Scaling bat wingbeat frequency and amplitude
1 43 Murray Drive, Hillarys, Western Australia, Australia 6025
2 Department of Conservation and Land Management, PO Box 51, Wanneroo,
Western Australia, Australia, 6065
* Author for correspondence (e-mail: bullen2{at}bigpond.com)
Accepted 6 June 2002
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Summary |
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The following relationship predicts wingbeat amplitude to within
±15° from flight speed and wing area (SREF,
m2) at all flight speeds:
w=56.92+5.18V+16.06log10SREF.
This equation is based on data up to and including speeds that require maximum
wingbeat amplitude to be sustained. For most species, the maximum wingbeat
amplitude was 140°.
Key words: bat, scaling, wingbeat, frequency, amplitude
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Introduction |
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In mammals, the duration of a muscle's contraction is adapted to its
function, and the contraction performance of the muscle is affected by the
resistance that it works against (Guyton
and Hall, 1996). If there were gross variations in the speed of
operation of the muscles driving the wingbeat of bats with different
phylogenetic relationships, foraging strategies or microhabitats, then we
would expect the relationships between wingbeat frequency
(fw) and airframe variables (such as mass, wing area and
wing span) to be complex.
In this study, we measure wingbeat frequency and amplitude across a range of flight speeds for 23 species representing all families of insectivorous, frugivorous and carnivorous bats that occur in tropical and temperate regions of Western Australia. We then propose a general model linking these variables to various airframe attributes and flight speed.
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Materials and methods |
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Chalinolobus gouldii Grey; Chalinolobus morio Grey; Chalinolobus nigrogriseus Gould; Hipposideros ater, Templeton; Macroderma gigas Dobson; Mormopterus planiceps Peters; Miniopterus schreibersii Kuhl; Nyctophilus arnhemensis Johnson; Nyctophilus geoffroyi Leach; Nyctophilus gouldi Tomes; Nyctophilus timoriensis Geoffroy; Pteropus poliocephalus Temminck; Pteropus scapulatus Peters; Rhinonycteris aurantius Grey; Scotorepens balstoni Thomas; Saccolaimus flaviventris Peters; Scotorepens greyii Grey (previously Nycticeius balstoni caprenus Troughton); Tadarida australis Grey; Taphozous georgianus Thomas; Taphozous hilli Kitchener; Vespadelus finlaysoni Kitchener, Jones and Caputi (previously Eptesicus pumilis Gray); Vespadelus regulus Thomas.
Data for two different populations of Nyctophilus timoriensis (Ntg and Nts/w) are presented and treated as separate species. The arid population Ntg inhabits the Coolgardie woodlands and has a mean mass of 11 g, whereas the mesic population Nts/w is endemic to the forests of southwestern Western Australia and has a mean mass of 14.2 g.
Data collection
Relevant aspects of species foraging ecologies and airframe measures were
collected from existing literature. Only publications using a consistent
measurement technique were used. This measurement protocol, relevant formulae
and a discussion of aerodynamic mechanisms and implications are provided in
Bullen and McKenzie (2001).
Capture and release techniques were used to collect a library of video
recordings complete with flight speed measurements. Bat flight was filmed
using video cameras (Sony Video8 Professional CCD-V100E in VHS format and Sony
digital Beta-cam model DVW-709WSP at a shutter speed of 1/250 s), both running
at 24 frames s-1. Wingbeat frequency and amplitude values were
determined from a frame-by-frame playback. Note that the Beta format video
actually showed two clear images of the bats wing position when replayed
via VHS because of the differences in the recording protocols of the
two video standards. It gives an effective frame rate of 48 frames
s-1 for test points recorded with the DigiBeta camera. Limited data
were also collected in an indoor observation chamber at high frame rates using
a cine camera running at 200 frames s-1 (Photosonics; Burbank, CA,
USA; model 61-1100).
Flight speeds were measured continuously in all cases using a hand-held K-band radar gun (model TS3, Municipal Electronics, UK, calibrated for a speed range of 1-28 m s-1). These speeds were `called' into a hand-held recorder and, if applicable, into the audio feature of the video camera while each test animal was being filmed. For each test, the angle between the gun's line of sight to the bat and the bat's line of flight was estimated by the operator and `called' into the recorder. A cosine correction was applied to the measured flight speeds to correct for this angle. Data corresponding to angles greater than 45° were ignored.
The mean angle of the wing between shoulder and tip, above or below the
body axis reference dorsal plane, was estimated within ±5° for each
frame in sequence. Note that this method is different from that used by
Pennycuick (1996) on birds.
Pennycuick (1996
) estimated
the angle created by the shoulder-to-wrist joint line only. A bat's hand wings
reaches higher positional angles than its arm wing at the end of the stroke,
so our method gives higher amplitude values. These were plotted against time
(Fig. 1) and fitted with
splines using Microsoft EXCEL. A minimum of three complete wingbeat cycles was
required to calculate frequency, fw, and amplitude,
w, reliably. At 24 frames s-1, a family of lines
can be fitted to the sequences, differing in their frequencies by a factor of
3. Given that bats with a mass of less than 50 g are known to use
fw in the range of 3-12 Hz
(Carpenter, 1985
;
Van Den Berg and Rayner,
1995
), the curve with the lowest fw was used
for all species because its frequency always fell within this range. This also
agrees with our own high frame rate data. The fw
information was then deduced directly from the time histories. The spline for
each low frame rate (VHS at 24 frames s-1) test point was reviewed
to obtain amplitude. The maximum and minimum amplitudes were then averaged and
compared to give
w values for each test point. Given that
the test points were all taken during periods approximating steady level
flight, peaks that were clearly out of phase with the sequence were ignored in
this average (see Fig. 1B).
This method is expected to give maximum and minimum
w values
slightly lower in magnitude than those obtained from high frame rate (>100
frames s-1) cine cameras. Because of the impracticability of
extensive use of high-frame-rate cine in the majority of our field
experimental situations, this was not attempted, and low-frame-rate video was
used to maximise data collection. See discussion below of the effect of this
procedure on the results.
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Sub-adults, pregnant females and animals with damaged wings, or that were visibly distressed or considered significantly underweight, were excluded. The methods used did not result in injury to or the death of the bats tested.
Four strategies were used to collect wingbeat data over a wide range of
flight speeds. First, bats were flown in a flight chamber to collect low-speed
data. Individual adult bats were released to fly around in a large, well-lit
room (11 m long, 5 m wide and 3.2 m high). All species were able to maintain
continuous level flight in this room. Although Mormopterus planiceps,
Chalinolobus gouldii and Tadarida australis did not achieve
their typical in-field flight speeds (see
Bullen and McKenzie, 2001),
they were flying 0.3-3 m s-1 (1-10 km h-1) above their
usual minimum steady level flight speed (R. D. Bullen and N. L. McKenzie,
unpublished data). Thus, they had a considerable margin of power for
manoeuvring. The floor, ceiling and walls of the room were painted in shades
of white or cream, which contrasted with the brown and black colours of the
fur and wing membranes of the bats. This method gave excellent coverage of the
lower speed range of the bats.
Second, free-air hand releases in daylight were used to collect mid- and high-speed data. The same video and speed measuring equipment was used. The bats were prone to escape after release by accelerating to high speed. Results were most readily obtained when the released bat was filmed against bright, monotonous backgrounds such as grass or sky. The initial period of 2-3 s, while bats accelerated from rest to their normal flight speed range, was excluded from data analysis. If, during the test point, the speed of the bat varied marginally (typically less than ±1 m s-1), then the speed at the mid-point of the run was taken as the average value for that run. If the speed varied by more that ±1 m s-1 then the run was broken into two or more test points. All readings for a species were pooled.
Third, daytime free-flying data were collected from large pteropodids as they commuted from roost to roost. Because of their size, it was possible to film the bats in flight in full daylight and to record their speed. Again, cosine corrections were applied to the measured flight speeds, as described above, to account for the off line-of-sight measurement errors.
Fourth, night-time free-flight data were also collected to supplement the
first two strategies and to check whether different wingbeat values were
obtained in a natural situation. Echolocation recordings were taken from
free-flying bats in situations while the foraging bat could be seen, lines of
flight estimated and speeds measured. An Anabat II ultrasound detector (Titley
Electronics, Australia) was used with its output stored directly onto
audiocassette tapes using a Sony Walkman Professional (WMD6C) tape recorder.
The species identity and fw values were then derived from
the recorded call sequences using COOL EDIT 2000 (Syntrillium Software, USA).
The species was identified by reference to a library of reference calls, and
the fw data were derived based on a direct correlation of
the wingbeat frequency with the echolocation call rate
(Lancaster et al., 1995).
These sequences were not filmed and did not provide data on
w.
Of the 23 species represented in this study, 11 provided data over the flight speed range of 3-9 m s-1 that is the majority of their speed range. Seven species provided data over the range less than 6 m s-1, covering their low-speed range only, and five provided data at speeds greater than 5 m s-1, which is their high-speed range only. Despite having scant or incomplete data sets, these last two categories were included to assess whether the generalised scaling model applied to all types of bat.
The fw and w data for each species
were then plotted against flight speed (refer, for an example plot, to
Fig. 2) and the plots reviewed
for a general pattern.
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Previously published wingbeat data for a number of other species are
included for comparison: Eidolon helvum (m=315 g),
high-speed cine data (Carpenter,
1986); Hypsignathus monstrosus (m=260 g),
high-speed cine data (Carpenter,
1986
); Myotis dasycneme (m=20 g), stroboscopic
flash data at 30 Hz (Britton et al.,
1997
); Noctilio leporinus (m=70 g), synchronised
cameras at 20 frames s-1
(Schnitzler et al., 1994
);
Pipistrellus pipistrellus (m=5 g), high-speed video at 250
frames s-1 (Thomas et al.,
1990
); Pteropus poliocephalus (m=700 g), manual
and high-speed cine data (Carpenter,
1985
); Rhinolophus ferrumequinum (m=22 g),
stroboscopic flash data at 100 and 200 Hz
(Aldridge, 1986
); Rousettus
aegyptiacus (m=180 g), high-speed cine data
(Carpenter, 1986
).
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Results |
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An example of the fw and w data for
Chalinolobus gouldii is given in
Fig. 2 and a second example for
fw, Mormopterus planiceps, is given in
Fig. 6. For all 11 species
assessed over a wide range of flight speed, fw initially
decreased with increasing speed until a mid-range speed was reached.
fw then remained relatively constant until high speeds
were achieved. This pattern can also be seen clearly in Figs
2A and
6. Regarding amplitude,
Fig. 2 also illustrates a
pattern common to all 11 species:
w increased with speed,
although there was very wide scatter.
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Estimation of wingbeat frequency
A summary plot of the relationship at low flight speeds between
fw and mass (m) is presented in
Fig. 3A. The
fw values correspond to Vmr that is
approximately midway through the flight-speed region that exhibits the
distinct reduction of fw. The line of best fit for
fw is also given. Fig.
3A shows good correlation between fw at low
speed and mass. The relationship is convenient to use for estimating
fw in bats at low speed and is given for
V=Vmr by:
![]() | (1) |
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At high flight speeds, mass was again a good predictor of
fw (Fig.
3B). Excluding the two outliers (discussed below), the line of
best fit for V<6 ms-1 was:
![]() | (2) |
Provided bat mass is known, the wingbeat frequency for any bat can be estimated at low or high flight speed to within ±1.5 Hz.
To improve the fidelity of the estimate and to provide a general
relationship across all speeds, we applied a multiple-parameter least-squares
regression analysis to the full data set and a range of morphological
variables. Using fw, m and flight speed
(V), a linear model explained 65.0 % of the variation
(P<0.00001, ±1.24 Hz). Including span and area in this
model improved the fit slightly to explain 73.0 % of the variation
(P<0.00001, ±1.09 Hz). A scatterplot of
fw versus flight speed suggested that more of the
variation would be explained by using a non-linear model. We therefore
evaluated polynomial and logarithmic fits. Including
log10m and log10V increased the
r2 values to 0.748. Including span and area did not
significantly improve the fit. Wingbeat frequency is then given over the full
flight speed and mass ranges by:
![]() | (3) |
This model is presented in Fig. 4.
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Equation 3 is presented in Fig. 5 for Chalinolobus gouldii, for comparison with the data of Fig. 2.
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The two outliers in Fig. 3B
have high-speed fw values significantly below the line
represented by this equation. They are Mormopterus planiceps (this
study) and Noctilio leporinus
(Schnitzler et al., 1994),
which have fw values corresponding to approximately 65 %
of the value predicted by Equation 2. Our empirical data on Mormopterus
planiceps are presented in Fig.
6. The Mormopterus planiceps outlier is a series of low
fw points treated separately. The upper series in
Fig. 6 is accurately
represented by the scaling equations.
A summary of the data included in the study is given in Table 4.
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Estimation of wingbeat amplitude
When variation in wingbeat amplitude was evaluated against flight speed and
morphological variables, flight speed (V) and wing area
(SREF) explained most of the variation
(Fig. 7A).
![]() | (4) |
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Equation 4 is presented in Fig. 5 for Chalinolobus gouldii for comparison with the data of Fig. 2.
The maximum and minimum values of w recorded during the
study are presented in Table 5
for each species. They are much higher than the extrapolations based on
Equation 4. This is to be expected given that pectoral girdle anatomy clearly
permits very high
w values to be used for extreme speeds
beyond our data set as well as during manoeuvres and periods of high
acceleration when extra power is required. For the reasons given in the legend
to Fig. 2, low flight speed
test points at amplitudes approaching the maximum and minimum values were
observed (typically +40 and -80°), but were not included in the derivation
of Equation 4.
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Discussion |
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Strong family-level phylogenetic relationships are apparent in our results.
Vespertilionidae are at one end of the relationship, Emballonuridae in the
middle and Pteropodidae at the opposite end. Given the high selective
pressures on morphological characters associated with modes of nutrition and
foraging, phylogeny in the absence of ecologically driven aerodynamic function
would be highly unlikely to produce optimum aerodynamic functionality. The
similarity in aerodynamic optimisations apparent in the various families of
bats presented here is consistent with the assumption that the model linking
morphological variables to flight speed and wingbeat kinematics is
functionally based, rather than an aerodynamically trivial artefact of
phylogenetic relationships (Felsenstein,
1982; McKenzie et al.,
1995b
).
The pooled species data included in this study show fw
values ranging from 3 to 12 Hz. Species with fw values up
to 12 Hz probably have downstroke and upstroke muscles composed of the fast
fibre type (Guyton and Hall,
1996). In species with fw values down to 3 Hz,
it is possible that the slow fibre type dominates. This hypothesis is based on
our observation that the larger bats with slow wingbeat frequencies are those
that are known to travel long distances (Pteropus poliocephalus,
refer to Churchill, 1998
;
Saccolaimus flaviventris, refer to
Strahan, 1995
; Tadarida
australis, refer to Churchill,
1998
) or migrate on an annual cycle. Further work is recommended
to confirm this suggestion.
The variation of fw with flight speed for these 23
species shows a two-stage characteristic that is reflected by the need for a
logarithmic equation (e.g. Fig.
6). At low speeds, fw changes with flight
speed, whereas at higher speeds it is nearly constant. This is consistent with
the rule that only a limited range of wingbeat frequencies are available to a
species (Rayner, 1985) to
provide the endurance required for long sustained flights. Initially,
fw decreases as flight speed increases, until cruising
speed ranges are reached (see mode speed data in
Table 3). The reduction is
small, approximately 2 Hz. At and above cruising speed, fw
appears to remain almost constant until the bats reach their extreme high
speed (e.g. Figs 2 and
6). In contrast to bats, bird
flight is more variable. Some birds show no change with speed
(Tobalske and Dial, 1996
),
others show a more-or-less linear relationship with speed
(Tobalske, 1995
) and still
others show a U-shaped relationship, with an initial fall in
fw with increasing speed followed by an increase of
fw at even higher speeds
(Bruderer et al., 2001
;
Park et al., 2001
). The
two-stage relationship in bats differs from published bird data. This almost
constant relationship between fw and flight speed at high
speed in the bats we have assessed may be due to use of their most efficient
muscle contraction frequency in the flight speed region of rapidly increasing
`opposing loading' and, therefore, metabolic power requirement. The opposing
loading applied to the muscles is due to the rapidly increasing drag
airloads.
For low-speed fw data, the model predicts a value that
lies in the centre of the scatter of the available empirical data for 19
species (Fig. 3A). In fact,
empirical values are within 1 Hz (approximately 1 S.D.) of the fitted model
across the speed range. Three of the remaining species show a bias when the
model is compared with the data (Mormopterus planiceps, Noctilio
leporinus and Nyctophilus timoriensiss/w). The model
underestimates fw data for Nyctophilus
timoriensiss/w (this study) and overestimates Noctilio
leporinus (Schnitzler et al.,
1994) by approximately 1 Hz. However, our data on Nyctophilus
timoriensiss/w are scant and the apparent bias may disappear
with more data. The model overestimates the frequency data for the `low range'
of Mormopterus planiceps (referred to as `outliers' in the results).
For high-speed fw data, the model predicts the empirical
data for 18 of 19 species, including the high fw range of
Mormopterus planiceps. The low fw range of
Mormopterus planiceps and Noctilio leporinus
(Schnitzler et al., 1994
) are
substantially overestimated. In both species, empirical wingbeat frequencies
are approximately 65 % of the value predicted by the model.
Mormopterus planiceps was unique among the 23 species assessed
during this study. The empirical fw data are arrayed in
two parallel series across the full flight speed range
(Fig. 6). The model predicted
the main (higher) frequency series. The other series averaged 3 Hz lower and,
provided that it is not a sampling artifact, would reduce the resultant
airspeed past the wing during the down- and upstroke. Preliminary calculations
indicate a consequent reduction in the profile power fraction of the wing of
approximately 4 % and in the inertial power fraction of 67 %. Preliminary
dissections by the authors also revealed that Mormopterus planiceps
has a very low flight muscle mass to total mass fraction of wing down-stroke
and up-stroke muscle groups (Vaughan,
1970; Hermanson and Altenbach,
1985
), approximately 7.5 % of total mass compared with a more
typical range of approximately 9-11 % for similar insectivores. Taken
together, these observations suggest a particular optimisation of this tiny
interceptor, in which the upper fw series is used for
acceleration to speed and for manoeuvring to intercept prey, while the lower
fw range is used for efficient cruising/commuting.
By comparison with birds, bats have a 50 % higher wingbeat frequency for a
given size range. Pennycuick
(1996) gives a model for bird
frequency based upon mass, wing span and wing area. This model is compared
with our high-speed bat data in Fig.
8.
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We had empirical data on w for 24 species (including
Rhinolophus ferrumequinum
(Aldridge, 1986
)). No species
departed substantially from the general model given by Equation 4. However,
unlike fw, all species showed a high level of scatter
(±20°) in the raw amplitude data (e.g.
Fig. 2), forming effectively
two blocks that represent the low- and high-speed experimental data collection
strategies (see Fig. 2). The
bats were studied in free flight at all times, during which climbing,
descending, accelerating and decelerating flight would require differences in
lift, drag and thrust. It is possible for the bat to generate lift and thrust
by changing the mean wingbeat angle of attack (
), the average airspeed
over the wings (Vwing), the wingbeat frequency and/or
amplitude. At low angles of attack, lift is directly proportional to the
product of
and Vwing2. For level
flight, when lift exactly equals weight, the bat must change the flow velocity
over its wings by changing forward speed, fw and/or
w if it changes its mean wing
during the stroke,
otherwise, it will climb or descend. In addition, to increase speed in level
flight, the bat must generate more thrust by using higher
,
fw and/or
w to offset the increasing
drag. Given that we have shown that fw is relatively
constant across the full speed range of bats,
must decrease as flight
speed increases (for constant or increasing
w) otherwise the
bat will generate excess lift and climb. To this end, wingbeat amplitude must
increase to generate the increased thrust, resulting in an even higher mean
Vwing value and an even lower mean value of
.
Equation 4 therefore represents the steady level-flight wingbeat amplitude
independent of the experimental context in which the data were collected. Note
that the previous study (Aldridge,
1986
) published data on
w of bats over a narrow
range of speeds (2.7-4.8 m s-1) using a flight tunnel. These data
fall within the predictions of our equation.
The difference between data recorded at 24 frames s-1 compared
with higher frame rates is given in Table
6 and can be seen in Fig.
1. At high wingbeat frequencies (>9.5 Hz), VHS video camera or
a cine camera with 24 frames s-1 and a slow shutter speed give an
accurate representation of the extrema of the wing positional angle, because
they occur at a repetition rate that ensures a high probability of the extrema
coinciding with the relatively long open shutter/scan time period. These
extrema are then used as the frame w value. Similarly, 24
frames s-1 is sufficient to capture relatively slowly moving wings
at low fw (<7 Hz) within 5° of its maximum position
(see Fig. 1C). There is,
however, a range of wing frequencies (approximately 7-9.5 Hz) used by bats of
20-50 g to which neither of these situations applies. For these bats, care
must be taken to use only the 24 frames s-1 frame images that
clearly show a significant variation in angle from frame to frame. Frames that
do not fit this criterion should not be included in averaging
w values for the test point. This effect is apparent in the
Tadarida australis data of Table
6, which underestimate actual amplitude by 10-20°.
Underestimation occurred in four of our bats: Taphozous hilli, Taphozous
georgianus, Tadarida australis and Saccolaimus. flaviventris.
Even so, this bias is of the same order as the overall scatter in the data
collected, and the data for these species have therefore been included in the
overall regression analysis. Fig.
7C shows that including these data has little effect on the
regression result. Data from Fig.
1A show that this effect is reduced at frame rates of 50 frames
s-1 and is not apparent at rates beyond 100 frames
s-1.
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Field observations of bats `hand-released' in daylight
(Table 5) suggest that bats
approach their amplitude limits of approximately 50-60° above and
80-90° below the reference dorsal plane in extreme flight conditions. This
gives a theoretical maximum of 140-150° for wingbeat amplitude at the high
speed extreme of the flight speed range, compared with a more typical range of
40-80°. This theoretical maximum will be influenced by the back, shoulder,
elbow and wrist morphology of the various species. Given that maximum
efficiency in skeletal muscle ordinarily occurs when the velocity of
contraction is approximately 30% of maximum
(Guyton and Hall, 1996) and
that fw is virtually constant in the bat's upper speed
range, our result of a threefold increase in amplitude at extreme speeds is
consistent with constant fw and best use of muscle
efficiency in the bat's normal speed range.
The relationships between fw, w and
flight speed are defined by two simple equations involving mass and wing area.
The same equations fitted tropical as well as temperate species, megabats and
microbats, the six microbat families assessed and species with the full range
of foraging ecologies. One scaling model fitted all. Its simplicity implies
that a single theme underlies bat aerodynamics. This argument is not circular
because no bats showed substantial departures from the model, despite
differences in foraging niche, climatic range and phylogeny. In this respect
at least, the kinematics of bat flight is different from that of birds.
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List of symbols |
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Acknowledgments |
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References |
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