[a]Sadataka Furui

# Clifford Fourier Transforms in (2+1)D Lattice Simulations of Soliton
Propagations

Serge Dos Santos

###### Abstract

Monte Carlo simulation of solitonic phonon propagation on a 2D plane in Weyl
fermion-sea is analyzed. We assume materials are filled with Weyl spinors
located on $256\times 256$ 2D lattice points, which are expressed by
quaternions ${\bf H}$.

The topology of solitonic phonon propagation is defined by modifying fixed
point actions of 4D Quantum Chromo Dynamics action to (2+1)D action, replacing
Dirac fermions by Weyl fermions, and changing the electric charge current flow
to the energy flow.

We consider $A$ type loops whose path are on a 2D plane, and $B$ type loops
which contain two parallel links that connect two 2D plane on different time
slices. The length of loops are restricted to be less than or equal to 8
lattice units. At the moment spatial lattice unit and time lattice unit are
same. They can be chosen arbitrarily when one compares hysteresis effects with
experimental data

Using the quaternion expression of Porteous, we calculate the plaquette part
of loop actions and the link part of loop actions. Link actions of $A$ type
loops cancel with each other, but those of $B$ type loops depends on whether
the spin rotation is clockwise or that is counterclockwise.

In the present work we consider average of clockwise rotating and
counterclockwise rotating loop contributions.

##  1 Introduction

In non-destructive-testing (NDT) the time reversal (TR) based nonlinear
elastic wave spectroscopy (TR-NEWS) was successful. Use of time reversal
mirrors (TRM) for enhancing signal to noise ratios of ultrasonic phonetic
waves scattered in materials was proposed by Fink[1] and the technique of
focussing, decomposition of time reversal operator (DORT) was developed[2].

In TR-NEWS, transducers which emit phonons and their TR phonons and receivers
are placed on sides of a 2 dimensional plane[3, 4]. Propagation of nonlinear
phonetic waves in materials is recently reviewed in [5].

Haldane[6] revealed the spin quantum Hall effect (QHE) in space 2 dimensional
TR violating electronic system can be explained through Chern index of
algebraic topology. Kane and Mele[7] studied the QHE in TR symmetric systems
and showed that $Z_{2}$ topological order characterize presence of ordered
phase on the Fermi surface. Fidkowski and Kitaev[8] discussed that TR
invariant Majorana fermion spin 1 system has a trivial and the Haldane phase,
and the breaking of $Z$ symmetry to $Z_{2}$ symmetry can be interpreted by
real (R) edge spin states and quaternion (H) edge spin states. Ryu et al.[9]
studied electromagnetic and gravitational responses and anomalies in
topological insurators and superconductors by extending the theory of [7].
They proposed anomaly cancellation between gravitation and electro-weak
interaction.

In 1982, Banks and Zaks [10] showed that in massless fermions systems,
spontaneous chiral symmetry breaking does not occur. The chiral symmetry
breaking in Dirac fermion system was discussed in comparison with
superconductivity in electronic media by Nambu and Jona-Lassinio[11]. They
pointed out without nonzero mass $m$, the particle would become an eigenstate
of charge or chirality, and in the absence of Coulomb interactions the
excitations become phonon-like.

The convolution of a Khokhlov-Zaboltskaya solitonic wave obtained by Lapidus
and Rudenko[12], and its TR solitonic wave show presence of anomalous zero
mode[16], and detailed analysis of TR-NEWS convolution data will present hints
on the gravitational anomaly suggested in [9].

We consider a pair of Weyl spinor and its TR symmetric Weyl spinor in charge
neutral media which violates spin rotational symmetry. When a soliton with
effective mass is produced in fermion-sea, a zero-mode and energy gap in the
infrared region could appear and the spontaneous chiral symmetry breaking can
occur. If it does not originate from electro-weak interaction, the zero-mode
need not be cancelled by gravitational anomaly. Since the media that a phonon
propagates is not necessarily charged, we construct a model of Weyl fermions
represented by quaternions sitting on $(2+1)D$ lattices[18, 19]. Instead of
electric flow that propagates in the background of gravity, we consider the
energy flow described by the DORT method.

To fix the path of phonons, we adopt the Fixed Point (FP) actions in the
momentum space. They consist of 7 $A-$type loops on a 2D plane and 13 $B-$type
loops that contain loops on two 2D planes on different time slices and two
parallel links that are perpendicular to the 2D planes. We restrict ourselves
to consider one loop graphs and total length of a loop is less than or equal
to 8 lattice units. Hasenfratz and Niedermayer[13] showed, in d=2 $O(3)$ non-
linear $\sigma$ model, it is possible to simulate classical actions by
discretized asymptotic free FP actions.

At each cite a quaternion is placed and products of quaternions along the loop
define the energy flow of the nonlinear waves. Since we consider situations in
which phonetic solitons and TR solitons are emitted from the wall on the left
hand side, the first step of the loop is chosen to be $e_{1}\Delta$, where
$\Delta$ is the lattice unit $\frac{1}{T_{c}N_{t}}$, where $T_{c}$ is the
critical temperature, $N_{t}$ is the lattice size in the time direction,
$e_{1}$ is the unit vector along the $x_{1}$ axis. Loops that start with
$ne_{1}\Delta(n\geq 2)$ yield larger actions and we do not consider. The path
of solitonic phonons are defined by minimizing the sum of 20 actions with
optimum weight functions determined by the Monte Carlo (MC) simulation. The
optimal superposition of TR-NEWS phonetic signals using memristor is
demonstrated in [21] and the principle of reservoir computing adopted in
memristors was reviewed in [22]. The motivation of applying Clifford algebra
to NDT is presented in [20] and we present here details of lattice simulations
and relations to the field theory.

##  2 Weyl fermions and Clifford algebra

The Weyl spinor and Majorana spinor in 4D are defined as

$\Psi\frac{1}{2}(1+\gamma_{0}{\bf u}),\quad\Psi\in Cl(1,3)$ (1)

where ${\bf u}=\pm\gamma_{3}$ for Weyl and ${\bf u}=\pm\gamma_{1}$ for
Majorana. ($\gamma_{0}\gamma_{1}$ operator induces charge conjugations.)

The Clifford Fourier Transform (CFT) in 2D is defined as

${\mathcal{F}}\\{f\\}(\omega)=\int_{R^{2}}f({\bf x})e^{-i_{2}\omega\cdot{\bf
x}}d^{2}{\bf x},\quad d^{2}{\bf x}=\frac{dx_{1}\wedge dx_{2}}{i_{2}},$ (2)

where $i_{2}$ is the unit bivector $e_{1}\wedge e_{2}$.

The nonlinear energy flow $f({\bf x})$ expressed by quaternions are Fourier
transformed[31],

$\hat{f}({\bf u})=\int_{R^{2}}e^{-{\bf i}x_{1}u_{1}}f({\bf x})e^{-{\bf
j}x_{2}u_{2}}d^{2}{\bf x}.$ (3)

The inverse Clifford transform (ICFT) of ${\mathcal{F}}\\{f\\}\in
L^{1}(R^{2};{\mathcal{G}}_{2})$ is

$f({\bf
x})=\frac{1}{(2\pi)^{2}}\int_{R^{2}}{\mathcal{F}}({\bf\omega})e^{i_{2}\omega}d^{2}{\omega}\\\
$

where $\omega=u_{1}e_{1}+u_{2}e_{2}+\omega_{12}e_{12}$, ${\bf
x}=x_{1}e_{1}+x_{2}e_{2}$,

$\displaystyle F=\\{f_{1}({\bf x,u}),f_{2}({\bf x,u})\\},$ $\displaystyle
f_{1}({\bf x,u})=2\pi e_{1}x_{1}u_{1},f_{2}({\bf x,u})=2\pi e_{2}x_{2}u_{2},$
(4)

is supposed to be invertible.

We let ${\bf u}$ be a $(2+1)D$ vector projected on $2D$ represented by
quaternions, and transformation $\tau$ yields real vector space ${\bf
x}=\tau({\bf u})$, where ${\bf u}\sim p_{L}({\bf u})$, where $L$ distinguishes
20 loops, that corresponds to 7 $A$ type loops and 13 $B$ type loops.

The normalizing flow proposed by Papamakarios et al.[27] for the convolutional
neural network (CNN) can be applied to the search of 7 dimensional optimal
weight functions of $A$ type actions.

We transform the momentum space distribution $p_{L}({\bf
u})=\sum_{k}^{E}p^{k}_{L}({\bf u})$ into random number distribution
$(0,1)^{E}$, where $E$ is the number of epochs. The distribution at the epoch
$k$ is related to that at the epoch $k-1$ by ${\bf u}_{k}=T_{k}({\bf
u}_{k-1})$. The inverse is ${\bf u}_{k-1}=T^{-1}_{k}({\bf u}_{k})$.

To every distribution $u_{i}$, we consider the Clifford Fourier transform
(CFT) $x_{i}=\tau(u_{i},{\bf h}_{i})$ where ${\bf h}_{i}=c_{i}({\bf u}_{<i})$,
and $\tau$ is the transformer and $c_{i}$ is the conditioner at the $i$th
epoch[31].

The position space transformation from $\bf u$ to $\bf x$, which is the
inverse CFT is defined as

$h({\bf x})={\mathcal{F}}^{-1}\\{{\bf x}\\}({\bf
u})=\frac{1}{(2\pi)^{2}}\int_{R^{2}}e^{fx_{1}u_{1}}{\mathcal{F}}\\{h\\}({\bf
u})e^{gx_{2}u_{2}}d^{2}{\bf u}$ (5)

where $d^{2}{\bf u}=du_{1}du_{2}$, $f,g\in{\bf H}\sim Cl(0,2),f^{2}=g^{2}=-1$,
or $f,g\in Cl(2,0)$.

##  3 Fixed point actions of one loop

The plaquette actions and link actions of loops on Eucledian $(2+1)D$ space
are mapped in quaternion space $\bf H$ as shown by Porteous[29]. A unit
quaternion $q\in\bf H$ satisfies $q^{2}=-1$ and for a vector $x$ there is
anti-involution $x^{-}$ and a vector element is represented as

$\left(\begin{array}[]{cc}x&xx^{-}\\\
1&x^{-}\end{array}\right)=\left(\begin{array}[]{cc}x&0\\\
0&x^{-}\end{array}\right)+\frac{1}{2}(1+xx^{-})\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)+\frac{1}{2}(1-xx^{-})\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right).$ (6)

Its transformation can be expressed as

$\left(\begin{array}[]{cc}a&c\\\
b&d\end{array}\right)\left(\begin{array}[]{cc}x&xx^{-}\\\
1&x^{-}\end{array}\right)\left(\begin{array}[]{cc}d^{-}&c^{-}\\\
b^{-}&a^{-}\end{array}\right)=\lambda\left(\begin{array}[]{cc}x^{\prime}&x^{\prime}x^{\prime-}\\\
1&x^{\prime}\end{array}\right).\\\ $ (7)

We interpret the real eigenvalue $\lambda=(bx+d)(bx+d)^{-}$ yields the
plaquette action and $xx^{-}$ yields the link action.

The matrix $\left(\begin{array}[]{cc}a&c\\\ b&d\end{array}\right)$ represents
a special conformal transformation and is called Vahlen representation[30].

In the case of $A$ type loops, we consider simple $2D$ QFT to obtain $f({\bf
x})$, from ${\bf\omega}=\omega_{01}e_{1}+\omega_{02}e_{2}$ as in [31].
However, in the case of $B$ type loops, we include the scalar term and
consider the energy-time transformation. We regard phonons as shock waves
produced on Fermi surfaces and calculate plaquette actions derived from the
fixed point action in $4D$[17] at $(2+1)D$ lattice points with the asymptotic
high momentum action subtracted [18, 19, 16]. Link actions of $B$ type loops
for large momenta are asymptotically zero, but they depend on the clockwise
rotating (f-link) or counterclockwise rotating (e-link). The relative strength
of actions of e-link and f-link type depends on the random number sets used in
MC. The average of e-link action and f-link action which are negative in the
infrared region, plus plaquette action which are similar to that of $A$ type,
are used. A solution of the traveling salesman problem[23] says that, $A-$
type loops contain altogether 13 orderings of 7 weights, and the $B-$ type
loops contain altogether 49 orderings of 13 weights.

In the case of $256\times 16$ lattices, the maximum and minimum of the total
action of $u_{2}/\Delta=0$ was about twice of those of $u_{2}/\Delta=1$. This
property reflects properties of CFT for $d=2,3$(mod 4) remarked by Hitzer[31].
For a multi-vector $A_{r}\in{\mathcal{G}}_{d}$ of odd grade $r=2s+1$ or even
grade $r=2s$

$A_{2s+1}i_{d}=-i_{d}A_{2s+1},\quad A_{2s}i_{d}=+i_{d}A_{2s}.$ (8)

The qualitative difference of e-link actions and f-link actions does not
change by the lattice size of $u_{2}$ direction. The grade dependence observed
in $256\times 16$ lattices was, however due to the asymmetry of $u_{1}$ vs
$u_{2}$. We checked the grade dependence by calculating the convolution of the
plaquette part and the link part of the spinor and its TR spinor for each loop
on $256\times 256$ lattices by using Mathematica[33], and calculated their
Fast Fourier Transform by using the the Machine Learning (ML) module
torchfft[25].

##  4 Convolution of Weyl spinor and its TR spinor

The convolution of $a,b\in L^{1}(R^{2,0};Cl(2,0))$ is

$(a*b)({\bf x})=\int_{R^{2,0}}a({\bf y})b({\bf x}-{\bf y})d^{2}{\bf y}$ (9)

We calculated plaquette actions and link actions of 20 FP actions of
$256\times 256$ lattices. The plaquette part of the convolution of the action
and its TR action of $L1,L2$ and $L18$ of $x_{1}/\Delta\leq 128$ are presented
in Fig.1. The loops projected on the 2D planes are shown in Fig.2. The loop
$L18$ starts from the middle of the lower edge, so the link of $2\Delta$
exists in the $u_{2}$ direction. The scale in the $(u_{1},u_{2})$ plane of
$L18$ plaquette is twice of that of $L1$, however the convolution in the
$(x_{1},x_{2})$ plane at largest $x/\Delta$ of $L18$ , $L1$ and $L2$ are
almost the same, due to our specific renormalization.

In most $B-$type loops f-link dominates over e-link. However, the convolution
of the link and its TR link of the $L8$ which contains two parallel links that
connect the upper 2D plane and the lower 2D plane with distance $\Delta$, as
shown in Fig.4 has relatively large e-link as compared to the $L3$, as shown
in Fig.3,

For detecting anomalous scattering position, we make a superposition of the
inverse Clifford transformed loop wave functions with the optimum weight. A
solution of the traveling salesman problem says that, $A-$ type loops contain
altogether 13 orderings of 7 weights, and the $B-$ type loops contain
altogether 49 orderings of 13 weights. Fig.5 shows average of $A$ type actions
which are almost loop independent, and $B$ type total actions of $L8$ (upper
boundary) and $L10$ (lower boundary) and the average.

The loop dependence of $B$ type total actions at $u_{2}=0$ as a function of
$u_{1}$ is shown in Fig.6. The line $p0_{8}$ and $p0_{10}$ mean $L8$ and $L10$
at $u_{2}=0$. There are pairs which have the overlapping path on a projected
2D plane but position of the red circle or the blue circle is shifted, like
$L4-L26$, $L8-L10$, $L13-L15$ and $L16-L17$. Except $L8-L10$, there appear a
point where the action at $(u_{1},u_{2})$ coincide as shown in Fig.7. We
expect only the $L8-L10$ action pair is coherent, and other pairs may not play
important roles in detection of hysteresis effects.

The action is large when the distance between the red circle and the blue
circle is large. When $x_{2}=0$, the convolution of each total action and TR
total action has a peak at around $x_{2}/\Delta\sim 22$. The dependence of
weight function of the fixed point $L8$ total action given by random number
samples is not significant as shown in Fig.8.

Figure 1:  The convolution of the plaquette action and its TR action of $L1$
(blue line) v.s. $L18$ (red line) (left) and those of $L1$ (blue line) v.s.
$L2$ (red line) as a function of $x_{1}/\Delta$ at $x_{2}=0$ (left). .

Figure 2: The $L1$ and the $L2$ loop (left) and the $L18$ loop (right) on a 2D
lattice. .

Figure 3:  The convolution of the e-link action and its TR action (orange) and
that of the f-link action and its TR action (paleblue) of the $L3$ loop (left)
and those of the $L8$ loop (right).

Figure 4: The $L3$ loop (left) and the $L8$ loop (right) on a 2D lattice. The
red circle of $L8$ shifted to the right by $\Delta$ yields the $L10$ loop

Figure 5: The sum of $A-$type plaquette actions (left) and $B-$type
plaquette+link actions of $L8$, $L10$ and the average (right) in momentum
space $u_{1}/\Delta=[0,256]$, $u_{2}=0$.

Figure 6: The loop dependence of $B-$type total action of $L3-L13$ (left) and
$L14-L27$ (right) in momentum space $u_{1}/\Delta=[0,256]$, $u_{2}=0$.

Figure 7: The sum of plaquette+link actions of $L4$, $L26$ and the average in
momentum space at $u_{2}=0$.

Figure 8: The weight function dependence of the convolution of $L8$ .
$x_{2}=0$, $0\leq x_{1}/\Delta\leq 128$. Red points are ’optimum’ $\gamma$
sampe and blue points are a non-optimum sample.

##  5 Conclusion and discussion

For optimization of the weight function in ML, we first apply the CNN using
the gradient decent method[25, 32]. We take the parameter space $(y,{\bf x})$,
where $y$ specifies the random number set and ${\bf x}$ specifies the 20
loops. Instead of shuffling in ML, we choose paths given by the solution of
traveling salesman problem and train the system to find the minimum action as
a combination of the 20 FP actions. Actions of each loop are calculated by
Mathematica and stored in the workstation

One tries to minimize the loss function $L(y,F({\bf x}))$ where the function
$F(\bf{x})$ is calculated step wise as $F_{m}({\bf x})=F_{m-1}({\bf
x})+\eta\gamma_{m}$ where $\eta$ is the learning rate, $\gamma_{m}$ is given
by the loss function of the previous step.

In usual ML, minimizing differences of predicted values and correct values are
tried by minimizing the loss function[32]. In our case, correct values
correspond to the path with minimal action, whose solution is not known.
Therefore, different from usual CNN, which was successful in pattern
recognition etc. we need to do unsupervised trainings. Search of the optimum
weight function of 20 FP actions is underway.

When the optimum weight function is obtained, CFT from $\bf u$ space to $\bf
x$ space yields information of the anomalous scattering position of the phonon
waves. Application to the medical research becomes in sight[21].

When the mass gap of phonons remain, it becomes solitonic. The same will
happen for neutrinos other than Majorana neutrinos. Solitonic wave
propagations are made not only from charged Dirac spinors but also charge
neutral Weyl spinors. TR symmetric solitonic wave in $(2+1)D$ system shows
explicit chiral symmetry violation.

## Acknowledgments

S.F. thanks RCNP of Osaka University for the support of using super computer
SQUID and Prof. M. Arai for the support of using a workstation in his
laboratory.

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| fast | ${\rm\nu_{c}^{\rm ssc}<\nu<\nu_{m}^{\rm ssc}}$ | $\beta(q-1)$ | $\beta(q-1)$ | 0 | 0.00%
|  | ${\rm\nu_{m}^{\rm ssc}<\nu}$ | $\beta(q+1)-1$ | $\frac{\beta(q-2)+q+2}{2}$ | 3 | 3.44%
ISM | slow | ${\rm\nu_{m}^{\rm ssc}<\nu<\nu_{c}^{ssc}}$ | $\frac{3\beta(q+2)+5q-6}{4}$ | $-$ | 0 | 0.00%
|  | ${\rm\nu_{c}^{\rm ssc}<\nu}$ | $\frac{3\beta(q+2)+q-6}{4}$ | $-$ | 0 | 0.00%
| fast | ${\rm\nu_{c}^{\rm ssc}<\nu<\nu_{m}^{\rm ssc}}$ | $\frac{\beta(5q-6)}{4}$ | $\frac{\beta(5q-6)}{4}$ | 0 | 0.00%
|  | ${\rm\nu_{m}^{\rm ssc}<\nu}$ | $\frac{3\beta(q+2)+q-6}{4}$ | $-$ | 0 | 0.00%
Table 9: Results from MCMC simulations of an off-axis afterglow model evolving in a constant circumstellar medium. | GRB160625B
---|---
$\mathrm{log_{10}(E/erg)}$ | $53.89_{-0.49}^{+0.55}$
$\mathrm{log_{10}(n/cm^{-3})}$ | $0.99_{-0.50}^{+0.53}$
$\mathrm{log_{10}(\varepsilon_{e})}$ | $-1.90_{-0.50}^{+0.55}$
$\mathrm{log_{10}(\varepsilon_{B})}$ | $-2.68_{-0.45}^{+0.50}$
$\mathrm{log_{10}(\Gamma_{0})}$ | $2.50_{-0.49}^{+0.59}$
$\mathrm{\theta_{j}}$ | $0.56_{-0.47}^{+0.50}$
$\mathrm{\theta_{obs}}$ | $1.81_{-0.51}^{+0.46}$
$\mathrm{p}$ | $2.84_{-0.47}^{+0.46}$
$\mathrm{q}$ | $0.22_{-0.50}^{+0.57}$

Figure 1: Upper panels: Evolution of the radiative parameter in the stellar-
wind (left) and uniform-density (right) medium considering the synchrotron and
expansion timescales. Lower panels show the evolution of energy radiated away
(left) and relation between the radiative parameters $\epsilon$ and $s$
(right). The uniform-density and stellar-wind environments are shown in red
solid and green dashed lines, respectively. The term $E_{0}$ corresponds to
the initial equivalent kinetic energy. We use the values of $\Gamma_{0}=60$,
$E_{0}=10^{52}\,{\rm erg}$, $\varepsilon_{e}=0.5$, $\varepsilon_{B}=5\times
10^{-2}$, $\chi_{\rm e}=1$, $A_{W}=0.1$, $n=1\,{\rm cm^{-3}}$.

Figure 2: The panels show the CR of synchrotron off-axis scenario in all
regimes for constant-density (above) and stellar-wind (below) environment for
$\epsilon=0.5$ without energy injection with the spectral and temporal indexes
from 2FLGC. The purple ellipses are displayed when the CR are satisfied within
1 sigma error bars and lines in red and cyan colors correspond to the expected
CRs.

Figure 3: The same as Fig. 2, but for SSC off-axis scenario.

Figure 4: The panels show the closure relations of synchrotron off-axis model
in all regimes for ISM (above) and wind (below) environment in the adiabatic
regime with energy injection ($q=0.5$) with the spectral and temporal indexes
from 2FLGC. The purple ellipses are displayed when the CR are satisfied within
1 sigma error bars and lines in red and cyan colors correspond to the expected
CRs.

Figure 5: The same as Fig 4, but for SSC off-axis scenario.

Figure 6: The 4-D parameter space (for the forward shock model with $\rm k=0$)
of the expected values of $\varepsilon_{\rm B}$ as a function of typical
values of $\varepsilon_{\rm e}$, $n$ and $E_{0,j}$ with two different values
of $p$, for the cooling break of synchrotron (row 1) and SSC (row 2). We
consider typical values in the ranges $10^{-2}\leq\varepsilon_{\rm e}\leq 1$,
$10^{-3}\leq n\leq 1\,{\rm cm^{-3}}$ and $10^{47}\leq E_{\rm 0,j}\leq
10^{51}\,{\rm erg}$ for $p=2.2$ (columns one and three) and $2.6$ (columns two
and four). Columns one and two corresponds to $q=0.7$ and columns three and
four to $q=0.3$.

Figure 7: The same as Fig. 6, but for wind-like scenario ($\rm k=2$) with
$10^{-3}\leq A_{\rm W}\leq 1$. Figure 8: The 3-D parameter space of the
expected values of $t_{\rm pk}$ as a function of $\Delta\theta$ and the
parameter ${\rm b}$ with two distinct values of $\epsilon$ and $E_{\rm 0}$ for
$k=0$ (above) and $k=2$ (below). Column one ($\epsilon=0.8$) and two
($\epsilon=0.2$) are for $E_{\rm 0}=5\times 10^{51}\,{\rm erg}$ and column
three ($\epsilon=0.8$) and four ($\epsilon=0.2$) are for $E_{\rm 0}=5\times
10^{53}\,{\rm erg}$. We consider the parameter $b$ in the range of $0.3\leq
b\leq 3$ for ${\rm n=1\,{\rm cm^{-3}}}$ and $A_{\rm w}=0.1$. Figure 9: The 3-D
parameter space of the expected values of $t_{\rm pk}/t_{\rm j}$ as a function
of $\Delta\theta/\theta_{\rm j}$ and the parameter ${\rm b}$ with two distinct
values of $\epsilon$ for $k=0$ (above) and $k=2$ (below). For each value of
$\epsilon$ we consider the ranges of $0.3\leq b\leq 1$ and $1<b\leq 3$.
Columns one and two correspond to $\epsilon=0.8$ and columns three and four to
$\epsilon=0.2$. Figure 10: The same as Fig. 8, but for two values of $q$.
Column one ($q=0.8$) and two ($q=0.2$) are for $E_{\rm 0}=5\times
10^{51}\,{\rm erg}$ and column three ($q=0.8$) and four ($q=0.2$) are for
$E_{\rm 0}=5\times 10^{53}\,{\rm erg}$. Figure 11: The same as Fig. 9, but for
two values of $q$. Columns one and two correspond to $q=0.8$ and columns three
and four to $q=0.2$.

Figure 12: Upper Panel: Multiwavelength observations of GRB 160625B with the
best-fit light curves. The light curves that describe the early observations
(LAT and optical for ${\rm t<10^{3}\,s}$) are generated with the synchrotron
and SSC off-axis afterglow model, and the light curves that explain the late
observations (X-ray and optical for ${\rm t>10^{3}\,s}$) are taken from Fraija
et al. (2017b). Lower Panel: All the photons with energies $>100$ MeV and
probabilities $>90$% of being associated with GRB 160625B. The magneta line is
the maximum photon energy radiated by the synchrotron afterglow model. Figure
13: The corner plot displays the outcomes of our MCMC parameter estimate for
the synchrotron and SSC off-axis model applied to GRB 160625B. The histograms
on the diagonal display the marginalized posterior densities for each
parameter and the median values are shown by red lines.