text:
\section{INTRODUCTION}

Over the last decade there have been several studies investigating the fate of relativistic symmetries at the Planck scale (see, {\it e.g.}, Refs.\cite{amelino2013quantum,magueijo2003generalized,amelino2012fate}). Particular interest has been devoted to scenarios such that the (inverse of the) Planck scale would set the minimum allowed value for wavelengths, and notably it has emerged that such a feature could be implemented within a fully relativistic picture \cite{amelino2001testable,amelino2011principle}. This is the realm of the so-called doubly-special (or deformed-special) relativistic (DSR) theories \cite{amelinoDSRijmpd1135,amelino2001testable,amelino2010doubly,kowalski2003non} where the Planck scale plays the role of a second relativistic invariant, in addition to the speed-of-light scale. Evidently these DSR scenarios require the adoption of deformed Poincar\'e transformations connecting inertial frames \cite{amelinoDSRijmpd1135}, and the associated new invariant laws may include indeed an oberver-independent minimum-wavelength law and/or an observer-independent modification of the dispersion/on-shellness relation (MDR). Another major implication of the DSR deformations of Poincar\'e transformations is that the laws of composition of momenta are no longer linear: in order to preserve their relativistic covariance they must be deformed matching the deformation of the Poincar\'e transformations.
 The mathematical formalism of Hopf algebras has been found to be a natural possibility\footnote{Note however that examples of DSR-relativistic scenarios not
 involving Hopf algebras have been provided in Refs.\cite{freidelDESITTERSNYDER,balena}.} for formalizing DSR-relativistic scenarios \cite{agostini2004hopf}, since Hopf algebras can be deformations of standard Lie algebras in the form of non-linear deformations of both commutation relations among symmetry generators and the so-called coproducts (on which the laws of composition of momenta are based; see later).

 So far all of these studies (with the only exception of the exploratory analysis in Ref.\cite{gacmixingold})
 focused on \textit{universal} deformations of the special-relativistic symmetries, {\it i.e.} deformations that affect identically all particle types.
Here we explore the possibility that the description of Planck-scale physics might require an additional level of complexity, namely different particles' kinematics might be dictated by different relativistic symmetries, {\it i.e.} we wonder whether, within the DSR framework, it is possible to consistently formulate relativistic theories that attribute different laws of kinematics to different particles, \textit{non-universal} deformations of relativistic symmetries. For the case of broken relativistic symmetries (preferred-frame scenarios) particle-dependent effects where first considered by Coleman and Glashow in Ref.\cite{coleman1997cosmic} and there were several developments of that research direction (see, {\it e.g.}, Ref.\cite{smeREVIEW}), but the possibility of particle-dependent properties within a relativistic picture has been so far only explored preliminarily by one of us in Ref.\cite{gacmixingold}, providing most results only
 at leading order in the deformation scale. We shall here report results valid to all orders in the deformation scale and consider a rather wide class of possible properties attributed to different types of particles. We also intend to show that these scenarios
 can be built using rather mild modifications of standard Hopf-algebra techniques. As we shall discuss extensively in the following sections, the key building block for our scenarios is a novel mathematical tool which we call \textit{mixing coproduct}, a generalization of
 the standard Hopf-algebra notion of coproduct. \\
 We shall here mostly postpone the analysis of the phenomenological implications of our
\textit{non-universal} deformations of relativistic symmetries, but the careful reader will notice how that ultimate objective guides
our technical efforts. In particular, while \textit{non-universal} deformations of relativistic symmetries
do not necessarily require modifications of the on-shell relation, in all of our case studies there is at least
one type of particle governed by modified on-shellness.
Indeed, one of the main reasons of interest in DSR-relativistic theories has been their rich phenomenology
when they involve modified on-shellness, a rare case of Planck-scale effect testable with
presently-available technologies \cite{grbgac1998,gacSMOLINprd2009,amelino2015icecube,jacob2007neutrinos}.
Of particular interest for the analysis we here report is that fact that some recent tests of modifications of on-shellness
for neutrinos have led to preliminarily encouraging results \cite{amelino2015icecube,Amelino-Camelia:2016ohi,amelino2016icecube,paperbyMA},
for values of the symmetry-deformation scale of about a tenth of the Planck scale, whereas for photons
other analyses have led to apparently robust bounds on the symmetry-deformation scale going all the way up to
the order of the Planck scale. In spite of the very preliminary nature of the mentioned neutrino studies, one cannot
avoid to wonder what would have to be the relativistic picture if it was actually true that modifications
of relativistic symmetries are stronger for neutrinos than for photons, leading us indeed to speculate
about non-universality of the deformation scheme \cite{abdo2009limit,aharonian2008limits}.

In addition to the present preliminary assessment of data on dispersion for photons and neutrinos further motivation
for non-universality is found upon contemplating macroscopic bodies (like soccerballs and planets) in a DSR picture:
it is easy to see \cite{amelino2011relative} that a phenomenologically viable DSR picture should ensure that the deformation
of relativistic properties fades away for macroscopic bodies, and there are known relativistic mechanisms
to enforce this property (see, {\it e.g.}, \cite{amelino2011relative} and references therein);
however it is still unclear what should be the DSR-relativistic kinematics applicable to processes involving a
fundamental particle and a more macroscopic systems. Assuming that indeed the DSR-deformation
of relativistic properties is, for example, stronger for the electron than for the bucky ball, what are the conservation laws
that one should apply for collisions between an electron and a bucky ball?

Our paper is organized as follows. In Section II we briefly review a well-known example of \textit{universal} deformation of the Poincar\'e symmetries, based on the $\kappa$-Poincar\'e Hopf algebra.
In Section III we start showing how it is possible to generalize such a setup in order to allow for models where
different particles obey different symmetry laws.
The first case we consider is that of having two (or more) deformed Poincar\'e algebras with different
deformation scales, e.g. $\ell$ and $\ell'$. This gives us the possibility to introduce
the notion of \textit{mixing coproduct} for the formalization of the kinematics of such \textit{non-universal} deformed-symmetry models. In Section IV we show that a scenario previously advocated by Magueijo and Smolin \cite{magueijo2003generalized} can be equivalently
reformulated in terms of a specific mixing coproduct. In Section V we add a further element of complexity and explore the possibility to define a mixing coproduct between different algebras, focusing on the case of mixing coproducts involving the $\kappa$-Poincar\'e algebra and the standard Poincar\'e Lie algebra. In our closing Section VI we offer a perspective on our results and
some observations on possible future developments.

\section{Universal coproduct: $\kappa$-Poincar\'{e} case of study}

In preparation for discussing DSR scenarios with nonuniversal relativistic properties, we find
useful to briefly review the most studied DSR scenarios with universal relativistic properties, which is based on mathematical structures
found in the $\kappa$- Poincar\'e Hopf algebra (specifically the so-called \textit{bi-cross-product basis}
of the $\kappa$- Poincar\'e Hopf algebra). When written in the \textit{bi-cross-product basis}, the $\kappa$- Poincar\'e commutators are
\begin{align}\label{bicross}
\begin{split}
&[P_\mu, P_\nu ]= 0, \quad [ R_i, P_0 ]= 0, \quad [ R_i,P_j ]= i\epsilon_{ijk}P_k, \\
&[ R_i, R_j ]= i\epsilon_{ijk} R_k, \quad [ R_i, N_j ]= -i\epsilon_{ijk} N_k, \\
&[N_{i},P_{0}]=iP_{i} \, \quad [N_{i},N_{j}]= -i\epsilon_{ijk}R_{k} ,\\
&[N_{i},P_{j}] = i\delta_{ij}\left(\frac{1-e^{-2\ell P_{0}}}{2\ell}+\frac{\ell}{2}{\vec{P}}^{2}\right)-i\ell P_{i}P_{j} ,
\end{split}
\end{align}
and, as a consequence, the mass Casimir gets deformed into
\begin{equation}
 \left(\frac{2}{\ell} \sinh\left(\frac{P_0}{2\ell}\right)\right)^2 - e^{\ell P_0}P_i P^i .
\end{equation}
The modification of the commutators between boosts and translation generators also
requires a deformation of the laws of composition of momenta. Indeed, one can easily check that under the action of $N_i$,
 give in terms of the commutators \eqref{bicross}, one would have that
\begin{equation}
[N^i_{[p,k]}, p_\mu+k_\mu] = [N^i_p + N^i_k, p_\mu + k_\mu] \neq 0 ,
\end{equation}
even if $p_\mu+ k_\mu = 0$. This means that the usual conservation laws would not be covariant.
As one can check by using the relations in Eqs. \eqref{bicross}, the correct modification (in order to achieve covariance) is
\begin{equation}
p_\mu \oplus_{\ell} k_\mu =\begin{cases} p_{0}+k_{0}\\ p_{i}+e^{- \ell p_{0}}k_{i}\end{cases} ,
\end{equation}
for momenta, while for the boosts one has
\begin{equation}
N^i_p \oplus N^i_k = N^i_p + e^{- \ell p_{0}} N^i_k + \ell\epsilon_{ijn} p^j R^n_k  .
\end{equation}

These modifications are such that now the condition of covariance of the composition law is obeyed
\begin{equation}
 [N^i_p \oplus N^i_k, p_\mu \oplus k_\mu] = 0 ,
\end{equation}
as the reader can easily verify.

In the formalism of Hopf algebras these observations can be formalized by saying that the co-algebra, i.e. the set of relations that define the action of the generators on the product of fields, is non-primitive and, in particular, is modified as follows
\begin{equation}
\Delta P_i = P_i \otimes 1 + e^{-\ell P_0} \otimes P_i \, , \quad \Delta N_i = N_i \otimes 1 + e^{-\ell P_0} \otimes N_i + \ell \epsilon_{ijk} P_j \otimes R_k
\end{equation}
 The coproducts of $R^i$ and $P_0$ have not been written down explicitly because they remain primitive, i.e. they are dictated by the Leibniz rule. \\
To sum up, we have seen that there are two key ingredients needed by a deformed relativistic picture, i.e. the deformed algebra closed by the generators of non-linearly deformed symmetry transformations has to be compatible with both the form of the mass Casimir (or, equally, the associated on-shell relation) and the conservation laws for the associated charges. We stress that for compatibility we mean that the boost generator must leave invariant the on-shell relation and must transform covariantly the composition law for momenta.

\section{Mixing coproduct: $\kappa$-Poincar\'{e} case of study}
\label{sec:mixingl-l'}

We are now ready to contemplate a further generalization of the notion of relativistic-symmetry deformation by allowing for \textit{non-universal} scenarios.
In this Section we introduce the notion of \textit{mixing coproduct} and provide a suitable formalization of it. As already mentioned, this would allow us to formulate a relativistic model where different particles (i.e. particles with different quantum numbers) do not follow the same relativistic laws. In particular, within the formalism of Hopf algebras, we can infer how to compose particles' four momenta from the coalgebraic sector. Thus, the mixing coproduct would be a mathematical object that gives us ``mixed" composition laws where the momenta we compose represent the charges associated to translation generators belonging to different algebras (for instance a Lie algebra and a Hopf algebra, or two different Hopf algebras) or, in some cases, to different ``bases" of the same Hopf algebra. We shall explain in this section what we mean exactly by these different cases.

Let us start by considering three Hopf algebras $H$, $H'$ and $H''$ and two maps $\phi$ and $\phi'$ defined as
\begin{equation}
\phi:H \to H'', \qquad \phi':H'\to H''
\end{equation}
 with inverse given by $\phi^{-1}$ and $\phi'^{-1}$ respectively. Then, it is possible to define a mixing coproduct \cite{gacmixingold} by composing these two maps as follows
\begin{align}\label{mixing coproduct}\begin{split}
&H'' \xrightarrow[]{\Delta''} H''\otimes H'' \xrightarrow[]{\phi^{-1} \otimes \phi'^{-1}} H\otimes H'.\\
\end{split}\end{align}

Thanks to the mixing coproduct $\phi^{-1} \otimes \phi'^{-1}$, we are here composing the momenta of two particles, whose symmetries are dictated by $H$ and $H'$ respectively, and the resulting particle with momentum given by the sum follows again a distinct symmetry group, i.e. $H''$. (i.e. it is the corresponding charge of the generator of translations in $H''$). It is not difficult to realize that, introducing analogous maps, it would be possible to have also mixing coproducts with target space either $H \otimes H''$ or $H' \otimes H''$. \\
In order to gather some confidence with this novel object and the related formalism, we shall discuss a couple of relevant examples. \\
As first example we consider the case in which $H,H',H''$ are three $\kappa$-Poincar\'{e} algebras $\kappa\mathcal{P}$, $\kappa'\mathcal{P}$ and $\kappa''\mathcal{P}$ which differ for the magnitude of the deformation parameter, $\ell$, $\ell'$, and $\ell''$ respectively. By means of the maps $\phi$ and $\phi'$, which in this case are simply morphisms, we can define two different ways to compose momenta in a mixed way
$$\oplus_{l\ell'}:M\oplus M' \to M'',$$
$$\oplus_{\ell'l}:M'\oplus M \to M'',$$
where here $M$, $M'$ and $M''$ stand for the three momentum spaces. This can be done by writing down explicitly the actions of the morphisms $\phi$ and $\phi'$ over the algebras $\kappa\mathcal{P}$ and $\kappa'\mathcal{P}$ respectively. \\
It is known that a given Hopf algebra can be written, as far as explicit formulas are concerned, in some rather different ways, depending on the conventions adopted \cite{kowalski2003non}. In fact, for Hopf algebras one must allow both linear and non-linear maps between the generators, mapping one ``basis" into another \cite{kowalski2003non}.

If we express the three Hopf Algebras all in the bicrossproduct basis, then the simplest way to define the morphisms $\phi$ and $\phi'$ is given by the following expressions
\begin{equation}\label{isol-l'}
\begin{alignedat}{3}
&\phi(P_{\mu})={\ell'' \over \ell }P_{\mu}'', \qquad &&\phi'(P_{\mu}')={\ell'' \over \ell'}P_{\mu}'', \\
&\phi(R_{i})=R_{i}'', \qquad &&\phi'(R_{i}')=R_{i}'', \\
&\phi(N_{i})=N_{i}'', \qquad &&\phi'(N_{i}')=N_{i}''.
\end{alignedat}
\end{equation}
Here $G \, \in \, \kappa\mathcal{P}$, $G' \, \in \, \kappa'\mathcal{P}$, and $ G'' \, \in \, \kappa''\mathcal{P}$ are used to denote the symmetry generators (in the bicrossproduct basis) of the three $\kappa$-Poincar\'{e} algebras characterized by different deformation parameters $\ell$, $\ell'$, and $\ell''$. In particular, it is possible to prove that these maps define isomorphisms between the Hopf algebras. For the sake of brevity, let us focus only on the morphism $\phi$. The reader can straightforwardly verify that
\begin{align*}
&\phi([P_{\mu},P_{\nu}])=0=[\phi(P_{\mu}),\phi(P_{\nu})],\\
&\phi([R_i,R_j])=\epsilon_{ijk}R_{k}''=[\phi(R_i),\phi(R_j)],\\
&\phi([N_i,N_j])=-\epsilon_{ijk}R_{k}''=[\phi(N_i),\phi(N_j)],\\
&\phi([R_i,N_j])=\epsilon_{ijk}N_k''=[\phi(R_i),\phi(N_j)] .\\
&\phi([R_i,P_0])=0=[\phi(R_i),\phi(P_0)],\end{align*}
It is also rather simple to verify the following equalities
\begin{align*}
&\phi([R_i,P_j])=\epsilon_{ijk}{\ell'' \over \ell }P_k''={\ell'' \over \ell }[R_i'',P_j'']=[\phi(R_i),\phi(P_j)],\\
&\phi([N_i,P_0])={\ell'' \over \ell }P_i''={\ell'' \over \ell }[N_i'',P_0'']=[\phi(N_i),\phi(P_0)]
\end{align*}
and finally
 \begin{align*}
 \phi([N_i,P_j]) &=\phi\Bigl[\delta_{ij}\Bigl({1-e^{-2 \ell P_{0}} \over 2\ell}+{\ell \over 2}|\bar{P}|^2\Bigr)- \ell P_{j}P_{k}\Bigr]\\
 &=\delta_{ij}\Bigl({1-e^{ -2 \ell { \ell'' \over \ell }P_{0}''} \over 2 \ell}+{\ell \over 2}{\ell''^2 \over \ell^2}|\bar{P}''|^2\Bigr)- \ell {\ell''^2 \over \ell^2}P_{j}''P_{k}''\\&={\ell'' \over \ell}
 \Bigl[\delta_{ij}\Bigl({1-e^{-2\ell''P_{0}''} \over 2\ell''}+{\ell'' \over 2} \ell \bar{P}''|^2\Bigr)-\ell''P_{j}''P_{k}''\Bigr]\\&=[\phi(N_i),\phi(P_j)].
 \end{align*}
These observations establish the isomorphism at the level of the algebra sector (i.e. commutators). Then, we need to look also at the coalgebra. For the coproducts we indeed find
\begin{align*}
 & \phi \otimes \phi ( \Delta ( P_0 ) ) = { \ell'' \over \ell } P_0 '' \otimes \mathds{1} + \mathds{1} \otimes { \ell'' \over \ell} P_0 = \Delta'' ( \phi ( P_0 )) , \\
 &\phi\otimes\phi(\Delta(P_i))={\ell'' \over \ell} P_0'' \otimes\mathds{1}+e^{-\ell{\ell'' \over\ell }P_0}\otimes{\ell'' \over \ell}P_0=\Delta''(\phi(P_i)),\\
 &\phi\otimes\phi(\Delta(R_i))=R_i''\otimes\mathds{1}+\mathds{1}\otimes R_i''=\Delta''(\phi(R_i))
\end{align*}
and
\begin{align*}
\phi\otimes\phi(\Delta(N_i)) & =N_i''\otimes\mathds{1}+e^{-\ell{\ell'' \over \ell}P_0}\otimes N_i''+\ell\epsilon_{ijk}{\ell'' \over \ell}P_j''\otimes R_k'' = \Delta''(\phi(N_i)).
\end{align*}
The last check we need concerns the compatibility of the map with the antipodes (then the compatibility with the counits follows straightforwardly), and also in this case it is easy to verify that
\begin{align*}
&\phi(S(P_0))=-{\ell'' \over \ell}P_0''=S''(\phi(P_0)),\\
&\phi(S(P_i))=-e^{-\ell{\ell'' \over \ell}P_0''}{\ell'' \over \ell}P_i''=S''(\phi(P_i)),\\
&\phi(S(R_i))=-R_i''=S''(\phi(R_i))\\
\end{align*}
and finally
$$\phi(S(N_i))=-e^{ \ell {\ell'' \over \ell }P_0''}N_i''+ \ell \epsilon_{ijk}e^{ \ell {\ell'' \over\ell }P_0''}{\ell'' \over\ell }P_j''R_k''=S''(\phi(N_i)).$$

We therefore established that the morphisms $\phi$ and $\phi'$ are actually isomorphisms connecting $\kappa\mathcal{P}$ with $\kappa' \mathcal{P}$ and $\kappa'\mathcal{P}$ with $\kappa'' \mathcal{P}$. These isomorphisms also has an inverse map, which for example
for $\phi$ is given by
\begin{align*}
&\phi^{-1}(P_{\mu}'')={ \ell \over \ell''}P_{\mu}, \\
&\phi^{-1}(R_{i}'')=R_{i}, \\
& \phi^{-1}(N_{i}'')=N_{i} .
\end{align*}
Of course, the inverse also constitutes an isomorphism between Hopf algebras. Indeed inverting the isomorphism
 simply amounts to exchange of roles between deformation scales, for example exchanging the roles of $\ell$ and $\ell''$. \\
Using the two morphisms $\phi$ and $\phi'$ we can construct, as anticipated in \eqref{mixing coproduct},
the mixing coproducts involving these three $\kappa$-Poincar\'{e} algebras as follows
\begin{align}\label{isol-l'2}\begin{split}
&p \oplus_{\ell\ell'}'' q=\begin{cases}{\ell'' \over\ell }p_{0}+{\ell'' \over \ell'}q_{0}\\{\ell'' \over\ell }p_{i}+{\ell'' \over \ell'}e^{- \ell p_{0}}q_{i}\end{cases},\\
&q \oplus_{\ell'\ell}'' p=\begin{cases}{\ell'' \over \ell'}q_{0}+{\ell'' \over\ell }p_{0}\\ {\ell'' \over \ell'}q_{i}+{\ell'' \over\ell }e^{-\ell'q_{0}}p_{i}\end{cases} .\\
\end{split}\end{align}

These relations give us two possible composition laws between momenta that belong to different momentum spaces. In fact, here $p\in M$, $q\in M'$ while, by definition,
$p \oplus_{l\ell'}'' q , \, q \oplus_{\ell' \ell}'' p \, \in M''$.
Consequently, this simple framework provides us a first example of a deformed relativistic theory where we are able to compose particles' momenta that live on different momentum spaces or, in other words, represent the charges associated to the symmetry transformations of different Hopf algebras. These two sums of momenta \eqref{isol-l'2} differ only for the order of the addenda.

 It should be noticed that these laws are not well defined when one of the three deformation parameters vanishes, i.e. we have to impose that $\ell \neq 0 \,$ , $\ell' \neq 0 \,$ , $\ell'' \neq 0$. This means that the above introduced morphisms cannot be used to compose a particle with symmetries described by the $\kappa$-Poincar\'{e} group with another whose momenta follow the standard Poincar\'{e} symmetries. As we shall see later, within our framework, this can be done with another class of maps. However, let us point out that this
 is consistent with the fact that we have proven they are isomorphisms between Hopf algebras and, therefore, they could not relate the Poincaré Lie algebra with a Hopf algebra since there is no isomorphism connecting them. Finally, let us notice that, according to \eqref{isol-l'2}, given two momenta $p$ and $q$ of two particles in $\kappa\mathcal{P}$ and in $\kappa'\mathcal{P}$ respectively, then the system of equations represented by the condition $p\oplus q=0$ (i.e. the condition of the conservation of momenta) does not depend on the specific choice of the target momentum space $M''$ nor on the order of addenda.

A second interesting example that allows us to study the properties as well as the meaning of mixing coproducts has been first studied in Ref.\cite{barcaroli2014relative}. In this case, one considers the map
\begin{equation}
\psi:\kappa{\mathcal{P}}\to\kappa{\mathcal{P}} ,
\end{equation}
which is explicitly given by the following formulas
\begin{align}\begin{split}\label{mappaLeo}
&P_0=\psi(K_0)=K_0,\\
&P_i=\psi(K_i)=e^{{ \ell \over 2}K_0}K_i,\\
&R_{i}=\psi(M_{i})=M_i,\\
&N_{i}=\psi(B_i)=e^{{ \ell \over 2}K_0}(B_i-{ \ell \over 2}\epsilon_{ijk}K_jM_k),
\end{split}\end{align}
where $(P_\mu , R_i, N_j)$ are the generators of the $\kappa$-Poincaré algebra in the bicrossproduct basis while $(K_\mu, M_i, B_i)$ are the generators of the $\kappa$-Poincaré algebra in the so-called classical basis \cite{kosinski1994classical}. In the classical basis we have
\begin{align}\label{class}
\begin{split}
&[B_i, K_0] = iK_i \, , \quad [B_i, K_j] = i\delta_{ij}K_0 \, , \quad [B_i, B_j] = -i \epsilon_{ijk}M_k , \\
&[M_i, K_0] = 0 \, , \quad [M_i, K_j] = i\epsilon_{ijk}K_k \, , \quad [M_i, M_j] = i \epsilon_{ijk}M_k , \\
&[K_\mu, K_\nu] = 0 \, , \quad [M_i, B_j] = i\epsilon_{ijk}B_k ,
\end{split}
\end{align}
while the coproducts are rather complicated and lengthy, so we do not report them but they can be found in \cite{Borowiec} or in references therein. \\
Thanks to this map $\psi$ and following steps similar to those we did above, one can write for instance the composition law
\begin{align}\begin{split}\label{mixingLeo}
&p \oplus_{B-S}^B k=\begin{cases}p_{0}+k_{0}\\p_{i}+e^{- \ell p_{0}}e^{{\ell \over 2}k_0}k_{i}\end{cases}\\
\end{split}\end{align}
that mixes particles $p$ and $k$ obeying the same symmetry group but expressed in two different bases (bicrossproduct the former and standard the latter), while giving back a momentum in the $\kappa$-Poincar\'{e} bicrossproduct basis.

While in the first part of this section we "mixed" Hopf algebras with different deformation parameter but described in the same basis,
here we are "mixing" two copies
of the same Hopf algebra (same deformation parameter) but adopting two different bases.
From a phenomenological point of view, one of the main points of interest resides in the fact that in the bicrossproduct basis one has on-shellness of the type
\begin{eqnarray}
{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}p_0\Bigr) = e^{\ell p_0}p_i p^i
\end{eqnarray}
whereas for the classical basis the on-shellness is undeformed
\begin{eqnarray}
k^2_0 = k_i k^i .
\end{eqnarray}

\section{Mixing coproduct for the Magueijo-Smolin DSR picture}
\label{MS}

Before moving on to more ambitious implementations of the notion of mixing coproduct,
we find appropriate to offer a brief aside showing that the mixing-coproduct techniques introduced
in the previous section
can also shed light on results previously obtained without abstracting the notion of mixing coproduct.
Our main objective for this small aside it to show that a composition law introduced
by Magueijo and Smolin in Ref.\cite{magueijo2003generalized} can be rephrased in the language of mixing coproducts. The key
ingredient is an operator
$$U[\ell]\equiv \exp(\ell P_0 D), $$
acting on the Poincar\'{e} algebra, with
$$D\equiv P_0{\partial \over \partial P_0}+P_1{\partial \over \partial P_1}.$$
Then, taking the generators transformed by this map
$$X[\ell]\equiv U^{-1} X U,$$
one can easily check that $X[\ell]$ still obey the Poincar\'{e} commutation rules. Thus, we have again the Poincar\'{e} algebra but with a modified coalgebric sector and with generators of infinitesimal translations given by
$$P_i[\ell]\equiv U[\ell] P_i = {P_i \over 1+\ell P_0},$$
and its inverse
$$P_i={P_i[\ell] \over 1-\ell P_0[\ell]}.$$

We also define the map $u[\ell]$
$$X \mapsto U X[\ell] U^{-1},$$
which for the generators $P_i$ reads
$$P_i \mapsto {P_i[\ell] \over 1-\ell P_0[\ell]}.$$

Following the formalism used in the previous section for the mixing coproduct, we construct the chain of maps
\begin{equation}
\label{cat1}
P\Bigl[{\ell \over 2}\Bigr]\xrightarrow{u^{-1}\bigl[{\ell \over 2}\bigr]}P\xrightarrow{\Delta}P\otimes P\xrightarrow{u[\ell]\otimes u[\ell]}P[\ell]\otimes P[\ell],
\end{equation}
which, when applied backwards, gives the momentum space the Magueijo-Smolin composition relation \cite{magueijo2003generalized}
$${p_i \over 1- {\ell \over 2} p_0}={q_i \over 1-\ell q_0}+{k_i \over 1-\ell k_0},$$
i.e.,
\begin{equation}
p_i={q_i+k_i-\ell q_0 k_i-\ell q_i k_0 \over 1-{\ell \over 2}(q_0+k_0)}.
\label{eq:MS_composition_rule}
\end{equation}

As a further aside it is amusing to notice that there is also another route for obtaining the Maueijo-Smolin
composition law. We start doing that by noticing that we already have a map linking $P\bigl[{\ell \over 2}\bigr]$ to $P[\ell]$:
$$P\Bigl[{\ell \over 2}\Bigr]\xrightarrow{u^{-1}\bigl[{\ell \over 2}\bigr]}P\xrightarrow{u[\ell]}P[\ell].$$
We call this map $w\bigl[{\ell \over 2},\ell\bigr]$
\begin{equation}
\label{w}
w\Bigl[{\ell \over 2},\ell\Bigr] \triangleright P_{i}\Bigl[{\ell \over 2}\Bigr] = {P_{i}[\ell] \over 1-{\ell \over 2}P_0[\ell]}.
\end{equation}

Now we need to prove that the chain in Eq. \eqref{cat1} is indeed the same chain given by
\begin{equation}
\label{cat2}
P\Bigl[{\ell \over 2}\Bigr]\xrightarrow{\Delta\bigl[{\ell \over 2}\bigr]}P\Bigl[{\ell \over 2}\Bigr]\otimes P\Bigl[{\ell \over 2}\Bigr]\xrightarrow{w\bigl[{\ell \over 2},\ell\bigr]\otimes w\bigl[{\ell \over 2},\ell\bigr]}P[\ell]\otimes P[\ell].
\end{equation}

First we observe that the coproduct of $P_i\bigl[{\ell \over 2}\bigr]$ can be obtained using the relations
$$P_i\Bigl[{\ell \over 2}\Bigr]={P_i \over 1+{\ell \over 2}P_0}, \quad P_i={P_i[{\ell \over 2}] \over 1-{\ell \over 2}P_0[{\ell \over 2}]},$$
from which it follows that
\begin{align}
\label{cop}
\begin{split}
\Delta\Bigl[{\ell \over 2}\Bigr]P_i\Bigl[{\ell \over 2}\Bigr]= & {P_i \otimes \mathbb{I}+\mathbb{I}\otimes P_i \over 1+{\ell \over 2}(P_0 \otimes \mathbb{I}+\mathbb{I}\otimes P_0)} \\
= & {{P_i\bigl[{\ell \over 2}\bigr] \over 1-{\ell \over 2}P_0\bigl[{\ell \over 2}\bigr]} \otimes \mathbb{I}+\mathbb{I}\otimes {P_i\bigl[{\ell \over 2}\bigr] \over 1-{\ell \over 2}P_0\bigl[{\ell \over 2}\bigr]} \over 1+{\ell \over 2}({P_0\bigl[{\ell \over 2}\bigr] \over 1-{\ell \over 2}P_0\bigl[{\ell \over 2}\bigr]} \otimes \mathbb{I}+\mathbb{I}\otimes {P_0\bigl[{\ell \over 2}\bigr] \over 1-{\ell \over 2}P_0\bigl[{\ell \over 2}\bigr]})}.
\end{split}
\end{align}

At this point given two particles with momenta $q$ and $k$ respectively in $P[\ell]$ and applying backwards the chain in Eq. \eqref{cat2}, we obtain the transformation
$$(q_i,k_i) \mapsto \Bigl({q_i \over 1-{\ell \over 2}q_0},{k_i \over 1-{\ell \over 2}k_0}\Bigr)$$
which has to be replaced in the coproduct of Eq. \eqref{cop}.
In order to do so we first notice that if
$$x_i\mapsto {x_i \over 1-{\ell \over 2}x_0}$$
then
$${x_i \over 1-{\ell \over 2} x_0} \mapsto {{x_i \over 1-{\ell \over 2}x_0} \over 1-{\ell \over 2}{x_0 \over 1-{\ell \over 2}x_0}}={x_i \over 1-\ell x_0},$$
Thus, substituting $\Bigl({q_i \over 1-{\ell \over 2}q_0},{k_i \over 1-{\ell \over 2}k_0}\Bigr)$ in the coproduct of Eq. \eqref{cop} we have
\begin{align*}
{{q_i \over 1-\ell q_0}+{k_i \over 1-\ell k_0} \over 1+{\ell \over 2}\Bigl({q_0 \over 1-\ell q_0}+{k_0 \over 1-\ell k_0}\Bigr)} ={q_i+k_i-\ell q_0 k_i-\ell q_i k_0 \over 1-{\ell \over 2}(q_0+k_0)},
\end{align*}
which is exactly the Magueijo-Smolin composition rule of Eq. \eqref{eq:MS_composition_rule}.

The map $w\bigl[{\ell \over 2},{\ell}\bigr]:P\bigl[{\ell \over 2}\bigr]\to P[\ell]$
is indeed like a rescaling map between two $\kappa$-Poincar\'{e} algebra with two distinct deformation scales $\ell$ and ${\ell \over 2}$, i.e.,
$${P_i\bigl[{\ell \over 2}\bigr] \over 1-{\ell \over 2}P_0\bigl[{\ell \over 2}\bigr]}\mapsto{{P_{i}[\ell] \over 1-{\ell \over 2}P_0[\ell]} \over 1-{\ell \over 2}{P_{0}[\ell] \over 1-{\ell \over 2}P_0[\ell]}}={P_{i}[\ell] \over 1-\ell P_0[\ell]}.$$

The equivalance between the chains in Eq. \eqref{cat1} and \eqref{cat2} can also be interpreted as the equivalence between maps
$$\Delta\circ u^{-1}[\ell]=(u^{-1}[\ell]\otimes u^{-1}[\ell])\circ \Delta[\ell],$$
which shows the compatibility between the coproducts of $P$ and $P[\ell]$ algebras, and more generally between the coproducts of $P[\ell']$ and $P[\ell]$ algebras.

\section{Mixing coproducts between Poincar\'{e} and $\kappa$-Poincar\'{e} algebras}
\label{sec:classificazione}

So far we focused on coproducts mixing pairs of algebras that were isomorphic to one another (or two bases of the same algebra). In this
section we show that one can consistently introduce mixing coproducts also for non-isomorphic algebras,
focusing on the case of mixing the standard Poincar\'{e} (Lie) algebra and the $\kappa$-Poincar\'{e} Hopf algebra.
 It should be noticed that this is rather challenging even though the Poincar\'{e} (Lie) algebra
 is obtained from the $\kappa$-Poincar\'{e} Hopf algebra in the limit in which the deformation parameter is removed.
 In fact, the mixing coproducts we analyzed in Section III all involve maps which are not analytic as the deformation parameters are removed ($\ell \, \rightarrow 0$). In this section we shall truly need a new type of mixing coproduct. For definiteness and simplicity we
 focus on the case of a 1+1-dimensional spacetime.

 We start by introducing notation for the most general composition law\footnote{Here and in the rest of this paper, if not otherwise specified, we denote with $\boxplus$ the mixing coproducts obtained following this procedure.}
\begin{equation}
p \boxplus k = \begin{cases} \epsilon(p_0,p_1,k_0,k_1)p_0+\zeta(p_0,p_1,k_0,k_1)k_0 \\ f(p_0,p_1,k_0,k_1)p_1+g(p_0,p_1,k_0,k_1)k_1 \end{cases},
\label{eq:comp1}
\end{equation}
where $p$ is the momentum of a $\kappa$-Poincar\'{e} particle and $k$ is the momentum of a Poincar\'{e} particle\footnote{By stating
that $p$ is the momentum of a $\kappa$-Poincar\'{e} particle we mean that if $p'$ is another particle momentum of the same type, then
$$p\oplus p'=\begin{cases}p_0+p_0',\\p_1+e^{-\ell p_0}p_1'\end{cases},$$
whereas of course if $k$ and $k'$ are two Poincar\'{e} particle momenta then obviously
$$k+k'=\begin{cases}k_0+k_0',\\k_1+k_1'\end{cases}.$$}, and we require that under a suitable boost generator $N_{[p,k]}$
\begin{equation}\label{conservazione_quadrimpulso}p \boxplus k=0 \Rightarrow [N_{[p,k]},p \boxplus k]=0.\end{equation}

This requirement guarantees the covariance of the composition rule in \eqref{eq:comp1}. In \eqref{eq:comp1} $\epsilon$, $\zeta$, $f$ and $g$ are general functions of the momenta $p,k$, which for $\ell \to 0$ have to be equal to 1.
We shall consistently denote by $p$ momenta of the $\kappa$-Poincar\'{e} type and by $k$ momenta of the Poincar\'{e} type.

Firstly, let us focus on two ``complementary" proposals which are respectively
\begin{equation}\label{sistemaEp0,p1}p \boxplus k=\begin{cases}\varepsilon(p_0,p_1)p_0+k_0 \\ f(p_0,p_1)p_1+k_1\end{cases}\end{equation}
and
\begin{equation}\label{sistemaEk0,k1}p \boxplus k=\begin{cases}p_0+\tilde{\varepsilon}(k_0,k_1)k_0 \\ p_1+\tilde{f}(k_0,k_1)k_1\end{cases},\end{equation}
where the boost must be of the form $N_{[p,k]}=h(p_0,p_1)N_{[p]}+N_{[k]}$ for the former case, and $N_{[p,k]}=N_{[p]}+\tilde{h}(k_0,k_1)N_{[k]}$ for the latter.

As we explicitly show in Appendix B, given the compatibility condition \eqref{conservazione_quadrimpulso} all the solutions of these systems, i.e. \eqref{sistemaEp0,p1} and \eqref{sistemaEk0,k1}, are of the type
\begin{equation}\label{box eps p0p1}p \boxplus k=\begin{cases} \varepsilon p_0+k_0 \\\sqrt{\varepsilon^2p_0^2-{4 \over \ell^2}\sinh^2\Bigl({\ell\over 2}p_0\Bigr)+e^{ \ell p_0}p_1^2}{p_1 \over | p_1|}+k_1\end{cases}\end{equation}
and
\begin{equation}\label{box eps k0k1}p \boxplus k=\begin{cases} p_0+\tilde{\varepsilon}k_0 \\\ p_1+e^{{\ell \over 2}\tilde{\varepsilon}k_0}\sqrt{{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}\tilde{\varepsilon}k_0\Bigr)-k_0^2+k_1^2}{k_1 \over | k_1|}\end{cases}\end{equation}
respectively. Thus, we would be left with just one undetermined function: $\varepsilon$ for the first system, and $\tilde{\varepsilon}$ for the second.

However, it is rather easy to understand that not all the choices for $\varepsilon$ and $\tilde{\varepsilon}$ can be acceptable. Indeed, if we take a look at Eqs. \eqref{box eps p0p1} and \eqref{box eps k0k1}, then it follows that we must have \footnote{Notice that these two disequalities can be obtained by imposing zero spatial momentum either for the particle $p$ or for the particle $k$ respectively.}
\begin{equation}
\varepsilon p_0^2 \ge {4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}p_0\Bigr) \quad \to \quad \varepsilon \ge {2 \over \ell p_0}\sinh\Bigl({ \ell \over 2}p_0\Bigr) ,
\end{equation}
for the first case, and
\begin{equation}
{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}\tilde{\varepsilon}k_0\Bigr)\ge k_0^2 \quad \to \quad \tilde{\varepsilon} \ge {2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)
\end{equation}
in the second situation. Essentially, this is a direct consequence of the functional form of the expressions. Moreover, it is natural to require that $\varepsilon$ and $\tilde{\varepsilon}$ have to be both surjective and injective for any value of, respectively, $p_1$ and $k_1$. They must also go to $1$ for $\ell$ going to zero and, finally, need to have the same sign of the energy ($p_0$ or $k_0$) in order to guarantee that the above introduced mixing coproducts actually represent a good and consistent choice for composing the momenta (the energies) of two particles.

Given that, we can now deduce several facts about the mixing coproducts we are trying to construct. For instance, if we concentrate on Eq. \eqref{box eps p0p1} and consider the case in which $\varepsilon$ depends only on $p_0$ and enjoys the aforementioned properties, then $\varphi(p_0)=\varepsilon p_0$ will have an inverse $\tilde{\varphi}(k_0)$ with the same characteristics. We can then choose
 $\tilde{\varepsilon}={\tilde{\varphi} \over k_0}$ in Eq. \eqref{box eps k0k1}. Notice that, with these choices, when we impose momentum conservation in a scattering process (i.e. $p\boxplus (-k)=0$), both composition laws give the same relations between $p$ and $k$. From the former we find
\begin{equation}
p \boxplus (-k)=0 \quad \to \quad \begin{cases}k_0=\varphi(p_0) \\ k_1=\sqrt{\varphi^2(p_0)-{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}p_0\Bigr)+e^{\ell p_0}p_1^2}{p_1 \over | p_1|}\end{cases}
\end{equation}
and
\begin{equation}
p\boxplus (-k)=0 \quad \to \quad \begin{cases}\tilde{\varphi}(k_0)=p_0\\ k_1=\sqrt{k_0^2-{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}\tilde{\varphi}(k_0)\Bigr)+p_1^2e^{\ell \tilde{\varphi}(k_0)}}{p_1 \over |p_1|}\end{cases}
\end{equation}
from the latter. The fact that we obtained two identical systems suggests that, as we shall show later, these two composition laws can be regarded one as the inverse of the other. This is a remarkable feature since for the consistency of this composition laws we must always have
\begin{align*}&\varphi\equiv\varepsilon p_0 \ge {2 \over \ell}\sinh\Bigl({\ell \over 2}p_0\Bigr),\\
& \tilde{\varphi}\equiv\tilde{\varepsilon} k_0 \ge {2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr).\end{align*}

From the former of the above disequalities we find
\begin{equation}
\varphi^{-1}\varphi(p_0)=\tilde{\varphi}\varphi(p_0)=p_0 \ge \tilde{\varphi}\Bigl[{2 \over \ell}\sinh\Bigl({\ell \over 2}p_0\Bigr)\Bigr],
\end{equation}
or equivalently
\begin{equation}
\tilde{\varphi}(k_0) \le {2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr),
\end{equation}
where we have introduced the notation $k_0={2 \over \ell}\sinh\Bigl({\ell \over 2}p_0\Bigr)$. Due to the properties we imposed on the functions, the same relation must hold also when we exchange the two sides and thus
\begin{equation}
\tilde{\varphi}(k_0) = {2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)
\end{equation}
and also
\begin{equation}
\varphi(p_0)={2 \over \ell}\sinh\Bigl({\ell \over 2}p_0\Bigr).
\end{equation}

With this procedure we thus conclude that there is only \textit{one} possible pair of mixing coproducts that deforms the momenta and verifies a sort of inverse relation. This is given by
\begin{equation}\label{(p+k)P p0}p\boxplus_{\mathcal{P}}k=\begin{cases}{2 \over \ell}\sinh\Bigl({\ell \over 2}p_0\Bigr)+k_0 \\ e^{{\ell \over 2}p_0}p_1+k_1\end{cases}\end{equation}
and
\begin{equation}\label{bla}p\boxplus k=\begin{cases}p_0+{2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr) \\ p_1+\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)k_1\end{cases}.\end{equation}

Let us now consider a particle of mass $m$ with momentum $p$ whose symmetries are described by the $\kappa$-Poincaré algebra. If we define
\begin{equation}
E={2 \over \ell}\sinh\Bigl({\ell \over 2}p_0\Bigr), \qquad \Pi=e^{{\ell \over 2}p_0}p_1
\end{equation}
then
\begin{equation}
E^2-\Pi^2={4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}p_0\Bigr)-e^{\ell p_0}p_1^2=m^2 .
\end{equation}

This tells us that the map $(p_0,p_1)\to(E,\Pi)$ ``transforms'' the particle $p$ into a particle with standard Poincaré symmetries\footnote{Notice that this is true also for the mixing coproducts of Eq. \eqref{box eps p0p1}.}. Given that, it is possible to regard the mixing coproduct of Eq. \eqref{(p+k)P p0} as a trivial sum of momenta once we deform the momentum coordinates $(p_0 , p_1)$, just as we saw already in the previous sections. On the other hand, we can not give a similar interpretation for the composition law in Eq.\eqref{bla}. In other words, we would like to have a sort of complementary map that transforms the momentum of a particle with Poincaré symmetries into the momentum associated to the $\kappa$-Poincaré algebra. This is not possible since if $k$ is a Poincaré particle
with mass $m$ and we define
\begin{equation}
\tilde{E}={2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr), \qquad \tilde{\Pi}=\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)k_1
\end{equation}
then
$${4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}\tilde{E}\Bigr)-e^{\ell \tilde{E}}\tilde{\Pi}^2 \ne m^2,$$
and the spatial part of the coproduct \eqref{bla} is not of the form $p_1+e^{-\ell p_0}\tilde{\Pi}$, regardless of the choice of $\tilde{\Pi}$. However, we can show that a suitable modification of Eq. \eqref{bla} actually allows to overcome these obstructions and to have a composition law that admits an interpretation analogous to
Eq. \eqref{(p+k)P p0}. To this end, let us consider a general mixing coproduct of the form
\begin{equation}\label{p0k0}p\boxplus k=\begin{cases}\varepsilon(p_0,k_0)p_0+k_0 \\ f(p_0,k_0)p_1+k_1\end{cases},\end{equation}
with $N_{[p,k]}=hN_{[p]}+N_{[k]}.$ Then we have
\begin{equation}
[N_{[p,k]},\varepsilon p_0+k_0]=h\Bigl({\partial \varepsilon \over \partial p_0}p_1p_0+\varepsilon p_1\Bigr)+{\partial \varepsilon \over \partial k_0}k_1p_0+k_1
\end{equation}
and
\begin{equation}
[N_{[p,k]},fp_1+k_1]=h\Bigl[{\partial f \over \partial p_0}p_1^2+f\Bigl({1-e^{-2 \ell p_0} \over 2 \ell}-{\ell \over 2}p_1^2\Bigr)\Bigr]+{\partial f \over \partial k_0}k_1p_1+k_0.
\end{equation}

Consequently $\varepsilon$ and $f$ must satisfy the following system of differential equations
\begin{equation}\label{sistema p0k0}\begin{cases}h\Bigl({\partial \varepsilon \over \partial p_0}p_1p_0+\varepsilon p_1\Bigr)-fp_1\Bigl({\partial \varepsilon \over \partial k_0}p_0+1\Bigr)=0 \\ h\Bigl[{\partial f \over \partial p_0}p_1^2+f\Bigl({1-e^{-2 \ell p_0} \over 2 \ell}-{\ell \over 2}p_1^2\Bigr)\Bigr]-f{\partial f \over \partial k_0}p_1^2-\varepsilon p_0=0 \\ \varepsilon p_0+k_0=0\end{cases}.\end{equation}

We redirect the reader to the Appendix A for the detailed analysis of this system. The solution is given by any pair of function $(f,\varepsilon)$ that reduces to Eqs. \eqref{(p+k)P p0} when $\varepsilon p_0+k_0=0$, provided that, as always, they also have the correct behavior for $\ell \rightarrow 0$ and do not present any sort of singularity. Notice that the fact that $\varepsilon={2 \over \ell p_0}\sinh\Bigl({\ell \over 2}p_0\Bigr)$
 when $\varepsilon p_0+k_0=0$ allows us to rewrite the constraints as
\begin{equation}
{2 \over \ell }\sinh\Bigl({\ell \over 2}p_0\Bigr)+k_0=0,
\end{equation}
or also
\begin{equation}
p_0+{2 \over \ell }\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)=0.
\end{equation}

Thus, when the constraint is satisfied, we have
\begin{equation}
\varepsilon={2 \over \ell p_0}\sinh\Bigl({\ell \over 2}p_0\Bigr)={k_0 \over {2 \over \ell }\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)}
\end{equation}
and
\begin{equation}
f=e^{{\ell \over 2} p_0}=e^{-\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)}={1 \over {\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}}.
\end{equation}

Finally, defining $\tilde{\varepsilon}={1 / \epsilon}, \quad \tilde{f}={1 / f}$, we can rewrite the system of Eq. \eqref{p0k0} as
\begin{equation}\label{p0k0invmolt}p\boxplus k=\begin{cases}\varepsilon(p_0,k_0)(p_0+\tilde{\varepsilon}(p_0,k_0)k_0) \\ f(p_0,k_0)(p_1+\tilde{f}(p_0,k_0)k_1)\end{cases},\end{equation}
where now $\tilde{\varepsilon}$ and $\tilde{f}$ reduce to \eqref{bla} on the constraint surface. \\
It is rather easy to prove that, given a general mixing coproduct of the form
\begin{equation}
p\boxplus k=\begin{cases}\tilde{\zeta}(p_0+\tilde{\varepsilon}k_0) \\ \tilde{g}(p_1+\tilde{f}k_1)\end{cases}
\end{equation}
or also
\begin{equation}
p\boxplus k=\begin{cases}\zeta(\varepsilon p_0+k_0) \\ g(fp_1+k_1)\end{cases} ,
\end{equation}
then the solutions $\tilde{\varepsilon}$, $\tilde{f}$, $\varepsilon$ and $f$ do not depend on the common factors $\tilde{\zeta}$, $\tilde{g}$, $\zeta$ and $g$, respectively. In fact, considering for instance the former case and acting with a boost $N_{[p,k]}=\tilde{i}(N_{[p]}+\tilde{h}N_{[k]})$, we find that $N_{[p,k]} \triangleright (p\boxplus k)_0$ has the following form
\begin{equation}
[\tilde{i}(N_{[p]}+\tilde{h}N_{[k]})\triangleright \tilde{\zeta}](p_0+\tilde{\varepsilon}k_0)+\tilde{\zeta}\tilde{i}(N_{[p]}+\tilde{h}N_{[k]})\triangleright(p_0+\tilde{\varepsilon}k_0).
\end{equation}

If, as usual we ask that $N_{[p,k]}\triangleright p\boxplus k=0$ when $p\boxplus k=0$, then $p_0+\tilde{\varepsilon}k_0=0$ in the above equation, and as a result it is easy to realize that both $\tilde{\zeta}$ and $\tilde{i}$ do not play any role in the identification of $\tilde{\varepsilon}$ (or $\tilde{h}$). Given that, we can ignore the functions $\varepsilon(p_0,k_0)$ and $f$ in Eq. \eqref{p0k0invmolt}, which we can then rewrite as
\begin{equation}\label{p0k0inv}p\boxplus k=\begin{cases}p_0+\tilde{\varepsilon}(p_0,k_0)k_0 \\ p_1+\tilde{f}(p_0,k_0)k_1\end{cases}\end{equation}
with $\tilde{\varepsilon}={2 \over \ell k_0}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)$ and $\tilde{f}={\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}$, on the constraint solutions. This allows us to modify the mixing coproduct in Eq. \eqref{bla} in the way we needed. Indeed, keeping $\tilde{\varepsilon}$ unmodified with respect to the case in Eq. \eqref{bla} while changing $\tilde{f}$ as
\begin{equation}
\tilde{f}=e^{-\ell(p_0+\tilde{\varepsilon}k_0)}\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)={e^{-\ell p_0} \over {\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}}
\end{equation}
we eventually obtain the mixing coproduct
\begin{equation}\label{(p+k)kP_k0}p\boxplus_{\kappa\mathcal{P}} k=\begin{cases}p_0+{2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr) \\ p_1+e^{-\ell p_0}{k_1 \over {\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}}\end{cases}.\end{equation}

If we assume that $k$ is a particle with mass $m$ and define
\begin{equation}
\tilde{E}={2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr), \qquad \tilde{f}={k_1 \over {\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}}
\end{equation}
we actually find that
\begin{equation}
{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}\tilde{E}\Bigr)-e^{\ell\tilde{E}}\tilde{\Pi}^2 = m^2:
\end{equation}
i.e. the particle $k$ has been ''deformed'' into a $\kappa$-Poincaré particle. It is worth noting that Eq. \eqref{(p+k)kP_k0} can be interpreted as a non-commutative law in the sense that we can hypothesize that by exchanging $p$ with $k$ one would have to write down
\begin{equation}\label{(k+p)kP k0}k\boxplus_{\kappa\mathcal{P}} p=\begin{cases}{2 \over \ell}\ln\Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)+p_0 \\ {k_1 \over {\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}}+{p_1 \over \Bigl({\ell \over 2}k_0+\sqrt{{\ell^2 \over 4}k_0^2+1}\Bigr)^2}\end{cases},\end{equation}
where the last term in the spatial part of the composition law can be rewritten simply as $e^{-\ell\tilde{\varepsilon}k_0}p_i$. Notice that
 we only multiplied the spatial part of the mixing coproduct \eqref{bla} by the function $e^{-\ell\tilde{\varepsilon}k_0}$, which as aforementioned does not alter the covariance of the composition law.

In summary, the composition laws in Eqs. \eqref{(p+k)P p0} , \eqref{(p+k)kP_k0} (or also \eqref{(k+p)kP k0}) define a Poincaré-like sum and a $\kappa$-Poincaré, respectively, with total momenta associated to the Poincaré or the $\kappa$-Poincaré algebra respectively.

Without the need to provide all the details of the proof (which follows the line of reasoning used to derive Eq. \eqref{p0k0}), we can generalize the above discussions to the cases
\begin{equation}
p\boxplus k=\begin{cases}\varepsilon(p_0,p_1,k_0,k_1)p_0+k_0\\f(p_0,p_1,k_0,k_1)p_1+k_1\end{cases}
\end{equation}
and
\begin{equation}
p\boxplus k=\begin{cases}p_0+\tilde{\varepsilon}(k_0,k_1,p_0,p_1)k_0\\ p_1+\tilde{f}(k_0,k_1,p_0,p_1)k_1\end{cases},
\end{equation}
where, on the constraint solutions, the above functions must coincide with either \eqref{box eps p0p1} or \eqref{box eps k0k1}.
By doing so, for any $\tilde{\varepsilon}$ we also have the non-commutative law
\begin{align*}
&p\boxplus_{\kappa\mathcal{P}}k=\begin{cases}p_0+\tilde{\varepsilon}k_0\\
p_1+e^{-\ell p_0}e^{-{\ell \over 2}\tilde{\varepsilon}k_0}\sqrt{{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}\tilde{\varepsilon}k_0\Bigr)-k_0^2+k_1^2}{k_1 \over |k_1|}\end{cases},\\
&k\boxplus_{\kappa\mathcal{P}}p=\begin{cases}\tilde{\varepsilon}k_0+p_0\\e^{-{\ell \over 2}\tilde{\varepsilon}k_0}\sqrt{{4 \over \ell^2}\sinh^2\Bigl({\ell \over 2}\tilde{\varepsilon}k_0\Bigr)-k_0^2+k_1^2}{k_1 \over |k_1|}+e^{-\ell \tilde{\varepsilon}k_0}p_1\end{cases}.\end{align*}
This concludes the classification of all the mixing coproducts involving the Poincaré and the $\kappa$-Poincaré algebras, which are of the form
\begin{equation}
p\boxplus k=\begin{cases}\varepsilon p_0+\tilde{\varepsilon} k_0\\ f p_1+\tilde{f}k_1\end{cases}.
\end{equation}

\section{Summary and outlook}

The main goal of this work was to provide illustrative examples of
consistent scenarios with non-universal deformation of relativistic symmetries.
Clearly the key challenge for such scenarios concerns the introduction of suitable mixing products, of which we provided several examples.
Our results preliminarily suggest that scenarios with non-universal deformation of relativistic symmetries
could in principle be realized in Nature and deserve dedicated experimental testing,
such as the mentioned studies looking for effects in the neutrino sector at a level of magnitude which is already
excluded for photons.
While we feel that such tests should not be postponed,
there are clearly still several tasks to be faced before fully establishing the consistency
of these scenarios. For example, it would be interesting to establish whether they
are compatible with the setup of (possibly deformed) quantum field theories, though one should perhaps view this
as a long-term goal, since even the understanding of quantum field theories with universal deformation of relativistic symmetries has still
not reached full maturity.

Our mixing coproducts are in principle directly applicable to interactions between composite and fundamental particles
(when governed by different relativistic properties). However, in some cases it should be possible to apply a constructive approach
for such mixing coproducts: ideally there might be cases in which one only needs to postulate
some universally-deformed relativistic properties for fundamental particles, then deriving the implied
(different) relativistic properties of various types of composite particles, and in such cases also the mixing coproducts
could be derived, rather then requiring a dedicated postulate. We feel that performing such derivations, even just in some
particularly simple toy model, would be an important contributions to the further development
of the investigations we here reported.

For what concerns the broader picture of phenomenology it will be interesting to identify some characteristic observable differences
between the new scenarios of non-universal deformations of relativistic symmetries, on which we here focused, and the
scenarios in which relativistic symmetries are broken in particle-dependent manner, which have already been studied
for several years\cite{coleman1997cosmic,smeREVIEW}.

\section*{Acknowledgements}

We are grateful to Leonardo Barcaroli for contributing to the initial stages of this project.

redpajama_set_name: RedPajamaArXiv


text:
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Could an All-Electric Corvette Crossover Be in the Works?
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BMW Takes U.S. Luxury Sales Crown for 2020
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Remembering Mr. Hudson: Jack Carl Miller Dies at 82
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Many new car dealers love the brands they sell. But few, if any, continue to sell the brand once it's failed. That wasn't true of Jack Carl Miller, who died Mon., Dec. 28, 2020, in Ypsilanti, Michigan. Miller was 82, and known for running the business his father...
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Like What You Drive and How it's Equipped? Thank the Military
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Shortly, the Economic Council and others will be spending over $200,000 to convince you that the county sales tax should be raised to 7%. You will hear that the infrastructure is crumbling, that the children are sweating in their classrooms with broken air conditioners, that the roads have potholes and the bridges are falling down. You will hear that a sales tax is good because 25% of it will come from tourists, and that tens of thousands of jobs will be created to rebuild those roads and bridges, county buildings, the jail and the parks.
Don't be fooled. This 17% increase in the sales tax will generate much more revenue than is arguably needed to repair the infrastructure that was neglected by conscious choices of county staff and commission. Over the last 5 years, the ad-valorem budget has grown 33%, yet Engineering and Public Works only saw a 3% growth. At the same time, the Sheriff's budget grew by 28% and county employees saw 12% in across the board raises (3% / year for 4 years). These conscious choices indicate that those running our county and school system were willing to defer maintenance until a pitch could be made for a new source of revenue.
A bond issue could have funded just enough to meet the critical needs. Instead, they want a sales tax that will generate $2.7B over 10 years whether it is needed or not. Do not doubt that they will spend every penny.
A 17% increase in the sales tax is a net tax increase of $270M per year, with no offsets to property taxes.
It is regressive and will affect low income residents the hardest.
It is not subject to the scrutiny applied to the annual ad-valorem budget.
It creates an incentive to purchase outside the county(Both Broward and Martin are at 6%, many internet retailers do not collect sales tax).
It is not an "infrastructure maintenance tax" but includes many new capital projects.
Unlike an infrastructure bond that would raise just enough money for critical needs, this granular tax generates a specific amount of money, and low priority projects will have to be funded in order to spend it all.Like previous proposals, it is a grab-bag of projects, many of which would never be done without a "must spend" windfall.
Many of the municipalities (PBG, Boca) didn't want the money.
It comes on top of the largest ad-valorem tax haul at the county level in history, up 8.2% over last year and up 33% since 2012. If passed, the 2017 equivalent tax hike would be 18%.
Over the last 5 years, the county has consistently underfunded engineering and public works(+3%), while increasing the Sheriff's budget by 28% and giving across the board raises to employees of 12% (3%/year for 4 years). When the overall ad-valorem budget increased by 33%, engineering saw a total of 3% in 5 years. This was a conscious choice.
VOTE NO ON THE SALES TAX ON THE NOVEMBER BALLOT.


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Granite Staters Weigh in on Renewable Energy Versus Drilling: Environmental Quality of Life Ranks High Across Party Lines
Since the fall of 2001, the University of New Hampshire's Survey Center has been conducting the Granite State Poll—a statewide, scientific survey of public opinion and behavior concerning policy issues—via telephone interviews with random samples of New Hampshire residents about four times each year.
Community, Environment, and Climate Change, New Hampshire Environment, New Hampshire, Politics and Elections, Public Opinion Publication
Half of Women in New Hampshire Have Experienced Sexual Harassment at Work
Sexual harassment in the workplace is a serious problem affecting workers across the United States and in New Hampshire. Nationwide, approximately four in ten women and more than one in ten men have been victims of workplace sexual harassment in their lifetimes.
New Hampshire, Vulnerable Families Research Program Demography, Employment, Gender, New Hampshire, Women Publication
Hard Times Made Harder: Struggling Caregivers and Child Neglect
Poverty is only one of many challenges tied to a report of child neglect. The analysis in this brief finds that neglected children whose caregivers struggle with substance abuse and mental health problems are at significant risk for out-of-home placement. Risk factors for out-of-home placement for neglected children are discussed, as well as a multifaceted approach to services to prevent neglect and out-of-home placement.
Vulnerable Families Research Program Child Care, Children, Health, Poverty Publication
Health Conditions and an Older Population Increase COVID-19 Risks in Rural America
In this brief, author Kenneth Johnson discusses the likely influence that the age structure and the incidence of pre-existing health conditions have on the risks of those exposed to COVID-19 in rural and urban counties in the United States.
COVID-19, Demography COVID-19, Demography, Health, Rural, Seniors, Urban Publication
Health Insurance Among Young Adults Rebounds Post Recession: More Become Dependents on a Parent's Plan After ACA Extends Coverage to Adult Children
While much of the existing research explores young adults' insurance only in the post-recession period (that is, 2010 to present), authors Michael Staley and Jessica Carson assess young adults' rates of coverage within and beyond the context of the recession by examining changes across the entire 2007 to 2012 period.
Vulnerable Families Research Program Children, Employment, Health, Health Insurance, Young Adults Publication
Help in a Haystack: Youth Substance Abuse and Mental Health Services in the North Country
A new brief from Nordblom Fellow Meghan Mills at the Carsey Institute finds that youth in New Hampshire's North Country have challenges in accessing support for substance abuse and mental health issues. Mills also finds that the providers face unique challenges, from getting referrals to hiring professionals, all while working without a functional network.
New Hampshire Coös Youth Study, Health, New Hampshire, Young Adults Publication
Hispanic Children Least Likely to Have Health Insurance
This policy brief examines health insurance coverage of Hispanic children and its relationship to their citizenship status, their parents'1 citizenship status, parents' insurance coverage, language spoken at home, and their state's Medicaid expansion policies. Hispanic Children Are Least Likely to Have Health Insurance In 2014, 94 percent of U.S. children had health insurance.2 Although this is a record high for children's coverage, 4.3 million children still remain without health insurance, and Hispanic children make up a disproportionate share of this group. Hispanic children have historically had the highest rates of uninsurance among children of any racial/ethnic group.3 In 2014, the most recent year for which data are available, 95.4 percent of non-Hispanic white children, 95.3 percent of black children, and 94.4 percent of multiracial children had health insurance coverage. In comparison, only 90.3 percent of Hispanic children were covered, leaving more than 1.7 million Hispanic children uninsured. Hispanic children in rural areas are less likely to have health insurance than Hispanic children in urban areas (9.4 percent versus 12.2 percent, respectively).4 Nearly 40 percent of all uninsured children are Hispanic, although Hispanic children make up only 24.3 percent of children in the United States (see Figure 1). By contrast, though nearly 52 percent of U.S. children are non-Hispanic white, they comprise only 40 percent of uninsured children. Black, non-Hispanic children account for 13.6 percent of children in the United States but just 10.8 percent of uninsured children.
Vulnerable Families Research Program Children, Health Insurance, Hispanics Publication
Home Care Workers: Keeping Granite Staters in Their Homes as They Age
Using data from the New Hampshire Direct Care Workforce Survey, this brief shows that New Hampshire's demand for home-based care workers outpaces supply because its population is aging at a faster rate than the national average. These workers play a critical role and face many challenges, including low pay, little or no paid time off, and lack of access to health insurance.
New Hampshire Employment, New Hampshire, Seniors, Wages Publication
Homeless Teens and Young Adults in New Hampshire (co-publication with the Children's Alliance of New Hampshire)
More than 1,000 adolescents and young adults in New Hampshire are homeless, and their numbers are growing. The brief, co-published with the Children's Alliance of New Hampshire, provides an estimate of homeless youth in New Hampshire calculated from national and state data and describes the needs of homeless youth based on interviews and a survey of providers of homeless services in the state.
Evaluation, New Hampshire, Vulnerable Families Research Program Family, Housing, New Hampshire, Young Adults Publication
Household Reports of Energy Assistance Receipt Increased 48 Percent During Recession: Proposed Cuts Threaten Vulnerable Families
This brief examines heating assistance usage and the implications of President Obama's 2012 budget proposal to cut $2.5 billion from the $5.1 billion energy assistance fund for low-income families at a time when families are struggling with higher energy costs amid a difficult economy. The federal Low-Income Home Energy Assistance Program (LIHEAP) assists vulnerable families in paying their home heating and cooling bills. Nationwide, from the winter of 2006/2007 to the winter of 2009/2010, there was a 48 percent increase in households receiving energy assistance. This growth appears to have accelerated with the recession, particularly in the rural Northeast and Midwest. Many more families are eligible than receive assistance. Brief author Jessica Carson discusses how proposed cuts would have a concrete and immediate impact on families, particularly those in rural areas and in harsh winter climates.
Vulnerable Families Research Program Family, Housing, Poverty, Safety Net Publication
How Far Would You Drive for Fresh Food? How Some Rural New Hampshire Residents Navigate a Dismal Food Landscape
Lack of access to food stores with healthy and affordable food is one of the central obstacles to eradicating hunger in America. Approximately 23.5 million Americans live more than a mile from a supermarket, which makes accessing healthy food more challenging. Among low income populations, especially those with young children and limited transportation, this distance can severely limit access to affordable and healthy foods. This brief reports the challenges that eighteen rural New Hampshire mothers face to secure healthy, affordable, and quality foods and suggests ways to help address these challenges.
New Hampshire, Vulnerable Families Research Program Family, Food Assistance, Low Income, New Hampshire, Rural, Safety Net Publication
How Yoopers See the Future of their Communities: Why Residents Leave or Stay in Michigan's Upper Peninsula
According to a Community and Environment in Rural America survey, Michigan's Upper Peninsula residents, often called "Yoopers," said that ties to community and the area's natural beauty were significant factors for those who planned on staying in this rural area, which comprises about a third of Michigan's land mass but only 4 percent of its population. Those planning on leaving cited employment opportunities and energy costs as the most important factors in their decision.
Demography Community, Demography, Environment, Public Opinion, Rural Publication
In this perspectives brief, authors Lawrence Hamilton and Thomas Safford discuss the enormous effect that the extent to which governments and individuals respect the recommendations of science and science-based public agencies is having on the impact of COVID-19.
Community, Environment, and Climate Change, COVID-19 COVID-19, Health, Public Opinion, Trust Publication
Immigration to Manchester, New Hampshire
This brief analyzes immigration and refugee resettlement in Manchester and the effects on the city's demographic composition, as well as the implications for its future. Authors Sally Ward, Justin Young, and Curt Grimm report that Manchester, New Hampshire, like the nation, is experiencing a new wave of immigration.
Demography, New Hampshire Community, Demography, Immigration, New Hampshire Publication
Improving Staff Communication for Small Non-Profits
In this white paper, author Elizabeth Schwaner discusses challenges faced by Girls' Empowerment Program, a non-profit organization based in the northeast U.S. that serves at-risk girls in its community through a year-round mentoring program paired with a residential social summer camp. Despite its undeniable strengths, issues with communication are a persistent challenge. Like many small organizations staffed by a few paid employees and many volunteers, identifying specific practices that lead to miscommunication or helpful informational pathways can be a challenge. This case study elucidates some of those challenges and opportunities for Girls' Empowerment, but with the expectation that other small, mixed-staff organizations may benefit from these lessons learned.
Leadership, New England, Young Adults Publication
Increased Reliance on Wives as Breadwinners during the First Year of the Recession
Among low-income families, the wages of employed wives account for the majority of family earnings, according to this Carsey brief. The analysis finds that in 2008, women contributed 56 percent of total family earnings, up from 51 percent in 2007. Also, husbands' education level and race are factors in how much wives contribute to family earnings.
Vulnerable Families Research Program Employment, Family, Gender, Wages, Women Publication
Indicators of New Hampshire Youth Well-Being (co-publication with the Children's Alliance of New Hampshire)
According to a new study, New Hampshire youth, ages 13 to 24, are more likely to complete school, be employed, and have lower obesity rates than their peers nationwide but fare worse in national measures of alcohol and substance abuse. This brief, a co-publication with the Children's Alliance of New Hampshire, provides an overview of youth well-being in New Hampshire calculated from national and state data and compares Granite State youth with peers across the country.
Evaluation, New Hampshire Demography, Education, Family, Health, New Hampshire, Poverty, Young Adults Publication
Informal Kinship Care Most Common Out-of-Home Placement After an Investigation of Child Maltreatment
This fact sheet examines differences between urban and rural areas in foster care placement with informal kin caregivers. The data for this analysis come from a national sample of children who had a maltreatment report that resulted in an investigation: the second National Survey of Child and Adolescent Well-Being.
Vulnerable Families Research Program Children, Family, Health Publication
Innovation in Food Access Amid the COVID-19 Pandemic
The COVID-19 pandemic has triggered income losses and rising demand for food-related support, while social distancing requirements have complicated access to usual nutrition support sites. In response, government agencies, private retailers, nonprofit organizations, and volunteer networks are undertaking innovative efforts to ensure food access by vulnerable populations. By highlighting strategies that are unfolding in real time, this brief shares an array of potential approaches for private, public, and nonprofit stakeholders to use in deploying their resources.
COVID-19, Vulnerable Families Research Program COVID-19, Food Assistance, Safety Net Publication
Intimate Partner Violence Among LGBTQ+ College Students
Drawing from a survey of 391 college students in same-sex relationships, this brief documents the rates and patterns of intimate partner violence, and responses to it among LGBTQ+ youth.
Vulnerable Families Research Program Education, Gender, Health, Young Adults Publication


Power wheel batteries and riding on toys batteries, we have verity of power wheels batteries, we stock kids ride on replacement batteries offering longer running times and reliability, and our ride-toy batteries are the exact fit of original replacement battery packs. Our most sealing brand is fisher-price power wheel batteries and kid trax ride on power wheel batteries, most of the power wheels batteries are 12 volt and 24 volt battery pack. Visit out battery store, batteries shack for all your battery needs.
© Batteries Shack. All rights reserved.


"I've finally given in; my eyes are shut under the blindfold.
I can hear and smell, but I cannot see. My hands are clammy I feel cold yet I am warm.
It is what I wanted, but I am now unsure, dubious yet at the same time excited and curious.
I would like to think I feel a bit like Alice just before she fell down the hole into the rabbit hole.
This entry was posted in Mind Fragments, The Journey and tagged Belief, Faith, hope, life, Love, photography. Bookmark the permalink.


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Intro to Motion
1.5 credits
DIG 205: Intro to Motion
Learn the ABCs of animation
How does animation actually work? In this online introductory animation course, you'll learn the 12 essential principles of animation including squash and stretch, anticipation, and exaggeration. Working with a professional animator, you'll explore how fundamental concepts in physics and motion are used to create sophisticated, convincing motion art in sequential art, video, 2D animation, and 3D animation, and more. Practical projects include stop-motion and flipbook-style animation.
Credit-bearing: 1.5 Credits
Classes start January, April, and August
Course developed by: Greg Marlow
Digital Media Undergraduate Certificate Digital Media Associate Degree
Analyzing Moving Objects
All art is communication, and the struggle of motion art is to communicate the story of objects on the go. In the first lecture, we examine the pivotal concepts of physics that help us understand how to recreate real world motion in the minds of our viewers. We'll explore different forms of energy and force and the relationships between them. We'll also touch on Newton's three laws of motion, terminal velocity, and equilibrium.
The Motion Artist's Media
Once we start to understand how motion works, we face the choice of how to communicate it. This week, we'll explore different categories of motion art, including motion in a single image, sequential art, low frame rate motion, and full motion video. We'll then look at techniques for achieving motion in those categories, highlighting the manipulation of position, scale, rotation, and distortion, as well as options for combining multiple techniques.
The Shoulders of Giants
After studying why and how things move, and the different forms at our disposal for communicating movement in art, we will finish up the course by discussing time-tested principles for achieving believable and appealing motion art. We'll look at 12 canonized principles of animation, and we'll then examine how to apply these principles to actual animation workflows, with definitions and explanations of key terms and concepts like poses, blocking, key frames, breakdowns, and tweening.
For sample work, visit student gallery
Creative projects explore motion capture, sequential art, stop motion, and animation principles.
Computer with Internet connection (broadband recommended).
Adobe Premiere, Photoshop, and Animate or equivalent programs.
Access to a webcam or digital video camera and ability to connect it to a computer to capture footage.
Relate concepts of physics (including energy, force, and motion) to the communication of movement in art.
Analyze the physics at work in examples of live motion.
Distinguish between four overarching categories of motion art: motion in a single image, sequential art, low frame rate motion, and full motion video.
Achieve motion on a 2D plane using the techniques of position, scale, rotation, and distortion, as well as combination techniques, such as parallax.
Use a webcam or digital camera to create single image, sequential, and full motion stop motion animation.
Build animated GIFs and full motion videos in MOV and MP4 formats.
Apply 12 traditional principles of animation in both 2D and 3D motion art.
Set up a pose-to-pose animation workflow with key frames, breakdowns, and tweening.
Create appealing and believable animations based on traditional techniques.
1.5 Credit Course
Project-Based: Exercises, Discussions, and Critiques
Tuition $510/credit
Registration Fee* $200
Our courses are project-based. In each course you'll enjoy a series of lectures, projects, discussions, and critiques designed to stretch your creative skills, earning college credit for your completed course.
Classes start January, April, and August, and this course can be completed as part of a 15-week term in 8 weeks. College credit from this course can be applied to a range of degree and certificate level programs at Sessions College.


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Rep. Jim Jordan
Ohio · District 4
Rep. Jim Jordan represents Ohio's 4th district in the House. He began spreading lies about the election on November 14, 2020 in a rant on Fox News: "Why did it seem like on Election Night, all the important swing states that took a several hour pause in the count, the President ended up losing?"
He later claimed that "the Democrats knew that Trump was going to win on Election Night, but they wanted to keep looking for and counting ballots until they got over the top." After talking about the impressive performance of downballot Republicans in another Fox News interview, he joked, "somehow President Trump lost to a guy who would hold an event… but only five people would show up! It makes no sense." He then went on to object to the counting of Electoral College votes from Arizona and Pennsylvania on Jan. 6.
Even as recently as Jan. 12, he refused to concede that Biden had won a free and fair election.
Rep. Jordan did not vote in support of the bipartisan commission proposal to investigate the Jan. 6 insurrection.
"Resign" Billboard
Advertiser-Tribune: GOP group wants Jordan out


This weekend I attended another Abney Park concert. I think it's my SIXTH, and this one was absolutely wonderful. Definitely my second favorite (my first favorite will always be my first AP concert.) The energy was just electric, and everyone was having such an awesome time. Whenever I looked around, there was the biggest smile on everyone's face.
The venue itself was amazing – it was the first live performance ever at the Bellingham Spark museum, and what perfect surroundings for steampunk! Before the concert we toured the museum, and there were tons of cool displays, many interactive.
The opening band was Adrian H and the Wounds. I enjoyed their music, although probably not well enough to buy a CD.
But then Abney Park came on stage. And wow. This was my first time seeing the new guys in person (Josh and Titus replacing Nathaniel who left to start his own band.) They couldn't match Nathaniel's flamboyance (who could?) but they were excellent musicians and nicely interactive with the crowd, as are all the other band members. During several songs one or more of them would jump off the stage and come play, sing, or dance with us.
As always, I try to film more of the between-songs moments, rather than the songs themselves! For one thing, I am always pressed up against the stage, as close as I can get, and my camera doesn't record sound well when I'm that close to the equipment. For another, I love the between-songs moments…you never know what you're going to get!
Between AP sets, the museum put on a Tesla coil demonstration. Very cool. It was amazing to see light bulbs lighting up just from being held close to the coil! Unfortunately for AP, this demonstration fried their computer's motherboard – but it wouldn't be an AP concert if something didn't break/go horribly wrong. It's all part of the fun, as Jody led a raid on the bar while Robert was trying to save the computer. The problem was solved, and the concert continued to its glorious ending….far too soon.
After the show, I met up with Robert and got him to sign my copy of his book The Wrath of Fate to me. I'd ordered a signed/numbered copy of his book the instant it was released, and had been surprised that I'd gotten copy #2 out of 300. Robert was impressed with my purchasing prowess; so impressed that he called Kristina over to see it. She said she was always afraid she was going to screw up the numbers when she was writing them in, and he said he was always afraid that the #2 book was going to get lost in the mail, and then what would he do? Send out another and say "Here you are – another #2!!!" I had him sign the book to my nickname "Lis" and he said that was his sister's name, and he'd never met another person called that. So that was cool, too.
And of course I got a pic with him.
I'm wearing my Steampunk NeoBedouin costume – which is actually based off the description in Robert's book. I had meant to tell him that, but we were so busy talking about other things, that I never got around to it. As much as I would've enjoyed it, I couldn't monopolize him all night!
There is more about my costume, including more pics and a making-of, here, here, and here, on my costuming blog. The costume did get recognized several times by fellow fans who would come up to me and shout "NeoBedouin!!!!". Which was awesome.
All-in-all, it was a fantastic night, and it definitely will not be my last AP concert.


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A return to Burns Lake
Joan McEwen has explored the dispute between Olympic boss John Furlong and journalist Laura Robinson.
Joan McEwen visited Burns Lake in October and conducted twenty interviews concerning abuse.
"Laura didn't feed us information," said hereditary chief Ronnie Alec, "and she didn't make us feel worse. She just listened, and we told her the truth."
If one considers the importance of Joan McEwen's book about how and why Ivan Henry was wrongfully imprisoned for 27 years, she would certainly deserve to be in the running for 'BC Author of the Year,' if such a contest existed.
Lawyer and author Joan McEwen
Joan McEwen went where most mainstream journalists would dare not go; or, more correctly, where most journalists would not be employed to go. She reached out to a person who is largely assumed to be an unsavoury character because it was the right thing to do.
McEwen first heard about the plight of Ivan Henry in 2010 following his acquittal by the BC Court of Appeal. Her book, Innocence on Trial: The Framing of Ivan Henry (Heritage House $22.95) detailed how failings of the justice system—police, lawyers, prosecutors, judges, politicians, etc.—have ruined much of Henry's life.
Now she's again raising her voice to suggest that society ought to afford more scrutiny to a controversial legal situation.
On November 26, 2015, the National Observer published lawyer Joan McEwen's article entitled "Waiting to be Heard" about the ongoing legal battles arising from allegations made by journalist Laura Robinson, first aired in Georgia Straight, about the actions of 2010 Olympics head honcho John Furlong when he worked in northern B.C. among First Nations children.
Again, Joan McEwen has gone where mainstream journalists have not been encouraged to go. She has visited the Burns Lake area to meet First Nations people in order to investigate the validity of accusations that were raised in Laura Robinson's controversial article.
Here is her report that aired in the National Observer.
First Nations v. John Furlong: "Waiting to be Heard"
by Joan McEwen
In September 18, 2015, Madam Justice Wedge of the B.C. Supreme Court dismissed an action brought by investigative journalist Laura Robinson against John Furlong, CEO of the Vancouver Organizing Committee (VANOC) for the 2010 Olympics.
Furlong declared himself vindicated and, on October 30, Geoff Plant, the former Attorney-General of B.C., wrote in the Globe and Mail that a door had closed on a particularly nasty part of Furlong's life—nasty in that he had been unfairly vilified by Robinson.
Opinions differ on the judgment, but one thing is clear: The voices of the First Nations alleged victims have never been heard. Yet, though not a single one of these people has testified in court, they have been discredited—both by the judge and in the court of public opinion.
A thumbnail sketch of the chronology is in order.
Beverly Abraham remembers the arrival of John Furlong at Immaculata Elementary School.
In September 2009, shortly before the February 2010 Olympics, Robinson received a tip that Furlong had worked as a Catholic missionary somewhere in Northern B.C. when he originally came to Canada. Finding nothing on the Internet about that, she dropped the matter.
In February 2011, while writing a review for his book Patriot Hearts, co-authored by journalist Gary Mason, it struck her that certain information regarding his background, arrival in Canada, and work experience appeared to be missing or did not make sense. Her research revealed that he had come to Canada as a Frontier Apostle missionary, and had taught at a Catholic elementary school in Burns Lake (1969-1970) and, from there, a Catholic high-school in Prince George.
In April 2011, Robinson wrote the review as well as an op-ed for the Danish organization, Play the Game.
After interviewing by phone eight people, Robinson travelled from Ontario to Burns Lake in April 2012. She took statements from eighteen individuals who alleged abuse by Furlong at one or both of the schools.
On September 26, 2012, the Georgia Straight published her article, "John Furlong Biography Omits Secret Past in Burns Lake"—featuring allegations of psychological and physical abuse (only) against Furlong.
Mollie Charlie with pot of soup welcomed Joan McEwen along with Mollie's sister Virginia (left) and Cherill Greening of the Tsimshian First Nation.
On November 27, 2012, Furlong sued Robinson for defamation (Furlong v. Robinson). In his Notice of Civil Claim, Furlong asserted that the First Nations' allegations of abuse were without merit. In Robinson's Defence to Civil Claim, she relied upon the statutory declarations sworn to by eight of the former students.
On March 31, 2015, Furlong dropped his lawsuit against Robinson.
Meanwhile, Robinson, having sued Furlong in January 2014 for defaming her (Robinson v. Furlong), pressed forward with her suit. The two-week trial (which I attended) took place in June 2015. (In that second action, Furlong raised, as part of his defence, the disposition of the one criminal complaint and three civil suits brought against him in respect of alleged sexual abuse. Those matters, however, formed no part of Robinson's Georgia Straight article, the article on which Furlong's lawsuit was based.)
Among my many concerns with Madam Justice Wedge's decision is this. Notwithstanding the narrow issue at trial—Did Robinson's attack on Furlong justify his counter-attack?—the judge remarked at length on the overall unreliability, given the many years that had passed, of the First Nations' accounts of alleged abuse.
Geoff Plant described the judge's decision as "careful" and "methodical"; said that it amounts to a "textbook on how not to do investigative journalism."
However, a central conclusion of the decision was that Robinson, prior to going to Burns Lake for the first time, had "telegraphed her intentions" by causing a notice to be posted "announcing the subject of her investigation—childhood abuse—its location, its timing, and the identity of the alleged abuser." The judge also accepted the testimony of Dr. John Yuille, the memory expert called by Furlong, who said that, by arriving an hour late in Burns Lake, Robinson gave those who were waiting the chance to "contaminate" each other's memories.
Richard Perry, hereditary chief, says he appreciated that Laura Robinson was prepared to listen.
Memory contamination is a common concern where, for example, multiple victims of sexual assault, usually strangers whose only connection was being victimized in this way, are left together to discuss what happened to them. However, the alleged victims in this case—many of whom are related—lived in close proximity on a reserve for over forty years following the conduct in issue. Can it seriously be argued that being notified ahead of time and being left alone for an hour "contaminated" their memories?
Geoff Plant further wrote that "they (the RCMP) discovered no reliable evidence to support the claims [sic] of abuse." In fact, the RCMP looked into only one case—the criminal complaint referred to above.
After reading the Supreme Court judgment, I resolved to travel to Burns Lake myself—to hear, first-hand, the stories of the former students. Given the incendiary nature of the sexual abuse allegations, and because they were not part of Robinson's Georgia Straight article, I chose not to explore that issue in my discussions.
In the near-two dozen interviews I conducted between October 4 and 8, I heard accounts consistent in every respect with those contained in the eight statutory declarations filed in the Furlong and Robinson aborted proceedings.
The stories of abuse I heard brought tears to my eyes.
Stories that, to this day, are deeply etched on the distraught faces of the people telling them.
Without exception, I found each of the First Nations persons with whom I spoke to be thoughtful, reflective, and believable. (I am a lawyer and have practiced for the past twenty-five years as a professional labour arbitrator.)
From their telling, the pain flowed not only from the abuse itself, but also from the fact that, apart from Robinson, no one cared enough to listen (Canadaland's Jesse Brown has just posted affidavits on his website).
When told about the judgment, they were shocked. How was it possible, they asked, that the judge could reach such a decision without ever hearing from them? No-one in the media asked for their reaction. It felt, to them, as though they did not exist.
Plant wrote that Robinson's announcement that she would not be appealing the decision was "both graceless and unrepentant."
On October 19, Robinson announced that, although she had received legal advice supporting an appeal, "Appealing the decision will not accomplish my original goal of sharing the stories of Indigenous people." What she said was true. When Furlong discontinued, in March 2015, his action against Robinson, he removed the opportunity for her to tell those very stories; to give their version in response to Furlong's blanket denial.
Geraldine and Maurice Joseph welcomed Joan McEwen into their home.
Was Furlong's decision to discontinue his action any less "graceless and unrepentant" than Robinson's decision not to proceed with an appeal?
More significantly, Plant stated that Robinson caused "brutal harm" to "those former students who were victimized by her zealotry."
After reading the article, I contacted the First Nations people to ask for their comments. Without exception, they said that Robinson was there to "help them talk about their stories;" that, finally, they had "someone who cared enough about us to want to hear our stories."
Joan McEwen with Cathy Woodgate, who drafted and signed a letter to the Justice Minister asking that John Furlong step down from Own the Podium pending an investigation into what happened when he was a missionary in Burns Lake, 1969-70.
Hereditary chief Richard Perry said he appreciated that Robinson was prepared to listen. Hereditary chief Ronnie Alec said that, though his people have always tried to avoid talking about their abuse at the hands of the Oblates, seeing Furlong on TV (before the Olympics) "stirred up bad memories, gave us flashbacks about what he did to us… Laura didn't feed us information, and she didn't make us feel worse. She just listened, and we told her the truth." Cathy Woodgate described her as "caring, willing to listen, and very sensitive."
Indeed, the members of the Lake Babine Nations were so grateful for Robinson's efforts that, when she returned to Burns Lake a third time, they congratulated her on her story.
It is obvious that Plant has never spoken to, let alone met, any of Furlong's former students. If he had wished to speak for the First Nations claimants, he should have talked to them first.
Indeed, anyone wishing to speak in the future on their behalf would be well-advised to do likewise.
[This article by Joan McEwen first appeared in National Observer. Republished with permission from the author.]
Meanwhile McGill-Queen's University Press has undertaken to publish its own edition of the six volumes of the final report of the Truth and Reconciliation Commission in English and French, after the TRC officially releases the report in mid-December. The final report is expected to be over 2 million words with contributions from over 6,750 survivor and witness statements from across the country, after a century of abuse at Indian Residential Schools.
The volumes are published in MQUP's prestigious Native and Northern Series, founded by the late Bruce Trigger and currently edited by Arthur J. Ray and Sarah Carter, and are available in print and E-book versions.
Commented Philip Cercone, Executive Director of McGill-Queen's University Press: "This is one of the most important documents to emerge in the history of Canada. These ground-breaking volumes are destined to work towards healing the breach of silence and ignorance that has surrounded these issues for more than a century."
Canada's
The History, Part 1
Origins to 1939
The Final Report of the Truth and Reconciliation
Commission, Volume 1
$39.95 CAD & USD paper
6.75 x 9.75 1024pp
34 tables, 2 graphs,
6.75 x 9.75 864pp
32 tables, 6 graphs, 76 photos
The Inuit and
Northern Experience
41 b&w photos, 10 tables
The Metis Experience
The Final Report of the
6.75 x 9.75 96pp
16 b&w photos
Missing Children and Unmarked Burials
11 graphs, 6 tables
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6 Responses to "A return to Burns Lake"
Jan Slakov says:
I'm so grateful for this article, which gives us a real window into what happened in Burns Lake many years ago (and what is going on now, right among us).
Joan McEwen is a hero for working for justice in cases where others would rather look away. Laura Robinson too, just read an article she wrote years ago: http://www.playthegame.org/news/news-articles/2002/crossing-the-line-violence-and-sexual-assualt-in-canadas-national-sport/
And I once got to speak with Cathy Woodgate – It is wonderful to see, in the photo above, the strength (and compassion) I heard in her voice.
Sylvia Smith says:
Thank you Joan for your most valuable contributions to paying witness to the ongoing colonial assaults that First Nations, Metis, and Inuit people are experiencing to this day. And look what happens when settler people, like Laura Robinson, have empathy and want to see justice done. Thank you Joan, for assisting and shedding more light on injustice after injustice. And thank you to the settler professionals who, seeing the deceipt within their own ranks (I'm thinking of Fay Brunning who is taking on senior lawyers in Canada's justice department for their withholding of documents for the survivors of St. Anne's IRS in Ontario), are willing to go to bat for the victims.
anne cameron says:
It's always encouraging and educational to have the opportunity to learn from open-minded and compassionate people such as yourself, Ron Hyde.
I too remember some harrowing details about that first sad wife, now silent. A silence that speaks loudly…and as is so often the case, people with stories to tell were somehow pushed to the roadsides, the shadows. Another parable of power and the ability to shut out the people who might enlighten us about that complicated period. Thanks to Joan McEwen for pursuing this story.
Ron Hyde says:
You appear to be willing to believe or listen to anyone that is female or aboriginal.
Four of the young girls proclaiming to be abused by Furlong, never wne to court because they had made a BAREFACE LIE as they didn't even attend the school where he worked. So much for "the professional journalist Laura Robinson". She didn't bother to check into the band villages where the children came from as undoubtedly that is where the abuse was, not at the school.
Now we have a bleeding heart lawyer (notice also a woman) who will repeat talking to the same people about the same thing but heaven forbid she might actually look into problems from where the children were raised. No, just prattle to a few alcoholics that are happy to agree with anything that is said.
Save us all the story or book that mirrors the soap operas on television. Justice you say, "it is to make you puke" when one has to listen to the type of vitriol that Anne Cameron spews. There will be no justice for anyone in Canada until the aboriginals are treated the same as other Canadians instead of blaming absolutely everything on the residential schools. The aboriginals have found the secret to everything is the blame game.
What about the Grand Chiefs and their "heritage villages" why does no one ask what they are doing about alcoholism, suicides, physical and sexual abuse of their children instead of acting like a bobble-head when the reason is the residential schools. This excuse doesn't seem to bother them pillaging the band funds for personal use while the ordinary band members live in poverty. Unfortunately 'work' never seems to be something that is discussed.
It would interesting if all the bleeding hearts involved in this charade would be that interested if the person in question was female instead of John Furlong. Maybe start looking at things in the real world instead of trying to make it into a soap opera.
I am more willing to believe what the FN survivors have to say than I am to believe what a professional talk master is saying. His own family has denied and scoffed at his self-serving blarney regarding the tragic death of his cousin, and, somehow, what his common-law wife has to say has been expunged from public record. I find it interesting that Laura Robinson never accused him of sexual abuse but he, himself, has made a big thing of denying it. Wasn't it Willy Shakespeare who wrote of "doth protest too much" ?
I'm beyond "glad" that Joan McEwen has looked into this. My feeling is that she will find Laura Robinson was unfairly hung out to dry by a bunch of high profile influential people with their own agenda.
Justice? For aboriginal people in Canada? It is to make you puke!