††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Coalescent dynamics of planktonic communities

Paula Villa Martín  Anzhelika Koldaeva  Simone Pigolotti
<EMAIL_ADDRESS>Okinawa Institute of Science and Technology Graduate
University, Onna, Okinawa 904-0495, Japan

###### Abstract

Planktonic communities are extremely diverse and include a vast number of rare
species. The dynamics of these rare species is best described by individual-
based models. However, individual-based approaches to planktonic diversity
face substantial difficulties, due to the large number of individuals required
to make realistic predictions. In this paper, we study diversity of planktonic
communities by means of a spatial coalescence model, that incorporates
transport by oceanic currents. As a main advantage, our approach requires
simulating a number of individuals equal to the size of the sample one is
interested in, rather than the size of the entire community. By theoretical
analysis and simulations, we explore the conditions upon which our coalescence
model is equivalent to individual-based dynamics. As an application, we use
our model to predict the impact of chaotic advection by oceanic currents on
biodiversity. We conclude that the coalescent approach permits to simulate
marine microbial communities much more efficiently than with individual-based
models.

##  I Introduction

Ecological communities are made up of a large number of species. Their
diversity varies in space and time as a result of the ecological forces they
are subject to [1]. In diverse ecological communities, one typically
encounters a few very abundant species and many rare species, some of which
are represented by just a few individuals. Because of these rare species, the
diversity of ecological communities is best described by spatially explicit
individual-based models (IBMs) [2, 3], rather than models based on species
concentration or densities. IBMs have contributed to rationalize fundamental
biodiversity patterns in terrestrial ecosystems, such as the scaling of the
average number of encountered species with the size of the sampled area [4, 5,
6, 2, 3].

In prototypical spatially explicit IBMs such as the multi-species voter model
[7, 4, 6, 3], individuals are placed on a two-dimensional lattice and
stochastically reproduce, die, and disperse. Even simple spatial IBMs are
challenging to solve analytically [3]. A significant advancement in their
understanding originates in the concept of duality [7, 4]. In this context,
“duality” is a mapping between the IBM and a different model, whose dynamics
proceeds backward in time. For this reason, we often refer to the original
model as the “forward” model and its dual as the “backward” model. The
backward model considers a sample of the population of the forward model at a
very long time and seeks to reconstruct its species composition. Individuals
in the sample are represented as particles that evolve backward in time. If
two particles happen to be on the same site, they can coalesce, signaling that
the two corresponding individuals have a common ancestor and are therefore
conspecific. Due to these events, the backward model is also termed
“coalescence” model. By tracking coalescence events, the backward dynamics
reconstructs the species composition of the original sample.

In short, duality maps a spatial IBM into a system of coalescing random
walkers. The advantage of this mapping is twofold. First, mathematical results
have been obtained for systems of coalescing random walkers [8, 9], leading to
exact predictions of biodiversity patterns [3]. Second, backward models are
much more efficient than forward models to simulate on a computer [5, 6, 2,
3]. One reason is that backward simulations describe samples of a virtually
infinite population, so that one does not have to worry about finite-size
effects.

IBMs have also been used to study microbial planktonic communities [10, 11,
12, 13, 14]. These models have been used to predict, for example, how fluid
flows affect the fate of mutants characterized by a reproductive advantage
[13, 14, 15, 16, 17] or by a different diffusivity [18, 19, 20, 21]. However,
when used to predict biodiversity patterns, these models face severe
computational limitations. Even state-of-the-art approaches are usually
limited to communities of tens of thousands of individuals. For comparison, a
liter of oceanic water can contain tens or hundreds of millions of planktonic
cells [22]. To encompass this problem, it has been suggested that each
individual in a IBM can be considered as a representative of an entire
subpopulation [23]. It is however unclear whether this interpretation can
account for the dynamics of very rare species.

With this motivation in mind, we recently proposed a coalescence model for the
dynamics of microbial planktonic communities [24], which encompasses the
limitation of IBMs. In this paper, we study the dynamics of this coalescence
model and argue that, under certain conditions, the coalescence model is dual
to an IBM. We support these mathematical predictions with extensive numerical
simulations of both models. We finally present an application our modeling
approach to observational data from protist communities [24].

The paper is organized as follows. In Section II, we introduce the (forward)
IBM and the (backward) coalescence model. We prove that, in the weak noise
limit, these two models are dual. In Section III, we briefly introduce the
main observables that are commonly employed in ecology to quantify
biodiversity. In Section IV, we test our theory by extensive numerical
simulations of the forward and backward models, both in the presence and
absence of chaotic advection. We also compare the model prediction with
metabarcoding data. Section V is devoted to conclusions and perspectives.

##  II Models

In this section we introduce two approaches for modeling the diversity of
planktonic populations.

The first approach is via an IBM, in which an initial population
stochastically evolves forward in time as a result of reproduction events,
competition among individuals, advection–diffusion of individuals in space,
and mutation/speciation events that give rise to new species. The forward
model can be seen as the multi-species version of a two-species competition
model [13, 14] that does not include mutations. The second approach is based
on a backward (coalescence) model [24]. The backward model considers a sample
of $N_{s}$ individuals and seeks to reconstruct its species composition by
tracing the ancestry of the individuals backwards in time.

We conclude the section by defining two regimes (weak and strong noise)
characterizing the forward model. We then demonstrate that duality between the
forward and the backward models rigorously holds in the weak noise regime.

###  II.1 Forward model

A microbial population inhabits a two-dimensional square area $A=L\times L$,
representing an aquatic environment. Initially, individuals are homogeneously
distributed. They can belong to different species, but for simplicity we
neglect their species identity for the time being. Individuals can
stochastically die, reproduce, and displace. As a result of these events, the
total number of individuals $N(t)$ fluctuates in time. Each individual
asexually reproduces at rate $\lambda$. When a reproduction occurs, the
daughter individual is placed at a random position in a square of side $l$
centered on the mother position. From now on, we refer to this square as the
“neighborhood” of an individual. Individuals die in a density-dependent way
with rate $\lambda\,\hat{n}$, where $\hat{n}$ is the number of other
individuals in their neighborhood. The dependence of the death rate on the
local density represents competition.

Each individual moves in space according to the advection-diffusion equations

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}x$ $\displaystyle=$ $\displaystyle
v_{x}(x,y,t)+\sqrt{2D}\xi_{x}(t),$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}y$ $\displaystyle=$ $\displaystyle
v_{y}(x,y,t)+\sqrt{2D}\xi_{y}(t)\,,$ (1)

Figure 1: Normalized average number of individuals $\langle N(t)\rangle/N_{0}$
in (a) absence and (b) presence of advection. The stationary population size
$N_{0}$ is varied by tuning the neighborhood linear size $l$. The dashed line
marks the theoretical condition $\tilde{D}=1$ separating the weak noise regime
from the strong noise regime. In all simulations, we fixed $L=\lambda=1$. In
both panels, two dots at $\tilde{D}=0.1$ and $\tilde{D}=10$ (with
$N_{0}=16384$) mark the set of parameters chosen for our further analysis.

where $x$ and $y$ are the coordinates of the given individual. The terms
proportional to $\sqrt{2D}$ represent diffusion. The functions
$\xi_{x}(t),\xi_{y}(t)$ are white noise sources satisfying
$\langle\xi_{i}(t)\rangle=0$,
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$
where $i\in\\{x,y\\}$ and $\langle\dots\rangle$ denotes an average over
realizations. For the time being, we impose periodic boundary conditions. The
functions $v_{x}(x,y,t)$ and $v_{y}(x,y,t)$ represent an advecting fluid flow.
In this paper we only consider incompressible velocity fields,
$\vec{\nabla}\cdot\vec{v}=0$ where $\vec{v}=(v_{x},v_{y})$ and
$\vec{\nabla}=(\partial/\partial x,\partial/\partial y)$. Indeed, two-
dimensional velocity fields in the oceans are incompressible to a very good
approximation, although in some local regions vertical currents can alter this
picture [25], potentially impacting population dynamics [13, 26, 27].

In the absence of birth and death dynamics and thanks to incompressibility,
Eqs. (II.1) predicts a homogeneous stationary distribution of individuals. We
tentatively assume that this distribution remains homogeneous in the presence
of birth and death processes. We call $N_{0}$ the average population size
under this hypothesis. By imposing that birth events statistically balance
death events, we find that

$N_{0}=L^{2}/l^{2},$ (2)

i.e. the average population size is equal to the ratio between the system area
and the area of the interaction neighborhood, see Appendix A. In practice,
this means that each interaction neighborhood contains one individual, on
average. In the following, we take this value as the initial population size,
$N(0)=N_{0}$.

We want to understand whether the assumption of homogeneous density holds. To
this aim, we study a macroscopic description of our IBM, see Appendix B. We
find that the populations remain homogeneous in the case when the stochastic
fluctuations induced by birth and death processes are relatively small. In
this case, the average number of individuals does not significantly deviate
from $N_{0}$.

In two dimensions, the relative strength of fluctuations is controlled by the
dimensionless parameter

$\tilde{D}=DN_{0}/(\lambda L^{2}),$ (3)

see [14]. For $\tilde{D}\gg 1$, stochastic fluctuations are small. In
contrast, for $\tilde{D}\ll 1$, fluctuations dominate the dynamics. In the
following, we refer to the $\tilde{D}>1$ and $\tilde{D}<1$ cases as the “weak
noise” and “strong noise”regime, respectively. .

Simulations of the model confirm that, in the strong noise regime, the average
number of individuals significantly differ from $N_{0}$, see Fig. 1. This
means that our assumption that the birth-death dynamics does not affect the
average number of individuals breaks down.

In particular, in the absence of advection, our simulations show that the
average number of individuals exceeds $N_{0}$, see Fig. 1a. This result
contrasts with that in [14], where a reduction of the average number of
individuals was observed in the strong noise regime. This discrepancy can be
explained by a difference in the way birth is implemented in the two models:
here, the daughter cell is placed at a random position in the neighborhood of
her mother, whereas in [14] the daughter cell is placed at the same position
as the mother.

In contrast, in the presence of advection, the average population size
decreases in the strong noise regime, see Fig. 1b. A qualitative explanation
is that advection effectively precludes individuals to visit some regions,
thereby increasing effective competition.

Hereafter, we fix $\lambda=1$, $L=7.5$, and $N_{0}=16384$. The value of $l$
corresponding to these choices is obtained from Eq. (2).

Figure 2: Events in the forward and backward models. (a) In the forward model,
individuals reproduce at rate $\lambda$ within a neighborhood, displace, and
die with rate $\lambda\,\hat{n}$, where $\hat{n}$ is the number of other
individuals in their square neighborhood of linear size $l$. (b) In the
backwards model individuals displace and coalesce with rate $\bar{\lambda}$
with individuals within their neighborhood. In the weak noise regime, the two
models are dual under the condition $\lambda=\bar{\lambda}$.

###  II.2 Backward model and duality

The backward model (or coalescence model) describes the dynamics of $N_{s}$
Lagrangian tracers. These tracers represent a sample of individuals from a
final population, i.e. from a population that evolved according to the forward
model at a time $t_{f}\gg 0$. We assume that the sample is homogeneously
distributed in a sample area $L_{s}\times L_{s}$, with $L_{s}\leq L$. We trace
back in time the evolution of the tracers in the sample. The coordinates of
the tracers evolve according to Eq. (II.1), which we integrate with negative
time increments.

While evolving back in time the coordinate of a given tracer, we might reach
the time at which the individual represented by the tracer was born. From that
instant, the tracer represents the position of the mother of the chosen
individual. This means that, at a given time, each tracer represents either an
individual in the final population or one of its ancestors. If the mother of
the chosen individual (or one of her ancestors) is alive in the final
population, this implies that the two tracers must be in the neighborhood of
each other at the time in which the birth event occurs. If this is the case,
we say that a “coalescence” has occurred, and the two tracers are merged into
one.

The events occurring in the forward and backward models are summarized in Fig.
2a and Fig. 2b, respectively.

To fully specify the coalescence model, we need to determine the rate
$\bar{\lambda}$ at which two individuals coalesce if they are in the
neighborhood of each other. We do so by considering a situation in which
$N_{s}=N_{0}$, i.e., the number of tracers is equal to the average number of
individuals in the forward model. In this scenario we are tracing all
individuals, therefore the total rate of birth $\lambda N_{0}$ in the forward
model must match the total rate of coalescence in the backward model. This
latter rate is equal to $\bar{\lambda}N_{s}$ by the same argument made in
Appendix A. Imposing that the two total rates must be equal and using
$N_{s}=N_{0}$ leads to $\bar{\lambda}=\lambda$, i.e. the coalescence rate for
individuals in the same neighborhood should match their birth rate in the
forward model.

We remark that this argument relies on the assumption that the dynamics of the
forward model is in the weak noise regime. In the strong-noise regime, some of
the assumptions underlying duality do not hold. First, we can not assume that
a sample of individuals in the final population is homogeneously distributed.
Second, since in the strong-noise regime $\langle N(t)\rangle\neq N_{0}$, we
can not easily draw a correspondence between the rates of the forward model
and the backward model.

###  II.3 Mutations and multi-species dynamics

We now add to the forward and backward models the notion of species identity.
In the forward model, we assume that a newborn individual belongs to a new
species with probability $\mu$. In ecological terms, the probability $\mu$ can
either be interpreted as a speciation probability, or as a probability for
individuals to be replaced by individuals belonging to new species immigrating
from outside the community. If the model is used to interpret metabarcoding
studies, $\mu$ represents the probability for individuals to accumulate
sufficient mutations to be considered as a new operational taxonomic unit
(OTU).

Figure 3: Ancestry trees. (a) Dynamics of the forward model represented as a
tree. An initial set of $N_{0}$ individuals reproduce and die over time until
all remaining individuals share the same common ancestor. Individuals that die
before the final time (dark red dots) are not relevant for the final
population, and are therefore removed. Individuals that do not have descendant
in the final population (light gray dots) are removed as well. (b) individuals
that have one direct descendant in the remaining tree (orange dots) are
removed; their number are saved as variables $d_{i}$ for each branch $i$. (c)
Mutations (marked with Xs) are introduced at each tree branch $i$ with a
probability $p_{i}=1-(1-\mu)^{d_{i}}$, see [3]. (d) Ancestry tree of the
backward model. In this case, we consider a sample of $N_{s}$ individuals from
the final population and reconstruct its ancestry. Individuals can coalesce or
mutate until one individual remains. This approach directly ignores branches
corresponding to individuals that do not belong to the sample.

When simulating the forward model, one can keep track of species identity by
assigning each newborn individual to a new species with probability $\mu$. We
however adopt an equivalent, but more efficient strategy, see [3, 28]. We
evolve individuals without keeping track of their species identity, until all
individuals descend from a single individual in the original population, see
Fig. 3a. The collection of descendents of this individual constitutes the
ancestry tree. Individuals that do not belong to the ancestry tree (light grey
in Fig. 3a) do not affect the diversity of the final population and can
therefore be ignored. A further simplification is to remove individuals that
have only one descendant in the ancestry tree (yellow in Fig. 3a). This can be
done by keeping track of the number of duplications $d_{i}$ occurred in each
branch $i$ separating the remaining individuals, see Fig. 3b. Since at each
duplication event the probability of a mutation is equal to $\mu$, the total
probability that at least a mutation has occurred in a branch $i$ is equal to
$p_{i}=1-(1-\mu)^{d_{i}}$. In this way, we assign the mutations a posteriori,
by drawing from the probabilities that a mutation has occurred in each branch,
see Fig. 3c.

In the backward model, mutations occur continuously at a stochastic rate
$\mu\lambda dt$. The reason is that in the backward model we do not track
individual birth events, unless they lead to coalescence. As anticipated, the
advantage of the coalescence model is that it is possible to reconstruct the
identity of a sample of $N_{s}$ individuals in the final population being a
subset of the total population. The corresponding ancestry tree is a subset of
the corresponding tree in the forward model which includes all ancestors of
the individuals in the sample up to their most recent common ancestor, see
Fig. 3d. In practice, mutations can be assigned on the branches of this tree
in a similar way as for the forward model. However, this is not necessarily an
efficient procedure in this case. The reason is that, for large system size
and in the presence of advection, the time it takes for all tracers to
coalesce can be exceedingly long. At the same time, it is not necessary to
track tracers backward in time before a mutation, since events occurred before
the mutations do not affect the diversity of the final sample. For this
reason, we evaluate the probability of mutations at each time step of the
backward dynamics and we simply eliminate tracers that have mutated.

We compare predictions of both models for a sample of $N_{s}$ individuals
taken at a final large time $t_{f}$. We fix $N_{s}<N_{0}$. In the absence of
advection, since we might have $\langle N(t)\rangle\gtrsim N_{0}$, we fix
$N_{s}=16000$. In the presence of advection, population sizes are $\langle
N(t)\rangle\lesssim N_{0}$, and we consider $N_{s}=8192$ to ensure
$N_{s}<N_{0}$.

##  III Diversity measures

Figure 4: Diversity measures predicted by the models with and without
advection in the strong (a, b, c, d) and weak (e, f, g, h) noise regimes. (a,
e) Value of $\alpha$-diversity as a function of the parameter $\tilde{D}$.
Measures for forward model (FM) (darker colors) and backward model (BM)
(lighter colors) are shown. (b, f) $\beta-$diversity, (c, g) Species-Area
$S(A)$, and (d, h) species abundance distribution for $\tilde{D}=0.1,10$. In
these panels, dark/light colors show curves for the forward/backward model for
$\tilde{D}=10$; these curves nearly coincide. Dashed lines of panels (d) and
(h) show the analytical prediction for a well-mixed system, Eq. (5). In all
panels, the mutation probability is equal to $\mu=5\cdot 10^{-3}$.

Quantifying biodiversity by simply counting the number of competing species is
not always appropriate. The reason is that most ecological communities,
including planktonic ones, are composed of relatively few abundant species and
a large number of rare species. The definition of biodiversity in terms of
number of species is insensitive to this distinction. This definition also
relies on the possibility of observing all these rare species, which is often
impossible in practice. To address this issue, other measures of biodiversity
have been proposed in the literature [29]. These alternative definitions give
different weight to abundant and rare species.

As anticipated, a basic biodiversity measure is the number of species present
in the community. Other measures take into account more explicitly spatial
distributions and compositional heterogeneity. Specifically, at equal number
of species, a population can be equally distributed among species or be
dominated by a few of them. Moreover, the manner that species distribute in
space, i.e. whether individuals of a species are grouped within specific areas
or highly dispersed in space, is an important characterization of
biodiversity. Here, we review the most common measures of biodiversity:

  * •

$\alpha$-diversity. The $\alpha$-diversity is the first and simplest measure.
It is defined as the total average number of species in the community [30]. We
measure the $\alpha$-diversity by simply averaging the total number of species
over multiple realizations of the forward and backward models.

  * •

$\beta$-diversity. The $\beta$-diversity quantifies spatial correlations
within species. It describes how species composition changes from local to
larger scales. There exist slightly different definitions of $\beta$-diversity
in the literature [31, 32]. We define $\beta$-diversity as the probability
that two random individuals at a given distance $r=\sqrt{x^{2}+y^{2}}$ belong
to the same species

$\beta(r)=\displaystyle\frac{\sum_{i}N_{s_{i},s_{i}}(r)}{\sum_{i,j}N_{s_{i},s_{j}}(r)}$
(4)

where $N_{s_{i},s_{j}}(r)$ is the number of pairs $i,j$ of species
$s_{i},s_{j}$ at distance $r$ [3]. At equal number of species, highly grouped
communities are characterized by a steeper $\beta$-diversity than more
dispersed ones.

  * •

Species-area relation. The species-area relation is defined as the average
number of species $S$ found in an area $A$ [33]. The species-area relation has
been fitted by mathematical relations of the form $S(A)\propto A$ for small
and large scales, and $S(A)\propto A^{z}$ for intermediate scales [34, 33, 35,
36]. We average the final number of species $S$ over several realizations of
our models for increasing sampling areas of size $A=L_{s}\times L_{s}$ located
at the center of our system.

  * •

Species-abundance distribution. The species-abundance distribution (SAD)
quantifies the the compositional heterogeneity of a population in terms of the
relative abundance of species. It is defined as the frequency $P(n)$ of
species with abundance $n$ in a sample. In a well-mixed system of population
size $N_{0}$ and with mutation probability $\mu$, the species-abundance
distribution has the form

$P(n)=\displaystyle\frac{N_{0}\mu e^{-\mu n}}{n},$ (5)

see [35, 37]. We expect our spatially-explicit model to generate a similar
species-abundance distributions, at least in the high diffusion limit. Apart
from this limiting case, analytical predictions for the species-abundance
distribution in spatially explicit models are rather hard to obtain [3]. The
species-abundance distribution is normalized to the $\alpha$-diversity, i.e.,
the average total number of species in the community:

$S\approx\int_{1}^{\infty}dn\,\frac{N_{0}\mu e^{-\mu n}}{n}.$ (6)

##  IV Results

In this section, we numerically verify that the forward and backward models
are equivalent in the weak noise regime, and test whether this equivalence
extends in the strong noise regime. We also study whether this equivalence is
affected by the value of the mutation probability $\mu$. To these aims, we
extensively simulate both systems for a broad parameter range and compute
diversity measures of the final population averaged over $10^{3}$
realizations. We perform these comparisons both in the presence and in the
absence of advection.

###  IV.1 Biodiversity measures in the absence of advection

We consider the two models in the weak and strong noise regimes and first
focus on the $\alpha$-diversity for different diffusion rates $D$. As
expected, the predictions of the forward and backward models present a small
by significative discrepancy in the strong noise regime, see Fig. 4a. We
quantify this discrepancy by the absolute relative difference

$\Delta\alpha=\frac{|\alpha_{f}-\alpha_{b}|}{\alpha_{f}+\alpha_{b}},$ (7)

where $\alpha_{f},\alpha_{b}$ are the $\alpha$-diversities measured in the
forward and backward dynamics, respectively. In the weak noise regime, we
observe compatible values of the $\alpha$-diversity, see Fig. 4e. This
analysis reveals that, for example for $\tilde{D}\approx 0.1$, the discrepancy
is on the order of $10\%$, see inset of Fig. 4a. In particular, the backward
model predicts a higher number of species than the forward model. As discussed
in section II for the number of individuals, the differences between the
forward and backward models in the strong noise limit can depend on model
details, such as the microscopic implementation of mutations.

Comparisons of the $\beta$-diversity, Fig. 4b,f, species-area relation, Fig.
4c,g, and species-abundance distribution, Fig. 4d,h lead to similar
conclusions. In the weak noise regime, the forward and backward models yield
nearly identical predictions for all these quantities, as expected. In the
strong-noise regime, we observed some discrepancies, of comparable magnitude
as for the $\alpha$-diversity.

###  IV.2 Biodiversity measures in the presence of chaotic advection

We now move to the case with advection. We define a two-dimensional
incompressible advecting field in terms of a stream-function $\phi(x,y)$. The
components of the field are related with the stream function by

$\displaystyle v_{x}(x,y;t)$
$\displaystyle=-\frac{\partial\phi(x,y;t)}{\partial y}$ $\displaystyle
v_{y}(x,y;t)$ $\displaystyle=\frac{\partial\phi(x,y;t)}{\partial x}.$ (8)

This definition automatically guarantees the incompressibility condition
$\vec{\nabla}\cdot\vec{v}=0$. We choose a dimensionless stream function that
generates a chaotic vortex in the vicinity of the meandering jet [38]:

$\phi(x,y)=-\tanh\left(\frac{y-B(t)\cos(kx)}{\sqrt{1+k^{2}B^{2}(t)\sin^{2}(kx)}}\right)+cy,$
(9)

where $B(t)=B_{0}+\epsilon\cos(wt+\Phi)$. We fix $k=2\pi/L$, $c=0.12$,
$L=7.5$, $B_{0}=1.2$, $\epsilon=0.3$, $w=0.4$, $\Phi=\frac{\pi}{2}$. We impose
periodic boundary conditions. We numerically solve the differential equations
(II.1) with the advecting field (8) using a 4th order Runge-Kutta method.

As in the case without advection, we observe a significant discrepancy in the
$\alpha$-diversity between the forward and backward models in the strong noise
regime, see Fig. 4a. The discrepancy progressively decreases for bigger
$\tilde{D}$, see Fig. 4e. The discrepancy tends to be smaller in the presence
of advection rather than in the absence of it, see inset of Fig.4a. One
possible explanation for this difference is that, at equal diffusivity, the
system subject to chaotic advection presents a different effective diffusivity
[39]. Other diversity measures show the expected behavior, with a close
correspondence in the weak noise regime and appreciable differences in the
strong noise regime, see Fig. 4b-d, and Fig. 4f-h.

A comparison between Fig. 4a and Fig. 4e. shows that advection tends to reduce
the average number of species. The choice of periodic boundary conditions
crucially impacts this result.

###  IV.3 Effect of mutation probability

We now study the dependence of the $\alpha$-diversity on the mutation
probabilities $\mu$ for both the forward and backward model. In the strong
noise regime and for a low mutation probability, the forward model predicts a
lower $\alpha$-diversity than the backward model, see Fig. 5a and Fig. 5b. The
relative difference between forward and backwards models weakly depends on
$\mu$. In the weak noise regime, models are compatible for all values of the
parameter $\mu$, see Fig. 5a,b. These behaviors are qualitatively similar in
the presence and in the absence of advection.

Figure 5: $\alpha$-diversity as a function of the mutation probability in the
strong (a), and weak (b) noise regimes. Dashed lines show the analytical
prediction for well-mixed systems, see Eq. (6). Figure 6: Individual
trajectories modeled with the coalescence model in the (a) absence and (b)
presence of advection. At the initial time $t=0$, the populations are
homogeneously distributed in a square of size $L\times L$. We simulate the
coalescence model with open boundary conditions. In (b), we employ the
velocity field given in Eqs. (8) and (9). Parameters are specified in Section
IVB. (c) Species abundance distributions of individual protist populations
sampled in oceans and lakes. (d) Dashed curves represent average abundance
distributions of protist populations sampled in oceans and lakes (c); solid
curves correspond to numerical simulations of the backward models with open
boundary conditions without and with advection, where we fixed $\tilde{D}=1$.

###  IV.4 Metagenomic data

In this section, we demonstrate that the coalescence model is able to predict
species-abundance distributions observed in aquatic environments.

We numerically simulate the coalescence model in the presence and absence of
advection, see Fig. 6a, b. In this case, we adopt open boundary conditions as
these are more relevant for the ocean, where a sample is embedded in a very
large area. We do not perform a comparison with the forward dynamics, since
the forward IBM can not be easily formulated with open boundaries. Individuals
are homogeneously distributed in a square at $t=0$. In the case with
advection, we set $c=0.12$, $B_{0}=1.2$, $w=0.5$, and $\epsilon=4$.

We compare the model prediction for the SAD with observational data. To this
aim, we employ two metabarcoding datasets: one from the TARA Oceans expedition
[40], and one from freshwater lakes [41]. Metabarcoding techniques permit to
sample planktonic diversity at unprecedented resolution [42]. In metabarcoding
studies, one obtains from a sample DNA fragments corresponding to a highly
conserved region of the genome, in this case a portion of the 18S ribosomal
RNA gene. Since this region is highly conservffed, sequences in the sample
with high degree of genetic similarity (at least 97%, in this case) are likely
to come from individuals within the same taxonomic group. These groups,
identified by genetic similarity, are called operational taxonomic units
(OTU). We plot the abundance distributions of the observed OTUs from each
dataset, see Fig. 6c. The data demonstrates that oceanic currents promote
abundance of rare species [24].

Our coalescence model predicts that the effect of oceanic currents leads to
SAD curves characterized by a steeper decay, see Fig. 6d and [24]. This effect
is also observed in the data, see Fig. 6d, although the slopes in the log-log
plot are slightly different in the data. As extensively discussed in [24], the
quantitative value of the slope is affected by microscopic details of the
model, and aspects of the metabarcoding analysis such as the similarity
threshold chosen to identify OTUs.

##  V Conclusions

In this paper, we developed a model for the dynamics of microbial aquatic
communities based on the idea of coalescence. Our coalescence model predicts
the diversity of a sample of organisms embedded in a very large, spatially
extended populations. It encompasses the limitations of individuals-based
model in describing communities made up of huge number of individuals. Our
model has the potential to bridges the gap between ecological dynamics at the
individual level and large-scale spatial dynamics.

We have shown that, in the weak noise regime, the model is equivalent to an
individual-based model proceeding forward in time. In the strong-noise regime,
this correspondence is only approximate, but both models predict qualitatively
similar biodiversity patterns. Due to its advantages, the coalescence model
presented in this paper provides a a versatile and powerful tool to predict
biodiversity observed in metabarcoding studies of planktonic communities [24].

Although, for simplicity, we focus on a simple model of microbial competition
dynamics, our approach can be extended to more general ecological settings and
to other communities. Such generalizations, combined with high-throughput
sequencing data, have the potential to shed light on the main ecological
forces determining community dynamics.

###### Acknowledgements.

We thank Massimo Cencini for comments on a preliminary version of this
manuscript. We thank Florian Pflug for helping us to debug the code. We thank
Ales Bucek and Tom Bourguignon for the analysis of metabarcoding data and
stimulating discussions. We are grateful for the help and support provided by
the Scientific Computing and Data Analysis section of Research Support
Division at OIST.

##  Appendix A Stationary number of individuals in the weak noise limit

In this Appendix, we compute the stationary number of individuals $N_{0}$ in
the limit in which individuals are homogeneously distributed. The numer of
individuals $n$ inside each neighborhood is a Poisson random variable with
average $\mu=N_{0}/M$, where $M=(L/l)^{2}$ is the total number of
neighborhoods.

To estimate $N_{0}$, we impose that the total average birth and death rates
should balance. The total average birth rate is simply $\lambda N_{0}$.
Individuals die with a rate proportional to the number of other individuals in
each neighborhood. Therefore, the total average death rate is equal to

total average death rate $\displaystyle=\lambda M\langle n(n-1)\rangle$
$\displaystyle=\lambda M\sum_{n}\frac{n(n-1)\mu^{n}e^{-\mu}}{n!}$
$\displaystyle=\lambda M\sum_{n}\frac{\mu^{n}e^{-\mu}}{(n-2)!}$
$\displaystyle=\lambda M\mu^{2}=\frac{\lambda N_{0}^{2}}{M}$
$\displaystyle=\frac{\lambda N_{0}^{2}l^{2}}{L^{2}}.$ (10)

Imposing that the total average death rate must be equal to the total average
birth rate leads to the condition $L=l\sqrt{N_{0}}$, or equivalently
$N_{0}=M$.

##  Appendix B Macroscopic description of the forward model

In this Appendix, we study a macroscopic description of our IBM. We introduce
the density of individuals $n(x,y;t)$, defined so that its integral over a
given area yields the number of individuals in that area at time $t$. We also
define the concentration $c(x,y;t)=(L^{2}/N_{0})n(x,y;t)$. The normalization
factor $L^{2}/N_{0}$ ensures that the average concentration is equal to one,
if the assumption of homogeneous density holds.

The dynamics of the concentration $c(x,y;t)$ can be derived in the small noise
limit, i.e. by assuming that the stochastic fluctuations induced by birth and
death processes are relatively small. Under this assumption, the concentration
is described by the stochastic Fisher-Kolmogorov equation

$\frac{\partial}{\partial
t}c(x,y;t)=\lambda(c-c^{2})-\vec{\nabla}\cdot[\vec{v}c]+D\nabla^{2}c+\sigma(c)\xi(x,y;t),$
(11)

where $\xi(x,y,t)$ is a noise field satisfying $\langle\xi(x,y,t)\rangle=0$
and
$\langle\xi(x,y,t)\xi(x^{\prime},y^{\prime},t^{\prime})\rangle=\delta(x-x^{\prime})\delta(y-y^{\prime})\delta(t-t^{\prime})$.
The multiplicative noise is interpreted using the Ito prescription; its
amplitude is equal to $\sigma(c)=\sqrt{\lambda L^{2}c(1+c)/N_{0}}$. Equation
(11) can be derived using a Kramers-Moyal expansion, see [14] and chapter 13
in [43]. In the derivation, we neglected contributions to the noise coming
from the diffusion operator, see [14].

In this case without advection, $\vec{v}=0$, the concentration $c(x,y;t)$ is
subject to two competing effects: the noise term in Eq. (11) which creates
fluctuations around the average solution $\langle c(x,y;t)\rangle=1$, and the
diffusion term which smoothens these fluctuations.

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An efficient method for solving equations in generalized quaternion and
octonion algebras

Cristina Flaut and Vitalii Shpakivskyi

Abstract. Quaternions often appear in wide areas of applied science and
engineering such as wireless communications systems, mechanics, etc. It is
known that are two types of non-isomorphic generalized quaternion algebras,
namely: the algebra of quaternions and the algebra of coquaternions. In this
paper, we present the formulae to pass from a basis in the generalized
quaternion algebras to a basis in the division quaternions algebra or to a
basis in the coquaternions algebra and vice versa. The same result was
obtained for the generalized octonion algebra. Moreover, we emphasize the
applications of these results to the algebraic equations and De Moivre’s
formula in generalized quaternion algebras and in generalized octonion
division algebras.

AMS Subject Classification: 17A35;

Keywords: generalized quaternion algebras; De Moivre’s formula.

0\. Introduction

Let $\gamma_{1},\gamma_{2}\in\mathbb{R}\setminus\\{0\\},$ let
$\mathbb{H}\left(\gamma_{1},\gamma_{2}\right)$ be the generalized quaternion
algebra with basis $\\{1,e_{1},e_{2},e_{3}\\}~{}$and $\mathbb{H(}1,1)$ be the
quaternion division algebra with basis $\\{1,i,j,k\\}$. The multiplication
table is given below:

$\cdot$ | $1$ | $e_{1}$ | $e_{2}$ | $e_{3}$
---|---|---|---|---
$1$ | $1$ | $e_{1}$ | $e_{2}$ | $e_{3}$
$e_{1}$ | $e_{1}$ | $-\gamma_{1}$ | $e_{3}$ | $-\gamma_{1}e_{2}$
$e_{2}$ | $e_{2}$ | $-e_{3}$ | $-\gamma_{2}$ | $\gamma_{2}e_{1}$
$e_{3}$ | $e_{3}$ | $\gamma_{1}e_{2}$ | $-\gamma_{2}e_{1}$ | $-\gamma_{1}\gamma_{2}$

The algebra $\mathbb{H}(1,-1)$ is called the algebra of coquaternions [2] or
also called the algebra of para-quaternions [8], or the algebra of split-
quaternions [5], [16], or the algebra of anti-quaternions, or the algebra of
pseudo-quaternions [18, p. 389], or hyperbolic quaternions [1]. We denote by
$\\{1,i_{1},i_{2},i_{3}\\}$ the basis of coquaternion algebra.

Proposition 1. ([10], Proposition 1.1) The quaternion algebra
$\mathbb{H}\left(\beta_{1},\beta_{2}\right)$ is isomorphic with quaternion
algebra $\mathbb{H}\left(x^{2}\beta_{1},y^{2}\beta_{2}\right),$ where $x,y\in
K^{\ast}.$ $\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

The real octonion division algebras are a non-associative and non-commutative
extension of the algebra of quaternions. Among all the real division algebras,
octonion algebra forms the largest normed division algebra.

Let $\mathbb{O}(\alpha,\beta,\gamma)$ be a generalized octonion algebra over
$\mathbb{R},$ with basis $\\{1,f_{1},...,f_{7}\\}\vskip 6.0pt plus 2.0pt minus
2.0pt\,\,$and the multiplication given in the following table:

$\cdot$ $1$ $\,\,\,f_{1}$ $\,\,\,\,\,f_{2}$ $\,\,\,\,f_{3}$ $\,\,\,\,f_{4}$
$\,\,\,\,\,\,f_{5}$ $\,\,\,\,\,\,f_{6}$ $\,\,\,\,\,\,\,f_{7}$ $\,1$ $1$
$\,\,\,f_{1}$ $\,\,\,\,f_{2}$ $\,\,\,\,f_{3}$ $\,\,\,\,f_{4}$
$\,\,\,\,\,\,f_{5}$ $\,\,\,\,\,f_{6}$ $\,\,\,\,\,\,\,f_{7}$ $\,f_{1}$
$\,\,f_{1}$ $-\alpha$ $\,\,\,\,f_{3}$ $-\alpha f_{2}$ $\,\,\,\,f_{5}$ $-\alpha
f_{4}$ $-\,\,f_{7}$ $\,\,\,\alpha f_{6}$ $\,f_{2}$ $\,f_{2}$ $-f_{3}$
$-\,\beta$ $\,\,\beta f_{1}$ $\,\,\,\,f_{6}$ $\,\,\,\,\,f_{7}$ $-\beta f_{4}$
$-\beta f_{5}$ $f_{3}$ $f_{3}$ $\alpha f_{2}$ $-\beta f_{1}$ $-\alpha\beta$
$\,\,\,\,f_{7}$ $-\alpha f_{6}$ $\,\,\,\beta f_{5}$ $-\alpha\beta f_{4}$
$f_{4}$ $f_{4}$ $-f_{5}$ $-\,f_{6}$ $-\,\,f_{7}$ $-\,\gamma$ $\,\,\,\gamma
f_{1}$ $\,\,\gamma f_{2}$ $\,\,\,\,\,\gamma f_{3}$ $\,f_{5}$ $\,f_{5}$ $\alpha
f_{4}$ $-\,f_{7}$ $\,\alpha f_{6}$ $-\gamma f_{1}$ $-\,\alpha\gamma$ $-\gamma
f_{3}$ $\,\alpha\gamma f_{2}$ $\,\,f_{6}$ $\,\,f_{6}$ $\,\,\,\,f_{7}$
$\,\,\beta f_{4}$ $-\,\beta f_{5}$ $-\gamma f_{2}$ $\,\,\,\gamma f_{3}$
$-\beta\gamma$ $-\beta\gamma f_{1}$ $\,\,f_{7}$ $\,\,f_{7}$ $-\alpha f_{6}$
$\,\beta f_{5}$ $\alpha\beta f_{4}$ $-\gamma f_{3}$ $-\alpha\gamma f_{2}$
$\beta\gamma f_{1}$ $-\alpha\beta\gamma$

The algebra $\mathbb{O}(\alpha,\beta,\gamma)$ is non-commutative, non-
associative but it is alternative (i.e. $x^{2}y=x\left(xy\right)$ and
$yx^{2}=\left(yx\right)x,\forall x,y\in\mathbb{O}(\alpha,\beta,\gamma)$),
flexible (i.e. $x\left(yx\right)=\left(xy\right)x,\forall
x,y\in\mathbb{O}(\alpha,\beta,\gamma)$), power-associative (i.e. for each
$x\in\mathbb{O}(\alpha,\beta,\gamma)$ the subalgebra generated by $x$ is an
associative algebra).

If $a\in\mathbb{O}(\alpha,\beta,\gamma),$
$a=a_{0}+a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7}$
then
$\bar{a}=a_{0}-a_{1}f_{1}-a_{2}f_{2}-a_{3}f_{3}-a_{4}f_{4}-a_{5}f_{5}-a_{6}f_{6}-a_{7}f_{7}$
is called the conjugate of the element $a.$ Let
$A=\mathbb{O}(\alpha,\beta,\gamma)$ and $a\in A.$ We have that
$t\left(a\right)=a+\overline{a}\in K$ and

$\,N\left(a\right)=a\overline{a}=a_{0}^{2}+\alpha a_{1}^{2}+\beta
a_{2}^{2}+\alpha\beta a_{3}^{2}+\gamma a_{4}^{2}+\alpha\gamma
a_{5}^{2}+\beta\gamma a_{6}^{2}+\alpha\beta\gamma a_{7}^{2}\in K.$

These elements are called the trace, respectively, the norm of the element
$a\in$ $A.$ It follows
that$\,\,\left(a+\overline{a}\right)a\,=a^{2}+\overline{a}a$=$a^{2}+n\left(a\right)\cdot
1$ and $a^{2}-t\left(a\right)a+N\left(a\right)=0,\forall a\in A,$ therefore
the generalized octonion algebra is quadratic.

The subset $\,\,A_{0}$=$\\{x\in A\,\,\,\mid\,t\left(x\right)$=$0\\}$ of
$A\,\,\,$is a subspace of the algebra $A$. It is obvious that $A=K\cdot
1\oplus A_{0},$ therefore each element $\,\,x\in\,\,A\,\,$ has the form
$x=x_{0}\cdot 1+\overrightarrow{x},\,$with $\,x_{0}\in
K\,\,\,\,$and$\,\,\,\overrightarrow{x}\in A_{0}.$ For $K=\mathbb{R},$ we call
$x_{0}$ the scalar part and $\overrightarrow{x}$ the vector part for the
octonion $x.$

If for $x\in A,$ the relation $N\left(x\right)=0$ implies $x=0$, then the
algebra $A$ is called a division algebra. For other details, the reader is
referred to [19].

In the papers [6], [7] are considered some algebraic equations in generalized
quaternion and octonion algebras. Due to Proposition 1, in the present paper
we reduced the study of an algebraic equation in an arbitrary algebra
$\mathbb{H}\left(\gamma_{1},\gamma_{2}\right)$ with
$\gamma_{1},\gamma_{2}\in\mathbb{R}\setminus\\{0\\}$ to study of the
corresponding algebraic equation in one of the following two algebras:
division quaternion algebra or coquaternion algebra. Moreover, De Moivre’s
formula and Euler’s formula in generalized quaternion algebras, founded in
[11], was proved using this new method, for $\gamma_{1},\gamma_{2}>0.$ With
this technique, the above mentioned results were also obtained for the
octonions.

1\. An isomorphism between the algebras
$\mathbb{H}(\gamma_{1},\gamma_{2}),$with $\gamma_{1},\gamma_{2}>0,$ and
$\mathbb{H}(1,1)$

Everywhere in this section, we will consider $\gamma_{1},\gamma_{2}>0$. An
isomorphism between the algebras $\mathbb{H}(\gamma_{1},\gamma_{2})$ and
$\mathbb{H}(1,1)$ is given by the operator $A$ and its inverse $A^{-1}$, where

$A:\quad e_{1}\mapsto i\sqrt{\gamma_{1}},\quad e_{2}\mapsto
j\sqrt{\gamma_{2}},\quad e_{3}\mapsto k\sqrt{\gamma_{1}\gamma_{2}}.$

It is easy to prove the following properties for the operator $A:$ 1)
$A\left(\lambda x\right)=\lambda
A\left(x\right),\forall\;\lambda\in\mathbb{R},\forall\;x\in\mathbb{H}\left(\gamma_{1},\gamma_{2}\right);$
2)
$A\left(x+y\right)=A\left(x\right)+A\left(y\right),\forall\;x,y\in\mathbb{H}\left(\gamma_{1},\gamma_{2}\right);$
3)
$A\left(xy\right)=A\left(x\right)A\left(y\right),\forall\;x,y\in\mathbb{H}\left(\gamma_{1},\gamma_{2}\right).$
From here, it results that the operators $A$ and $A^{-1}$ are additive and
multiplicative.

Proposition 1.1. The operators $A$ and $A^{-1}$ are continuous and their norms
are equal with $1$.

Proof. We denote by $\|\cdot\|_{\mathbb{H}(\gamma_{1},\gamma_{2})}$ the
Euclidian norm in $\mathbb{H}\left(\gamma_{1},\gamma_{2}\right)$. Since the
spaces $\mathbb{H}\left(\gamma_{1},\gamma_{2}\right)$ and $\mathbb{H}(1,1)$
are normed spaces, then the continuity of $A$ is equivalent with the
boundedness of $A,$ i.e. there is a real constant $c$ such that for all
$x\in\mathbb{H}\left(\gamma_{1},\gamma_{2}\right),$ we have

$\frac{\left\|A\left(x\right)\right\|_{\mathbb{H}\left(1,1\right)}}{\left\|x\right\|_{\mathbb{H}\left(\gamma_{1},\gamma_{2}\right)}}\leq
c.$

It results that

$\frac{\left\|x_{0}+x_{1}i\sqrt{\gamma_{1}}+x_{2}j\sqrt{\gamma_{2}}+x_{3}k\sqrt{\gamma_{1}\gamma_{2}}\right\|}{\left\|x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}\right\|}=$
$=\frac{\sqrt{x_{0}^{2}+x_{1}^{2}\gamma_{1}+x_{2}^{2}\gamma_{2}+x_{3}^{2}\gamma_{3}}}{\sqrt{x_{0}^{2}+x_{1}^{2}\gamma_{1}+x_{2}^{2}\gamma_{2}+x_{3}^{2}\gamma_{3}}}=1.$

$\,\Box\vskip 12.0pt plus 4.0pt minus 4.0pt$

2\. The algebra $\mathbb{H}(\gamma_{1},\gamma_{2}),$ with
$\gamma_{1},\gamma_{2}<0$ or $\gamma_{1}\gamma_{2}<0$

In this situation, the algebra $\mathbb{H}(\gamma_{1},\gamma_{2})$ is
isomorphic with $\mathbb{H}(1,-1).$ We suppose first that
$\gamma_{1},\gamma_{2}<0$. An isomorphism between the algebra
$\mathbb{H}(\gamma_{1},\gamma_{2}),$ where $\gamma_{1},\gamma_{2}<0,$ and the
algebra $\mathbb{H}(1,-1)$ is given by the operator $B$ and its inverse
$B^{-1}$, where

$B:\quad e_{1}\mapsto i_{3}\sqrt{-\gamma_{1}},\quad e_{2}\mapsto
i_{2}\sqrt{-\gamma_{2}},\quad e_{3}\mapsto i_{1}\sqrt{\gamma_{1}\gamma_{2}}.$

For $\gamma_{1}>0,\gamma_{2}<0$, an isomorphism between the algebra
$\mathbb{H}(\gamma_{1},\gamma_{2})$ and the algebra $\mathbb{H}(1,-1)$ is
given by the operator $C$ and its inverse $C^{-1}$, where

$C:\quad e_{1}\mapsto i_{1}\sqrt{\gamma_{1}},\quad e_{2}\mapsto
i_{2}\sqrt{-\gamma_{2}},\quad e_{3}\mapsto i_{3}\sqrt{-\gamma_{1}\gamma_{2}}.$

For $\gamma_{1}<0,\gamma_{2}>0,$ an isomorphism between the algebra
$\mathbb{H}(\gamma_{1},\gamma_{2})$ and the algebra $\mathbb{H}(1,-1)$ is
given by the operator $D$ and its inverse $D^{-1}$, where

$D:\quad e_{1}\mapsto i_{3}\sqrt{-\gamma_{1}},\quad e_{2}\mapsto
i_{1}\sqrt{\gamma_{2}},\quad e_{3}\mapsto i_{2}\sqrt{-\gamma_{1}\gamma_{2}}.$

The properties of the operators $B,B^{-1},C,C^{-1},D,D^{-1}$ are similarly
with the properties of the operator $A$.

Since each algebra $\mathbb{H}(\gamma_{1},\gamma_{2})$ is isomorphic with
algebra of quaternions or coquaternions, it results that the above operators
provide us a simple way to generalize known results in these two algebras to
generalized quaternion algebra.

3\. Application to the algebraic equations

Let
$x=x_{0}+x_{1}e_{1}+x_{2}e_{3}+x_{3}e_{3}\in\mathbb{H}(\gamma_{1},\gamma_{2})$
and let
$f:\mathbb{H}(\gamma_{1},\gamma_{2})\rightarrow\mathbb{H}(\gamma_{1},\gamma_{2})$
be a continuous function of the form
$f(x)=f_{0}(x_{0},x_{1},x_{2},x_{3})+f_{1}(x_{0},x_{1},x_{2},x_{3})e_{1}+f_{2}(x_{0},x_{1},x_{2},x_{3})e_{2}+f_{3}(x_{0},x_{1},x_{2},x_{3})e_{3}$.
Let $F$ be the one of the operators $A,B,C$ or $D$, depending on the signs of
$\gamma_{1}$ and $\gamma_{2}$. We define the operator $\mathfrak{F}$ which for
any continuous function $f,$ taking values in
$\mathbb{H}(\gamma_{1},\gamma_{2}),$ maps it in the continuous function
$\mathfrak{F}f,$ taking values in $\mathbb{H}(1,1)$ or $\mathbb{H}(1,-1)$ by
the rule:

$\mathfrak{F}f:=f_{0}+f_{1}\,F(e_{1})+f_{2}\,F(e_{2})+f_{3}\,F(e_{3}).\vskip
6.0pt plus 2.0pt minus 2.0pt$

Theorem 3.1. Let $x^{0}\in\mathbb{H}(\gamma_{1},\gamma_{2})$ be a root of the
equation $f(x)=0$ in $\mathbb{H}(\gamma_{1},\gamma_{2})$. Then $F(x^{0})$ is a
root of the equation $\mathfrak{F}$$f\big{(}F(x)\big{)}=0$ in
$\mathbb{H}(1,1)$ or $\mathbb{H}(1,-1)$, depending on the signs of
$\gamma_{1}$ and $\gamma_{2}$. The converse is also true.

Proof. Let $\gamma_{1},\gamma_{2}>0$. Applying operator $A$ to the equality
$f(x^{0})=0$ and using the continuity of $A$, we obtain

$A\big{(}f(x^{0})\big{)}=Af\big{(}A(x^{0})\big{)}=A(0)=0.\ $

To prove the converse statement we apply the operator $A^{-1}$ to the equality
$f(x^{0})=0$. The remaining cases can be proved similarly. $\Box$

Therefore, all results from quaternionic equations and from coquaternionic
equations can be generalized in $\mathbb{H}(\gamma_{1},\gamma_{2}).$

It is known that any polynomial of degree $n$ with coefficients in a field $K$
has at most $n$ roots in $K$. If the coefficients are in $\mathbb{H}(1,1),$
the situation is different. For the real division quaternion algebra over the
real field, there is a kind of a fundamental theorem of algebra: If a
polynomial has only one term of the greatest degree, then it has at least one
root in $\mathbb{H}(1,1)$. ([21], Theorem 65; [4], Theorem 1).

We consider the polynomial of degree $n$ of the form

$f(x)=a_{0}xa_{1}x\ldots a_{n-1}xa_{n}+\varphi(x),$ (3.1.)

where $x,\,a_{0},\,a_{1},\ldots,a_{n-1},\,a_{n}\in\mathbb{H}(1,1),$ with
$a_{\ell}\neq 0$ for $\ell\in\\{0,1,\ldots,n\\}$ and $\varphi(x)$ is a sum of
a finite number of monomials of the form $b_{0}xb_{1}x\ldots b_{t-1}xb_{t}$
where $t<n$. From the above, it results that the equation $f(x)=0$ has at
least one root. Applying operator $A^{-1}$ to this last equality, the equation
$\big{(}A^{-1}f\big{)}\big{(}A^{-1}(x)\big{)}=0,$ with
$x=x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3},$ has at least one root in
$\mathbb{H}(\gamma_{1},\gamma_{2})$. Therefore, we proved the following
result:

Theorem 3.2. In the generalized quaternion algebra
$\mathbb{H}(\gamma_{1},\gamma_{2})$ with $\gamma_{1},\gamma_{2}>0$ any
polynomial of the form $(3.1)$ has at least one root.$\,\Box$

In the following, we consider the equation $x^{2}+ax+b=0,$ where
$\mathbb{H}(\gamma_{1},\gamma_{2})$ with $\gamma_{1},\gamma_{2}>0$. We say
that a root
$x_{0}=a_{0}+a_{1}i_{1}+a_{2}i_{2}+a_{3}i_{3}\in\mathbb{H}(\gamma_{1},\gamma_{2})$,
$\gamma_{1},\gamma_{2}>0$ is a ellipsoidal root if every element
$x_{1}\in\mathbb{H}(\gamma_{1},\gamma_{2})$ of the form
$x_{1}=a_{0}+b_{1}i_{1}+b_{2}i_{2}+b_{3}i_{3}$ such that
$\gamma_{1}a_{1}^{2}+\gamma_{2}a_{2}^{2}+\gamma_{1}\gamma_{2}a_{3}^{2}=\gamma_{1}b_{1}^{2}+\gamma_{2}b_{2}^{2}+\gamma_{1}\gamma_{2}b_{3}^{2}$
is also a root of this equation.

Using Theorem 3 from [12] and Theorem 3.1, we just proved the following
theorem:

Theorem 3.3. In the generalized quaternion algebra
$\mathbb{H}(\gamma_{1},\gamma_{2})$ with $\gamma_{1},\gamma_{2}>0$ the
equation $x^{2}+ax+b=0$ has ellipsoidal root if and only if $a$ and $b$ are
real numbers and $a^{2}-4b<0$. $\Box$

More results on the structure of roots of the quadratic quaternionic equations
can be found in see [22], [13], [14], [15].

In the following, we apply the above results to the coquaternion algebra. We
will consider one of the three cases: $\gamma_{1},\gamma_{2}<0$;
$\gamma_{1}<0,\gamma_{2}>0$; $\gamma_{1}>0,\gamma_{2}<0$. We say that a root
$x_{0}=a_{0}+a_{1}i_{1}+a_{2}i_{2}+a_{3}i_{3}\in\mathbb{H}(\gamma_{1},\gamma_{2})$
is a hyperboloidal root if every element
$x_{1}\in\mathbb{H}(\gamma_{1},\gamma_{2})$ of the form
$x_{1}=a_{0}+b_{1}i_{1}+b_{2}i_{2}+b_{3}i_{3}$ such that
$\gamma_{1}a_{1}^{2}+\gamma_{2}a_{2}^{2}+\gamma_{1}\gamma_{2}a_{3}^{2}=\gamma_{1}b_{1}^{2}+\gamma_{2}b_{2}^{2}+\gamma_{1}\gamma_{2}b_{3}^{2}$
is also a root of this equation. Using Theorem 2.5 of [17] and the above
Theorem 3.1, we proved:

Theorem 3.4. Suppose that coefficients $r_{k}$, $k=0,1,\ldots,n$ of the
polynomial equation $r_{n}x^{n}+r_{n-1}x^{n-1}+\ldots+r_{0}=0$,
$x\in\mathbb{H}(\gamma_{1},\gamma_{2})$ are real numbers. Therefore all its
non-real roots are hyperboloidal. $\Box$

Solutions of linear equations in coquaternionic algebra can be found in [9],
[5] and solutions of linear equations in quaternion algebra can be found, for
example, in [23], [20].

Let $x=x_{0}+x_{1}i+x_{2}j+x_{3}k\in\mathbb{H}(1,1)$,
$x_{0},x_{1},x_{2},x_{3}\in\mathbb{R}.$ We denoted by

$\widetilde{x}=A^{-1}(x)=x_{0}+x_{1}\frac{e_{1}}{\sqrt{\gamma_{1}}}+x_{2}e_{2}\frac{e_{2}}{\sqrt{\gamma_{2}}}+x_{3}e_{3}\frac{e_{1}}{\sqrt{\gamma_{1}\gamma_{2}}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$

Example 3.5. Consider the general polynomial equation in $\mathbb{H}(1,1):$

$\underset{p=1}{\overset{n}{\sum}}\left(\underset{l=1}{\overset{m_{p}}{\sum}}a_{p,l,1}xa_{p,l,2}x...a_{p,l,p}xa_{p,l,p+1}\right)+c=0.$

This equation is ”equivalent” to the following equation in
$\mathbb{H}(\gamma_{1},\gamma_{2}),$ with $\gamma_{1},\gamma_{2}>0$:

$\underset{p=1}{\overset{n}{\sum}}\left(\underset{l=1}{\overset{m_{p}}{\sum}}\widetilde{a}_{p,l,1}\widetilde{x}\,\widetilde{a}_{p,l,2}\,\widetilde{x}\ldots\widetilde{a}_{p,l,p}\,\widetilde{x}\,\widetilde{a}_{p,l,p+1}\right)+\widetilde{c}=0.$

Therefore, the algebraic equations $f(x)=0,$ with $f$ continuous, in usual
quaternions $\mathbb{H}(1,1)$ can be reduced to the similar equations in all
algebras $\mathbb{H}(\gamma_{1},\gamma_{2})$ for $\gamma_{1},\gamma_{2}>0$ and
vice versa. A similar result can be found for different signs of $\gamma_{1}$
and $\gamma_{2}$.

4\. De Moivre’s formula

In the following, we will use some ideas and notations from [3]. Let
$q=q_{0}+q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{3}\in\mathbb{H}(\gamma_{1},\gamma_{2})$,
$\gamma_{1},\gamma_{2}>0$, $q_{0},q_{1},q_{2},q_{3}\in\mathbb{R}$ and

$|q|=\sqrt{q_{0}^{2}+\gamma_{1}q_{1}^{2}+\gamma_{2}q_{2}^{2}+\gamma_{1}\gamma_{2}q_{3}^{2}}.$

Consider the sets

$\mathcal{S}_{G}^{3}=\\{q\in\mathbb{H}(\gamma_{1},\gamma_{2}),\gamma_{1},\gamma_{2}>0:\,|q|=1\\},\newline
$
$\mathcal{S}_{G}^{2}=\\{q\in\mathbb{H}\left(\gamma_{1},\gamma_{2}\right),\gamma_{1},\gamma_{2}>0:\,q_{0}=0,|q|=1\\}.$

Any $q\in\mathcal{S}_{G}^{3}$ can be expressed as
$q=\cos\theta+\varepsilon\sin\theta,$ where

$\cos\theta=q_{0},\;\;\varepsilon=\frac{q_{1}e_{1}+q_{2}e_{2}+q_{3}e_{3}}{\sqrt{\gamma_{1}q_{1}^{2}+\gamma_{2}q_{2}^{2}+\gamma_{1}\gamma_{2}q_{3}^{2}}}.$

Using Proposition 2 from [3] and applying the operator $A^{-1}$ we will find
De Moivre’s formula for
$\mathbb{H}\left(\gamma_{1},\gamma_{2}\right),\gamma_{1},\gamma_{2}>0$.

Theorem 4.1. Let
$q=\cos\theta+\varepsilon\sin\theta\in$$\mathcal{S}$${}_{G}^{3}$,
$\theta\in\mathbb{R}$. Then $q^{n}=\cos n\theta+\varepsilon\sin n\theta$ for
every integer $n$.

Theorem 4.1 is the same with Theorem 7 from the paper [11], obtained with
another proof.$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Using Corollary 3 from [3] and Theorem 3.1, we obtain the next statement.

Proposition 4.2. i) In $\mathbb{H}(\gamma_{1},\gamma_{2}),$
$\gamma_{1},\gamma_{2}>0$ the equation $x^{n}=1$ whit $n$ integer and $n\geq
3$ has infinity of roots, namely

$q=\cos\frac{2\pi}{n}+\varepsilon\sin\frac{2\pi}{n}\in\mathcal{S}_{G}^{3},\
\varepsilon\in\mathcal{S}_{G}^{2}.$

ii) In $\mathbb{H}(\gamma_{1},\gamma_{2}),$ $\gamma_{1},\gamma_{2}>0$ the
equation $x^{n}=a$, $n\in\mathbb{N}$, $a\in\mathbb{R}$ has infinity of roots,
namely $\sqrt[n]{a}\,q,$ where
$q=\cos\frac{2\pi}{n}+\varepsilon\sin\frac{2\pi}{n}\in\mathcal{S}_{G}^{3},$
with $\varepsilon\in$$\mathcal{S}_{G}^{2}.$ If $n$ is even it is necessary
that $a>0$.$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

5.

De Moivre’s formula and Euler’s formula for octonions

In the following, we will generalize in a natural way De Moivre formula and
Euler’s formula for the division octonion algebra
$\mathbb{O}\left(1,1,1\right).$ For this, we will use some ideas and notations
from [3]. We consider the sets

$\mathcal{S}^{3}=\\{a\in\mathbb{O}(1,1,1):\;N(a)=1\\},$
$\mathcal{S}_{G}^{3}=\\{a\in\mathbb{O}(\alpha,\beta,\gamma):\;N(a)=1\\},$
$\mathcal{S}^{2}=\\{a\in\mathbb{O}(1,1,1):\;t(a)=0,N\left(a\right)=1\\}.$

$\mathcal{S}_{G}^{2}=\\{a\in\mathbb{O}(1,1,1):\;t(a)=0,N\left(a\right)=1\\}.$

We remark that for all elements $a\in\mathcal{S}^{2},$ we have $a^{2}=-1.$ Let
$a\in\mathcal{S}^{3},a=a_{0}+a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7}.$
This element can be write under the form

$a=\cos\lambda+w\sin\lambda,$

where $\cos\lambda=a_{0}~{}$ and

$w=\frac{a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7}}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}+a_{7}^{2}}}=$
$=\frac{a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7}}{\sqrt{1-a_{0}^{2}}}.$

Since $w^{2}=-1,$ we obtain the following Euler’s formula:

$\displaystyle e^{\lambda w}$ $\displaystyle=$
$\displaystyle\overset{\infty}{\underset{i=1}{\sum}}\frac{\left(\lambda
w\right)^{n}}{n!}=\overset{\infty}{\underset{i=1}{\sum}}\frac{\left(-1\right)^{n}\lambda^{2n}}{\left(2n\right)!}+w\overset{\infty}{\underset{i=1}{\sum}}\frac{\left(-1\right)^{n-1}\lambda^{2n-1}}{\left(2n-1\right)!}=$
$\displaystyle=$ $\displaystyle\cos\lambda+w\sin\lambda.$

Proposition 5.1. The cosinus function is constant for all elements in
$\mathcal{S}^{2}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof. Indeed, $\cos
w=\overset{\infty}{\underset{i=1}{\sum}}\frac{\left(-1\right)^{n}\lambda^{2n}}{\left(2n\right)!}=\cos
i.\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proposition 5.2. For $w\in\mathcal{S}^{2},$ we have
$(\cos\lambda_{1}+w\sin\lambda_{1})(\cos\lambda_{2}+w\sin\lambda_{2})=\cos(\lambda_{1}+\lambda_{2})+w\sin(\lambda_{1}+\lambda_{2}).\vskip
6.0pt plus 2.0pt minus 2.0pt$

Proof. By straightforward calculations $\Box\vskip 6.0pt plus 2.0pt minus
2.0pt$

Proposition 5.3. (De Moivre formula for octonions) With the above notations,
we have that

$a^{n}=e^{n\lambda w}=\left(\cos\lambda+w\sin\lambda\right)^{n}=\cos
n\lambda+w\sin n\lambda,$

where $a\in\mathcal{S}^{3},$ $n\in\mathbb{Z}$ and $\lambda\in\mathbb{R}.\vskip
6.0pt plus 2.0pt minus 2.0pt$

Proof. For $n>0,$ by induction. We obtain

$\displaystyle a^{n+1}$ $\displaystyle=$
$\displaystyle\left(\cos\lambda+w\sin\lambda\right)^{n+1}=$ $\displaystyle=$
$\displaystyle\left(\cos\lambda+w\sin\lambda\right)^{n}\left(\cos\lambda+w\sin\lambda\right)=$
$\displaystyle=$ $\displaystyle(\cos n\lambda+w\sin
n\lambda)\left(\cos\lambda+w\sin\lambda\right)=$ $\displaystyle=$
$\displaystyle\cos(n+1)\lambda+w\sin(n+1)\lambda.$

Since $a^{-1}=\cos\lambda-w\sin\lambda=\cos(-\lambda)+w\sin(-\lambda),$ it
results the asked formula for all $n\in\mathbb{Z}.\Box\vskip 6.0pt plus 2.0pt
minus 2.0pt$

Remark 5.4. It is known that any polynomial of degree $n$ with coefficients in
a field $K$ has at most $n$ roots in $K$. If the coefficients are in
$\mathbb{O}\left(1,1,1\right)$ there is a kind of a fundamental theorem of
algebra: If a polynomial has only one term of the greatest degree, then it has
at least one root in $\mathbb{O}\left(1,1,1\right)$ (see [21], Theorem 65).

Theorem 5.5. Equation $x^{n}=a,~{}$where
$a\in\mathbb{O}\left(1,1,1\right)\setminus\mathbb{R},$ has $n$ roots.

Proof. The octonion $a$ can be written under the form
$a=\sqrt{N\left(a\right)}\frac{a}{\sqrt{N\left(a\right)}}.$ The octonion
$b=\frac{a}{\sqrt{N\left(a\right)}}$ is in $\mathcal{S}^{3},$ then we can find
the elements $w\in\mathcal{S}^{2}$ and $\lambda\in\mathbb{R}$ such that
$b=\cos\lambda+w\sin\lambda.$ From Proposition 5.3, we have that the solutions
of the above equation are
$x_{r}=\sqrt[n]{Q}\left(\cos\frac{\lambda+2r\pi}{n}+w\sin\frac{\lambda+2r\pi}{n}\right),$
where $Q=\sqrt{N\left(a\right)}$ and $\ r\in\\{0,1,...,n-1\\}.\Box\vskip 6.0pt
plus 2.0pt minus 2.0pt$

Corollary 5.6. If $a\in\mathbb{R},$ therefore the equation $x^{n}=a$ has an
infinity of roots.

Proof. Indeed, if $a\in\mathbb{R}$, we can write $a=a\cdot 1=a\left(\cos
2\pi+w\sin 2\pi\right),$ where $w\in\mathcal{S}^{2}$ is an arbitrary element.
$\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Remark 5.7. The rotation of the octonion $x\in\mathbb{O}\left(1,1,1\right)$ on
the angle $\lambda$ around the unit vector $w\in\mathcal{S}^{2}$ is defined by
the formula

$x^{r}=\overline{u}xu,$

where $u\in\mathcal{S}^{3},u=\cos\frac{\lambda}{2}+w\sin\frac{\lambda}{2}$ and
$\overline{u}=\cos\frac{\lambda}{2}-w\sin\frac{\lambda}{2}.$

Using the form $x=x_{0}\cdot 1+\overrightarrow{x},$ $y=y_{0}\cdot
1+\overrightarrow{y}\ \ \ $for the octonions
$x,y\in\mathbb{O}\left(1,1,1\right),$ we obtain the following expression for
the product of two octonions:

$\displaystyle xy$ $\displaystyle=$ $\displaystyle\left(x_{0}\cdot
1+\overrightarrow{x}\right)\left(y_{0}\cdot 1+\overrightarrow{y}\right)=$
$\displaystyle=$ $\displaystyle x_{0}y_{0}\cdot
1+x_{0}\overrightarrow{y}+y_{0}\overrightarrow{x}+<\overrightarrow{x},\overrightarrow{y}>+\overrightarrow{x}\times\overrightarrow{y},$

where $<\overrightarrow{x},\overrightarrow{y}>$ is the inner product of two
octonionic-vector and $\overrightarrow{x}\times\overrightarrow{y}$ is the
cross product. From here, we obtain that $x^{r}=\overline{u}xu=$
$=\left(\cos\frac{\lambda}{2}-w\sin\frac{\lambda}{2}\right)\left(x_{0}\cdot
1+\overrightarrow{x}\right)\left(\cos\frac{\lambda}{2}+w\sin\frac{\lambda}{2}\right)=$
$=x_{0}+\overrightarrow{x}\cos\lambda-w<w,\overrightarrow{x}>\left(1-\cos\lambda\right)-\left(w\times\overrightarrow{x}\right)\sin\lambda.$
It results that that rotation does not transform the octonion-scalar part, but
the octonion-vector part $\overrightarrow{x}\ $is rotated on the angle
$\lambda$ around $w$.

6\. An isomorphism between the algebras $\mathbb{O}(\alpha,\beta,\gamma)$ and
some its applications

In the following, we will consider the generalized real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ and the algebras $\mathbb{O}(1,1,1)$ and
$\mathbb{O}(1,1,-1)$. Let $\\{1,f_{1},\ldots,f_{7}\\}$ be a basis in
$\mathbb{O}(\alpha,\beta,\gamma)$, and
$\\{1,\widetilde{f_{1}},\ldots,\widetilde{f_{7}}\\}$ be the canonical basis in
$\mathbb{O}(1,1,1)$, and $\\{1,\widehat{f_{1}},\ldots,\widehat{f_{7}}\\}$ be
the canonical basis in $\mathbb{O}(1,1,-1)$.

We prove that the algebra $\mathbb{O}(\alpha,\beta,\gamma)$ with
$\alpha,\beta,\gamma\in\mathbb{R}\setminus\\{0\\}$ is isomorphic with algebra
$\mathbb{O}(1,1,1)$ or $\mathbb{O}(1,1,-1)$ and indicate the formulae to pass
from one basis to another basis. Thus, if $\alpha,\beta,\gamma>0$ then the
real octonion algebra $\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with
algebra $\mathbb{O}(1,1,1)$ and this isomorphism is given by the relations:

$\displaystyle A_{1}:\quad$ $\displaystyle
f_{1}\mapsto\widetilde{f_{1}}\sqrt{\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widetilde{f_{2}}\sqrt{\beta},\;\;f_{3}\mapsto\widetilde{f_{3}}\sqrt{\alpha\beta},$
$\displaystyle f_{4}\mapsto\widetilde{f_{4}}\sqrt{\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widetilde{f_{5}}\sqrt{\alpha\gamma},\;\;f_{6}\mapsto\widetilde{f_{6}}\sqrt{\beta\gamma},\;\;f_{7}\mapsto\widetilde{f_{7}}\sqrt{\alpha\beta\gamma}\,.$

If $\alpha,\beta>0,\gamma<0,$ then the real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with algebra
$\mathbb{O}(1,1,-1)$ and this isomorphism is given by the relations:

$\displaystyle A_{2}:\quad$ $\displaystyle
f_{1}\mapsto\widehat{f_{1}}\sqrt{\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widehat{f_{2}}\sqrt{\beta},\;\;f_{3}\mapsto\widehat{f_{3}}\sqrt{\alpha\beta},$
$\displaystyle f_{4}\mapsto\widehat{f_{4}}\sqrt{-\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widehat{f_{5}}\sqrt{-\alpha\gamma},\;\;f_{6}\mapsto\widehat{f_{6}}\sqrt{-\beta\gamma},\;\;f_{7}\mapsto\widehat{f_{7}}\sqrt{-\alpha\beta\gamma}\,.$

If $\alpha,\gamma>0,\beta<0,$ then the real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with algebra
$\mathbb{O}(1,1,-1)$ and this isomorphism is given by the relations:

$\displaystyle A_{3}:\quad$ $\displaystyle
f_{1}\mapsto\widehat{f_{1}}\sqrt{\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widehat{f_{4}}\sqrt{-\beta},\;\;f_{3}\mapsto\widehat{f_{5}}\sqrt{-\alpha\beta},$
$\displaystyle f_{4}\mapsto\widehat{f_{2}}\sqrt{\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widehat{f_{3}}\sqrt{\alpha\gamma},\;\;f_{6}\mapsto\widehat{f_{6}}\sqrt{-\beta\gamma},\;\;f_{7}\mapsto\widehat{f_{7}}\sqrt{-\alpha\beta\gamma}\,.$

If $\alpha>0,\beta,\gamma<0,$ then the real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with algebra
$\mathbb{O}(1,1,-1)$ and this isomorphism is given by the relations:

$\displaystyle A_{4}:\quad$ $\displaystyle
f_{1}\mapsto\widehat{f_{1}}\sqrt{\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widehat{f_{4}}\sqrt{-\beta},\;\;f_{3}\mapsto\widehat{f_{5}}\sqrt{-\alpha\beta},$
$\displaystyle f_{4}\mapsto\widehat{f_{6}}\sqrt{-\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widehat{f_{7}}\sqrt{-\alpha\gamma},\;\;f_{6}\mapsto\widehat{f_{2}}\sqrt{\beta\gamma},\;\;f_{7}\mapsto\widehat{f_{3}}\sqrt{\alpha\beta\gamma}\,.$

If $\alpha<0,\beta,\gamma>0,$ then the real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with algebra
$\mathbb{O}(1,1,-1)$ and this isomorphism is given by the relations:

$\displaystyle A_{5}:\quad$ $\displaystyle
f_{1}\mapsto\widehat{f_{4}}\sqrt{-\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widehat{f_{1}}\sqrt{\beta},\;\;f_{3}\mapsto\widehat{f_{5}}\sqrt{-\alpha\beta},$
$\displaystyle f_{4}\mapsto\widehat{f_{2}}\sqrt{\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widehat{f_{6}}\sqrt{-\alpha\gamma},\;\;f_{6}\mapsto\widehat{f_{3}}\sqrt{\beta\gamma},\;\;f_{7}\mapsto\widehat{f_{7}}\sqrt{-\alpha\beta\gamma}\,.$

If $\alpha,\gamma<0,\beta>0,$ then real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with algebra
$\mathbb{O}(1,1,-1)$ and this isomorphism is given by the relations:

$\displaystyle A_{6}:\quad$ $\displaystyle
f_{1}\mapsto\widehat{f_{4}}\sqrt{-\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widehat{f_{1}}\sqrt{\beta},\;\;f_{3}\mapsto\widehat{f_{5}}\sqrt{-\alpha\beta},$
$\displaystyle f_{4}\mapsto\widehat{f_{6}}\sqrt{-\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widehat{f_{2}}\sqrt{\alpha\gamma},\;\;f_{6}\mapsto\widehat{f_{7}}\sqrt{-\beta\gamma},\;\;f_{7}\mapsto\widehat{f_{3}}\sqrt{\alpha\beta\gamma}\,.$

If $\alpha,\beta<0,\gamma>0,$ then the real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with algebra
$\mathbb{O}(1,1,-1)$ and this isomorphism is given by the relations:

$\displaystyle A_{7}:\quad$ $\displaystyle
f_{1}\mapsto\widehat{f_{4}}\sqrt{-\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widehat{f_{5}}\sqrt{-\beta},\;\;f_{3}\mapsto\widehat{f_{1}}\sqrt{\alpha\beta},$
$\displaystyle f_{4}\mapsto\widehat{f_{2}}\sqrt{\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widehat{f_{6}}\sqrt{-\alpha\gamma},\;\;f_{6}\mapsto\widehat{f_{7}}\sqrt{-\beta\gamma},\;\;f_{7}\mapsto\widehat{f_{3}}\sqrt{\alpha\beta\gamma}\,.$

If $\alpha,\beta,\gamma<0,$ then the real octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ is isomorphic with algebra
$\mathbb{O}(1,1,-1)$ and this isomorphism is given by the relations:

$\displaystyle A_{8}:\quad$ $\displaystyle
f_{1}\mapsto\widehat{f_{4}}\sqrt{-\alpha},$
$\displaystyle\;\;f_{2}\mapsto\widehat{f_{5}}\sqrt{-\beta},\;\;f_{3}\mapsto\widehat{f_{1}}\sqrt{\alpha\beta},$
$\displaystyle f_{4}\mapsto\widehat{f_{6}}\sqrt{-\gamma},$
$\displaystyle\;\;f_{5}\mapsto\widehat{f_{2}}\sqrt{\alpha\gamma},\;\;f_{6}\mapsto\widehat{f_{3}}\sqrt{\beta\gamma},\;\;f_{7}\mapsto\widehat{f_{7}}\sqrt{-\alpha\beta\gamma}\,.$

It is easy to prove that the operators $A_{k}$, $k=\overline{1,8}$ is additive
and multiplicative. The following statement can be proved completely analogous
as Proposition 1.1.

Proposition 6.1. The operators $A_{k}$, $k=\overline{1,8}$ are continuous and
have norm  $1$.$\Box$

Let
$x=x_{0}+\sum\limits_{k=1}^{7}x_{k}f_{k}\in\mathbb{O}(\alpha,\beta,\gamma)$
and let
$g:\mathbb{O}(\alpha,\beta,\gamma)\rightarrow\mathbb{O}(\alpha,\beta,\gamma)$
be a continuous function of the form
$g(x)=g_{0}(x_{0},\ldots,x_{7})+\sum\limits_{k=1}^{7}g_{k}(x_{0},\ldots,x_{7})f_{k}$.
Let $L$ be one of the operators $A_{k}$, $k=\overline{1,8}$, depending on the
signs of $\alpha,\beta$ and $\gamma$. We define the operator $\mathfrak{L}$ by
the rule:

$\mathfrak{L}g:=f_{0}+\sum\limits_{k=1}^{7}g_{k}\,L(f_{k}).$

The operator $\mathfrak{L}$ for any continuous function $g,$ taking values in
$\mathbb{O}(\alpha,\beta,\gamma),$ maps it in the continuous function
$\mathfrak{L}g,$ taking values in $\mathbb{O}(1,1,1)$ or $\mathbb{O}(1,1,-1),$

The following statement can be analogously proved as Theorem 3.1.

Theorem 6.2. Let $x^{0}\in\mathbb{O}(\alpha,\beta,\gamma),$ be a root of the
equation $g(x)=0$ in $\mathbb{O}(\alpha,\beta,\gamma)$. Then $L(x^{0})$ is a
root of the equation $\mathfrak{L}$$g\big{(}L(x)\big{)}=0$ in
$\mathbb{O}(1,1,1)$ or $\mathbb{O}(1,1,-1)$, depending on the signs of
$\alpha,\beta,$ and $\gamma$. The converse is also true. $\Box$

Thus, the study of algebraic equations in an arbitrary algebra
$\mathbb{O}(\alpha,\beta,\gamma)$ with
$\alpha,\beta,\gamma\in\mathbb{R}\setminus\\{0\\}$ was reduced to study of the
corresponding algebraic equation in one of the following two algebras:
division octonion algebra $\mathbb{O}(1,1,1)$ or algebra $\mathbb{O}(1,1,-1)$.

Using the above notations, we can prove the following theorem.

Theorem 6.3. Equation $x^{n}=a,$ where
$a\in\mathbb{O}(\alpha,\beta,\gamma)\setminus\mathbb{R},\alpha,\beta,\gamma>0,$
has $n$ roots.

Proof. The octonion $b=\frac{a}{\sqrt{N\left(a\right)}}$ is in
$\mathcal{S}_{G}^{3},$ then there are
$w\in\mathcal{S}_{G}^{2},\,w=A_{1}^{-1}\left(\widetilde{w}\right),\widetilde{w}\in\mathcal{S}^{2}$
and $\lambda\in\mathbb{R}$ such that $b=\cos\lambda+\widetilde{w}\sin\lambda.$
From Proposition 5.3, we have that the solutions of the above equation are
$x_{r}=A_{1}^{-1}\left(\widetilde{x}_{r}\right)=\sqrt[2n]{N\left(a\right)}\left(\cos\frac{\lambda+2r\pi}{n}+\widetilde{w}\sin\frac{\lambda+2r\pi}{n}\right),$
where $r\in\\{0,1,\ldots,n-1\\}$ and $\widetilde{x}_{r}$ is a solution of the
equation $\widetilde{x}^{n}=\widetilde{a}$ in $\mathbb{O}(1,1,1).\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

Remark 6.4.  Using the operator $A_{1}^{-1},~{}$the rotation of the octonion
$x\in\mathbb{O}(\alpha,\beta,\gamma)$ on the angle $\lambda$ around the unit
vector $w\in\mathcal{S}_{G}^{2}$ is defined by the formula

$x^{r}=\overline{u}xu,$

where
$u\in\mathcal{S}_{G}^{3},w\in\mathcal{S}_{G}^{2},~{}u=\cos\frac{\lambda}{2}+w\sin\frac{\lambda}{2}$
and $\overline{u}=\cos\frac{\lambda}{2}-w\sin\frac{\lambda}{2}.$

By straightforward calculations, it results that rotation does not transform
the octonion-scalar part, but the octonion-vector part $\overrightarrow{x}\
$is rotated on the angle $\lambda$ around $w$.

Example 6.5. 1) Let $a\in\mathcal{S}^{3},$
$a=\frac{\sqrt{2}}{2}+\frac{1}{\sqrt{14}}\widetilde{f}_{1}+\frac{1}{\sqrt{14}}\widetilde{f}_{2}+\frac{1}{\sqrt{14}}\widetilde{f}_{3}+\frac{1}{\sqrt{14}}\widetilde{f}_{4}+\frac{1}{\sqrt{14}}\widetilde{f}_{5}+\frac{1}{\sqrt{14}}\widetilde{f}_{6}+\frac{1}{\sqrt{14}}\widetilde{f}_{7},$
we have $\cos\lambda=\frac{\sqrt{2}}{2},\sin\lambda=\frac{\sqrt{2}}{2}.$ It
results that $a=\cos\frac{\pi}{4}+v\sin\frac{\pi}{4},$ where
$v=\frac{1}{\sqrt{7}}(\widetilde{f}_{1}+\widetilde{f}_{2}+\widetilde{f}_{3}+\widetilde{f}_{4}+\widetilde{f}_{5}+\widetilde{f}_{6}+\widetilde{f}_{7}).$
The vector $a$ corresponds to the rotation of the space $\mathbb{R}^{8}$on the
angle $\frac{\pi}{2}$ around the vector
$v=\left(\frac{1}{\sqrt{7}},\frac{1}{\sqrt{7}},\ldots,\frac{1}{\sqrt{7}}\right)\in\mathbb{R}^{7}$
written in the canonical basis.

2) In the algebra $\mathbb{O}(2,4,7),\,$ for the above element
$a\in\mathcal{S}^{3},$ we have
$b$=$A^{-1}\left(a\right)$=$\frac{\sqrt{2}}{2}$+$\frac{1}{2\sqrt{7}}f_{1}$+$\frac{1}{2\sqrt{14}}f_{2}$+$\frac{1}{7\sqrt{2}}f_{3}$+$\frac{1}{4\sqrt{7}}f_{4}$+$\frac{1}{14}f_{5}$+$\frac{1}{14\sqrt{2}}f_{6}$+$\frac{1}{28}f_{7}\in\mathcal{S}_{G}^{3}$
and corresponds to the rotation of the space $\mathbb{R}^{8}$on the angle
$\frac{\pi}{2}$ around the vector
$v=\left(\frac{1}{2\sqrt{7}},\frac{1}{2\sqrt{14}},\frac{1}{7\sqrt{2}},\frac{1}{4\sqrt{7}},\frac{1}{14},\frac{1}{14\sqrt{2}},\frac{1}{28}\right)\in\mathbb{R}^{7}$written
in the basis $\\{f_{1},\ldots,f_{7}\\}.$

Case when $\alpha=\beta=1,\gamma=-1$

In this case, the octonion algebra $\mathbb{O}(1,1,-1)$ is not a division
algebra (is a split algebra). The norm of an octonion
$a\in\mathbb{O}(1,1,-1),$
$a=a_{0}+a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7},$
in this situation, can be positive, zero or negative. In the following, we
used definitions and propositions obtained for the split quaternions as in
[16] to generalized them to similar results for the split octonions. A split
octonion is called spacelike, timelike or lightlike if
$N\left(a\right)<0,N\left(a\right)>0$ or $N\left(a\right)=0.$ If
$N\left(a\right)=1,$ then $\ a$ is called the unit split octonion.

Spacelike octonions

Let $a\in\mathbb{O}(1,1,-1)$ such that $N\left(a\right)=-1,$ be a spacelike
octonion. For the octonion
$w=\frac{a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7}}{\sqrt{1+a_{0}^{2}}},$
we have $N\left(w\right)=$ $-1$ and $t\left(w\right)=0,$ therefore $w^{2}=1.$
Denoting $\sinh\lambda=a_{0}$ and
$\cosh\lambda=\sqrt{1+a_{0}^{2}},\lambda\in\mathbb{R},$ it results:

$a=e^{\lambda w}=\sinh\lambda+w\cosh\lambda.$

If $a\in\mathbb{O}(1,1,-1)$ with $N\left(a\right)<0,$ we have
$a=\sqrt{\left|N\left(a\right)\right|}(\sinh\lambda+w\cosh\lambda).$

Proposition 6.6. We have that
$a^{n}=(\sqrt{\left|N\left(a\right)\right|})^{n}(\sinh\lambda+w\cosh\lambda)$
for $n$ odd and
$a^{n}=(\sqrt{\left|N\left(a\right)\right|})^{n}(\cosh\lambda+w\sinh\lambda)$
for $n$ even. $\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Timelike octonions

Let $a\in\mathbb{O}(1,1,-1)$ such that $N\left(a\right)=1,$ be a timelike
octonion. If $1-a_{0}^{2}>0,$ for the octonion
$w=\frac{a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7}}{\sqrt{1-a_{0}^{2}}},$
we have $N\left(w\right)=$ $1$ and $t\left(w\right)=0,$ therefore $w^{2}=-1$
Denoting $\cos\lambda=a_{0}$ and
$\sin\lambda=\sqrt{1-a_{0}^{2}},\lambda\in\mathbb{R},$ it results:

Proposition 6.7. With the above notations, we have the Euler’s formula:

$a=e^{\lambda w}=\cos\lambda+w\sin\lambda.$

Proof. Indeed, $e^{\lambda
w}=\overset{\infty}{\underset{i=1}{\sum}}\frac{\left(\lambda
w\right)^{n}}{n!}$=$\overset{\infty}{\underset{i=1}{\sum}}\frac{\left(-1\right)^{n}\lambda^{2n}}{\left(2n\right)!}+w\overset{\infty}{\underset{i=1}{\sum}}\frac{\left(-1\right)^{n-1}\lambda^{2n-1}}{\left(2n-1\right)!}$=$\cos\lambda+w\sin\lambda.\Box$

If $a\in\mathbb{O}(1,1,-1)$ with $N\left(a\right)>0,$ it results
$a=\sqrt{N\left(a\right)}\left(\cos\lambda+w\sin\lambda\right).$

Proposition 6.8. We have that $a^{n}=(\sqrt{N\left(a\right)})^{n}(\cos
n\lambda+w\sin n\lambda).\Box$

Proposition 6.9.

1)If $a\in\mathbb{O}(1,1,-1),$ it results
$a^{n}=(\sqrt{N\left(a\right)})^{n}(\cos n\lambda+w\sin n\lambda).$

2) The equation $x^{n}=a$ has $n$ roots:
$\sqrt[2n]{\left|N\left(a\right)\right|}\left(\cosh\frac{\lambda}{n}+w\sinh\frac{\lambda}{n}\right).\Box$

If $1-a_{0}^{2}<0,$ we have
$w=\frac{a_{1}f_{1}+a_{2}f_{2}+a_{3}f_{3}+a_{4}f_{4}+a_{5}f_{5}+a_{6}f_{6}+a_{7}f_{7}}{\sqrt{a_{0}^{2}-1}},$
with $N\left(w\right)=$ $-1$ and $t\left(w\right)=0,$ therefore $w^{2}=1.$
Denoting $\cosh\lambda=a_{0}$ and
$\sinh\lambda=\sqrt{a_{0}^{2}-1},\lambda\in\mathbb{R},$ it results:

Proposition 6.10. With the above notations, we have Euler’s formula:

$a=e^{\lambda w}=\cosh\lambda+w\sinh\lambda.$

Proof. Indeed, $e^{\lambda
w}=\overset{\infty}{\underset{n=0}{\sum}}\frac{\left(\lambda
w\right)^{n}}{n!}$=$\overset{\infty}{\underset{n=0}{\sum}}\frac{\lambda^{2n}}{\left(2n\right)!}+w\overset{\infty}{\underset{n=0}{\sum}}\frac{\lambda^{2n+1}}{\left(2n+1\right)!}$=$\cosh\lambda+w\sinh\lambda.\Box$

If $a\in a\in\mathbb{O}(1,1,-1)$ with $N\left(a\right)<0,$ it results
$a=\sqrt{\left|N\left(a\right)\right|}(\cosh\lambda+w\sinh\lambda).$

Proposition 6.11.

1) If $a\in\mathbb{O}(1,1,-1),$ then
$a^{n}=(\sqrt{\left|N\left(a\right)\right|})^{n}(\cosh n\lambda+w\sinh
n\lambda).$

2) The equation $x^{n}=a$ has only one root:
$\sqrt[2n]{\left|N\left(a\right)\right|}\left(\cosh\frac{\lambda}{n}+w\sinh\frac{\lambda}{n}\right).\Box\vskip
6.0pt plus 2.0pt minus 2.0pt$

Remark 6. 12. Using the above technique, De Moivre’s formula and Euler’s
formula can be easy proved for the octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma),$ with
$\alpha,\beta,\gamma\in\mathbb{R}\setminus\\{0\\}$ such that
$\mathbb{O}(\alpha,\beta,\gamma)$ is split. Thus, the study of algebraic
equations in an arbitrary algebra $\mathbb{O}(\alpha,\beta,\gamma)$ with
$\alpha,\beta,\gamma\in\mathbb{R}\setminus\\{0\\}$ was reduced to study of the
corresponding algebraic equation in one of the following two algebras:
division octonion algebra $\mathbb{O}(1,1,1)$ or algebra $\mathbb{O}(1,1,-1)$.

Conclusion. In this paper, we used isomorphism between the real quaternion
algebras $\mathbb{H}(\gamma_{1},\gamma_{2})$ and $\mathbb{H}(1,1)$ or
$\mathbb{H}(1,-1)$ to reduced the study of some algebraic equations in an
arbitrary algebra $\mathbb{H}\left(\gamma_{1},\gamma_{2}\right)$ with
$\gamma_{1},\gamma_{2}\in\mathbb{R}\setminus\\{0\\}$ to study of the
corresponding algebraic equation in one of the following two algebras:
division quaternion algebra or coquaternion algebra. The same result was
obtained for the generalized octonion algebra
$\mathbb{O}(\alpha,\beta,\gamma)$. De Moivre’s formula in generalized
quaternion algebras and generalized octonion division algebras was proved
using this new method.

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Cristina FLAUT

Faculty of Mathematics and Computer Science, Ovidius University,

Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA

http://cristinaflaut.wikispaces.com/; http://www.univ-ovidius.ro/math/

e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Vitalii SHPAKIVSKYI

Department of Complex Analysis and Potential Theory

Institute of Mathematics of the National Academy of Sciences of Ukraine,

3, Tereshchenkivs’ka st., 01601 Kiev-4, UKRAINE

http://www.imath.kiev.ua/~complex/; e-mail<EMAIL_ADDRESS>