FERMILAB-PUB-23-414-CSAID current page.north east -2in -0.5in

Reweighting Monte Carlo Predictions and Automated Fragmentation Variations in
Pythia 8

Christian Bierlich1$\spadesuit$, Phil Ilten2$\dagger$, Tony Menzo2$\star$,
Stephen Mrenna2,3$\maltese$, Manuel Szewc2$\parallel$, Michael K.
Wilkinson2$\perp$, Ahmed Youssef2$\ddagger$, and Jure Zupan2$\mathsection$

1 Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden

2 Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,USA

3 Scientific Computing Division, Fermilab, Batavia, Illinois, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

This work reports on a method for uncertainty estimation in simulated
collider-event predictions. The method is based on a Monte Carlo-veto
algorithm, and extends previous work on uncertainty estimates in parton
showers by including uncertainty estimates for the Lund string-fragmentation
model. This method is advantageous from the perspective of simulation costs: a
single ensemble of generated events can be reinterpreted as though it was
obtained using a different set of input parameters, where each event now is
accompanied with a corresponding weight. This allows for a robust exploration
of the uncertainties arising from the choice of input model parameters,
without the need to rerun full simulation pipelines for each input parameter
choice. Such explorations are important when determining the sensitivities of
precision physics measurements. Accompanying code is available at
gitlab.com/uchep/mlhad-weights-validation.

###### Contents

  1. 1 Introduction
  2. 2 Method
    1. 2.1 Standard Accept-Reject Algorithm
    2. 2.2 Modified Accept-Reject Algorithm
    3. 2.3 Variation Details
  3. 3 Validation
    1. 3.1 Validation simulations
    2. 3.2 Timing
  4. 4 Conclusions

##  1 Introduction

Almost all collider tests of the Standard Model (SM) of particle physics rely
on predictions obtained using event generators [1, 2]. An important part of
these tests is the estimation of uncertainties on those predictions, which can
often be obtained by varying input parameters to the event generator. An
important and practical consideration is the efficiency of the algorithms used
for uncertainty estimations. Already, efficient reweighting methods exist for
the hard process and the parton shower [3, 4, 5, 6, 7] and various weighting
techniques are used to match and merge the hard process and parton shower,
although typically not in the context of variations [8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20]. Similar methods for estimating the uncertainties in
hadronization have, up to now, remained elusive. The standard procedure to
handle hadronization uncertainties, prior to this work, was to perform
repeated simulations with different sets of values for the relevant
hadronization model’s parameters, where the values are chosen such that the
model’s predictions remain compatible with the reference data [21, 22, 23]. In
this manuscript, we remedy this by introducing an efficient solution for
reweighting kinematic parameters of hadronization and provide an
implementation for the complete hadronization model of the Pythia 8 Monte
Carlo event generator.

The uncertainties in the prediction for the hard process, based on matrix
elements, are typically estimated by varying the factorization and
renormalization scales, although this does not capture, of course, our full
ignorance of the importance of missing, higher-order terms in the calculation.
Additional uncertainties in hard-process calculations can arise from the
choice of couplings and the parton distribution functions, including the scale
at which they are evaluated. An event weight can then be calculated from the
ratio of the hard process calculated with the modified choices to the baseline
ones. For parton shower uncertainties, the situation is more complicated
because the probability distributions must be evaluated many times and must
preserve unitarity. Here, a modification of the Monte-Carlo sampling algorithm
is necessary to account correctly for accepted-and-rejected trial emissions.

The standard procedure to handle hadronization uncertainties is to perform
repeated simulations with different sets of values for the relevant
hadronization model’s parameters, where the values are chosen such that the
model’s predictions remain compatible with the reference data. Statistical
comparisons can then be made on observables relevant to a particular analysis.
While this is straightforward, it is also computationally expensive,
especially if the predictions are further simulated at the detector level
(material interactions, detector response, etc.). It is advantageous to use
instead only one sample of events in the detector simulation, and then compute
relative probabilities for different hadronization parameter choices.

Conceptually, it is not difficult to calculate an alternative probability for
a given accepted hypothesis. In practice, it can be technically challenging to
re-organize an existing Monte Carlo method to do so. In this paper, we
describe a method to calculate relative probabilities for predictions based on
a veto algorithm and apply it to uncertainty estimates in simulations of
hadronization based on the Lund string fragmentation model [24]. The presented
method is similar to the one used previously for parton shower uncertainty
estimates [15, 3, 5, 6, 7], and is implemented here for kinematic-based
parameters of the string model, not flavor-based parameters.111The flavor
selector in Pythia 8 does not use the same accept-reject algorithm as for
kinematics, and is as such an entirely separate challenge. A key difference
between the parton shower and hadronization is that the parton shower uses a
veto algorithm; a no-emission probability up to a given scale $Q$ is
calculated, and then an emission is produced at that scale. For hadronization,
the scale of the next emission is always at the scale determined from the
previous emission. While both processes are Markov chains, this distinction
critically changes how the varied parameter uncertainties are propagated.

While this manuscript provides an efficient method to compute fragmentation
uncertainties in Pythia 8 specifically, it is also applicable more broadly. It
could, for instance, be applied to cluster hadronization-models [25, 26, 27,
28, 29, 30, 31]; various machine-learning based hadronization-models, such as
those described in Ilten:2022jfm,Chan:2023ume,Ghosh:2022zdz,refs. ref. [32,
33, 34]; or the multiparton interaction model within Pythia 8 itself [35]. For
instance, because the multiparton interaction (MPI) model in Pythia 8 produces
additional strings in a parton-shower-like fashion, the method presented here
is already directly applicable to the variation of MPI parameters.
Furthermore, the variation of kinematic hadronization parameters, described
here, already works “as is” when the MPI model is switched on. However, it is
important to note that as the number of interactions per event increases, the
spread in the produced weights will also increase, and so care must be taken
to ensure the reweighted sample has sufficient statistical power.

The paper is organized as follows: in section 2, we provide a detailed
presentation of the proposed method for fast uncertainty estimation in
hadronization simulations. Then, in section 3, we validate the effectiveness
of the method by applying it to two distinct data samples. Finally, in section
4, we summarize our findings and draw conclusions.

##  2 Method

An event produced by an event generator, like Pythia 8, begins from a small
number of partons that evolve through various stages. At each stage the color
quantum numbers are tracked in the large color $N_{c}$ limit, such that each
new color is assigned a new color index. In this limit, only planar color
flows are retained, and colored partons can be assigned a unique pair of
integers to represent color and anticolor. After the perturbatively-motivated
evolution of the parton shower, one of the last stages in the event
development is hadronization. Prior to this step, the collection of quarks,
antiquarks, and gluons can be partitioned into color-singlet objects (strings)
based on their color quantum numbers. The Lund string model of hadronization
[24, 36, 37] is then applied to reduce strings into the observed hadrons. The
string represents a flux tube of the non-perturbative strong force between a
quark and an antiquark that successively breaks into hadrons, represented by
stable oscillating string states characterized by their four-momentum $p_{h}$
and flavor. The full probability of a given fragmentation can be split into a
flavor selection, a transverse momentum sampling, and a longitudinal momentum
sampling, which are all combined to ensure a physical emission. A detailed
discussion of the Lund fragmentation function as implemented in Pythia 8 can
be found in Bierlich:2022pfr,refs. ref. [38]. Here, we summarize those
elements needed for the uncertainty estimation of the hadronization.

The Lund symmetric fragmentation function, or scaling function, determines the
probability for a hadron to be emitted with longitudinal lightcone
momentum222The lightcone momentum is defined as $E\pm p_{z}$ for a parton
moving in the $\pm z$ direction. fraction $z$. The momentum fraction is
defined as the fraction of the remaining lightcone momentum taken away by the
hadron, with the remaining $1-z$ lightcone momentum fraction available for the
subsequent production of hadrons. The fragmentation function has the following
form:

$f(z)\propto\frac{1}{z^{1+r_{Q}bm^{2}_{Q}}}\left({1-z}\right)^{a}\exp\left(-\frac{bm^{2}_{\perp}}{z}\right)\,,$
(1)

where $Q$ is the quark flavor, $m_{Q}$ is the quark mass, $m_{\perp}^{2}\equiv
m^{2}+p_{T}^{2}$ is the square of the transverse mass, where $m$ is the hadron
mass and $p_{T}$ is the transverse momentum of the hadron, and $r_{Q}$, $a$,
and $b$ are the constant parameters fixed by fits to experimental data.333The
default parameter names and values as implemented in Pythia 8 are
StringZ:aLund = 0.68, StringZ:bLund = 0.98, StringZ:rFactC = 0, and
StringZ:rFactB = 0.855 for $a$, $b$, $r_{c}$, and $r_{b}$, respectively. The
Bowler modification $z^{-r_{Q}bm^{2}_{Q}}$ in eq. 1 is only included for heavy
quarks, i.e., $r_{Q}=0$ unless $Q\in\\{c,b\\}$ [39]. Pythia 8 also allows for
modifications to the $a$-parameter to be used in splittings involving strange
quarks $s$ or diquarks $D$, parameterized by the form $a_{i}^{\prime}=a+\delta
a_{i}$, where $\delta a_{i}$ represents an adjustable parameter444The default
parameter names and values as implemented in Pythia 8 are StringZ:aExtraSQuark
= 0 and StringZ:aExtraDiquark = 0.97, for $s$ and $D$ respectively. within
Pythia 8 with $i\in\\{s,D\\}$; the form of $f(z)$ is also modified from eq. 1,
accounting for the fact that the emitted quarks can be of a different flavor
than the endpoints of the original string. The maximum of $f(z)$, denoted
$f_{\max}$, can be determined analytically for a given set of input parameter
values, denoted $c_{i}$. Sampling $z$ from $f(z)$ is done by selecting a
pseudo-random number $x$ until one satisfies $x<f(z)/f_{\max}\leq 1$, a method
known as the accept-reject algorithm, further described in section 2.1.

In the default model of Pythia 8, the transverse momentum $p_{T}$ of each
emitted hadron is obtained by first generating a back-to-back transverse
momentum of each new $q\bar{q}$ pair string break, sampling from a Gaussian
distribution. The physics origin of the Gaussian is the calculation of the
tunneling probability of the $q\bar{q}$ pair, through a classically forbidden
region, calculable in the WKB approximation as [40]

$\frac{\mathrm{d}\mathcal{P}}{\mathrm{d}^{2}p_{T}}\propto\exp\left(-\pi
m_{q}^{2}/\kappa\right)\exp\left(-\pi p^{2}_{T}/\kappa\right)\,,$ (2)

where $\kappa$ is the string tension, and $m_{q}$ and $p_{T}$ are the quark
masses and transverse momenta respectively. The quark mass (and thus flavor)
and $p_{T}$ can therefore be generated separately. In this paper we will
introduce reweighting for the parameter controlling the $p_{T}$ distribution,
and address the reweighting of the flavor parameters in a future paper.

The $q$ from a particular string break combines with a $\bar{q}$ from an
adjacent string break to produce a hadron. In principle, the generation of the
hadron $p_{T}$ should be parameter free, with $\kappa$ known, based on the
arguments above. In practice, however, generating Gaussian $p_{T}$ kicks with
$\sigma_{p_{T}}^{2}=\kappa/\pi\approx$ (0.25 GeV)2 produces too soft hadron
spectra. Therefore, $\sigma_{p_{T}}$ is left as a free parameter555Within
Pythia 8, $\sigma_{p_{T}}$ is set with the parameter name StringPT:sigma.. The
end effect is that the hadron $p_{T}$ is composed of the quark pair $p_{T}$,
each generated from a Gaussian distribution:

$P(p_{x},p_{y},\sigma_{p_{T}})=\frac{1}{2\pi\sigma_{p_{T}}^{2}}\exp\left[-\big{(}p_{x}^{2}+p_{y}^{2}\big{)}/\big{(}2\sigma_{p_{T}}^{2}\big{)}\right]\,.$
(3)

Such Gaussian distributions can be sampled with complete efficiency, e.g.,
using the Box–Muller transform [41].

Our key interest is to calculate uncertainties arising from different choices
of the parameters $a$, $a_{s}^{\prime}$, $a_{D}^{\prime}$, $b$, $r_{c}$,
$r_{b}$, and $\sigma_{p_{T}}$ as they enter into eqs. 1 and 3. In the
following, we first review the accept-reject algorithm so as to later
introduce a modified version of it, best suited for the uncertainty estimation
on the parameters of eq. 1. We also explain how to perform uncertainty
estimation for $\sigma_{p_{T}}$ by taking advantage of the direct sampling
from eq. 3.

It should be noted that the hadronization algorithm described above is used
while the mass of the remaining string is sufficiently large, such that
suitable phase space exists to produce a hadron and a remaining string. When
the remaining string reaches a sufficiently low mass, a specialized splitting
is performed where two hadrons are produced without a remaining string, rather
than a hadron and the remaining string [42]. However, this splitting is not
always successful; if the remaining string has an $m_{\perp}$ smaller than the
summed $m_{\perp}$ of the two hadrons, then the entire hadronization of the
string is rejected and started over. This rejection only depends upon the
kinematics of the string, and does not directly depend upon the kinematic
hadronization parameters; the $b$ parameter is used to break the symmetry of
the system, but does not change the rejection rate. Consequently, this final
possible rejection is accounted for by including the weight of all rejected
strings, in addition to the weights of the accepted strings.

###  2.1 Standard Accept-Reject Algorithm

The accept-reject algorithm can be used to sample a probability distribution
when the maximum value of the probability distribution, or a reliable
overestimate thereof, is known. The algorithm for sampling the probability
distribution ${P(z,c_{i})}$ begins by defining an acceptance probability
$P_{\text{accept}}(z,c_{i})$ for a trial value of $z$,

$P_{\text{accept}}(z,c_{i})\equiv\frac{P(z,c_{i})}{\widehat{P}}\,\leq 1\,.$
(4)

Both the acceptance probability $P_{\text{accept}}(z,c_{i})$ and the
probability distribution ${P(z,c_{i})}$ depend on a set of parameter values
$c_{i}$, that we will later vary. The constant $\widehat{P}$ is chosen so that
the relation in eq. 4 is satisfied; it can be either the analytic maximum or a
numerically estimated overestimate. A trial value for $z$ is accepted only if
$P_{\text{accept}}$ is larger than a random uniform variate. If the trial
value of $z$ is rejected, with probability
$P_{\text{reject}}=1-P_{\text{accept}}$, a new trial $z$ is then selected. The
algorithm continues until a given $z$ value is accepted. That is, in the
standard accept-reject algorithm, the value of $z$ is selected with
probability $p$ given by the product of the final accept probability times a
factor accounting for all of the rejected trials:

$p(z)=P_{\text{accept}}(z)\sum_{n=0}^{\infty}A^{n}\,\text{, where
}A=\int_{0}^{1}\text{d}z^{\prime}\,\big{(}1-P_{\text{accept}}(z^{\prime})\big{)}\,,$
(5)

where the dependence on the chosen parameter values $c_{i}$ has been
suppressed for brevity. Summing the geometric series in $A$ gives,

$p(z)=\frac{P_{\text{accept}}(z)}{1-A}=\frac{P_{\text{accept}}(z)}{\displaystyle\int_{0}^{1}\text{d}z^{\prime}\,P_{\text{accept}}(z^{\prime})}=P(z)\,,$
(6)

showing that the algorithm yields the desired distribution. The exact value of
$\widehat{P}$, provided that $P_{\text{accept}}\leq 1$, only affects the
efficiency of the algorithm; the further $\widehat{P}$ is from the actual
maximum of $P(z,c_{i})$, the less efficient the sampling.

###  2.2 Modified Accept-Reject Algorithm

Next, we present a modification of the accept-reject algorithm that assigns
appropriate weights to the existing event, depending on how the parameter
values $c_{i}$ are varied. We refer to the original set of parameter values
$c_{i}$ as the baseline and the new set $c_{i}^{\prime}$ as the alternative.
If the event generated with the baseline parameters has weight $w$ (typically
in Pythia 8, $w=1$), the modified accept-reject algorithm calculates the
weight $w^{\prime}$ that corresponds to the alternative values of the
parameters. If $w^{\prime}>w$, the event is more probable given the
alternative parameter values; if $w^{\prime}<w$, it is less probable.

For the calculation of the weight $w^{\prime}$, one needs to keep track of all
the trial $z$ values in the standard accept-reject algorithm. For each $z$
that was rejected, $w$ is multiplied by $R_{\text{reject}}^{\prime}(z)$, while
for the accepted value of $z$, the multiplication is by
$R_{\text{accept}}^{\prime}(z)$. Here, $R_{\text{accept}}^{\prime}(z)$ is the
ratio of alternative and baseline acceptance probabilities,

$R_{\text{accept}}^{\prime}(z)=\frac{P^{\prime}_{\text{accept}}(z)}{P_{\text{accept}}(z)}=\frac{P^{\prime}(z)}{P(z)}\,\text{,
with
}P_{\text{accept}}^{\prime}(z,c^{\prime}_{i})=\frac{P^{\prime}(z,c^{\prime}_{i})}{\widehat{P}}\,,$
(7)

while $R_{\text{reject}}^{\prime}(z)$ is the ratio of the alternative and the
baseline rejection probabilities,

$R_{\text{reject}}^{\prime}(z)=\frac{P^{\prime}_{\text{reject}}(z)}{P_{\text{reject}}(z)}=\frac{1-P^{\prime}_{\text{accept}}(z)}{1-P_{\text{accept}}(z)}=\frac{\widehat{P}-P^{\prime}(z)}{\widehat{P}-P(z)}\,.$
(8)

The value of $\widehat{P}$ can always be chosen such that both
$P_{\text{accept}}^{\prime}\leq 1$ and $P_{\text{accept}}\leq 1$, albeit at
some loss of efficiency when the equality does not hold for the latter.
Explicitly, we can write the per-event hadronization weight as

$w^{\prime}=w\prod_{i\in}R_{i,\text{accept}}^{\prime}(z)\prod_{j\in\text{rejected}}R_{j,\text{reject}}^{\prime}(z),$
(9)

where $w$ is the baseline event weight, the first product is over accepted
trials of $z$, and the second product is over the rejected trials of $z$.

We can readily show that the weight $w^{\prime}$ corresponds to the correct
probability $p^{\prime}(z)$ for selecting the final trial-$z$ value using the
alternative parameter values $c_{i}^{\prime}$:

$p^{\prime}(z)=P_{\text{accept}}(z)R_{\text{accept}}^{\prime}(z)\sum_{n=0}^{\infty}A^{\prime
n}\,\text{, where
}A^{\prime}=\int_{0}^{1}\text{d}z^{\prime}\,\big{(}1-P_{\text{accept}}(z^{\prime})\big{)}R^{\prime}_{\text{reject}}(z^{\prime})\,.$
(10)

Summing the geometric series in $A^{\prime}$ gives

$p^{\prime}(z)=\frac{P^{\prime}_{\text{accept}}(z)}{1-A^{\prime}}=\frac{P^{\prime}_{\text{accept}}(z)}{\displaystyle\int_{0}^{1}\text{d}z^{\prime}\,P^{\prime}_{\text{accept}}(z^{\prime})}=P^{\prime}(z)\,,$
(11)

as desired.

A few considerations are worth mentioning. As in the case of parton-shower
variations, the modified rejection ratio in eq. 8 is inversely proportional to
the difference $\hat{P}-P$ and can become large if $\hat{P}\simeq P$, leading
to large weights. It is thus advantageous for $\hat{P}$ to not approximate the
maximum value of $P(z,c_{i})$ too closely, but to be larger by an
${\mathcal{O}}(1)$ factor. In practice, multiplying $\hat{P}$ by a factor of
ten typically leads to stable results.666This factor may be adjusted within
Pythia 8 by modifying the corresponding overSample parameter for each
alternative parameter, e.g., for parton-shower variations,
UncertaintyBands:overSampleFSR specifies the over-sample factor for QCD final-
state radiation enabled by the fsr:* set of variation keywords. The final
event weight $w^{\prime}$ can also become large in cases when the baseline and
alternative probability distributions have limited overlap, i.e., the baseline
distribution does not provide proper support for the alternative distribution.
Good indicators of the fidelity of the reweighting are the mean weight

$\mu\equiv\sum_{i=1}^{N}\frac{w^{\prime}_{i}}{N}$ (12)

(or, equivalently, the weight sum $\sum_{i}w^{\prime}_{i}$ ) and the effective
number of events

$n_{\text{eff}}\equiv\frac{(\sum_{i=1}^{N}w^{\prime}_{i})^{2}}{\sum_{i=1}^{N}w_{i}^{\prime
2}},$ (13)

where $N$ is the number of generated events. If the mean event weight is not
near unity, or if the effective number of events is significantly lower than
the actual number of simulated events, care should be taken when interpreting
the weighted results.

###  2.3 Variation Details

Currently, we have implemented variations for the $a,b,r_{c}$, and $r_{b}$
parameters of the Lund string fragmentation function $f(z)$ given by eq. 1,
and the hadron transverse momentum $\sigma_{p_{T}}$ of eq. 3. The variation
weight for one selection of $\sigma_{p_{T}}$ does not require the use of the
accept-reject algorithm but can be calculated directly using the Box–Muller
transform:

$w^{\prime}=\frac{\sigma^{2}}{\sigma^{\prime
2}}\exp\left(-\kappa\left(\frac{\sigma^{2}}{\sigma^{\prime
2}}-1\right)\right)\,,$ (14)

where $\kappa=(n_{1}^{2}+n_{2}^{2})/2$ and ${n}_{i}$ are normally distributed
random variates.

The two event weights arising from variations in eqs. 1 and 3 can be combined
into a single event weight by multiplication, due to the fact that we are
sampling in a sequential manner from $P(p_{x},p_{y})$ and $P(z|p_{x},p_{y})$,
i.e., $P(p_{x},p_{y})$ does not depend upon $z$. However, variations of the
parameters of $f(z)$ must be considered as a group. While a variation of the
$a$ parameter for a fixed $b$ parameter can be calculated and vice versa, the
product of weights from these two calculations is not equivalent to varying
both $a$ and $b$ simultaneously. One reason for this is that the maximum
weight $f_{\max}(a_{1},b_{1})$ is different from the maximum weights
$f_{\max}(a_{1},b_{0})$ and $f_{\max}(a_{0},b_{1})$. This applies to all of
the parameters that enter into eq. 1: $a,b,r_{c},$ and $r_{b}$.

##  3 Validation

The goal of the presented reweighting method is to enable the use of
alternative event weights $w^{\prime}$ to produce the desired distributions
using the original sample of events, rather than generating a new sample for
each alternative parameter value. Therefore, we validate the method by
generating samples of $10^{6}$ events using Pythia 8 configured with a set of
baseline parameter values. During this generation, we also calculate, using
the modified accept-reject algorithm, a per-event weight $w^{\prime}$
corresponding to an alternative set of parameter values. We then compare the
$w^{\prime}$-weighted distributions to those obtained by generating new
samples using Pythia 8 configured with the alternative parameter values as the
baseline and without using the modified accept-reject algorithm.

We do this for the Lund parameters $a$, $b$, and $r_{b}$, as well as the
fragmentation transverse-momentum width $\sigma_{p_{T}}$.777Although
validated, the reweighting method for the $a_{s}^{\prime}$ and
$a_{D}^{\prime}$ parameters is not shown within this paper for the sake of
brevity. The top panels in figs. 1, 2, 3 and 4 show that the observables of
interest, described in further detail below, are sensitive to changes in $a$,
$b$, $r_{b}$, and $\sigma_{p_{T}}$, respectively. The bottom panels in figs.
1, 2, 3 and 4 show the agreement between the $w^{\prime}$-weighted
distributions and those generated with the alternative values set as the
baseline. We also vary parameters $a$ and $b$ simultaneously, and fig. 5 shows
the analogous plots for these cases. Whenever not explicitly stated, the
parameters are set to their default values of $a=0.68$, $b=0.98$,
$r_{b}=0.855$, and $\sigma_{p_{T}}=0.350$ from the Monash tune [43].

###  3.1 Validation simulations

The event samples were all generated using a modified version of Pythia 8.310
[44]. For event samples in which $a$, $b$, or $\sigma_{p_{T}}$ were varied, we
simulated electron-positron collisions with a center-of-mass energy at the
measured $Z$-boson mass.888The relevant configuration parameters are Beams:idA
= 11, Beams:idB = -11, Beams:eCM = 91.189, PDF:lepton = off,
WeakSingleBoson:ffbar2gmZ = on, 23:onMode = off, and 23:onIfAny = 1 2 3 4. We
then applied the selections from the ALEPH analysis described in
ALEPH:1996oqp,refs. ref. [45] using the corresponding Rivet analysis [46],
finally obtaining a dataset consisting predominantly of $Z$-boson decays to
hadrons. For validating the variations in $r_{b}$, we instead simulated
proton-proton collisions with a center-of-mass energy of $13{\mathrm{\
Te\kern-1.00006ptV}}$ and applied the selections from the LHCb analysis
described in LHCb:2017llq,refs. ref. [47].999The relevant configuration
parameters are Beams:idA = 2212, Beams:idB = 2212, Beams:eCM = 13000,
PhaseSpace:pTHatMin = 15, HardQCD:HardBBbar = on, and PartonLevel:MPI = off.
Additionally, for efficient generation, all $b$-hadron decays which do not
explicitly contain a $J/\psi$ are switched off using the ID:onMode method.
Note that this configuration does not include $b$-hadron production from $g\to
b\bar{b}$ splittings, but still provides a useful proxy for the distribution.
These requirements provide a sample of jets that contain a $J/\psi$, have
transverse momentum $p_{T}(\text{jet})>20{\mathrm{\ Ge\kern-1.00006ptV}}$, and
lie in the pseudorapidity range $2.5<\eta(\text{jet})<4.0$.

Figure 1: Comparison of the distributions, shown in arbitrary units, of the
event charge multiplicity when the parameter $a$ is (top) explicitly set to
different values, or (bottom) when it is varied using different methods. In
the top panel, the lower row shows the ratios of the distributions generated
with various values of $a$ to that generated with $a=0.68$. In the bottom
panel, the distributions labeled $e$ were generated with the value of the
parameter $a$ explicitly set to (left) $0.30$, (middle) $0.55$, and (right)
$0.76$. The distributions labeled $w^{\prime}$ are all taken from the same
sample generated with $a=a^{\text{base}}=0.68$, but with different sets of
alternative event weights, calculated using the accept-reject algorithm
applied according to the alternative values of $a$. The bottom row shows the
ratios of the latter distributions to the former.

Figure 2: Comparison of the distributions, shown in arbitrary units, of the
event charge multiplicity when the parameter $b$ is explicitly set to (top)
different values or (bottom) when it is varied using different methods. In the
top panel, the lower row shows the ratios of the distributions generated with
various values of $b$ to that generated with $b=0.98$. In the bottom panel,
the distributions labeled $e$ were generated with the value of the parameter
$b$ explicitly set to (left) $0.58$, (middle) $0.80$, and (right) $1.07$. The
distributions labeled $w^{\prime}$ are all taken from the same sample
generated with $b=b^{\text{base}}=0.98$, but with different sets of
alternative event weights, calculated using the accept-reject algorithm
applied according to the alternative values of $b$. The bottom row shows the
ratios of the latter distributions to the former.

Figure 3: Comparison of the distributions, shown in arbitrary units, of the
ratio of the transverse momentum of a $J/\psi$ meson to the transverse
momentum of the jet in which it is found, $z(J/\psi)$, when the parameter
$r_{b}$ is (top) explicitly set to different values, or (bottom) when it is
varied using different methods; see LHCb:2017llq,refs. ref. [47] for details
of this analysis. In the top panel, the lower row shows the ratios of the
distributions generated with various values of $r_{b}$ to that generated with
$r_{b}=0.855$. In the bottom panel, the distributions labeled $e$ were
generated with the value of the parameter $r_{b}$ explicitly set to (left)
$0.657$, (middle) $0.459$, and (right) $1.792$. The distributions labeled
$w^{\prime}$ are all taken from the same sample generated with
$r_{b}=r_{b}^{\text{base}}=0.855$, but with different sets of alternative
event weights, calculated using the accept-reject algorithm applied according
to the alternative values of $r_{b}$. The bottom row shows the ratios of the
latter distributions to the former.

Figure 4: Comparison of the distributions, shown in arbitrary units, of the
event charge multiplicity when the parameter $\sigma_{p_{T}}$ is (top)
explicitly set to different values, or (bottom) when the parameter
$\sigma_{p_{T}}$ is varied using different methods. In the top panel, the
lower row shows the ratios of the distributions generated with various values
of $\sigma_{p_{T}}$ to that generated with $\sigma_{p_{T}}=0.350$. In the
bottom panel, the distributions labeled $e$ were generated with the value of
the parameter $\sigma_{p_{T}}$ explicitly set to (left) $0.283$ and (right)
$0.360$. The distributions labeled $w^{\prime}$ are all taken from the same
sample generated with $\sigma_{p_{T}}=\sigma_{p_{T}}^{\text{base}}=0.350$, but
with different sets of alternative event weights, calculated using the accept-
reject algorithm applied according to the alternative values of
$\sigma_{p_{T}}$. The bottom row shows the ratios of the latter distributions
to the former.

Figure 5: Comparison of the distributions, shown in arbitrary units, of the
event charge multiplicity when the parameters $a$ and $b$ are (top) explicitly
set to various values, or (bottom) when and $a$ and $b$ are simultaneously
varied using different methods. In the top panel, the lower row shows the
ratios of the distributions generated with various values of $a$ and $b$ to
that generated with $a=0.68$ and $b=0.58$. In the bottom panel, the
distributions labeled $e$ were generated with the values of the parameters $a$
and $b$ explicitly set to (left) $a,b=0.30,0.80$ and (right) $a,b=0.76,0.98$.
The distributions labeled $w^{\prime}$ are all taken from the same sample
generated with $a=a^{\text{base}}=0.68$ and $b=b^{\text{base}}=0.58$, but with
different sets of alternative event weights, calculated using the accept-
reject algorithm applied according to the alternative values of $a$ and $b$.
The bottom row shows the ratios of the latter distributions to the former.

In general, the $w^{\prime}$-weighted distributions are in good agreement with
the distributions where the parameter values were set as the baseline. This
agreement breaks down if the Lund fragmentation function for the alternative
parameter values is large in a range where the Lund fragmentation function
approaches zero for the baseline parameter values, as shown in fig. 2 (bottom
left) and fig. 3 (bottom right). The reweighting then requires large weights
and samples the phase space poorly. To illustrate this point, fig. 6 shows
distributions of the Lund fragmentation function for different values of
$r_{b}$ and the corresponding distributions of event weights.

Figure 6: (Top) distributions of the Lund fragmentation function $f(z)$ for
different values of $r_{b}$ with $m_{\perp}=25{\mathrm{\ Ge\kern-1.00006ptV}}$
and all other parameters set to their default values. (Bottom) distributions
of the event weights $w^{\prime}$ for reweighting from
$r_{b}=r_{b}^{\text{base}}=0.855$, shown on a log-log scale with varying bin
sizes. Notice that when $r_{b}=1.792$, $f(0.575)\approx 1$, but
$f(0.575)\approx 0$ for $r_{b}=0.855$, resulting in weights far from unity
when reweighting from $r_{b}=r_{b}^{\text{base}}=0.855$ to $r_{b}=1.792$. The
weights when $r_{b}=r_{b}^{\text{base}}=0.855$ are exactly equal to 1, though
they may appear shifted due to the binning.

Therefore, care must be taken when selecting the baseline value of a parameter
to be varied, since the reweighting method may not successfully reproduce the
distributions, if the alternative parameter values are too different. This can
be checked by calculating the mean event weight $\mu$, as defined in eq. 12.
In the limit of infinite data, and because the events were generated according
to $P(z,c_{i})$, we can write

$\mu\equiv\sum_{i=1}^{N}w^{\prime}_{i}/N\approx\int\text{d}\mathcal{E}\,P(\mathcal{E},c_{i})w^{\prime}(\mathcal{E})\,,$
(15)

where $\mathcal{E}$ is an event composed of a series of accepted and rejected
values of $z$, with joint probability $P(\mathcal{E},c_{i})$ depending on the
generation baseline parameters. Explicitly, and with some abuse of notation,
we can integrate over the variable-length sets of accepted and rejected values
$\vec{z}$ and $\vec{z}_{\text{rejected}}$,

$\mu=\int\text{d}\vec{z}\,\text{d}\vec{z}_{\text{rejected}}\,w^{\prime}(\vec{z},\vec{z}_{\text{rejected}})\prod_{\mathrlap{i\in\vec{z}}}P_{i,\text{accept}}\prod_{\mathrlap{j\in\vec{z}_{\text{rejected}}}}P_{j,\text{reject}}\,.$
(16)

Introducing the expression for $w^{\prime}$ in eq. 9, the
$P_{i,\text{accept}}$ and $P_{j,\text{reject}}$ factors cancel out and

$\mu=\int\text{d}\vec{z}\,\text{d}\vec{z}_{\text{rejected}}\,\prod_{\mathrlap{i\in\vec{z}}}P^{\prime}_{i,\text{accept}}\prod_{\mathrlap{j\in\vec{z}_{\text{rejected}}}}P^{\prime}_{j,\text{reject}}=\int\text{d}\mathcal{E}\,P^{\prime}(\mathcal{E},c^{\prime}_{i})=1\,.$
(17)

Thus, if the generated events cover appropriately the phase space for both
$P(z,c_{i})$ and $P^{\prime}(z,c^{\prime}_{i})$, the weights have an
expectation value $\mu$ consistent with one. If $\mu\not\approx 1$, the
reweighting method is unlikely to reproduce the distributions well. This can
be caused by the baseline distribution providing insufficient coverage for the
alternative distribution, since generated datasets are limited by finite
statistics.

In addition to $1-\mu$, the ratio of the effective sample size
$n_{\text{eff}}$ (defined in eq. 13) to the number of generated events $N$
(equivalent to the square of the ratio of the relative statistical uncertainty
of the unweighted sample to that of the weighted sample) is also a useful
metric, as it describes the statistical power of the reweighted alternate
distribution. If $n_{\text{eff}}/N\ll 1$, it may be necessary to adjust the
baseline distribution to be closer to the alternate distribution. For example,
when reweighting the $b$ parameter, a number of baseline values could be
chosen and the reweighting technique of this paper can then be used to sample
between these distributions to ensure $n_{\text{eff}}$ remains sufficiently
large. However, while $n_{\text{eff}}/N$ can indicate when the base
distribution differs significantly from the alternate distribution, it cannot
capture whether the baseline distribution provides full support for the
alternate distribution, i.e., if the alternate distribution enters a region of
phase space that is not generated from the baseline distribution.
Consequently, table 1 provides the mean event weights, as well as
$n_{\text{eff}}/N$, for different values of $a$, $b$, $r_{b}$, and
$\sigma_{p_{T}}$.

If one compares the mean event weight in table 1 to the distributions in figs.
1, 2, 3, 4 and 5, one can see that the proximity of the mean event weight to
unity is a good predictor of the similarity of the distributions.

Table 1: Difference of mean weight $\mu$ from one and the ratio of the effective sample size $n_{\text{eff}}$ to the number of generated events $N$ for the listed variations; see the text. A value of zero or one indicates that $\mu$ or $n_{\text{eff}}/N$, respectively, is exactly one, corresponding to the base case where all weights are one. variation | $1-\mu$ | $n_{\text{eff}}/N$ | figure
---|---|---|---
$r_{b}^{\text{base}}=0.855$ | 0 | 1 | $\Biggr{\\}}$ figs. 3 and 6
$r_{b}=0.657$ | $\phantom{-}\left(0.1\pm 7.9\right)\times 10^{-4}$ | $6.2\times 10^{-1}$
$r_{b}=0.459$ | $\phantom{-}\left(1.2\pm 2.4\right)\times 10^{-3}$ | $1.9\times 10^{-1}$
$r_{b}=1.792$ | $\phantom{-}\left(1.1\pm 0.4\right)\times 10^{-1}$ | $7.3\times 10^{-4}$
$a^{\text{base}}=0.68$ | 0 | 1 | $\Biggr{\\}}$ fig. 1
$a=0.30$ | $-\left(0.5\pm 4.0\right)\times 10^{-3}$ | $5.8\times 10^{-2}$
$a=0.55$ | $-\left(2.4\pm 4.7\right)\times 10^{-4}$ | $8.2\times 10^{-1}$
$a=0.76$ | $-\left(1.7\pm 2.6\right)\times 10^{-4}$ | $9.4\times 10^{-1}$
$b^{\text{base}}=0.98$ | 0 | 1 | $\Biggr{\\}}$ fig. 2
$b=0.58$ | $\phantom{-}\left(4.0\pm 2.0\right)\times 10^{-2}$ | $2.3\times 10^{-3}$
$b=0.80$ | $\phantom{-}\left(1.4\pm 1.4\right)\times 10^{-3}$ | $3.4\times 10^{-1}$
$b=1.07$ | $-\left(3.4\pm 3.7\right)\times 10^{-4}$ | $8.8\times 10^{-1}$
$\sigma_{p_{T}}^{\text{base}}=0.350$ | 0 | 1 | $\Biggr{\\}}$ fig. 4
$\sigma_{p_{T}}=0.283$ | $\phantom{-}\left(1.2\pm 0.8\right)\times 10^{-2}$ | $1.4\times 10^{-2}$
$\sigma_{p_{T}}=0.360$ | $-\left(4.9\pm 3.1\right)\times 10^{-4}$ | $9.1\times 10^{-1}$
$a^{\text{base}}=0.68$, $b^{\text{base}}=0.58$ | 0 | 1 | $\Biggr{\\}}$ fig. 5
$a=0.30$, $b=0.80$ | $\phantom{-}\left(4.6\pm 1.3\right)\times 10^{-2}$ | $5.7\times 10^{-3}$
$a=0.76$, $b=0.98$ | $\phantom{-}\left(1.2\pm 0.7\right)\times 10^{-2}$ | $2.1\times 10^{-2}$

###  3.2 Timing

A clear benefit of using the reweighting method is that it is universally
faster than generating new samples with the alternative parameter values set
explicitly. To demonstrate this, we generate a set of $10^{2}$ samples with
$10^{3}$ events each, using the same Pythia 8 settings described above, where
we calculate weights for an additional alternative parameter value in each
sample. We measure the time it takes to generate each event using a single
$2.5~{}\text{GHz}$ Intel Xeon CPU. Figure 7 shows the arithmetic mean of the
time spent to generate a single event as a function of the number of
alternative values calculated for Lund parameter $a$. As shown, the marginal
cost per additional parameter variation is $\approx 0.04\,\mathrm{ms}$, and it
takes $\approx 0.7\,\mathrm{ms}$ to generate an event with 10 alternative
values. Since it takes $\approx 0.3\,\mathrm{ms}$ to generate an event with no
alternative values, it would take $\approx 3\,\mathrm{ms}$ to generate 10
separate events with the alternative values set explicitly, more than 3 times
longer than using the modified veto algorithm. These savings vary, depending
on the Lund parameter in question, but in all cases, they increase
dramatically when one considers detector simulations, which often take
$\approx$ 1,000 times longer than the event generation.

Figure 7: Average time required to generate a single event as a function of
the number of alternative parameter values of Lund parameter $a$ calculated
during the generation. The error on each point is the standard error of the
mean. The amount of time required to generate a single event increases
linearly; the best-fit curve is shown in red, and its equation is given in the
legend.

##  4 Conclusions

In this study, we have introduced a robust mathematical framework and
validated its practical implementation for the fast estimation of
hadronization uncertainties in Monte Carlo simulations. By complementing the
existing algorithmically efficient uncertainty estimations for the hard
matrix-element calculations and parton shower calculations already implemented
in Pythia 8 [3, 5] and other event generators, our method now offers a rapid
estimate of parametric uncertainties for fully hadronized events. Accompanying
code is available at gitlab.com/uchep/mlhad-weights-validation, and the
reweighting code will be directly incorporated into the next Pythia 8 release.

It is important to acknowledge certain limitations of the method: if the
parameter variations result in acceptance probability distributions that are
far removed from the baseline, the modeling of the new distributions will be
poor due to lack of coverage, exemplified in extreme values for the weights.
We have found that the deviation of the mean event weight from one is a simple
and useful diagnostic tool to identify potential issues; one can also
explicitly check that the alternative acceptance probability distribution
sufficiently overlaps with the baseline acceptance probability distribution.
As long as this coverage condition is met, the presented method provides a
practical solution for fast uncertainty estimation in hadronization models,
especially in the context of full detector simulations.

#### Acknowledgments:

We thank L. Gellersen and T. Sjöstrand for careful reading and constructive
comments on the manuscript. The work was partially completed during the
Physics at TeV Colliders Workshop, Les Houches, June 2023. AY, JZ, MS, and TM
acknowledge support in part by the DOE grant de-sc0011784 and NSF OAC-2103889.
PI and MW are supported in part by NSF OAC-2103889 and NSF-PHY-2209769. SM is
supported by the Fermi Research Alliance, LLC under Contract No. DE-
AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office
of High Energy Physics. CB acknowledges support from the Knut and Alice
Wallenberg foundation, contract number 2017.0036.

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# Animal Inspired Application of a Variant of Mel Spectrogram for Seismic Data
Processing

Samayan Bhattacharya
Department of Computer Science and Engineering
Jadavpur University
188, Raja Subodh Chandra Mallick Rd,
Jadavpur University Campus Area,
Jadavpur, Kolkata,
West Bengal 700032, India
<EMAIL_ADDRESS>
Sk Shahnawaz
Department of Computer Science and Engineering
Jadavpur University
188, Raja Subodh Chandra Mallick Rd,
Jadavpur University Campus Area,
Jadavpur, Kolkata,
West Bengal 700032, India
<EMAIL_ADDRESS>

###### Abstract

Predicting disaster events from seismic data is of paramount importance and
can save thousands of lives, especially in earthquake-prone areas and
habitations around volcanic craters. The drastic rise in the number of seismic
monitoring stations in recent years has allowed the collection of a huge
quantity of data, outpacing the capacity of seismologists. Due to the complex
nature of the seismological data, it is often difficult for seismologists to
detect subtle patterns with major implications. Machine learning algorithms
have been demonstrated to be effective in classification and prediction tasks
for seismic data. It has been widely known that some animals can sense
disasters like earthquakes from seismic signals well before the disaster
strikes. Mel spectrogram has been widely used for speech recognition as it
scales the actual frequencies according to human hearing. In this paper, we
propose a variant of the Mel spectrogram to scale the raw frequencies of
seismic data to the hearing of such animals that can sense disasters from
seismic signals. We are using a Computer vision algorithm along with
clustering that allows for the classification of unlabelled seismic data.

> _"He did not notice the faint rumble that shook the earth"_ \- The Tiger in
> the Tunnel by Ruskin Bond
>

_Keywords_ Mel spectrogram $\cdot$ Computer vision $\cdot$ seismic monitoring
stations

##  1 Introduction

Understanding seismological data is important for disaster prediction as well
as understanding the physical properties of the earth’s crust. In the past
years, the drastic rise in the number of seismic monitoring stations has
transformed the field of seismological research from an observation-based to a
data-driven science [1]. In general, earthquake seismology problems fall into
three categories: probabilistic risk assessment [2, 3, 4, 5, 6], earthquake
recognition for data mining and early-earthquake detection [7, 8, 9, 10, 11],
and earthquake prediction for a warning system [12, 13]

Seismology studies are conducted with four objectives: (1) disaster
preparation or adjustment, (2) actions to reduce long-term the disaster of
earthquake, (3) disaster response strategies, and (4) post-disaster recovery
planning, which are known as the strategies of preparedness, mitigation,
response, and recovery, respectively. The challenges in the detection of every
earthquake from seismic data are: (1) a huge amount of noisy data in the
seismic records and (2) many earthquake events are undetected[11]. As a case
in point, the nonvolcanic tremors were first observed in southwestern Japan
only two decades ago[14] This is because the weak signal generated by these
tremors is hard to detect in certain regions. Hence, efficient detection of
seismic signals would allow us to better understand the processes of seismic
zones and hence predict earthquakes effectively. In recent years Machine
Learning algorithms have been demonstrated to be effective in classification
and prediction tasks. Unlike other AI algorithms where the features have to be
manually defined[15, 16, 17] machine learning enables the computer to select
relevant features on its own, it does require huge amounts of labeled data to
train successfully. This requires a lot of man-hours and is a major
disadvantage. A better alternative is the use of unsupervised learning, like
clustering algorithms, that allow training on unlabelled data. The computer
can separate the data into distinct classes based on similarities among
members of a particular class. Unsupervised learning has been applied to the
data from volcano monitoring systems[18, 19, 20, 21], induced seismicity[22,
23], global seismicity[9], and local vs distance earthquakes[24].

##  2 Proposed method

Mel spectrograms are widely used in speech recognition. The raw frequency data
is converted to mel scale using the following formula:

$f\textquoteright=2595.log(1+(f/700))$ (1)

The rationale behind this is, it has been empirically proven that humans do
not perceive frequencies linearly but logarithmically[25]

In this paper, we convert the raw seismological data using the formula

$f\textquoteright=c1.log(1+(f/c2))$ (2)

where, c1 and c2 are constants, derived empirically.

This is based on the fact that some animals can predict earthquakes from
seismic signals[26]. Therefore, it must help to adjust the spectrogram from
human hearing to the hearing of those animals. From a machine learning point
of view, this operation reduces the variance in data, unimportant for the
given task, and increases variance for the important data. Thus, it makes it
easier for the machine learning model to learn the important features for the
prediction or classification task.

The obtained frequencies are converted to a spectrogram using a short-time
Fourier transform and passed to the computer vision network.

In this experiment, we use a Convolutional Neural Network (CNN)[27] to extract
features from the spectrogram. The CNN is a form of deep neural network that
uses multiple filters with trainable weights to extract features, usually from
a 2D representation of the data. The CNN we used is a resnet[28]. Resnet
consists of layers of convolutions with skip connections between some layers,
along with activation layers (we used ReLU) and pooling layers. Each layer of
convolution applies a trainable filter to the data passed to it. We used a
combination of max and average pooling layers in our model. The output of the
resnet was fed into a clustering model.

Figure 1:  Used pipeline  Figure 2: Mel spectrogram and the proposed
spectrogram created from the same data

We used the Gaussian mixture model[29] for clustering, where the goal is to
find a set of K-normal distributions of mean $\mu_{k}$ and covariance
$\sigma_{k}$ (where k=1 to K) to best describe the overall data. As the final
output of the algorithm, a categorical variable is also inferred. The Gaussian
mixture clustering is a probabilistic and more flexible version of the K-means
clustering algorithm, in which the clusters can be unbalanced in terms of
internal variance, each covariance can be anisotropic, and where decision
boundary is soft. The negative likelihood of the data to be fully described by
the set of normal distributions is used as the clustering loss. The number of
clusters is inferred by our procedure. For the Gaussian mixture algorithm, we
initialize the number of clusters (K) = 10 and let the model train with
expectation minimization strategy[29]. We made clustering optional, once
clustering loss stagnated after 6000 epochs. We employed batch processing,
which randomly selects subsets of the whole dataset for faster training. This
also prevents the model from getting stuck in local minima. We trained the
model for 10000 epochs. The Clustering loss decreased by a factor of 5.

##  3 Data

For this experiment, we used seismic data (waveforms and related metadata),
publicly available on the IRIS data management center website. IRIS data
services is funded by the Seismological Facilities for the Advancement of
Geoscience and EarthScope (SAGE) Project, which is funded by the NSF under
Cooperative Agreement EAR-1261681. In addition, we used some simulation data
from USGS.

Figure 3:  M 7.5 Scenario Earthquake - Castle Mnt Earthquake2014-03-25
21:00:00 (UTC), 61.473°N 150.312°W ,10.0 km depth Credit: U.S. Geological
Survey  Figure 4:  2014 Seismic Hazard Map-Alaska Credit: U.S. Geological
Survey Department of the Interior/USGS

We also introduced some artificial data to the dataset. The artificial data
was generated by using data augmentation techniques (translation, reflection
about horizontal and vertical axis).

##  4 Results

Our method consistently outperformed other, commonly used models, with a
similar number of parameters. In addition, our model was able to train well on
smaller datasets. In the comparative study between the same model, trained
with our variant of Mel spectrogram and ordinary spectrogram, the one trained
on the variant of Mel spectrogram outperformed the one with normal spectrogram
by a substantial margin. [Table 1]

CNN | Clustering loss with ordinary spectrogram | Clustering loss with Mel spectrogram
---|---|---
Resnet 18 | 6.27 | 5.04
Resnet 50 | 5.62 | 4.50
Resnet 101 | 4.79 | 3.95
Resnet 152 | 3.56 | 2.90
Table 1: Clustering loss comparison for different depths of Resnets with
ordinary spectrogram and our spectrogram Figure 5:  Visualization of data pre
and post clustering

##  5 Discussion

Our method, thus, provides a valuable tool for remote seismic monitoring
centers, who don’t have access to a high amount of computational resources.
The clustering algorithms can detect very subtle patterns, which might be
missed by human seismologists, and group data items together. Thus, it is
particularly effective for data shown in [30,31]. The clustering also allows
the detection of seismological phenomena that might have been considered
regular and ignored by a seismologist. This is important for spotting
precursors to a natural disaster, like an earthquake or volcanic eruption. It
also enables a better understanding of the inner mechanics of the Earth’s
crust by detecting hard to detect phenomena, like nonvolcanic tremors, low-
frequency earthquakes, distant vs local earthquakes, etc. Animals like
elephants have been widely known to run inland from the coast just before an
earthquake, followed by a tsunami
(https://www.nationalgeographic.com/animals/article/news-animals-tsunami-
sense-coming, https://www.dailymail.co.uk/news/article-3614477/How-Ning-Nong-
elephant-saved-tsunami-incredible-bond-little-girl-baby-jumbo-inspired-
Michael-Morpurgo-s-sequel-War-Horse.html). Since hearing of animals has not
been empirically tested to generate proper constants for the Mel spectrogram,
we tried out different constants stochastically and reported the results for
the best one, so far, in this paper. We are continuing to test more
combinations of constants and better results could be forthcoming. If we get
close to the hearing efficiency of an elephant then every seismological
monitoring station would be able to detect an approaching earthquake and issue
an early warning of up to 20 minutes before the earthquake hits, saving lives
of people in the region by giving them sufficient time to get to safe places.

##  6 Conclusion

We have proposed an idea that, to our knowledge, has not been explored before.
It is far from perfect as the manual testing of constants for the Mel
Spectrogram variant is tedious and time-consuming. We are currently working on
automating the process. There is a lot of work to be done in the field of ML
for seismology, which due to lack of attention from the industry, remains
largely neglected. It is a very important area of study with the potential to
save thousands of lives each year.

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Equivariant Join and Fusion of Noncommutative Algebras

Equivariant Join and Fusion
of Noncommutative Algebras

Ludwik DA̧BROWSKI †, Tom HADFIELD ‡ and Piotr M. HAJAC §

L. Da̧browski, T. Hadfield and P.M. Hajac

† SISSA (Scuola Internazionale Superiore di Studi Avanzati),
† Via Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>
‡ G-Research, Whittington House, 19-30 Alfred Place, London WC1E 7EA, UK
<EMAIL_ADDRESS>
§ Institytut Matematyczny, Polska Akademia Nauk,
§ ul. Śniadeckich 8, 00-656 Warszawa, Poland<EMAIL_ADDRESS>http://www.impan.pl/~pmh/

Received June 30, 2015, in final form October 03, 2015; Published online
October 13, 2015

We translate the concept of the join of topological spaces to the language of
$C^{*}$-algebras, replace the $C^{*}$-algebra of functions on the interval
$[0,1]$ with evaluation maps at $0$ and $1$ by a unital $C^{*}$-algebra $C$
with appropriate two surjections, and introduce the notion of the _fusion_ of
unital $C^{*}$-algebras. An appropriate modification of this construction
yields the fusion comodule algebra of a comodule algebra $P$ with the coacting
Hopf algebra $H$. We prove that, if the comodule algebra $P$ is principal,
then so is the fusion comodule algebra. When $C=C([0,1])$ and the two
surjections are evaluation maps at $0$ and $1$, this result is a
noncommutative-algebraic incarnation of the fact that, for a compact Hausdorff
principal $G$-bundle $X$, the diagonal action of $G$ on the join $X*G$ is
free.

$C^{*}$-algebras; Hopf algebras; free actions

46L85; 58B32

##  1 Introduction

The join of topological spaces is a crucial concept in algebraic topology – it
is used in the celebrated Milnor’s construction of a universal principal
bundle [9]. The goal of this paper is to provide a noncomutative-geometric
version of the join of a Cartan compact Hausdorff principal bundle (no local
triviality assumed) with its structure group. In particular, we obtain this
way a noncommutative version of the diagonal $G$-action on the $n$-fold join
$G*\cdots*G$ of a compact Hausdorff topological group $G$, which is the first
step in Milnor’s construction.

To make this paper self-contained and to establish notation and terminology,
we begin by recalling the basics of classical joins, strong connections [6]
and principal comodule algebras [7]. In the first section, we define the join
and fusion $C^{*}$-algebra of an arbitrary pair of unital $C^{*}$-algebras.
For commutative $C^{*}$-algebras our join construction recovers Milnor’s
definition in the case of compact Hausdorff topological spaces. In the second
section, we deal with a join and fusion of a comodule algebra $P$ with the
coacting Hopf algebra $H$. To define an $H$-comodule algebra structure that
corresponds to the diagonal action in the classical setting, we provide a
“gauged” algebraic version of the join and fusion $C^{*}$-algebras. (The
gauged $C^{*}$-algebraic versions of the join and fusion are given
respectively in [2] and [5].) The main result of this paper is Theorem 5.3
concluding the principality of the fusion comodule algebra of $P$ and $H$ from
the principality of $P$.

##  2 Classical join construction

Let $I=[0,1]$ be the closed unit interval and let $X$ be a topological space.
The _unreduced suspension_ $\Sigma X$ of $X$ is the quotient of $I\times X$ by
the equivalence relation $R_{S}$ generated by

$\displaystyle(0,x)\sim(0,x^{\prime}),\qquad(1,x)\sim(1,x^{\prime}).$

Now take another topological space $Y$ and, on the space $I\times X\times Y$,
consider the equivalence relation $R_{J}$ given by

$\displaystyle(0,x,y)\sim(0,x^{\prime},y),\qquad(1,x,y)\sim(1,x,y^{\prime}).$

The quotient space $X*Y:=(I\times X\times Y)/R_{J}$ is called the _join_ of
$X$ and $Y$. It resembles the unreduced suspension of $X\times Y$, but with
only $X$ collapsed at 0, and only $Y$ collapsed at 1. In particular, if $Y$ is
a one-point space, the join $X*Y$ is the cone $CX$ of $X$. If $Y$ is a two-
point space with discrete topology, then the join $X*Y$ is the unreduced
suspension $\Sigma X$ of $X$.

If $G$ is a topological group acting continuously on $X$ and $Y$ from the
right, then the diagonal right $G$-action on $X\times Y$ induces a continuous
action on the join $X*Y$. Indeed, the diagonal action of $G$ on $I\times
X\times Y$ factorizes to the quotient, so that the formula

$\displaystyle([(t,x,y)],g)\longmapsto[(t,xg,yg)]$

makes $X*Y$ a right $G$-space. It is immediate that this continuous action is
free if the $G$-actions on $X$ and $Y$ are free.

Consider $CX\times Y$, $X\times CY$ and $X\times Y$ as $G$-spaces with the
diagonal $G$-actions. Note that there is a continuous surjection
$\pi_{I}\colon X*Y\ni[(t,x,y)]\mapsto t\in[0,1]$ such that
$\pi_{I}^{-1}([0,\frac{1}{2}])$ and $\pi_{I}^{-1}([\frac{1}{2},1])$ are
$G$-equivariantly homeomorphic to $CX\times Y$ and $X\times CY$ respectively.
Thus $X*Y$ is $G$-equivariantly homeomorphic to the gluing of $CX\times Y$ and
$X\times CY$ over $X\times Y$:

$\displaystyle X*Y\cong(CX\times Y)\underset{X\times Y}{\sqcup}(X\times CY).$

If $Y=G$ with the right action assumed to be the group multiplication, we can
construct the join $G$-space $X*Y$ in a different manner: at 0 we collapse
$X\times G$ to $G$ as before, and at 1 we collapse $X\times G$ to $(X\times
G)/R_{D}$ instead of $X$. Here $R_{D}$ is the equivalence relation generated
by $(x,h)\sim(x^{\prime},h^{\prime})$, where $xh=x^{\prime}h^{\prime}$. More
precisely, let $R_{J}^{\prime}$ be the equivalence relation on $I\times
X\times G$ generated by

$\displaystyle(0,x,h)\sim(0,x^{\prime},h)\qquad\text{and}\qquad(1,x,h)\sim(1,x^{\prime},h^{\prime}),\qquad\text{where}\quad
xh=x^{\prime}h^{\prime}.$ (2.1)

The formula $[(t,x,h)]k:=[(t,x,hk)]$ defines a continuous right $G$-action on
$(I\times X\times G)/R_{J}^{\prime}$. One can also easily check that the
formula

$\displaystyle
X*G\ni[(t,x,h)]\longmapsto\big{[}\big{(}t,xh^{-1},h\big{)}\big{]}\in(I\times
X\times G)/R_{J}^{\prime}$ (2.2)

yields a $G$-equivariant homeomorphism.

If we further specify also $X=G$ with the right action assumed to be the group
multiplication, then the $G$-action on $X*Y=G*G$ is automatically free.
Furthermore, since the action of $G$ on $X*G$ is free whenever it is free on
$X$, we conclude that the natural action on the iterated join of $G$ with
itself is also free. For instance, for $G={\mathbb{Z}}/2{\mathbb{Z}}$ we
obtain a ${\mathbb{Z}}/2{\mathbb{Z}}$-equivariant identification
$({\mathbb{Z}}/2{\mathbb{Z}})^{*(n+1)}\cong S^{n}$, where $S^{n}$ is the
$n$-dimensional sphere with the antipodal action of
${\mathbb{Z}}/2{\mathbb{Z}}$.

##  3 Strong connection and principal comodule algebra

Let $H$ be a Hopf algebra with coproduct $\Delta$, counit $\varepsilon$ and
bijective antipode $S$. Next, let $\delta\colon P\to P\otimes H$ be a coaction
making $P$ a right $H$-comodule algebra. We shall frequently use the
Heyneman–Sweedler notation (with the summation sign suppressed) for coproduct
and coaction:

$\displaystyle\Delta(h)=:h_{(1)}\otimes
h_{(2)},\qquad\delta(p)=:p_{(0)}\otimes p_{(1)}.$

######  Definition 3.1 ([7]).

Let $P$ be a right comodule algebra over a Hopf algebra $H$ with bijective
antipode, and let

$\displaystyle B:=P^{\operatorname{co}H}:=\\{p\in P\,|\,\delta(p)=p\otimes
1\\}.$

be the coaction-invariant subalgebra. The comodule algebra $P$ is called
principal if the following conditions are satisfied:

  1. 1)

the coaction of $H$ is Hopf–Galois, that is, the map

$\displaystyle\operatorname{can}_{P}\colon\
P\underset{B}{\otimes}P\longrightarrow P\otimes H,\qquad p\otimes q\longmapsto
pq_{(0)}\otimes q_{(1)},$

(called the canonical map) is bijective,

  2. 2)

the comodule algebra $P$ is right $H$-equivariantly projective as a left
$B$-module, i.e., there exists a right $H$-colinear and left $B$-linear
splitting of the multiplication map $B\otimes P\rightarrow P$.

######  Definition 3.2 ([3]).

Let $H$ be a Hopf algebra with bijective antipode. A _strong connection
$\ell$_ on a right $H$-comodule algebra $P$ is a unital linear map $\ell\colon
H\rightarrow P\otimes P$ satisfying the following bicolinearity (i.e., left
and right colinearity) and splitting conditions:

  1. 1)

$(\operatorname{id}\otimes\delta)\circ\ell=(\ell\otimes\operatorname{id})\circ\Delta$,
$(\delta^{L}\otimes\operatorname{id})\circ\ell=(\operatorname{id}\otimes\ell)\circ\Delta$,
where
$\delta^{L}:=(S^{-1}\otimes\operatorname{id})\circ\mathrm{f\/lip}\circ\delta$;

  2. 2)

$\widetilde{\operatorname{can}_{P}}\circ\ell=1\otimes\operatorname{id}$, where
$\widetilde{\operatorname{can}_{P}}\colon P\otimes P\ni p\otimes
q\mapsto(p\otimes 1)\delta(q)\in P\otimes H$.

We will use the Heyneman–Sweedler-type notation $\ell(h)=:\ell(h)^{\langle
1\rangle}\otimes\ell(h)^{\langle 2\rangle}$ with the summation sign
suppressed. One can easily prove (see [7, p. 599] and references therein) that
a comodule algebra is principal if and only if it admits a strong connection.
Here we need the following slight generalization of this fact:

######  Lemma 3.3.

Let $H$ be a Hopf algebra with bijective antipode. Then a right $H$-comodule
algebra $P$ is principal if and only if it admits a $($not necessarily
unital$)$ linear map $\ell\colon H\rightarrow P\otimes P$ satisfying the
bicolinearity and splitting conditions of Definition 3.2.

###### Proof.

Assume that $\ell\colon H\rightarrow P\otimes P$ is a linear map satisfying
the bicolinearity and splitting conditions. Let $\pi_{B}\colon P\otimes P\to
P\otimes_{B}P$ be the canonical surjection. Define

$\displaystyle L\colon\ P\otimes H\ni p\otimes
h\longmapsto\pi_{B}\big{(}p\ell(h)^{\langle 1\rangle}\otimes\ell(h)^{\langle
2\rangle}\big{)}\in P\underset{B}{\otimes}P.$

It follows immediately from the splitting property of $\ell$ that

$\displaystyle\operatorname{can}_{P}(L(p\otimes
h))=p\,\widetilde{\operatorname{can}_{P}}(\ell(h))=p\otimes h.$

Applying $\operatorname{id}\otimes\varepsilon$ to the splitting condition for
$\ell$, we obtain $m\circ\ell=\varepsilon$, where $m$ is the multiplication
map on $P$. Combining it with the left colinearity of $\ell$, which implies
that

$\displaystyle\delta\big{(}q_{(0)}\ell(q_{(1)})^{\langle
1\rangle}\big{)}\otimes\ell(q_{(1)})^{\langle
2\rangle}=q_{(0)}\ell(q_{(3)})^{\langle 1\rangle}\otimes
q_{(1)}S(q_{(2)})\otimes\ell(q_{(3)})^{\langle 2\rangle}$
$\displaystyle\hphantom{\delta\big{(}q_{(0)}\ell(q_{(1)})^{\langle
1\rangle}\big{)}\otimes\ell(q_{(1)})^{\langle
2\rangle}}{}=q_{(0)}\ell(q_{(1)})^{\langle 1\rangle}\otimes
1\otimes\ell(q_{(1)})^{\langle 2\rangle},$

we obtain

$\displaystyle L(\operatorname{can}_{P}(p\otimes
q))=\pi_{B}\big{(}pq_{(0)}\ell(q_{(1)})^{\langle
1\rangle}\otimes\ell(q_{(1)})^{\langle 2\rangle}\big{)}=\pi_{B}\big{(}p\otimes
q_{(0)}\ell(q_{(1)})^{\langle 1\rangle}\ell(q_{(1)})^{\langle
2\rangle}\big{)}$ $\displaystyle\hphantom{L(\operatorname{can}_{P}(p\otimes
q))}{}=\pi_{B}(p\otimes q).$

Finally, the bicolinearity of $\ell$ implies that the formula
$s(p):=p_{(0)}\ell(p_{(1)})$ defines a left $B$-linear right $H$-colinear
splitting of the multiplication map $B\otimes P\to P$. The reverse implication
(principality $\Rightarrow$ existence of a bicolinear $\ell$ with the
splitting property) follows from [3, Lemma 2.2]. ∎

##  4 Join and fusion of unital $\boldsymbol{C^{*}}$-algebras

Let $\otimes_{\min}$ denote the spatial (minimal) tensor product. Due to the
fact that minimal tensor products preserve injections (e.g., see [11,
Proposition 4.22] or [12, Section 1.3]), for any unital $C^{*}$-algebras
$A_{1}$ and $A_{2}$ the natural maps

$\displaystyle A_{1}\ni a\longmapsto a\otimes 1\in
A_{1}\underset{\min}{\otimes}A_{2}\qquad\text{and}\qquad A_{2}\ni a\longmapsto
1\otimes a\in A_{1}\underset{\min}{\otimes}A_{2}$

are injective. This allows us to view $A_{1}$ and $A_{2}$ as subalgebras of
$A_{1}\otimes_{\min}A_{2}$. We use this natural identification in what
follows.

######  Definition 4.1.

Let $A_{1}$ and $A_{2}$ be unital $C^{*}$-algebras, and let $C$ be a unital
$C^{*}$-algebra equipped with two $C^{*}$-ideals $J_{1}$ and $J_{2}$ such that
the direct sum $\pi_{1}\oplus\pi_{2}\colon
C\rightarrow(C/J_{1})\oplus(C/J_{2})$ of the canonical surjections
$\pi_{i}\colon C\to C/J_{i}$, $i\in\\{1,2\\}$, is surjective. We call the
unital $C^{*}$-algebra

$\displaystyle A_{1}\underset{\pi}{\circledast}A_{2}:=\big{\\{}x\in
C\underset{\min}{\otimes}A_{1}\underset{\min}{\otimes}A_{2}\,\big{|}\,(\pi_{i}\otimes\operatorname{id})(x)\in(C/J_{i})\otimes_{\min}A_{i},\,i\in\\{1,2\\}\big{\\}}$

the _fusion_ $C^{*}$-algebra of $A_{1}$ and $A_{2}$ over $C$. When $C$ is the
$C^{*}$-algebra $C([0,1])$ of all continuous complex-valued functions on the
unit interval and $\pi_{1}:=\mathrm{ev}_{1}$ and $\pi_{2}:=\mathrm{ev}_{0}$
are the evaluation maps respectively at $1$ and $0$, we call
$A_{1}\circledast_{\pi}A_{2}$ the _join_ $C^{*}$-algebra of $A_{1}$ and
$A_{2}\,$, and denote it by $A_{1}\circledast A_{2}$.

When $C$ is a commutative $C^{*}$-algebra, the projections $\pi_{i}$
correspond to inclusions of two disjoint closed subsets (playing the role of
points $0$ and $1$) into a compact Hausdorff space (playing the role of
$[0,1]$). Observe also that, if $A_{1}:=C(X)$ and $A_{2}:=C(Y)$ are the
$C^{*}$-algebras of continuous functions on compact Hausdorff spaces $X$ and
$Y$ respectively, then

$\displaystyle A_{1}\circledast A_{2}=C(X*Y).$

On the noncommutative side, it is worth mentioning that the join of matrix
algebras is used to construct the celebrated Jiang–Su $C^{*}$-algebra [8].

##  5 Join and fusion of a principal $\boldsymbol{H}$-comodule algebra with
$\boldsymbol{H}$

Since a diagonal coaction is not in general an algebra homomorphism, to obtain
an equivariant version of our noncommutative fusion and join constructions, we
need to modify Definition 4.1 in the spirit of (2.1), (2.2). To avoid
analytical complications that are tackled in [5], we also need to change the
setting from $C^{*}$-algebraic to algebraic. In particular, we replace
$\otimes_{\min}$ by $\otimes$.

######  Definition 5.1.

Let $\delta\colon P\to P\otimes H$ be a right coaction making $P$ a comodule
algebra, and let $C$ be a unital $C^{*}$-algebra equipped with two
$C^{*}$-ideals $J_{1}$ and $J_{2}$ such that the direct sum

$\displaystyle\pi_{1}\oplus\pi_{2}\colon\
C\longrightarrow(C/J_{1})\oplus(C/J_{2})$

of the canonical surjections $\pi_{i}\colon C\to C/J_{i}$, $i\in\\{1,2\\}$, is
surjective. We call the algebra

$\displaystyle P\underset{\pi}{\overset{\delta}{\circledast}}H:=\big{\\{}x\in
C\otimes P\otimes H\,\big{|}\,$
$\displaystyle\hphantom{P\underset{\pi}{\overset{\delta}{\circledast}}H:=\big{\\{}}{}(\pi_{1}\otimes\operatorname{id})(x)\in(C/J_{1})\otimes\delta(P),\,(\pi_{2}\otimes\operatorname{id})(x)\in(C/J_{2})\otimes\mathbb{C}\otimes
H\big{\\}}$

the _equivariant fusion_ algebra of $P$ and $H$ over $C$. When $C:=C([0,1])$
and $\pi_{1}:=\mathrm{ev}_{1}$, $\pi_{2}:=\mathrm{ev}_{0}$, we call
$P\circledast^{\delta}_{\pi}H$ the _equivariant join_ algebra of $P$ and $H$,
and denote it by $P\circledast^{\delta}H$.

######  Lemma 5.2.

The coaction $\operatorname{id}\otimes\operatorname{id}\otimes\Delta\colon
C\otimes P\otimes H\to C\otimes P\otimes H\otimes H$ restricts and corestricts
to $\Delta_{P\circledast{}^{\delta}_{\pi}H}\colon
P{\circledast}^{\delta}_{\pi}H\to(P{\circledast}^{\delta}_{\pi}H)\otimes H$
making $P{\circledast}^{\delta}_{\pi}H$ a right $H$-comodule algebra.

###### Proof.

Note first that for $i\in\\{1,2\\}$, we have

$\displaystyle\big{(}(\pi_{i}\otimes\operatorname{id}\otimes\operatorname{id})\circ(\operatorname{id}\otimes\operatorname{id}\otimes\Delta)\big{)}\\!\bigg{(}\sum_{j}f_{j}\otimes
p_{j}\otimes
h_{j}\bigg{)}=(\operatorname{id}\otimes\operatorname{id}\otimes\Delta)\\!\bigg{(}\sum_{j}\pi_{i}(f_{j})\otimes
p_{j}\otimes h_{j}\bigg{)}.$

Assume now that $\sum_{j}f_{j}\otimes p_{j}\otimes h_{j}\in
P\circledast^{\delta}_{\pi}H$. Then the above tensor belongs to

$\displaystyle(C/J_{1})\otimes\big{(}(\operatorname{id}\otimes\Delta)(\delta(P))\big{)}=(C/J_{1})\otimes\big{(}(\delta\otimes\operatorname{id})(\delta(P))\big{)}\subseteq(C/J_{1})\otimes\delta(P)\otimes
H$

for $i=1$ and to $(C/J_{2})\otimes\mathbb{C}\otimes H\otimes H$ for $i=2$. ∎

The main result of this paper is:

######  Theorem 5.3.

Let $C$ be a unital $C^{*}$-algebra equipped with two $C^{*}$-ideals $J_{1}$
and $J_{2}$ such that the direct sum $\pi_{1}\oplus\pi_{2}\colon
C\rightarrow(C/J_{1})\oplus(C/J_{2})$ of the canonical surjections
$\pi_{i}\colon C\to C/J_{i}$, $i\in\\{1,2\\}$, is surjective. Then, for any
principal right $H$-comodule algebra $P$, the equivariant fusion right
$H$-comodule algebra $P{\circledast}^{\delta}_{\pi}H$ is principal.

###### Proof.

Note first that, due to the surjectivity of $\pi_{1}\oplus\pi_{2}$, we can
choose $x\in C$ such that

$\displaystyle(\pi_{1}\oplus\pi_{2})(x)=(1,0).$

Since $\\{0,1\\}\subseteq\operatorname{spec}(x^{*}x)$, we can take a
continuous function $f\colon\operatorname{spec}(x^{*}x)\to[0,1]$ such that
$f(0)=0$ and $f(1)=1$, and define $t:=f(x^{*}x)\in C$. Then there exist
elements $\sqrt{t},\sqrt{1-t}\in C$ enjoying the properties
$\pi_{2}(\sqrt{t})=0=\pi_{1}(\sqrt{1-t})$.

Next, let $\ell\colon H{\to}P\otimes P$ be a strong connection on $P$. Then
the bicolinearity of $\ell$ implies that the linear map

$\displaystyle\tilde{\ell}\colon\ H\longrightarrow\big{(}C\otimes P\otimes
H\big{)}\otimes\big{(}C\otimes P\otimes H\big{)},$
$\displaystyle\tilde{\ell}(h):={\sqrt{t}}\otimes\ell(h_{(2)})^{\langle
1\rangle}\otimes S(h_{(1)})\otimes{\sqrt{t}}\otimes\ell(h_{(2)})^{\langle
2\rangle}\otimes h_{(3)}$
$\displaystyle\hphantom{\tilde{\ell}(h):=}{}+\sqrt{1-t}\otimes 1\otimes
S(h_{(1)})\otimes\sqrt{1-t}\otimes 1\otimes h_{(2)},$

corestricts to
$(P\circledast^{\delta}_{\pi}H)\otimes(P\circledast^{\delta}_{\pi}H)$. Indeed,

$\displaystyle(\pi_{1}\otimes\operatorname{id}\otimes\operatorname{id})\otimes(\operatorname{id}\otimes\operatorname{id}\otimes\operatorname{id})(\widetilde{\ell}(h))$
$\displaystyle\qquad{}=\big{(}[1]\otimes\delta\big{(}\ell(h_{(1)})^{\langle
1\rangle}\big{)}\big{)}\otimes\big{(}\sqrt{t}\otimes\ell(h_{(1)})^{\langle
2\rangle}\otimes
h_{(2)}\big{)}\in\big{(}C/J_{1}\otimes\delta(P)\big{)}\otimes\big{(}C\otimes
P\otimes H\big{)},$
$\displaystyle(\pi_{2}\otimes\operatorname{id}\otimes\operatorname{id})\otimes(\operatorname{id}\otimes\operatorname{id}\otimes\operatorname{id})(\widetilde{\ell}(h))$
$\displaystyle\qquad{}=\big{(}[1]\otimes 1\otimes
S(h_{(1)}\big{)}\otimes\big{(}\sqrt{1-t}\otimes 1\otimes
h_{(2)}\big{)}\in\big{(}C/J_{2}\otimes\mathbb{C}\otimes
H\big{)}\otimes\big{(}C\otimes P\otimes H\big{)}.$

Much in the same way,

$\displaystyle(\operatorname{id}\otimes\operatorname{id}\otimes\operatorname{id})\otimes(\pi_{1}\otimes\operatorname{id}\otimes\operatorname{id})(\widetilde{\ell}(h))\in\big{(}C\otimes
P\otimes H\big{)}\otimes\big{(}C/J_{1}\otimes\delta(P)\big{)},$
$\displaystyle(\operatorname{id}\otimes\operatorname{id}\otimes\operatorname{id})\otimes(\pi_{2}\otimes\operatorname{id}\otimes\operatorname{id})(\widetilde{\ell}(h))\in\big{(}C\otimes
P\otimes H\big{)}\otimes\big{(}C/J_{2}\otimes\mathbb{C}\otimes H\big{)}.$

The bicolinearity of the thus corestricted $\tilde{\ell}$ is evident. Finally,
taking advantage of the right colinearity and the splitting property of
$\ell$, we check that

$\displaystyle\widetilde{\operatorname{can}_{P\circledast^{\delta}_{\pi}H}}(\tilde{\ell}(h))=t\otimes\ell(h_{(2)})^{\langle
1\rangle}\ell(h_{(2)})^{\langle 2\rangle}\otimes S(h_{(1)})h_{(3)}\otimes
h_{(4)}$
$\displaystyle\hphantom{\widetilde{\operatorname{can}_{P\circledast^{\delta}_{\pi}H}}(\tilde{\ell}(h))=}{}+(1-t)\otimes
1\otimes S(h_{(1)})h_{(2)}\otimes h_{(3)}$
$\displaystyle\hphantom{\widetilde{\operatorname{can}_{P\circledast^{\delta}_{\pi}H}}(\tilde{\ell}(h))}{}=t\otimes\ell(h_{(2)})^{\langle
1\rangle}\ell(h_{(2)})^{\langle 2\rangle}\\!{}_{(0)}\otimes
S(h_{(1)})\ell(h_{(2)})^{\langle 2\rangle}\\!{}_{(1)}\otimes
h_{(3)}+(1-t)\otimes 1\otimes 1\otimes h$
$\displaystyle\hphantom{\widetilde{\operatorname{can}_{P\circledast^{\delta}_{\pi}H}}(\tilde{\ell}(h))}{}=t\otimes
1\otimes S(h_{(1)})h_{(2)}\otimes h_{(3)}+(1-t)\otimes 1\otimes 1\otimes h$
$\displaystyle\hphantom{\widetilde{\operatorname{can}_{P\circledast^{\delta}_{\pi}H}}(\tilde{\ell}(h))}{}=1\otimes
1\otimes 1\otimes h.$

Now the claim follows from Lemma 3.3. ∎

Observe that in the special case of the join construction, we can take $t$ to
be the inclusion map $[0,1]\to\mathbb{C}$. Moreover, as explained in the next
section, the equivariant join comodule algebra $P\circledast^{\delta}H$
becomes piecewise trivial, and Theorem 5.3 follows from [7, Lemma 3.2].
However, even in this special case of the equivariant join comodule algebra,
our non-unital strong-connection formula is differently constructed than the
complicated strong-connection formula in [7, Lemma 3.2]. This might be very
important for index pairing computations involving concrete strong-connection
formulas. Finally, let us remark that a fully fledged $C^{*}$-algebraic
approach to Theorem 5.3 can be found in [5].

######  Example 5.4.

In particular, taking $P=H$ to be the Hopf algebra of ${\rm SU}_{q}(2)$, we
can conclude the principality of Pflaum’s noncommutative instanton bundle
[10]. The fact that this bundle is not trivializable at the $C^{*}$-algebraic
level was proved in [1] using [4]. In the classical setting, except for the
trivial group, all joins $G*G$, where $G$ is a compact Hausdorff group, are
non-trivial as principal $G$-bundles. We conjecture the same is true for all
compact quantum groups [13].

##  6 Piecewise structure

This section is an adjustment of [7, Section 5.1] to the setting of
equivariant join construction. It is important to bear in mind that we cannot
expect to have such a piecewise (pullback) structure in the general case of
equivariant fusion comodule algebra.

Let $P$ be a principal right $H$-comodule algebra with a coaction $\delta$.
First we define the following algebras

$\displaystyle P_{1}:=\left\\{x\in C([0,1])\otimes P\otimes
H\,\big{|}\,(\mathrm{ev}_{0}\otimes\operatorname{id}\otimes\operatorname{id})(x)\in\mathbb{C}\otimes
H)\right\\},$ $\displaystyle P_{2}:=\left\\{x\in C([0,1])\otimes P\otimes
H\,\big{|}\,(\mathrm{ev}_{1}\otimes\operatorname{id}\otimes\operatorname{id})(x)\in\delta(P)\right\\}.$

The $P_{i}$’s become $H$-comodule algebras for the coactions obtained by the
restrictions and corestrictions of
$\operatorname{id}\otimes\operatorname{id}\otimes\Delta$, and the subalgebras
of $H$-coaction invariants are respectively

$\displaystyle B_{1}:=\\{y\in C([0,1])\otimes
P\,|\,(\mathrm{ev}_{0}\otimes\operatorname{id})(y)\in\mathbb{C}\\},$
$\displaystyle B_{2}:=\\{y\in C([0,1])\otimes
P\,|\,(\mathrm{ev}_{1}\otimes\operatorname{id})(y)\in
P^{\operatorname{co}H}\\}.$

Now one can identify $P$ with the pullback comodule algebra

$\displaystyle\\{(p,q)\in P_{1}\oplus
P_{2}\,|\,(\mathrm{ev}_{1}\otimes\operatorname{id})(p)=(\mathrm{ev}_{0}\otimes\operatorname{id})(q)\\}$

of the $P_{i}$’s along the right $H$-colinear algebra homomorphisms

$\displaystyle(\mathrm{ev}_{1}\otimes\operatorname{id})\colon\
P_{1}\longrightarrow P\otimes
H,\qquad(\mathrm{ev}_{0}\otimes\operatorname{id})\colon\ P_{2}\longrightarrow
P\otimes H.$

### Acknowledgements

All authors are grateful to Piotr M. Sołtan and Karen R. Strung for references
concerning the minimal tensor product and the Jiang–Su $C^{*}$-algebra
respectively. Ludwik Da̧browski and Piotr M. Hajac were partially supported by
PRIN 2010-11 grant “Operator Algebras, Noncommutative Geometry and
Applications” and NCN grant 2011/01/B/ST1/06474, respectively. Tom Hadfield
was financed via the EU Transfer of Knowledge contract MKTD-CT-2004-509794.
Also, Piotr M. Hajac is very thankful to SISSA for its hospitality.

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