{michielsen,deraedt}. In this approach, the key role is played by the time delay between particle arrivals at the detectors of a Bell-type experiment, so that coincidences are counted only if two particles arrive at roughly the same time. Such a time delay is heuristically designed and explained with the properties of the apparatus.
In the proposed model, the time-of-fly plays a similar role in determining coincidences to be counted.
However, it is a natural consequence of the rules of motion in the presence of magnetic forces and a stochastic quantity due to stochastic preparations.

The paper is organized as follows. In Sect. \ref{sec:summ} a summary of the three-dimensional spinless model is presented, with some additional considerations with respect to \cite{2017}.
Then in Sect. \ref{sec:spin} the additional rules for spin $1/2$ are introduced, both for homogeneous and inhomogeneous (SG) fields. These results are extended to higher spins in Sect.~\ref{sec:higher}. Two-particle spin entanglement is discussed in Sect. \ref{sec:entang}.

\section{Summary of the Local-Realistic Model for a Spinless Particle}\label{sec:summ}

In this section, we summarize the rules for particle emissions (Sect.~\ref{sec:emis}), microscopic motion (Sect.~\ref{sec:mot}), and how probability densities are derived from them (Sect.~\ref{sec:prob}). The reader is referred to \cite{2017} or its companion paper \cite{compa} for more detail.

\subsection{Lattice and particle emissions} \label{sec:emis}
The lattice is composed of three spatial dimensions $x=\{x_d\}\in \mathbb{Z}^3$, and one temporal dimension $n\in \mathbb{N}$. Each of the spatial dimensions is characterized by the same fundamental length and acts independently.

We describe ensembles of particles that are emitted at some sources after having been similarly prepared.
Source setting consists in defining the number of sources $N_s$, their location $x_0^{(k)}$, probabilities $P_0^{(k)}$ (such that $\sum_k P_0^{(k)}=1$), and phase $\epsilon^{(k)}$, with $k\in[1,N_s]$.

Each particle is emitted from one source, randomly chosen among those available according to their probabilities. Thus source preparation fixes the source position $x_0=\{x_{0d}\}\in \mathbb{Z}^3$ and the source phase $\epsilon=\{\epsilon_d\}\in\mathbb{Q}^3$.
Additionally, the source momentum $v_0=\{v_{0d}\}\in \mathbb{Q}^3$, and the source (momentum) polarization $\rho=\{\rho_d\}\in\mathbb{Q}^3$ are randomly attributed to the particle. The latter two quantities are further subject to the conditions $\sum_{d=1}^3\rho_d^2=1$, $\sum_{d=1}^3 v_{0d}^2\leq 1$.

\subsection{Microscopic motion} \label{sec:mot}
Microscopic motion is defined by a set of rules involving quantities carried by particles and quantities carried by lattice node (subscript $xt$).

The particle-carried quantities (or ``counters") are: its span $\ell=\{\ell_d\}\in\mathbb{Z}^2$, lifetime $t\in \mathbb{N}$, momentum $v=\{v_d\}\in \{-1,0,1\}^3$, momentum propensity $\bm{v}:=\mathbb{E}[v]$, ``energy" $e:=\mathbb{E}[v^2]$, quantum momentum $v_Q=\{v_{Qd}\}\in\mathbb{Q}^3$, momentum due to external forces $v_F=\{v_{Fd}\}\in\mathbb{Q}^3$.

Particles exchange momentum-mediating entities called ``bosons'' with the lattice, according to the mechanism illustrated below. External-force bosons (FB) carry a momentum $f=\{f_d\}\in\mathbb{Q}^3$, while quantum particle bosons (PB) carry momenta $w^{(\cdot)}\in\mathbb{Q}$ and their own lifetime $t^{(\cdot)}$.

The particle's motion rules are summarized as
\begin{eqnarray}
&& t[n]=t[n-1]+1, \quad t[n_0]=0 \label{eqn:tn}\\
&& \ell_d[n]=\ell_d[n-1]+v_d[n], \quad \ell_d[n_0] = 0 \label{eqn:ell}\\
&& x_d[n]=x_{0d}+\ell[n]
\label{eqn:xn}\\
&& \mathbb{P}(v_d[n]=\pm 1)=\frac{e_d[n]\pm \bm{v}_d[n]}{2}\;,\quad \mathbb{P}(v_d[n]=0)=1-e_d[n] \label{eqn:prv}\\
&& e_d[n]=\frac{1+\bm{v}_d^2[n]}{2} \label{eqn:ene}\\
&& \bm{v}_d[n]=v_{Qd}[n]+v_{Fd}[n]
\label{eqn:vprop}\\
&& v_{Fd}[n]=\sum_{n'=n_0+1}^n f_d(x[n'],n') \label{eqn:force}\\
&& v_{Qd}[n]=v_{0d}-\rho_d^2 \sum_\ell \sum_{\lambda\neq \ell} w^{(\ell\lambda)}[n] \label{eqn:vq}\\
&& w^{(\ell\lambda)}[n]= w^{(\ell\lambda)}[n-1] \cdot\left(1-\frac{1}{2t^{(\ell\lambda)}}\right) \label{eqn:pbm}
\end{eqnarray}
where $n_0$ is the iteration at which the emission has taken place.

Equations (\ref{eqn:tn})--(\ref{eqn:xn}) describe the increments of lifetime, span, and position as a function of momentum. Equations (\ref{eqn:prv})--(\ref{eqn:ene}) relate the probability distribution of momentum to momentum propensity. Equation (\ref{eqn:vprop}) states that momentum propensity is the sum of two contributions, due to quantum and external forces, respectively.
External forces are described by interactions with the lattice, where each node can be occupied by a force boson. When a particle visits the node, it captures the resident FB and incorporates its momentum as described in (\ref{eqn:force}). A new boson is then recreated at the node.
In (\ref{eqn:vq}), quantum momentum is initially set to the source momentum and then build up from an exchange of bosons and their momenta between the particle and the lattice. The dynamics of the PB-momentum is given in (\ref{eqn:pbm}).

Lattice-carried quantities are the span trace $\lambda_{xt}=\{\lambda_{d,xt}\}\in\mathbb{Z}^3$ and the phase trace, $\epsilon_{xt}\in\mathbb{Q}^3$, which represent the memory of the span and phase carried by the last particle that has visited the node $x$ with lifetime $t$. Additionally, the exchange with particles generate quantum lattice bosons (LB), carrying momenta $\omega_{xt}^{(\cdot)}\in\mathbb{Q}$, whose dynamics read
\begin{eqnarray}
&& \omega_{xt}^{(\ell\lambda)}[n]=\omega_{xt}^{(\ell\lambda)}[n-1]\cdot\left(1-\left(\frac{\omega_{xt}^{(\ell\lambda)}[n^{(\ell\lambda)}_{QR}]}{t^{(\ell\lambda)}}\right)^2\right)\;. \label{eqn:lbm}
\end{eqnarray}

Rules (\ref{eqn:tn})--(\ref{eqn:lbm}) are partially overcome in case of a quantum reset or an external reset. A Quantum Reset (QR) occurs when $\ell_d \neq \lambda_{d,xt}$ for at least one dimension $d$. If it is the case, the following exchanges apply:
\begin{eqnarray}
&& w^{(\ell\lambda)} \xLeftarrow[QR]{} \frac{\omega_{xt}^{(\ell\lambda)}}{\sum_{d=1}^3\rho_d^2 \delta_d^{(\ell\lambda)}} \label{eqn:pb}\\
&& \omega_{xt}^{(\ell\lambda)} \xLeftarrow[QR]{} \sum_{d=1}^3 \delta_d^{(\ell\lambda)} v_{Qd} -\epsilon^{(\ell\lambda)} \label{eqn:lb}\\
&& \ell_d \xLeftrightarrow[QR]{} \lambda_{d,xt} \label{eqn:swspan}\\
&& \epsilon \xLeftrightarrow[QR]{} \epsilon_{xt}  \label{eqn:sweps}
\end{eqnarray}
where $\delta_d^{(\ell\lambda)} := |\ell_d-\lambda_{d,xt}|$ is the path difference and $\epsilon^{(\ell\lambda)} := \epsilon-\epsilon_{xt}$ is the phase difference.

Rules (\ref{eqn:pb})--(\ref{eqn:lb}) state that the QR creates a new momentum-carrying LB, labeled $\ell\lambda$ to unambiguously identify the information carried by the particle, resp., the lattice node. The new LB replaces the old one of the same type, which is transferred to the particle and becomes a particle boson (PB). Rules (\ref{eqn:swspan})--(\ref{eqn:sweps}) describe the exchange of counters between the particle and the lattice.

An External Reset (ER) occurs when an external-force boson is captured, and is defined by the rules
\begin{eqnarray}
&& v_{d0} \xLeftarrow[ER]{} \bm{v}_d \label{eqn:erv0}\\ && \ell_d \xLeftarrow[ER]{} \ell_d-2f_d \frac{\sum_{d'=1}^3 \ell_{d'}f_{d'}}{\sum_{d'=1}^3 f_{d'}^2}
\label{eqn:er} \\ && \epsilon \xLeftarrow[ER]{} \epsilon+1\;.\label{eqn:ereps}
\end{eqnarray}
Although not necessary, rule (\ref{eqn:erv0}) is introduced here for the sake of model elegance. It states that each ER can be seen as a new emission, thus removing the special role of sources that are now seen as just the nodes where the last interaction has taken place.
Rule (\ref{eqn:er}) generalizes the 1D situation where the span's sign in inverted at each external interaction. Rule (\ref{eqn:ereps}) adds a $\pi$ phase angle after each interaction.

\subsection{Probability densities} \label{sec:prob}
The source position, momentum, and polarization are treated as random variables. In particular, the probability density function of the source momentum is  $\rho(v_{0d})=(1/2)$, $\forall d$.

Stochastic preparation implies that $\bm{v}$ and thus $x$ are random variables, too. We aim at evaluating the probability mass function of the position for an ensemble of similarly-prepared particles. Unfortunately, it is generally not possible to explicitly evaluate $\rho(x;t)$. However, as discussed in \cite{2017}, for sufficiently large times we can use the approximation $\rho(x;t)\approx \rho(\bm{x};t)$, where $\bm{x}\in\mathbb{R}$ is the expected value of the position\footnote{We shall generally denote expected values with bold letters.}.

We describe in the rest of this section the procedure to evaluate the joint pdf's $\rho(\bm{x};t)$ and $\rho(\bm{v}_Q)$ in the presence of quadratic potentials, potential barriers, or geometrical constraints.

\subsubsection{Quadratic potentials}

It was shown in \cite{2017} that, in case of a quadratic potential, the expected motion is given by
\begin{equation}
    \bm{x_d}=A_d(t)x_{0d}+B_d(t)\bm{v}_{Qd}+C_d(t) \label{eqn:xd}
\end{equation}
and
\begin{eqnarray}
\bm{v}_{Qd} &=& v_{0d}-\rho_d\sum_i \sum_{j\neq i} \sqrt{P_0^{(i) } P_0^{(j)}} \frac{\sin(arg)}{\pi\sum_{d=1}^3 \rho_d\delta_d^{(ij)}}
\label{eqn:vq}
\\
(arg) &=& \pi\sum_{d=1}^3 \delta_d^{(ij)}  \frac{\bm{x_{d}}-A_d(t)\frac{x_{0d}^{(i)}+x_{0d}^{(j)}}{2}-C_d(t)}{B_d(t)}-\pi\epsilon^{(ij)}\;.
\label{eqn:arg}
\end{eqnarray}
where $A(t)$, $B(t)$, and $C(t)$ are functions of lifetime whose form depends on the FB momentum (external force) $f(x,t)$. For example, a free particle is described by $A=1$, $B=t$, $C=0$; a free faller by $A=1$, $B=t$, $C=ft^2/2$; an harmonic oscillator by $A=\cos\Omega t$, $B=\sin\Omega t/\Omega$, $C=0$.

The joint pdf of the positions is found by applying the rule
\begin{equation}
    \rho(\bm{x};t)=\frac{1}{2}\left|\frac{\partial (v_{01},\ldots,v_{03})}{\partial(\bm{x}_1,\ldots,\bm{x}_3)}\right| \;,
\end{equation}
yielding
\begin{equation}
    \rho(\bm{x};t) = \frac{1+\sum_i \sum_{j\neq i} \sqrt{P_0^{(i) } P_0^{(j) } }  \cos \left(arg \right) }{\prod_{d=1}^3 2|B_d(t)|} \label{eqn:rhox3d}
\end{equation}
In \cite{2017} it was shown as well that the Schr\"{o}dinger equation and Born rule can be retrieved from (\ref{eqn:rhox3d}).

Similarly, the joint pdf of the momenta is evaluated as
\begin{equation}
    \rho(\bm{v}_Q;t)=\frac{1}{2^3} \left(1+\sum_i\sum_{j\neq i}\sqrt{P_0^{(i)} P_0^{(j)}}  \cos\left( \pi\sum_{d=1}^3 \delta_d^{(ij)}\bm{v}_{Qd}-\pi \epsilon^{(ij)}\right) \right) \;. \label{eqn:rhov}
\end{equation}

Note that the expected-motion equations (\ref{eqn:xd})--(\ref{eqn:arg}) can be used, alongside with the source settings, as an approximated version of the model to accelerate numerical simulations\footnote{As explained in \cite{compa}, the quantum momentum should be filtered by a first-order lag before inserted in (\ref{eqn:vq}) in order to avoid numerical instabilities and represent the non-instantaneousness of the training process.}.

\section{Spin \textonehalf} \label{sec:spin}

A spin mechanism is now superimposed to the microscopic motion mechanism treated in the previous sections.
The momentum quantities $v_0$ and $v$ have their spin counterparts $s_0$ and $s$. The (momentum) polarization $\rho$ has its spin counterpart in the (spin) polarization $\mu$.
The latter two quantities might actually coincide: at the present stage of the model development, we do not have reasons to distinguish them but for their naming.

\subsection{Microscopic motion} \label{sec:spinmot}
Particles are emitted with additional, randomly-attributed properties denoted as ``source spin", $s_0\in[-1,1]$ and ``source polarization", $\mu_0:=\{\mu_{0d}\}\in \mathbb{Q}^3$, such that $\sum_{d=1}^3 \mu_{0d}^2 =1$.

While $s_0$ remains constant during a particle's evolution, polarization is prone to change at each time the particle experiences a magnetic field, which is represented under the form $B:=\{\lambda_d \} B_M$, such that $\sum_{d=1}^3 \lambda_d^2=1$. Clearly, $\lambda$ represents the unit vector along which the physical field is directed. The quantity $B_M$ represents the magnitude of the magnetic field in lattice units.

The evolution of the polarization follows the rule
\begin{multline}
\mu_d [n+1]=\mu_d [n]-\gamma\mu_M B_M \left(\mu_{d+1} [n] \lambda_{d+2} [n]-\mu_{d+2} [n] \lambda_{d+1} [n] \right
\label{eqn:polarev}
\end{multline}
with $\mu_d [n_0]=\mu_{0d}$. The quantity $\mu_M$ represents the magnitude of the magnetic moment of the particle, and the dimension indexes must be taken as modulo three. The quantity $\gamma$ represents the gyromagnetic ratio.
Note that, if the $\lambda_d$'s are constant, the sum $\sum_d\mu_d^2$ is constant, too.
Similarly, the sum $M:=\sum_d\mu_d\lambda_d$ does not change with iterations if the field is constant\footnote{The quantity $M$ clearly corresponds to the cosine of the angle between the two directions $\mu_0$ and $\lambda$.}. We also define for later use $M_0:=\sum_{d} \mu_{0d} \lambda_d$.

The quantity denoted here as "spin" is a bivalued quantity $s\in\{-1,1\}$ that varies during the particle's evolution according to the rule
\begin{equation}
s[n] = \mathrm{sign}(s_0+M[n]) \;.
\label{eqn:s_def}
\end{equation}
Clearly, the expected value of spin $\mathbb{E}[s[n]]=M[n]$. For such reason, we shall denote the quantity $M$ as ``spin propensity" in the following.

Assuming that $\lambda$ is constant, the magnetic force (external boson momentum) due to spin is described in analogy to the classical expression,
\begin{equation}
f_d [n]=-\mu_M \sum_{d'=1}^3 \mu_{d'} [n] \frac{\partial B_M}{\partial x_d}[n] \lambda_{d'}[n]\;.
\label{eqn:magnforce}
\end{equation}
We shall assume for later use that $B_M$ is parameterizable as $\partial B_M/\partial x_d:=B_F \nu_d$, with $\sum_{d=1}^3 \nu_d^2 =1$. The force is thus directed along the $\nu:=\{\nu_d\}$ direction and we can define its magnitude as $f_\nu [n]=-\mu_M B_F M[n]$.

When an external boson is captured, besides the actualization of its momenta and spans (\ref{eqn:erv0})--(\ref{eqn:ereps}), the particle undergoes an External Reset (ER) of its polarization,
\begin{equation}
\mu_d \xLeftarrow [\mathrm{ER}]{} s \lambda_d \;.
\label{eqn:88}
\end{equation}
This reset causes an instantaneous variation of the spin propensity, that jumps to the current spin. In analogy with classical physics, we define a ``magnetic energy" $\mathcal{E}:=-\mu_MB_M M$. The variation of magnetic energy across the ER is compensated by a reset of momentum propensity in the direction perpendicular to force, which is described by the rules
\begin{eqnarray}
&& \bm{v}_d \xLeftarrow [\mathrm{ER}]{} \alpha \bm{v}_d+(1-\alpha)\nu_d \sum_{d=1}^3 \bm{v}_d \nu_d \;, \label{eqn:er_v}\\
    && \alpha^2 = \frac{\sum_d \bm{v}_d^2-\mu_MB_M(s-M)}{\sum_d \bm{v}_d^2} \;.
    \label{eqn:magnen}
\end{eqnarray}

\subsection{Probability densities}

We aim at evaluating the pmf
$\rho(s)$, which results from the particular preparation at the source and the nature of the magnetic field experienced by the ensemble of particles. We shall consider first a preparation (``pure state") for which the
source polarization has a definite value $\mu_{0}$ for all the particles of the ensemble.

\subsubsection{Homogeneous field}
If the magnetic field $B_d=B_M \lambda_d$ is homogeneous in space (though possibly variable with time), no magnetic force is experienced, thus no external reset occurs.
If the field is also constant, $M$ does not change with the iterations and thus is always equal to its initial value,  $M[n]\equiv M_0$.

From (\ref{eqn:s_def}), we have that $s$ is also constant as
\begin{equation}
s[n] = \left\{ \begin{array}{ll}
1\:, \quad & s_0 > -M_0 \\
-1\:, \quad & s_0 < -M_0 \\
\end{array} \right. \;.
\label{eqn:ss0}
\end{equation}
Since $s_0=U[-1,1]$, the probabilities of spins up and down are evaluated as
\begin{equation}
\rho(\pm 1) = \mathbb{P}(s=\pm 1)=\frac{1\pm M_0}{2}=
\frac{1\pm\cos (\mu_0,\lambda)}{2}\;,
\label{eqn:rho_up}
\end{equation}
and is easily generalized to the case of a variable field, in perfect agreement with QM prediction.

The meaning of the polarizations in the model can be now clarified. If the field is along one particular direction $d$, then $\lambda_d=1$, and consequently $\rho(1)=(1+\mu_d)/2$, $\rho(-1)=(1-\mu_d)/2$. The expectation of the spin is therefore $\bm{s}=(1)(1+\mu_d )/2+(-1)(1+\mu_d)/2=\mu_d$. Thus the $d$-polarization represents the standard QM quantity $\langle S_d \rangle$, that is, the expected value of the spin measured along the $d$ direction.

It should be also apparent that the standard QM spinor formulation of a spin state can be retrieved by defining the complex vector quantity
\begin{equation}
    \chi=\left(\sqrt{\frac{1+\mu_{3}}{2}},\frac{\mu_{1}-\iota\mu_{2}}{\sqrt{2} \sqrt{(1+\mu_{3})}}\right)^T\;.\label{eqn:spinor}
\end{equation}
from which all standard results can be obtained.

\subsubsection{1D-inhomogeneous field (Stern-Gerlach apparatus)}
We shall consider now the case where the prepared particles pass through a Stern-Gerlach apparatus. Inside this apparatus, the field has a prevalent magnitude $B_M$ along a constant direction $\lambda$ and some small inhomogeneity inducing a magnetic force of magnitude $\mu_MB_F$ along the constant direction $\nu$.

Outside the SG, $\lambda=0$, $\mu\equiv\mu_0$, and so is $M=0$. Thus, spins up and down are equally distributed.
Let us define $n_i$ as the iteration at which particles enter the SG. The spin propensity becomes $M[n_i]=M_0$. Thus, the probability distribution of $s[n_i]$ follows (\ref{eqn:rho_up}).

The presence of a magnetic force in the SG activates the External Reset condition. We shall assume, for the sake of discussion only, that each node inside the SG hosts a magnetic-force boson and thus the first ER occurs right at the SG entry at $n_i$. Then,
\begin{equation}
M \xLeftarrow [\mathrm{ER}]{} s\sum_{d=1}^3|\lambda_d^2=s  \;,
\end{equation}
that is, the spin propensity ``jumps" to the current spin value. At the immediately next iteration, the application of (\ref{eqn:s_def}) states that $\mathbb{P}(s[n_i+1]=\sigma)=(1+\sigma s[n_i])/2$. In other words, if $s[n_i]=1$, then $s[n_i+1]$ will be 1 with probability one. Inversely, if if $s[n_i]=-1$, then $s[n_i+1]$ will surely remain -1. At successive ER's, the situation does not change and thus the spin remains constant throughout the whole SG apparatus. The probability of having spins up or down, respectively, at the SG exit is thus still given by (\ref{eqn:rho_up}),
\begin{equation}
    \rho(s) = \frac{1+s\cos(\mu_0,\lambda)}{2}\;.
\end{equation}

\subsubsection{Cascade of SG's} \label{sec:cascade}
In textbook descriptions of spin, two or more SG apparatuses in series are often employed to illustrate its non-classical properties.

In the proposed model,
particles having spin $s_1=\pm 1$ at the output of the first SG have also polarization $\mu=\pm \lambda_1$. At the entry of the second SG, the spin propensity is thus $M=s_1\sum_{d} \lambda_{1d}\lambda_{2d}$. Using the result of the previous section, the probability of spins up or down at the exit of the second SG is evaluated as
\begin{equation}
    \rho(s_2|s_1) = \frac{1+s_2s_1\cos(\lambda_{1},\lambda_{2})}{2} \;,
\end{equation}
again in perfect agreement with standard QM calculation.

\subsection{SG simulation and numerical results} \label{sec:numer1}
We shall consider a magnetic field concentrated in a certain region of space along the propagation direction $x_2$ and oriented along the $x_3$ axis, with a one-dimensional inhomogeneity along the same direction, $B=(0,0,B_1 x_3)$. Even if this field does not satisfy Maxwell equation $\nabla \cdot B=0$, we choose it to simplify the notation. In fact, the literature has shown its equivalence to any ``physical" field where the inhomogeneity is along one constant direction, provided that the two directions are exchanged \cite{hsu}.

In the proposed model, particles are emitted at the source according to a Gaussian-wave preparation.
To represent a Gaussian wave with zero mean and variance $(N_{s}-1)/4$, $N_{s}$ sources are set at adjacent nodes centered at $x=0$. The source probability and phase are set as
\begin{equation}
P_{0}(x)=\displaystyle\frac{1}{2^{N_{s}-1}}{N_{s}-1 \choose x+\frac{N_{s}-1}{2}}, \quad \epsilon_d(x)= mx    \;,
\label{eqn:P0gauss}
\end{equation}
with $m\in[0,1]$ and $(\cdot)$ representing here a binomial coefficient.

To represent the SG scenario, we have set $m_1=m_3=0$, while $m_2$ would determine the average particle speed along the propagation direction.
We assume that the time spent before encountering the SG is sufficient to complete both the ``lattice training" and the ``particle training" processes discussed in \cite{2017}, and we simulate the expected-motion version of the model, equations (\ref{eqn:xd})--(\ref{eqn:arg}), as explained in Sect.~\ref{sec:prob}.
When (\ref{eqn:rhov}) is applied to (\ref{eqn:P0gauss}), one obtains a momentum pdf that approximates a Dirac delta centered around the phase momentum, $\rho(v_Q)\approx\delta(v_Q-m)$. From (\ref{eqn:xd}), it follows that $\bm{x}_d\approx m_dt$.
We shall take this approximation as exact for the direction of propagation, and posit $x_3=m_3t$, thus limiting our analysis to the plane perpendicular to propagation.

By virtue of the equivalence (\ref{eqn:spinor}), the initial polarizations are chosen as to represent an initial spin state $\chi_0=(\chi_1,\chi_2)$,
\begin{equation}
\mu_{03}=\chi_1\chi^*_1-\chi_2\chi^*_2, \quad \mu_{01}=\chi_1\chi_2^*+\chi_1^*\chi_2,\quad \mu_{02}=-\iota\left(\chi_1\chi_2^*-\chi_1^*\chi_2\right)
\;,
\end{equation}
where the asterisk denotes here complex conjugation.

Ensemble results are compared with those of quantum mechanics (theoretical values) obtained by using the two-component propagator \cite{hsu,reddy}
\begin{equation}
K^{(SG)} (x,t|x_0) = \frac{1}{(2\iota t)^{3/2}} \cdot \exp\left(\frac{\iota\pi(x-x_0 )^2}{2t}+\frac{\iota\pi(x_1+x_{01}) \sigma_3 \phi t}{2}-\frac{\iota\pi\phi^2 t^3}{24} \right)
\label{eqn:168}
\end{equation}
in lattice units, where $\sigma_3$ denotes here the third Pauli matrix.

The theoretically expected pdf is obtained numerically from the propagated spinor $\chi(x,t)$
as $\rho(x;t)=\chi^\dagger \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)\chi$. This pdf is to be compared with the frequency of particle arrivals at nodes $x$ after $t$ iterations computed by the average-motion model.
The theoretically expected spin density
is obtained as $\langle S_3 \rangle(x;t)=\chi^\dagger \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)\chi$. This quantity is to be compared with its counterpart in the proposed model, obtained as the difference between the frequency of arrivals of particles with $s=1$ ($\mu_3=\lambda_3=1$) and of those with $s=-1$ ($\mu_3=-\lambda_3=-1$).

Figures (\ref{fig:sg1})--(\ref{fig:sg2}) show the calculated spin density after $t=64$ iterations for a source scenario with $\mu_MB_F=0.1/\pi^2$, $d_1=d_2=1$, $\chi_1=\chi_2=1/\sqrt{2}$ (that is, $\mu_{01}=1$ in the proposed model). Globally, these result match the theoretical values, which clearly show the ``textbook" spin separation occurring along the inhomogeneity direction.

\begin{figure}[h]
\input{sterngerlach.tex}
\caption{Differential frequency of arrival (b) and theoretical spin density $\langle S_3 \rangle$ (a) for $N_p=10000$, $t=64$ as a function of position (Stern-Gerlach, Gaussian wave, $\mu_M B_F=0.1/\pi^2$, $d_1=d_3=1$, $\chi_1=\chi_2=1/\sqrt{2}$).}
\label{fig:sg1}
\end{figure}
\begin{figure}[h]
\centering
\input{sterngerlach_sigma2.tex}
\caption{Differential frequency of arrival (blue) and theoretical spin density (red) for $N_p=10000$, $t=64$ as a function of the inhomogeneity direction (Stern-Gerlach, Gaussian wave, $\mu_M B_F=0.1/\pi^2$, $d_1=d_3=1$, $\chi_1=\chi_2=1/\sqrt{2}$).}
\label{fig:sg2}
\end{figure}

\section{Higher spins} \label{sec:higher}

In this section we extend the model to particle having spin $S$ higher than \textonehalf. For particles with general spin number $S$ ($S=1/2, 1, 3/2, \ldots$), the quantity here denoted as spin ($s$) can take $2S+1$ values equispaced between +1 and -1 ($s\in[1,1-1/S,\ldots,-1]$).

\subsection{Microscopic motion}

In addition to their source spin and $\mu$-polarization, particles are emitted with a second polarization vector (third, including $\rho$) $\tau_0:=\{\tau_{0d}\}\in\mathbb{Q}^3$.
The constraint on polarization components is generalized as
\begin{equation}
  \sum_{d}\mu_{d0}^2 = \tilde{s}_0^2, \quad \sum_{d}\tau_{d0}^2 = \frac{S+1}{S}-\tilde{s}_0^2 \;,
  \label{eqn:mu0squared}
\end{equation}
where $\tilde{s}_0$ denotes the spin-rounded value of $s_0$, $\mathrm{Round}(S(1+s_0))/S-1$.
Clearly, for $S=1/2$, one retrieves  $\tilde{s}_0=\pm 1$ and $\sum_d\mu_{d0}^2=1$.

To describe the dynamics of polarization, equation (\ref{eqn:polarev}) still holds when applied to both vectors $\mu$ and $\tau$. In the presence of a constant magnetic field, these dynamics conserve the quantities $\sum_d\mu_d^2$ and $\sum_d\tau_d^2$.

The rules replacing (\ref{eqn:s_def}) determine at each iteration the particle's spin $s$ among the $2S+1$ possible values.
The general rule reads
\begin{equation}
    s[n] = -1+\frac{1}{S}\sum_{k=1}^{2S}\Theta(s_0-\tilde{s}_k[n]), \quad \tilde{s}_k:=-1+2\sum_{\sigma=-1}^{-1+\frac{k-1}{S}} P(\sigma)
\end{equation}
where $\Theta$ is the unit step function and $P(\sigma)=\mathbb{P}(s=\sigma)$ is the spin pmf.
Clearly, (\ref{eqn:s_def}) is retrieved for $S=1/2$, with $P(\pm 1)=(1\pm M)/2$ .

The pmf of spin (a $2S+1$-valued discrete random variable) is completely defined by its first $2S$ central moments.
The latter are prescribed by the model as functions of the quantities $M=\sum_d \mu_d\lambda_d$ and $T:=\sum_d \tau_d\lambda_d$.
The first two central moments read
\begin{eqnarray}
    && \mathbb{E}[s] = M \;, \label{eqn:esm} \\
    && \mathbb{E}\left[(s-M)^2\right] := V = \frac{\sum_d\tau_d^2-T^2}{2} \;, \label{eqn:var}

\end{eqnarray}
with $\mathbb{E}\left[(s-M)^3\right] = -\mathbb{E}[(s-M)^2]\cdot \mathbb{E}[s]$ etc.
Overall, we can write $s[n]=f(M[n],V[n],s_0)$.

The ER rule (\ref{eqn:er}) is still valid. Additionally,
\begin{equation}
\tau_d \xLeftarrow [\mathrm{ER}]{}     \lambda_d\sqrt{\frac{S+1}{S}-s^2}
    \label{eqn:tau}
\end{equation}
holds.
Note that, since $\sum_d\lambda_d^2=1$ by definition, both $\sum_d\mu_d^2$ and $\sum_d\tau_d^2$ are constant across an ER.

\subsection{Probability densities}

For a homogeneous and constant magnetic field, (\ref{eqn:rho_up}) is generalized as follows. Due to the polarization dynamics, the quantities $M$ and $T$ do not change with time, thus $M[n]\equiv M_0$, $T[n]\equiv T_0$. Similarly, $\sum_d\mu_d^2$ and $\sum_d\tau_d^2$ are constant. The transition probabilities are thus constant, too. Since $s_0$ is attributed to a particle once and for all, the spin $s$ will stay constant while crossing the magnetic field $\lambda$. Its probability distribution is thus uniquely determined by the values $M_0$ and $T_0$.

For a non-uniform magnetic field (SG apparatus), again the probability distribution of spin is determined at the first ER, that is, at the entry of the SG, and is a function of the quantities $M_0$ and $T_0$. With the ER (\ref{eqn:tau}), $M$ jumps to $s$ and $T$ jumps to $\sqrt{\frac{S+1}{S}-s^2}$. Consequently, the spin propensity $\mathbb{E}[s]$ is $s$ and the variance becomes zero. At the next iteration, the spin will remain constant with probability one. The probability of spins at the exit of the SG is thus uniquely determined by the values $M_0$ and $T_0$.

In the case of two SG in cascade,
we have $\mu_1=s_1\lambda_1$ and $\tau_1=\lambda_1\sqrt{\frac{S+1}{S}-s_1^2}$ at the exit of the first SG. Therefore, at the entry of the second SG, $M_1=s_1Y$ and $V_1=\left(\frac{S+1}{S}-s_1^2\right)\frac{1-Y^2}{2}$, having defined $Y:=\sum_d\lambda_{1d}\lambda_{2d}=\cos(\lambda_1,\lambda_2)$.
Accordingly, the spin takes a value $s_2=f(M_1,V_1,s_0)$. After the first ER, the spin propensity jumps to $s_2$ and the variance jumps to zero. Therefore, at the next iterations, the spin will remain equal to $s_2$. Overall, the probability of having a certain spin $s_2$ at the exit of the second SG depends on $Y$ and $s_1$.

As an example, for $S=1$, the spin pmf is evaluated from the first two moments as
\begin{equation}
    P(1)=\frac{V+M^2+M}{2}, \quad
    P(0) = 1-V-M^2, \quad P(-1)=\frac{V+M^2-M}{2}
    \label{eqn:s1}
\end{equation}
Thus the spin pdf at the exit of the second SG is
\begin{equation}
 \rho(s_2)=(1-s_2^2)+\frac{s_2}{2}M_1+\left(-1+\frac{3}{2}s_2^2\right)(V_1+M_1^2) \;.
\end{equation}
Using the expressions for $M_1$ and $V_1$ derived above, one obtains
\begin{equation}
    \rho(s_2|s_1)=\left(\frac{s_2^2-s_1^2}{2}-\frac{3}{4}s_1^2s_2^2\right)+\frac{s_1s_2}{2}Y+\left(1-\frac{3}{2}(s_1^2+s_2^2) +\frac{9}{4}s_1^2s_2^2 \right)Y^2 \; \label{eqn:rhoS1}
\end{equation}
which precisely match the QM results computed, e.g., in \cite{tekin}\footnote{Cascaded Stern-Gerlach probabilities for higher spin have been seldom studied in the literature. Equations (\ref{eqn:esm})--(\ref{eqn:var}) and, consequently, equations of the type (\ref{eqn:rhoS1}), are not present in \cite{tekin} but have been built upon the general result of that paper.}. Further, it can be verified that $\rho(s_2|s_1)=\rho(s_1|s_2)$, as it should be.

\section{Spin Entanglement} \label{sec:entang}
In \cite{2017} momentum entanglement has been described within the context of the Local-Realistic Model. Here we extend those results to the case of spin entanglement.

\subsection{Particle emission}
Entangled particles are emitted at sources as pairs ($n_R=2$) and denoted with a superscript $R\in\{I,II\}$.
In addition to assigning entangled momenta ($v_0^{(I)}+v_0^{(II)}=0$), the source preparation attributes anti-correlated entangled source polarizations and spins
, according to the rules $$\mu_0^{(I)}+\mu_0^{(II)}=0, \quad s_0^{(I)}+s_0^{(II)}=0\;.$$
We shall denote, without loss of generality, $\mu_0^{(I)}:=\mu_0$ and $s_0^{(I)}:=s_0$. Consequently, $\mu_0^{(II)}=-\mu_0$ and $s_0^{(II)}=-s_0$, where $\sum_d\mu_{0d}^2=1$ and $s_0=U[-1,1]$ as for $n_R=1$.

\subsection{Microscopic motion}
All rules described above remain the same in the case of entangled particles, except for the fact that the quantity $\tilde{M}^{(R)}:=\mathrm{sign}(M^{(R)})|M^{(R)}|^{n_R}$ replaces $M^{(R)}:=\sum_d \mu_d^{(R)} \lambda_d^{(R)}$ in the spin dynamics (\ref{eqn:s_def}), the magnetic force expression (\ref{eqn:magnforce}) and the magnetic energy (\ref{eqn:magnen}), which are thus generalized as
\begin{eqnarray}
s^{(R)}[n] &&= \mathrm{sign}\left(s_0^{(R)}+\tilde{M}^{(R)}[n]\right)\;, \\
f_\nu^{(R)}[n] &&=-\mu_MB_F\tilde{M}^{(R)}[n]\;, \\
\mathcal{E}^{(R)}[n] &&= -\mu_M B_M \tilde{M}^{(R)}[n]\;.
\label{eqn:sr}
\end{eqnarray}

Particularly relevant for what follows is the description of external resets, based on (\ref{eqn:er}) that is still valid. By virtue of its definition above, when $M^{(R)}$ jumps to $s^{(R)}$, $\tilde{M}^{(R)}$ jumps to $\mathrm{sign}(s^{(R)})|s^{(R)}|^2=s^{(R)}$.
From energy conservation, the reset of momentum in the direction of propagation, equation (\ref{eqn:er_v}), is now regulated by the coefficient
\begin{eqnarray}
&& \bm{v}_d^{(R)} \xLeftarrow [\mathrm{ER}]{} \alpha^{(R)} \bm{v}_d^{(R)}+(1-\alpha^{(R)})\nu_d^{(R)} \sum_{d=1}^3 \bm{v}_d^{(R)} \nu_d^{(R)} \;, \label{eqn:er_v} \\
&&\alpha^{(R)}=\frac{\sum_d (\bm{v}_d^{(R)})^2-\mu_MB_M(s^{(R)}-\tilde{M}^{(R)})}{\sum_d(\bm{v}_d^{(R)})^2}\;. \label{eqn:alpharr}
\end{eqnarray}
For $n_R=1$, equations of Sect.~\ref{sec:spin} are clearly retrieved.

\subsection{Probability densities} \label{sec:ent_prob}

We shall evaluate the joint pmf $\rho(s^{(I)},s^{(II)})$, representing the probability that two entangled particles
arrive at either of two ``detectors" opportunely placed downstream of their respective SG apparatuses in order to intercept the beams with $s^{(R)}=\pm 1$.

We note that, experimentally, this pmf and any other related statistics are obtained by counting the coincidences in arrivals at detectors. For this purpose, a data analysis procedure is required to group particles in pairs according to their arrival times, often using a time-coincidence window. A very thorough and enlightening discussion on this point can be found in \cite{deraedt,larsson,michielsen}.

Assuming, without loss of generality, that the settings of the two stations are identical, the arrival times are solely determined by the momentum acquired by the two ensembles of particles in the respective directions of propagation.
While source momenta are identical, particles in the two SG's undergo a different momentum reset in the direction of propagation, described by rules (\ref{eqn:er_v})--(\ref{eqn:alpharr}).
At the entering of the respective SG the spin propensity is $M^{(R)}=M_0^{(R)}:=
\sum_{d} \mu_{0d}^{(R)} \lambda_d^{(R)}$.
While the other parameters of influence are common to both particles, it is therefore the cosine term
$M_0^{(R)}=\cos(\mu_0^{(R)},\lambda^{(R)})$ that controls the momentum through (\ref{eqn:alpharr}) and ultimately defines the time of arrival at the respective detectors.

For this reason, a coincidence in arrival times is \textit{expectedly} recorded when the two particles have the same value of $M_0$,
that is, when
\begin{equation}
M_0^{(I)}=\sum_{d=1}^3 \lambda_d^{(I)} \mu_{0d}^{(R)} = \sum_{d} \lambda_d^{(II)} \mu_{0d}^{(R)}=M_0^{(II)}:=M_0
\label{eqn:smu}
\end{equation}
It is now easy to show that, for each selection of $\lambda^{(I)}$, $\lambda^{(II)}$, two values of $M_0$
fulfill (\ref{eqn:smu}), namely,
\begin{equation}
\hat{M}_0
=\pm \sqrt{\frac{1-\sum_{d=1}^3 \lambda^{(I)}_d \lambda^{(II)}_d}{2}} \;,
\label{eqn:smu1}
\end{equation}
having opposite signs and equal probability. Geometrically, these values correspond to the two unit $\mu_0$ vectors bisecting the angle between the two directions $\lambda^{(I)}$, $\lambda^{(II)}$ and co-planar to the same, that is
$$\angle\hat{\mu}_0=(\angle\lambda^{(I)}+\angle\lambda^{(II)})/2\;.$$
All other values of $\mu_0$ give rise to coincidences only with a very small probability and thus do not contribute to the joint pmf.

Similarly to the non-entangled case, inside the SG $s^{(R)}$ remains constant at the entry value $\mathrm{sign}(s_0^{(R)}+\mathrm{sign}(M_0)M_0^2)$.
The distributions of $s^{(R)}$ are therefore found as
\begin{equation}
s^{(I)} = \left\{ \begin{array}{ll}
1\;, \quad & s_0 > -\mathrm{sign}(M_0)M_0^2 \\
-1\;, \quad & s_0 < -\mathrm{sign}(M_0)M_0^2
\end{array} \right. \label{eqn:sI}
\end{equation}
and
\begin{equation}
s^{(II)} = \left\{ \begin{array}{ll}
1\;, \quad & s_0 < \mathrm{sign}(M_0)M_0^2 \\
-1\;, \quad & s_0 > \mathrm{sign}(M_0)M_0^2
\end{array} \right. \label{eqn:sII}
\end{equation}

\begin{figure}
\centering
\subfigure []{
\begin{tikzpicture} [scale=0.7]

\draw [->] (3,-0.5) -- (3,6.5); \node [left] at (3,6.5) {$s$};
\draw [->] (-0.5,3) -- (6.5,3); \node [below] at (6.5,3) {$s_0$};
\draw [gray] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0);
\draw [gray] (0,0) -- (6,6);
\draw [gray] (0,6) -- (6,0);

\draw [dashed] (2,0) -- (2,6);
\draw [dashed] (4,0) -- (4,6);

\node [below, fill=white] at (6,3) {$1$};
\node [below, fill=white] at (4,3) {$M_0^2$};
\node [below, fill=white] at (2,3) {$-M_0^2$};

\node [below, fill=white] at (0,3) {$-1$};

\draw [blue] (0,1) -- (5,6);
\draw [red] (1,6) -- (6,1);
\draw [<->] (1,2) -- (1,1);
\node [right] at (1,1.5) {$M_0^2$};

\end{tikzpicture}
}
\subfigure []{
\begin{tikzpicture} [scale=0.7]

\draw [->] (3,-0.5) -- (3,6.5); \node [left] at (3,6.5) {$s$};
\draw [->] (-0.5,3) -- (6.5,3); \node [below] at (6.5,3) {$s_0$};
\draw [gray] (0,0) -- (0,6) -- (6,6) -- (6,0) -- (0,0);
\draw [gray] (0,0) -- (6,6);
\draw [gray] (0,6) -- (6,0);

\draw [dashed] (2,0) -- (2,6);
\draw [dashed] (4,0) -- (4,6);

\node [below, fill=white] at (6,3) {$1$};
\node [below, fill=white] at (4,3) {$M_0^2$}
\node [below, fill=white] at (2,3) {$-M_0^2$};

\node [below, fill=white] at (0,3) {$-1$};

\draw [blue] (1,0) -- (6,5);
\draw [red] (0,5) -- (5,0);
\draw [<->] (5,4) -- (5,5); \node [left] at (5,4.5) {$M_0^2$}

\end{tikzpicture}
}
\caption{Illustration of the quantities $s_0^{(I)}+\mathrm{sign}(M_0)M_0^2$ whose sign is $s^{(I)}$ (blue) and $s_0^{(II)}+\mathrm{sign}(M_0)M_0^2$ whose sign is $s^{(II)}$ (red), as a function of $s_0$, for $M_0>0$ (a) and $M_0<0$ (b).} \label{fig:spin2}
\end{figure}

With the help of Fig. \ref{fig:spin2}, we further note that $\{s^{(I)},s^{(II)}\}=\{1,1\}$ for
$s_0\in[-M_0^2,M_0^2]$ if $M_0>0$.
Likewise, $\{s^{(I)},s^{(II)}\}=\{-1,-1\}$
for $s_0\in[-M_0^2,M_0^2]$ if $M_0<0$.
Regardless of the sign of $M_0$, $\{s^{(I)},s^{(II)}\}=\{-1,1\}$ for $s_0<-M_0^2$
and $\{s^{(I)},s^{(II)}\}=\{1,-1\}$ for
$s_0 > M_0^2$.

Finally, recalling that $\rho(s_0)=1/2$, the joint distribution is evaluated as
\begin{eqnarray}
&& \rho(1,1) =
\mathbb{P}(M_0>0)\cdot\frac{2M_0^2}{2}=\frac{M_0^2}{2}\\
&& \rho(-1,-1) = \mathbb{P}(M_0<0)\cdot\frac{2M_0^2}{2}=\frac{M_0^2}{2}\\
&& \rho(1,-1) = \frac{1-M_0^2}{2} = \rho(-1,1)\;.
\end{eqnarray}
Since from (\ref{eqn:smu1}) the only meaningful value is $\hat{M}_0^2=(1-\cos(\lambda^{(I)},\lambda^{(II)}))/2$, the joint distribution reads
\begin{equation}
\rho(s^{(I)},s^{(II)}) = \frac{1-s^{(I)} s^{(II)}\cos(\lambda^{(I)},\lambda^{(II)})}{4}\;, \label{eqn:rhosigma}
\end{equation}
and the expected value of the product $s^{(I)} s^{(II)}$ is $-\cos(\lambda^{(I)},\lambda^{(II)})$,
that is, precisely the QM prediction.

\subsection{Numerical Results} \label{sec:numer}
We aim at representing here a textbook two-channel Bell test experiment. A source produces pairs of entangled particles, sent in opposite directions. Each particle beam encounters a SG. Emerging particles from each channel are detected and coincidences in arrivals counted.
Similarly to the non-entangled scenario simulated in Sect.~\ref{sec:numer1}, we shall take $\lambda^{(R)}=\nu^{(R)}$, i.e., an inhomogeneity directed along the field in both SG. While the orientation $\lambda^{(II)}$ is fixed, $\lambda^{(I)}$ is varied between $-\pi$ and $\pi$ in the plane $x_1$--$x_3$. The direction of the two emitted beams is taken as $\pm x_2$.
Coincidences are registered when particles arrive at either of a pair of detectors
at a distance $\ell$ from the source
at the same time.
We assume that the detector sizes in the planes $x_1$--$x_3$ are sufficiently large to capture the whole beams leaving the SG regardless of the particles' deflection.

In the proposed model, particles are emitted at the respective sources according to a Gaussian-wave preparation, with $m_1=m_3=0$ (see Sect.~\ref{sec:numer1}), while $m_2$ would determine the average particle speed along the propagation direction.
The initial polarizations are randomly chosen at each emission between $N_\mu$ possible values, equally spaced between $\angle\hat{\mu}_0$ and $\angle\hat{\mu}_0+2\pi$.

Similarly to the single SG scenario, we assume both particle and lattice ``training" completed and we simulate the expected-motion version (\ref{eqn:vq})--(\ref{eqn:arg}) of the model\footnote{Contrarily to Sect.~\ref{sec:numer1}, we renounce using the expected-motion equation (\ref{eqn:xd}) and use the full model instead, in order to capture the dispersion of particle arrivals around the average beam centers.}.
In particular, we represent the momentum reset occurring at the ER, (\ref{eqn:er_v}), by attributing an expected momentum at SG entering that equals $m_2$.
Consequently, the expected time to arrival is evaluated as
\begin{equation}
  \bm{T}^{(R)} = \frac{m_2T_0}{\sqrt{m_2^2-\mu_BB_M(s^{(R)}-\tilde{M}_0^{(R)})}} \;.
\end{equation}
where $T_0=\ell/m_2$ is the expected arrival time in the absence of SG.
As anticipated in Sect.~\ref{sec:ent_prob}, the momentum jump and the consequent time delay depend only on $M_0^{(R)}$.

Assuming $\mu_BM_B\ll m_2^2$, the expected time delays among arrivals at the two stations are evaluated as
$$ \Delta T:=\bm{T}^{(I)}-\bm{T}^{(II)}\approx T_0\frac{\mu_BB_M}{2m_2^2}(s^{(I)}-s^{(II)})\;.$$
Accordingly, time arrivals at station I are corrected by adding the value $\Delta T$ before checking the coincidences.

Ensemble results are compared with QM prediction (\ref{eqn:rhosigma}).
Figure~\ref{fig:bell} shows the frequency of the four types of coincidences as a function of the angular difference between the two fields,
with $\mu_MB_F=0.04$, $N_p=1\cdot 10^4$, $N_\mu=16$, $\ell=210$. When compared with the QM predictions, these results confirm the substantial equivalence of the two models as anticipated in the previous section.

\begin{figure}
\centering
\input{bell1.tex}
\caption{Frequency of coincidences of arrivals $(1,1)$ (circles), $(-1,-1)$ (squares), $(1,-1)$ (stars), and $(-1,1)$ (triangles), with their theoretical values $\rho(1,1)=\rho(-1,-1)$ (red solid) and $\rho(1,-1)=\rho(-1,1)$ (blue solid) as a function of the angle (rad) between $\lambda^{(I)}$ and $\lambda^{(II)}$ (Bell experiment, Gaussian-wave beams, $\mu_MB_F=0.04$, $N_p=1\cdot 10^4$, $N_\mu=16$, $\ell=210$).} \label{fig:bell}
\end{figure}

\subsection{Discussion}

The previous sections have shown that the proposed model exactly reproduces the joint pmf of a spin-entangled quantum system where coincidences in spin are counted only if they are accompanied by coincidence in arrival time, that is, the regime that Bell test experiments aim to reach \cite{deraedt}.
Nevertheless, nowhere in the proposed model, particles, say, II ``know" about which magnetic field experience particles I, thus locality applies, certainly together with realism.

At this point, the reader may wonder why, although $\mu_0$ and $s_0$ play the role of ``hidden variables" as those explicitly discarded by Bell's theorem, Bell's inequalities are violated and the QM statistics are correctly reproduced.

The apparent answer is that the process leading to the pdf (\ref{eqn:rhosigma}) does not fall under the form postulated by this theorem\footnote{We use Bell's theorem for deterministic variables here. In fact, the stochastic nature of the model resides in the attribution of the hidden variables, while the relations $s^{(R)}=f(\mu_0^{(R)},s_0^{(R)},\lambda^{(R)})$ as given by (\ref{eqn:sI})--(\ref{eqn:sII}) are fully deterministic.}.
We must consider the fact that, for most choices of $s_0$ and $\mu_0$, no arrival coincidence can be detected.
For example, the expected value of the product $s^{(I)} s^{(II)}$ must be evaluated as
\begin{multline}
\mathbb{E}(s^{(I)} s^{(II)} )
\int_H s^{(I)} \left(\lambda^{(I)},s_0,\mu_0\right)\cdot s^{(II)}\left(\lambda^{(II)},s_0,\mu_0\right)\cdot \\ \cdot p_{coi}\left(\mu_0,\lambda^{(I)},\lambda^{(II)}\right) \cdot \rho(s_0)\rho(\mu_0)ds_0d\mu_0
\;,
\end{multline}
where $p_{coi}$ is the probability of coincidence counting, which depends on the respective arrival times of the two particles.

Indeed, (\ref{eqn:smu1}) states that $p_{coi}\approx\delta(\mu_0-\hat{\mu}_0(\lambda^{(I)},\lambda^{(II)}))$.
The emergence of a particular polarization $\hat{\mu}_0$ that is a function of both $\lambda^{(I)}$ and $\lambda^{(II)}$ results from the fact that for other values of $\mu_0$, the desired coincidence of $T^{(I)}$ and $T^{(II)}$ at the two stations is impossible or, at least, highly improbable.
Thus the counting of coincidences introduces a weighting factor $p_{coi}$ that ultimately gives rise to the spin-entanglement correlations.

When compared with the Bell definition of hidden variable theory,
$$C_h=
\int_H s^{(I)} (\lambda^{(I)},h)\cdot s^{(II)}(\lambda^{(II)},h)\cdot \rho(h|\lambda^{(I)},\lambda^{(II)})dh \;,$$
where $h:=\{s_0,\mu_0\}$ is our set of the \textit{independent} hidden variables\footnote{We use $h$ instead of the more common $\lambda$ to avoid confusion with the magnetic field components.}, we see that $\rho(h|\lambda^{(I)},\lambda^{(II)})=p_{coi}(\mu_0,\lambda^{(I)},\lambda^{(II)})\rho(s_0)\rho(\mu_0)$.

It becomes thus apparent that the Bell assumption $\rho(h|\lambda^{(I)},\lambda^{(II)})=\rho(h)$, often referred to as ``Measurement Independence", is not satisfied by the proposed model.
Therefore, the proposed model is not forbidden by Bell's theorem, which is based on Bell's assumptions, to violate Bell's inequalities.

However, the non-validity of this assumption cannot be explained with the usually  solutions (i.e., either non-localism or non-realism), nor with an equally-unpleasant form of (super-)determinism \cite{vervoort}.
This motivation to renounce to measurement independence should not either be confused with a detector inefficiency loophole.
In the proposed model, the detection process itself is 100\% efficient, although the coincidence counting makes that only a fraction of the total entangled pairs of particles concur in building the quantum statistics.

Note that a similar situation was discussed in \cite{2017} for momentum-entangled systems, although in that case the two observables (detector positions $x^{(I)}$ and $x^{(II)}$) are not binary functions. Still, a crucial role is played by the definition of coincidences, in this case based on the simultaneous (expected) arrival of the two particles at localized detectors.

\section{Conclusions}
The paper has shown how nonrelativistic QM including spin can be reproduced by realistic, stochastic, and localistic  rules applied to individual particles.
QM predictions are indeed retrieved as probability distributions of position, momentum, angular momentum, spin, etc. without appealing to the QM mathematical machinery itself.

To represent spin scenarios, such as Stern-Gerlach apparatuses or a Bell test experiment, the proposed model does not appeal to two-dimensional complex spinors and matrices but uses a relatively simple set of rules, implying that (1) spin is a dichotomic quantity carried on by particles whose value depends on a random source setting and a spin propensity; (2) the latter can vary at each iteration as a function of polarization, which is a three-dimensional attribute, and the magnetic field experienced; (3) polarization is randomly attributed during preparation and has its own rules of variation; it further concurs in determining how particles react to magnetic fields.

Several refinements of the model are still possible. For example, relativistic Newton's second law shall inspire a mechanism to prevent that the momentum propensity becomes larger than unity under the action of persistent forces. Multi-state and many-particle systems are yet to be fully studied, too.

redpajama_set_name: RedPajamaArXiv


The Westminster players relived a historic tale of fear, betrayal, and death through The Crucible, a play about the Salem witch trials in 1692. Directed by guest artist and Westminster alumnus Celia Quillian, the performance ran on Oct. 28 and Oct. 29. During the time of colonial New England, settlement in America had not yet expanded to the West. Living in an unknown land also meant exposure to foreign diseases. The strict moral and religious rules of the Puritan community in Salem combined with the fear of the unknown heightened the anxiety in the village, and the villagers turned to witchcraft as an explanation for their troubles. The play was written by American playwright Arthur Miller during the Cold War as a parallel to McCarthyism, a term derived from a campaign led by Senator Joseph McCarthy to search for suspected communists in the United States. The term refers to unjustified investigations and accusations made without proper evidence. The heavy historical significance of the play was what made The Crucible a distinctive performance for many of the students.
"I'd say what's different about The Crucible is it carries a message," said junior Andrew Oglesby. "There are a few sociopolitical aspects, and it was made in the 1950's. It carries a lesson about unnecessary accusations, or prejudice or vengeance. It also has a darker tone than most plays I've done." For Quillian, researching and teaching the history of The Crucible made directing it more worthwhile.
"It was really fun to work with that script," she said. "It let me hunker down and explore that text, and now I'm really excited, personally, about other plays that are set in a historic time period and based off historic events. Something that I love most about directing is doing the back research for it." Quillian graduated from Westminster in 2010, and returning to the school to direct a play renewed old memories.
From her experience in the Upper School, Quillian also knew that being a student at Westminster meant pressure, academically and socially. So during rehearsals, she packed in fun exercises that not only helped the students learn their parts, but also made the long rehearsal process more enjoyable. In addition, Quillian had the students rehearse the play several times by improvising their lines before they had them memorized. This way, the students could better understand the language of late 17th century.
Being a freshman, this was Kapasi's first play with the Upper School. With The Crucible, her high school theatre career began on a more serious tone, which also marked the heightened rigor of high school theatre.
In the play, Kapasi plays Ann Putnam, the wife of Thomas Putnam, one of the wealthiest people in Salem. But despite her affluence, she endures a very personal loss—seven of her eight children die after birth. Furthermore, her only remaining daughter becomes ill. In her sorrow and desperation, she blames witchcraft and demonic forces for her loss.
Similarly, Oglesby's character, Reverend John Hale, devotes much of his energy to investigating witchcraft. In the play, Hale wants to bring the justice of God through condemning witches, but as more and more innocent people are being accused, he realizes the corruption of both the court and the justice system. For Oglesby, Hale's strong sense of religious and moral obligation made the role sometimes difficult to portray.
As a man of high moral integrity, Hale's only motive is to administer justice and to serve God, which sets him apart from characters like Abigail. Abigail, played by junior Caroline Stewart, was a former maid of John and Elizabeth Proctor, until she was disgraced and fired for having an affair with John Proctor. In seeking vengeance, she accuses innocent people of witchcraft and gains power in the town and the court. Consequently, Abigail's antagonistic qualities presented a challenge for Stewart.
Power and influence drives other characters as well, such as Deputy Governor Thomas Danforth, played by senior Lauren Mobley. Danforth was the chief judge of the court and was responsible for convicting the villagers of witchcraft. Danforth holds the court's authority in high regard, and never changes his verdict despite the lack of evidence. He refuses to suspend the witch trial hangings in fear that his own authority would be questioned.
While the whole town is engrossed in fear, John Proctor, played by junior Ryan Costley, is one of the only characters who does not believe in the existence of Satanic arts. He is outspoken about his beliefs and questions the veracity of the court and the trials themselves.
"The biggest theme he raises is about understanding the whole story before you make assumptions, but also the idea of the scapegoating of people, people being blamed for something they're not involved in, and people getting scared and therefore targeting other people about those fears instead of assessing why those fears exist," said Costley.
In the play, fear has a death count. In total, 19 people are hanged for not confessing to witchcraft. What made The Crucible a difficult production for some of the students was the sheer number of lives lost and the staggering consequences of misguided fear.
The overarching themes in The Crucible of groundless accusations, morality, and fear don't just apply to witch trials. They have been relevant throughout history, and are still relevant today.
"This play was written during a time of political disarray, and it is often put on during times of political disarray," said Quillian. "In this particular election cycle, no matter what side you're on, I think we can all agree that there's been a lot of finger-pointing from the beginning. There was a lot conspiracy-throwing, people making up things on the Internet, people feeding off of false news of every side." Thus, even though The Crucible may seem archaic and removed from our time, the message still pertains to a modern audience.


text:
Virginia Code > Title 1 > Chapter 2.1 > Article 2 > § 1-254
Virginia Code 1-254: Town
"Town" means any existing town or an incorporated community within one or more counties which became a town before noon, July 1, 1971, as provided by law or which has within defined boundaries a population of 1,000 or more and which has become a town as provided by law.
Terms Used In Virginia Code 1-254
Town: means any existing town or an incorporated community within one or more counties which became a town before noon, July 1, 1971, as provided by law or which has within defined boundaries a population of 1,000 or more and which has become a town as provided by law. See Virginia Code 1-254
Code 1919, § 5; Code 1950, § 1-13; 1950, p. 24, § 1-13.29; 1971, Ex. Sess., c. 49; 1995, c. 155; 2005, c. 839.
Article 2 Contents
Valverde & Rowell, P.C.
Virgina Beach, VA Immigration and employment Law
1 CFR Part 425 - President'S Commission on White House Fellowships
CFR > Title 3 - The President
U.S. Code > Title 28 > Part II - Department of Justice
U.S. Code > Title 3 - The President
California Codes > Government Code > Title 2 > Division 3 - EXECUTIVE DEPARTMENT
Florida Regulations > Administration Commission
Florida Regulations > Executive Office of the Governor
Florida Statutes > Chapter 120 - Administrative Procedure Act
Florida Statutes > Title IV - Executive Branch
Texas Constitution > Article 4 - Executive Department
Texas Constitution Art. 4 sec. 1 - Officers Constituting Executive Department
Texas Constitution Art. 4 sec. 2 - Election of Officers of Executive Department
Texas Constitution Art. 4 sec. 3 - Returns of Election; Declaration of Election; Tie Votes; Contests
Texas Constitution Art. 4 sec. 3a - Death, Disability, or Failure to Qualify of Person Receiving Highest Vote for Governor

redpajama_set_name: RedPajamaCommonCrawl


Catalyses a step in the 3-hydroxypropanoate/4-hydroxybutanoate cycle, an autotrophic CO2 fixation pathway found in some thermoacidophilic archaea . The enzyme from Sulfolobus tokodaii does not act on either NADH or crotonyl-CoA . Different from EC 1.3.1.8, which acts only on enoyl-CoA derivatives of carbon chain length 4 to 16. Contains Zn2+.
Teufel, R.; Kung, J.; Kockelkorn, D.; Alber, B.; Fuchs, G.
Loder, A.J.; Han, Y.; Hawkins, A.B.; Lian, H.; Lipscomb, G.L.; Schut, G.J.; Keller, M.W.; Adams, M.W.; Kelly, R.M.
Reisch, C.; Crabb, W.; Gifford, S.; Teng, Q.; Stoudemayer, M.; Moran, M.; Whitman, W.


The secretary of Iran's Human Rights Council has described Supreme Leader Ayatollah Seyyed Ali Khamenei's message to the youth in Europe and North America as a shrewd step toward developing a mutual understanding with the world.
The leader's message to the youth, which has new features, deals with the West's expansionist policies and its showdown with Islam.
The Islamic Revolution is one major reason why Westerners are sensitive to the political Islam model we have championed. Islamic Awakening is one of the phenomena the Islamic Revolution set in motion in the world of Islam.
Confrontation with the Muslim faith is being pursued vigorously in the West. A victory for Iran in this confrontation would result in the revival of the faith the West is so afraid of. They were hopeful Iran's Islamic model would not find its way beyond Iranian borders.
In Egypt, for instance, the popular movement turned into a revolution followed by formation of a government. But the Muslim Brotherhood could not offer what Egyptians favored and anticipated. People were allowed to choose between two options: a secular system – like Turkey's – and a Salafi system. But they wanted neither of them. In other words, efforts to establish a government failed, but the movement has not failed yet.
That they say the Islamic Awakening movement has foundered is not true, because the yardstick on this question is Iran [whose Islamic Revolution continues to inspire people]. An Islamophobia project has been set in motion. Paris terror attacks are just one act in this long-running play. For sure, it's not the first one, nor will it be the last.
That's why people in the West, the youth in particular, should understand that Islam and freedom are not the main points here, and that the image they have painted of the Muslim faith has nothing to do with genuine Islam. What they are depicting is an American-style Islam. The late Imam [Khomeini] once said: They [the West] tarnish Islamic principles through their own agents.
You see that under a flag which bears "There is no god but Allah", which is the most popular slogan in history and is all about monotheism and freedom-seeking, [a reference to the black flag of ISIL under which members of the notorious terrorist group brutally behead their hostages in the name of Islam] the terrorists decapitate people as they shout 'God is the greatest'.
So when people in the West hear this slogan "There is no god but Allah", they automatically recall the beheading of an innocent person. These are new techniques through which they enslave people by engineering the public opinion.
When the Islamic Revolution emerged victorious, nobody would have imagined that its message would one day gain traction.
The Leader's message is very significant for the future of the West. If you take a look back at history, you will learn that the Great French Revolution [1789-1793] to which the Western history is greatly indebted had three key slogans.
Number one is equality which bans the exploitation of others at the hands of one group. But equality is non-existent in the West today. Those who are rich hold power.
The second slogan is liberty. In the West, liberty is treated as a petty issue, and you are not allowed to challenge or examine the rationality or foundations of things. "There is no compulsion in religion" says you should not impose an ideology on us; let us decide – as humans – to be laic or religious persons. There is no such opportunity to choose [in the West].
Fraternity is the third slogan which is nowhere to be seen in the West today. In one world war, for instance, as many as 56 million people were killed. Acts of violence – which were later committed during the Vietnam War and has since been repeated – civil unrest, racism, self-centeredness, and egomaniac behaviors in a country which claims to be the cradle of freedom are all the direct result of this civilization. Such a civilization is doomed to fail and its prospects are not bright.
We need to be patient [for the leader's message to sink in]. Words are like seeds which are sown; they grow gradually. When the Islamic Revolution emerged victorious, nobody would have imagined that its message would one day gain traction. Truthful words will definitely grow, especially through modern devices, and rational, simple, fluent and dialectical literatures.
Explaining well-founded knowledge in a simple language is an art. The holy Prophet (PBUH) spoke in a way that everyone could understand depending on their knowledge.
The content of this message [the Leader's] shows a new method has been utilized. It should be promoted. Perhaps it is the first time after the revolution that we use this language to develop a mutual understanding with the world.


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The Craze Behind That Ugly Christmas Sweater
By Enjoli Francis December 8, 2011
Image credit: Caitlin Proctor
It's usually the first thing to get canned on TLC's "What Not to Wear," but that ugly Christmas sweater has become quite the popular holiday item. Just ask those selling them online.
Ugly Christmas sweater designs range from the subtle to the outrageous - there's even one portraying a recent meme of an officer pepper spraying protesters in California. On Nordstrom.com, one sweater goes for $600. Singer Justin Bieber reportedly wore a $1,650 red-and-white Christmas sweater during a live performance.
The men behind UglyChristmasSweaterParty.com - Adam Paulson, Brian Miller and Kevin Wool,2004 and 2005 graduates of Indiana State University - started noticing an increase in ugly Christmas sweater parties in mid-2000. The parties are said to have started in 2001 in Vancouver, British Columbia.
"From talking to people, I think the consensus is that the majority of people spend 364 days out of the year worrying about what they look like," Paulson told ABC News today. "[The sweater party] gives them one night to look totally stupid, to feel like they fit in, no matter what they wear. All inhibitions are gone."
Miller bought the web domain name and started a blog to keep track of rising trends, for example, ugly Christmas sweaters. Then in 2009, after hearing that people were having a hard time finding ugly holiday sweaters, the three started an online business.
They bought about 50 sweaters from a local Goodwill. Paulson said he thought he'd wasted $75. But by the next day, "they were all gone," he said.
Nancy Schloetel of Mountlake Terrace, Wash., started her business Notyouraveragesweater.com during the 2010 holiday season. Schloetel said she got the sweaters from lots of different thrift stores. Although the 2010 season was quiet - she said people had trouble finding her online - this year business has been steady since early November.
"All my best stuff has gone out the door," she told ABC News. "I've been shipping stuff out all night long."
Schloetel said she had purchased about 360 sweaters this season and had sold half so far. Her sweaters range in price from $18 to $24, though some have gone for $40, if she makes a few design enhancements.
She told ABC News that her customized sweaters have included battery-operated lights and even a sewn-on Nativity scene.
"If they're too common, too normal, I try to do something to uglify them," she said, laughing.
This year, the guys behind UglyChristmasSweaterParty.com said they have 15,000 sweaters, which they now buy from a "sweater guy." Their inventory has grown from filling the 600-square-foot floor of a loft to a basement to an entire leased warehouse. The sweaters go for $24.
Charlene Sarmiento, a spokeswoman for Goodwill Industries International, said that she'd heard sweaters were going fast at the more than 2,600 Goodwill stores across the U.S. and Canada.
"They are flying off the shelves. Lots of people are looking for those type of festive, holiday sweaters, and they are selling very well," she told ABC News.
Anne Marie Blackman of Killington, Vt., started My Ugly Christmas Sweater in 2009. A former stay-at-home mom looking to make money, Blackman decided to embellish some sweaters that would definitely be contenders for most ugly at any holiday fete. She took her products to eBay and they sold.
"I was shocked," she told ABC News today, "and I decided to make it a business."
Blackman said that her site had 100,000 page views Wednesday and almost 5,000 hits on Facebook. This year she moved the sweaters from her home to a warehouse. Of the 2,500 sweaters - triple the number from last season - available for purchase, two-thirds have been sold, she said. Her sweaters range from $35 to $80 and sell all over the world.
"It's a very easy, fun way to have a theme party," she said. "It started out a young person's trend, people making fun of their mom and grandmom's sweaters. Now it's moved to those moms and grandmoms having their own parties."
Miller, Paulson and Wool - who also have day jobs - released a book in November advising readers of what to wear if they're invited to an ugly Christmas sweater party, and how to throw one. Paulson said the last half of the book included "the ugly Christmas sweater Hall of Fame."
Schloetel said that ugly Christmas sweaters were good conversation starters.
"For one thing, everyone just looks so silly," she said. "You can't help but smile. You don't have to worry about getting dressed up or getting ready. The worse you look, the better. It puts a real fun atmosphere in place."


text:
The New York State Military History Museum will be closed until further notice
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159th Regiment
William Tiemann Collection
Donated by the Tiemann and Farrar Families
'Amid the Many Toils and Dangers'
The Civil War Letters of William Francis "Frank" Tiemann
William Francis "Frank" Tiemann (1844-1926) served during the Civil War with the 159th Regiment of the New York State Volunteers, perhaps longer than any other member of the regiment. He was a recruiter for the regiment and three years later he was discharged when the regiment was disbanded.
He was wounded in the Battle of Irish Bend and fought in two other major engagements in Louisiana, at Port Hudson and Mansura. In Virginia at the Third Battle of Winchester (Opequon), he was captured and held prisoner for five months before being exchanged in February 1865. He rejoined the regiment and was in Augusta and other locations in Georgia until late September 1865. He was an infantry officer who also performed duties as an adjutant and provost marshal. After the war he was the regiment's historian.
Frank, as he was called, was the second son of Julius William Tiemann (1817-1903) and Jane Waugh (Stephenson) Tiemann (1815-1853). He was born on March 29, 1844, in the Manhattanville section of New York City. His father was a bookkeeper for a leading paint manufacturing company founded by his grandfather, Johann Anton (Anthony) Tiemann (1778-1862), and, beginning in 1848, was headed by a politically prominent uncle, Daniel F. Tiemann.
When he was growing up, Frank Tiemann lived with his parents and other relatives in several locations on Manhattan and at one time in Brooklyn.
In the summer of 1862 Frank and his older brother, Julius Henry Tiemann (1840-1906), who as known as Harry, were recruiters for a regiment then being organized in the state's Third Senatorial District (Brooklyn) by Edward Leslie Molineux (1833-1915). In 1850 the Molineux family and that of Julius and Jane Tiemann were neighbors in Brooklyn. At about that time Molineux, who apparently became a "big brother" to Frank, joined Daniel F. Tiemann & Company where Frank also worked. Molineux became a partner in the firm, but left it immediately after the Civil War.
Before the war, Harry Tiemann was an enlisted member of the New York 7th Regiment. In 1861, when that regiment was called to defend Washington, D.C., he was abroad studying in Germany. His place in the ranks of Company B for the 90-day tour of duty was voluntarily taken by Molineux.
Frank Tiemann enlisted on September 5, 1862, as an 18 year-old sergeant with the understanding that he would later receive a commission. The 159th was organized on October 28, 1862, and mustered into Federal service on November 1 "for three years, or during the war." Initially, Frank Tiemann was assigned to Company B; Harry Tiemann was a lieutenant in Company B.
Command of the regiment, part of the 19th Army Corps, was awarded to Molineux on November 25, and that day he was promoted to the rank of colonel. On December 3, the 159th sailed for Louisiana with an expedition led by Major General Nathaniel P. Banks. Shortly after the unit's arrival at Baton Rouge, Frank Tiemann received his commission and was reassigned to Company A. Later in the war he was promoted to captain and at war's end to major. The 159th left Louisiana on July 17, 1864, for duty in Virginia, initially at Bermuda Hundred and then in the lower Shenandoah Valley. At the close of the war the regiment was posted in Georgia.
The majority of Frank Tiemann's war-time letters was written to his father, Julius William Tiemann, and was addressed to him at 128 William Street, New York City, the Manhattan office of D. F. Tiemann & Company. Other letters were sent to his step-mother, Marie Antoinette Megie Tiemann (1841-1902), who he called Nettie.
Frank Tiemann had three younger sisters, Ella Antonia (1846-1920), Ida Jane (1848-1933) and Leila Annette (1850-1896). The infants he mentions are his step-brother, Hermann Newell Tiemann (1863-1957), and his step-sister, Margaret Livingston, who died in infancy in August 1865.
The letters dated December 9, 1862, through December 19, 1863, are today in the possession of Peter Tiemann Farrar of Andover, Vermont. Farrar is Frank Tiemann's great grandson. The letters dated January 5, 1864, through September 21, 1865, are today in the possession of Joyce Molineux Tiemann of Chester, Vermont. She is Frank Tiemann's granddaughter.
In his correspondence, Frank frequently mentions his brother Harry, who resigned his commission in December 1863, and Edward Leslie Molineux, who often is referred to by his military rank or informally as Edward or Ned. A cousin, John Anthony Tiemann (1841-1897), was also a member of the 159th, serving as a sergeant and later lieutenant. Occasionally, Frank Tiemann reports on seeing Joe Stephenson, a maternal cousin, and Will Tackaberry, another relative. The relationship, if any, of men identified as Leander, Drake, and Edgar is unknown.
The transcription of these letters follows Tiemann's idiosyncrasies and inconstancies of spelling, punctuation and the use of capital letters. In a few instances, insignificant errors in the spelling of proper names have been silently corrected.
In 1891, Frank Tiemann, then residing in Brooklyn, compiled and published the regimental history, "The 159th Infantry, New York State Volunteers, in The War of The Rebellion, 1862-1865." Perhaps at that time he penned his "Personal recollection of Army Life during the War of the Rebellion, 1862-1865," a narrative based largely on the letters he wrote home. The manuscript of "Personal recollections" is in the possession of Joyce Molineux Tiemann.
He also wrote two other unpublished monographs, "Reminiscences," an account of his growing up on Manhattan, and in 1894, "Prison Life in Dixie," an account of his wartime imprisonment. Typescript copies of both are in the New York State Library. "Prison Life" is also in the library of the Virginia Historical Society.
Frank Tiemann died May 2, 1926, in East Orange, New Jersey, one of the last survivors of the 159th New York State Volunteers.
Transcriber's Notes:
Two postwar letters sent by Frank Tiemann to his farther, Julius W. Tiemann, in 1866 – one from Savannah, Georgia, and the other from St. Louis, Missouri – are added at the end of his Civil War letters.
A smattering of letters sent by various individuals to Julius W. Tiemann is presented chronologically in the general run of letters sent by Frank Tiemann.
This transcription also includes two separate batches of letters sent to Julius W. Tiemann during the Civil War – a batch of three letters from Lyman Burnham of Brooklyn, the brother-in-law of Edward Leslie Molineux, and a batch of nine letters from Edward Leslie Molineux in which he discusses business affairs and Harry Tiemann's desire to leave military service.
-- Will Molineux, January 2009
Photographs from the collection of Peter Tiemann Farrar:
1. The photograph of Frank Tiemann was taken in the studio of K.W. Beniczky of No. 2 New Chambers Street, corner Chatham Street, New York City, in September 1862. The reverse is labeled "W.F. Tiemann, Co. B, 159th Regt, N.Y.S. Vols. Acting Sergeant Major in charge Recruit Camp on Clove Road, Brooklyn, N.Y."
2. Veterans of the 159th New York State Volunteers are pictured on June 11, 1884, when the Society of the Army of the Potomac paraded in Brooklyn as part of its 15th annual reunion. Standing in front center and holding a sword is General Edward Leslie Molineux. Frank Tiemann is believed to be the man standing behind Molineux's right shoulder with his hands holding a pair of gloves. It is likely that this picture was taken in front of Molineux's residence at 117 Fort Greene Place, Brooklyn, and, if so, that the boys on the stairway could be his sons, Leslie Edward and Cecil Sefton Molineux
To Julius W. Tiemann (Father)
To Marie Antoinette Megie Tiemann (Stepmother, Nettie)
To Harry Tiemann (Brother)
To Leila Tiemann (Sister)
To Colonel Edward Molineux
From Others to William F. Tiemann
Back to 159th Regiment During the Civil War
New York State Division of Military and Naval Affairs: Military History
URL: http://www.dmna.state.ny.us/historic/reghist/civil/infantry/159thInf/Tiemann/TiemannIndex.htm
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text:
Recorded crime in Scotland: handling offensive weapons
Safer Communities Directorate
This report presents information on handling offensive weapons recorded by the police.
This document is part of a collection
Recorded Crime in Scotland
2. Trends in crimes of handling weapons
3. Characteristics of Weapons Crime in Scotland
4. Changes to counting rules for handling offensive weapons
Table 1: Number of 'Handling offensive weapons' crimes recorded by the police and number sampled, April to September 2013 and April to September 2017.
Table 2: Gender of perpetrators (where identifiable) by crime type, April to September, 2013 and 2017.
Table 3: Age of perpetrators (where identifiable) by crime type, April to September, 2013 and 2017.
Table 4: Population of Scotland, by age group, 2013 and 2017.
Table 5: Weapons used, where identified, by crime type, April to September, 2013 and 2017.
Table 6: How crimes are reported / come to the attention of the police, by crime type, April to September, 2013 and 2017.
Table 7: Location where the crime was committed, where known, by crime type, April to September, 2013 and 2017.
Table 8: Time the crime occurred, by crime type, April to September, 2013 and 2017.
Table 9: Whether there was a mention of either drugs or alcohol, by crime type, April to September, 2013 and 2017.
Table 10: Gender of victims (where identifiable) by crime type, April to September, 2017.
Table 11: Age of victims (where identifiable) by crime type, April to September, 2017.
Table 12: Gender of victim, by gender of perpetrator and crime type, April to September, 2017.
Table 13: Relationship between victim and perpetrator (where identifiable), by crime type, April to September, 2017.
Table 14: Crimes committed against a person using the weapon, by crime type, April to September, 2017.
Table 15: Physical injuries sustained, by crime type, April to September, 2017.
Table 16: Quality of crime recordings, April to September 2017, by Police Division.
How to access background or source data
This report presents information on the handling of offensive weapons within public places in Scotland. This includes new details on the characteristics of these cases, based on a random sample of police recorded crimes.
The main sources of evidence all point to a substantial and long term reduction in the handling of offensive weapons in Scotland over the past decade.
As fewer cases of handling offensive weapons were being recorded by the police, resulting in fewer convictions – both the number of adults who believed knife carrying was common in their local area and the number of emergency hospital admissions due to assault with a sharp object also fell.
Over time, the vast majority of those convicted of handling an offensive weapon remain male but by 2016-17 they were older than their counterparts 10 years earlier (with their median age increasing from 24 to 29 years old).
Based on a random sample of police recorded crimes from April to September 2017
The overwhelming majority of police recorded crimes for handling an offensive weapon in a public setting were committed by males (89%). The median age of a perpetrator was 29 years old when the weapon wasn't used in other criminal activity, this is slightly older than in 2013 when the median age was 27. When the weapon was used in other criminal activity the median age was 27.
Between 2013 and 2017, where the weapon was not used in other criminal activity, there was a decrease in the proportion of perpetrators aged between 20 and 29 (from 34% to 24%) and an increase in those aged 40 or over (from 19% to 25%). Individuals aged 13 to 19, 20 to 29 and 30 to 39 were more likely to be found in possession of an offensive weapon than expected from their overall share of the population. This is the case whether the weapon was used in other criminal activity or not.
Almost a fifth (17%) of all victims were teenagers (i.e. aged between 13 and 19). Over a quarter (26%) were aged between 20 and 29 with further 24% aged between 30 and 39 and 30% aged 40 or over. Individuals aged 13 to 19, 20 to 29, and 30 to 39 are more likely to be victims of a weapon-based crime than would be expected given their share of the population as a whole.
Three-quarters of criminal acts involving the use of a weapon in a public setting had a male victim (75%). In 70% of all cases both the perpetrator and victim were male. Just under a fifth (19%) had a female victim and a male perpetrator. Victims of criminal acts involving the use of a weapon had an median age of 31 years old.
Perpetrators and victims were more likely than not to know each other (59% known compared to 41% not known). They were both more likely than average to live in urban areas and areas of higher deprivation.
The most common criminal act committed with a weapon in a public setting was threatening or abusive behaviour (44%), followed by common assault (32%) and serious assault (14%). Threatening or abusive behaviour was more likely to occur and serious assault less likely to occur when the victim was female, compared to male victims.
The majority of criminal acts committed with a weapon against a person in a public setting involved no physical injury to the victim (71%). This proportion was slightly higher when the weapon was a bladed or pointed article like a knife (74%).
In 29% of cases the victim of a weapon-based crime in a public setting sustained some degree of physical injury (including 26% of cases where the weapon was a bladed or pointed article and 31% for other offensive weapons).
When bladed or pointed articles were used against other people in a public setting, they caused a serious assault level injury in 21% of cases (such as stab wounds or other permanent disfigurement).
Criminal acts committed with a weapon against other people in public settings were more likely to involve no physical injury if the victim was female and if the victim did not know the perpetrator.
A person in possession of an offensive weapon whilst in a public place, school or prison, without reasonable excuse or lawful authority, is guilty of a crime under the Criminal Law (Consolidation) (Scotland) Act 1995. These findings relate to a sample of crimes recorded by the police under this act.
Justice Analytical Services


While English colonists had established a foothold in the middle of Wangunk country in 1635, it may have been as early as 1638 that colonial authorities persuaded the local sachem Sowheage to agree to an expansion of English settlement into other parts of his territory. Subsequently, the townsmen of Wethersfield, Connecticut purchased a tract of land six miles in breadth on both sides of the Connecticut River which expanded six miles deep to the west of the river and three miles deep to its east.1 Yet, plans for planting Mattabesset were in place at least by 1646, when William Phelps joined an already established committee for that purpose, [PRCC 1:146] and accelerated after Sowheage's death sometime around 1649.
In 1650 the committee tasked to investigate possible settlements within Wangunk country reported that the only suitable place it could find was one about a half mile from the Indian village at Mattabesset and that site would be able to support about fifteen colonial families. [Charles Adams, 5] In September of 1651, the colony ordered that Mattabesset, later renamed Middletown, should be formed as a town, and, starting the following year, representatives from there were elected to the General Court.
While the English settlement of Middletown had begun, plans were made to draw boundaries around Native and colonial properties. In 1650, Governors John Haynes and Edward Hopkins reserved about 350 acres in three tracts of land for Sowheage's heirs and descendants at Mattabesset and Wangunk but the limits of the reservation remained undefined for close to a dozen years.
In the summer of 1662, eleven Wangunks sold a one hundred and fifty square mile strip of land spread over six miles on both banks of the river, preserving for themselves the use of thirty acres. But settlement and land sales only aggravated tensions between settlers and the tribe. In 1665 colonial authorities organized a committee2 to investigate a "difference" between the town and the Indians living nearby with a view towards establishing the bounds of a Wangunk reservation. [PRCC 2:14] The Town of Middletown appointed its own exploratory team to assist the committee on May 4, 1666 [IP 1.2.137/1757.05.11.00], but nothing much was accomplished until 1670.3 In March, Middletown officials selected three townsmen to walk the bounds of the remaining common land granted by Haynes and Hopkins on the east side of the river. Three weeks later, the colonial survey revealed the Wangunk tribal property to be thirty-three acres of land butted and bounded by the river and a highway, nine more acres of grassland divided into parcels, and approximately six acres of land at Otter or Deer Island.
This did not stop individual tribal members from selling off their personal plots of land. The General Assembly, the body who regulated purchases of Indian property, allowed a number of settlers to buy land at Wangunk. [MLR 1:61, MLR 4:98, PRCC 4: 98] In 1697, it relaxed its normally strict regulations to permit individuals to acquire up to an acre of Wangunk meadowland, [PRCC 4:212] and many of Sowheage's descendants sold their properties on both sides of the Connecticut River. In some instances these sales were out of private pecuniary gain. Other times, they were to pay debts to colonists, as in the 1713 deed from the sons of the late sunksqua to settle her funeral expenses. [YIPP Wyllys Papers].
In any event, there was no shortage of English purchasers. Middletown farmers had used Wangunk meadows to grow crops and feed their animals since 1650. However, as their numbers increased, many colonists moved their households across the river permanently and called their settlement East Middletown or the Third Society. As Ives observes, Indian Hill "became a target of colonial industry." [Ives 67] In 1714 residents there petitioned for parish privileges. The General Assembly permitted Giles Hall, a mariner and local ship builder, to purchase land there in 1716. The following year Hall was tasked with planning access roads to transport materials to the river. Job Bates, a blacksmith, followed Hall and other families of rope makers, metal workers, and sail outfitters, all of whose livelihoods depending on ship building or the shipping industry. [Ives 67] By 1725 Middletown had become the largest port city between New York and Boston. Because of the shallowness of the Connecticut River at Middletown, the town became the final port on the river for many large ships, especially those in the West Indies trade.
Over all this development, in an expression of the tribe's sovereignty, Wangunk leadership maintained a watchful eye and acted decisively. In 1697, Papenum, the sachem of Haddam, in response to the growing colonial desire for Indian land, wrote a will that restricted the sale of her property to tribal members only. [Hermes, 153] Similarly in 1713, Sarah One-Penny directed that her property should be inherited only by her grandson Scippio Two-Shoes. [Hartford Probate] In 1728, tribal leadership, now in the hands of Cushoy, a descendant of Robin, monitored the expansion of colonial roads at Wangunk by being part of the survey and construction crews.
However, the General Assembly's decision to place the colonists' new meeting house strategically within the Wangunk reservation in the 1740s may have marked a turning point in tribal affairs. By then, many of the Wangunk were tempted to move elsewhere. In 1731, some of their Quinnipiac relatives were encouraged to resettle at Waterbury, Connecticut, and by 1747 a number of Quinnipiac had removed among the Tunxis at Farmington, Connecticut. By the middle of the eighteenth century, a Wangunk diaspora took tribal members to Indian communities at Mohegan, Farmington, Hartford, and New Hartford. At the same time, the presence of a growing congregation of colonial settlers in the middle of the Wangunk reservation must have been disconcerting. The remaining Wangunk responded to these pressures by consolidating their properties in the hands of tribal leaders instead of alienating them to colonial settlers. From 1740 to 1751, land deeds indicate that the Wangunk sold land to Cushoy or members of his family. [MLR 10: 546-8, MLR 10:548, MLR 13: 187, 612.] Land that was still held by absentee Wangunk were often leased, but it is unknown whether the leaseholders were Indian or colonists.
The first major challenge to this strategy was in 1754, when Jabez Hamlin, an agent for the colonists of the Third Society presented a memorial to the General Assembly [1754.05.13.00, 1754.05.14.00], complaining that the land around their new meeting house was surrounded by two hundred and fifty acres of Indian land, and requested that lots of "five or six acres." The proposal failed to pass though the Lower House, and the Upper House concurred in the negative outcome.
Two years later, the society adapted a new approach. On May 13, 1756 several of the society's residents—a number of military officers, farmers, and a blacksmith—wrote the General Assembly [1756.05.13.00] to say that the Indian land around their meeting place was well suited for development, either for ship building or tillage, and considering the small numbers of Indians who lived there, it would be most advantageous to be able to purchase the land or trade it for another parcel. They accompanied their request with a memorial from a less than impartial number of selectmen from Middletown [1756.05.15.00], a tax collector, two ship captains of the West Indies trade, and a town merchant heavily invested in the advancement of shipbuilding. The selectmen indicated that for a number of years they had supported Cushoy, now aged and infirm, and his son Tom in his final sickness. The town officials noted that the Indians were indigent but owned significant parcels of land, which they offered could be sold to cover the expense of the Indians' care. The Assembly commissioned an investigative committee [1756.05.00.00], and, encouraged by such progress, one of the selectmen, Samson Howe, hired Hartford County surveyor William Welles to map the two reservations. [1756.09.28.00] Despite a strong recommendation to sell from the committee [1756.10.11.00], the legislature once again failed to approve the request.
Building upon the success of Ranney, in October 1760, two Middletown residents, Job Bates and Isaac Waterman, applied to the General Assembly, requesting the partition and sale of the Wangunk reservation. [1760.10.06.00] Their argument consisted of explaining that the fifty acres at Indian Hill had been "used, worn out, and left useless almost by the Indians, and is of but little service to them." Bates and Waterman offered a supporting affidavit by five Wangunk proprietors, but the measure failed to win the support of both houses and died in committee.
In May of 1762, five male descendants of the original Wangunk proprietors, showing that their numbers were reduced and scattered, petitioned for a "wise and judicious" committee to examine the title and circumstances of the Indian land, indicating that "it would be much best said lands be sold and the avails thereof be divided" among them. [1762.05.18.00] Like Ranney, the petitioners were absentee tribal members, most of them Christianized and educated, living at Tunxis in Farmington. While there is no official record of the General Assembly's immediate action on the request, it is most likely that they tabled consideration on the matter.


Many of us who have gone through pregnancy and child birth recognize the term 'pelvic floor muscles'. During and after pregnancy we are advised to do pelvic floor training. Still many women after pregnancy experience issues with the pelvic floor. Trust me, I speak from experience.
There is a lack of conversations about this area. Only when I ask other women there seems to be a common experience. I think there is a taboo sphere around the intimate areas of the female body in general. Women try to solve these issues on their own and rarely talk about it with their partners, girlfriends or professionals. Others assume it is just something you have to deal with. That is why I try to open the conversation with other women. I sincerely hope this blog contributes to an open space to talk about women's health.
One solution is to go to the pelvic floor physiotherapist. Another choice is to do the Carin exercises. I spoke to Marta from Spain who is a practising physiotherapist in Mallorca (Spain). During her studies in physiotherapy, Marta encountered pelvic floor revalidation programmes and was fascinated by the effective and simple methods to cure pelvic issues.
For 16 years Marta has been dedicated to women's health and explains her motivation for the topic: "I want to contribute to solving a problem. Currently women are offered a choice between medication and operation. In each patient I see the challenge in helping them that helps me to develop and grow as a professional. Even after 16 years, I feel excited to go to work".
"It is important to be informed about new developments in the field. I often go to conferences and read scientific studies to help my patients with the newest and best solutions and techniques. The difficulty lies however in providing training for the pelvic area. Compared to other body parts, pelvic floor muscles are a difficult area to improve by training".
Exercising the pelvic floor muscles can be confusing and this might be one of the reasons why women find it difficult to do. Marta makes me aware of the positive developments. She tells me there are an increasing number of pelvic floor therapists active in the field. Also for women who want to train their pelvic floor muscles at home there are a variety of products available to choose from. But awareness and education of the topic is still going slow: "The lack of knowledge amazes me. In my view the biggest taboo in women's health is about this body part. This makes it difficult for our field to be acknowledged".
The most important advice Marta gives us is to be self-aware of our body: "Know your body, and in specific the lower 'pelvic' part. Awareness about your body and health is important to notice changes in your body. In regard to the pelvic floor area it is important to know how to prevent constipation and to urinate properly, and probably as important, to be prepared for pregnancy and to recover well after child birth. Pregnancy after all is the main cause for pelvic floor issues".
She advises to go to a pelvic floor therapist or you do the Carin exercises.
Kirsten van der Kolk asks expert's advise on women's health issues. In this edition she wants to know why pelvic floor exercise is important.
If you want to get in contact with Kirsten after reading this, simply send her a message: kirsten.vanderkolk@lifesense-group.com.