# Quantum Transport in Graphene Using Surface Acoustic Wave Resonators

Yawen Fang  Laboratory of Atomic and Solid State Physics, Cornell University,
Ithaca, NY, 14853, USA  Yang Xu  School of Applied and Engineering Physics and
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY,
USA  Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing, China  Kaifei Kang  School of
Applied and Engineering Physics, Cornell University, Ithaca, NY, USA  Benyamin
Davaji  Department of Electrical and Computer Engineering, Northeastern
University, Boston, MA, USA  Kenji Watanabe  National Institute for Materials
Science, Tsukuba, Japan  Takashi Taniguchi  National Institute for Materials
Science, Tsukuba, Japan  Amit Lal  Department of Electrical and Computer
Engineering, Cornell University, Ithaca, NY, USA  Kin Fai Mak  School of
Applied and Engineering Physics and Laboratory of Atomic and Solid State
Physics, Cornell University, Ithaca, NY, USA  Kavli Institute at Cornell for
Nanoscale Science, Ithaca, NY, USA  Jie Shan  School of Applied and
Engineering Physics and Laboratory of Atomic and Solid State Physics, Cornell
University, Ithaca, NY, USA  Kavli Institute at Cornell for Nanoscale Science,
Ithaca, NY, USA  B. J. Ramshaw<EMAIL_ADDRESS>Laboratory of Atomic
and Solid State Physics, Cornell University, Ithaca, NY, 14853, USA  Kavli
Institute at Cornell for Nanoscale Science, Ithaca, NY, USA

###### Abstract

Surface acoustic waves (SAWs) provide a contactless method for measuring the
wavevector-dependent conductivity. This technique has been used to discover
emergent length scales in the fractional quantum Hall regime of traditional,
semiconductor-based heterostructures. SAWs would appear to be an ideal match
for van der Waals (vdW) heterostructures, but the right combination of
substrate and experimental geometry to allow access to the quantum transport
regime has not yet been found. We demonstrate that SAW resonant cavities
fabricated on LiNbO3 substrates can be used to access the quantum Hall regime
of high-mobility, hexagonal boron nitride (hBN) encapsulated graphene
heterostructures. Our work establishes SAW resonant cavities as a viable
platform for performing contactless conductivity measurements in the quantum
transport regime of vdW materials.

Introduction.—Spectroscopic probes of frequency-dependent conductivity are
ubiquitous in condensed matter, but analogous techniques for measuring the
momentum-dependent conductivity are relatively scarce. The fundamental problem
is that photons have millimeter-scale wavelengths at the energy scales
relevant to emergent phenomena, but the length scales of these phenomena, such
as charge density waves or moiré potentials, are typically nanometers to
microns.

This problem is solved by replacing light with sound. Sound waves traveling
across the surface—surface acoustic waves (SAWs)—of a piezoelectric material
create an oscillating electric field at the sound wavelength: tens-of-microns
to tens-of-nanometers at MHz to GHz frequencies. This oscillating electric
field interacts with a conducting material placed in proximity to the SAW. By
measuring the resultant changes in sound velocity and attenuation, one
measures the material’s conductance at the SAW wavelength. This technique is
contactless, bulk, measures at finite wavelength, and is compatible with low
temperatures and high magnetic fields. In the sense that it is a bulk,
contactless technique, SAWs can be thought of as the transport counterpart to
the thermodynamic measurement of quantum capacitance [1].

SAWs have been used to great effect in GaAs/AlGaAs heterostructures, where
naturally-piezoelectric GaAs substrates generate SAWs that interact with the
2D electron gas [2]. SAWs were used to show that current-induced modification
of the bubble and stripe phases is a local phenomenon [3], to measure the
periodicity and energy-momentum dispersion of the stripe phase at filling
factor $9/2$ [4], and to measure the Fermi surface in the composite Fermi
liquid at filling factor $1/2$ [5].

Figure 1: Combining vdW materials with surface acoustic waves. (a) Schematic
of a SAW resonator driven by a radiofrequency (RF) source across the
interdigital transducers (IDTs). A vdW heterostructure is placed in the center
of the cavity and connected to gate electrodes. (b) The change in SAW velocity
($\delta v/v$) and attenuation ($\Gamma$) as a function of the longitudinal
sheet conductivity of a material placed in contact with the resonator. The
transition region, where the velocity change is largest and there is a peak in
the attenuation, is determined by the parameters of the particular
piezoelectric substrate (see SI). (c) Simulation of a SAW standing-wave
resonator on lithium niobate. A conductive sample in the middle of the cavity
is shown in grey. Black arrows indicate the direction and magnitude of the
electric field at the surface of the substrate. In the sample region, the
conducting sample screens the transverse component of the electric field: this
interaction with the electric field feeds back into the stiffness of the
substrate lattice, modifying the speed of sound and sound attenuation as shown
in panel (b).

SAWs would appear to be an ideal technique for van der Waals (vdW) materials,
where it can be difficult to make electrical contact, where there are emergent
and engineered length scales such as charge density wavelengths [6] and moire
periodicities [7], and where there are predictions for wavelength-dependent
phenomena [8, 9]. While there has been some work incorporating vdW
heterostructures and SAWs [10, 11], the quantum transport regime has not been
reached.

The two main challenges are finding a suitable substrate and increasing signal
size. First, a piezoelectric substrate compatible with high-mobility device
fabrication and electrostatic gating must be identified. Second, even
relatively large vdW heterostructures are two to four orders of magnitude
smaller than the 2DEGs used in the aforementioned studies of GaAs/AlGaAs. As
the SAW signal is directly proportional to the sample area, standard SAW
delay-lines are not sensitive enough for quantum transport experiments on vdW
heterostructures.

We have developed SAW resonant cavities on LiNbO3 substrates that are
compatible with high-mobility, hexagonal boron nitride (hBN)-encapsulated,
graphene heterostructures. We show that the resonant cavity geometry increases
signal-to-noise by two orders of magnitude over the traditional delay-line
geometry. We observe strong quantum oscillations of the cavity resonance
frequency in the quantum Hall regime of graphene. We comment on the present
limitations of the technique and suggest future design improvements.

Background—SAWs generate electric fields that extend approximately one
wavelength above the surface of a piezoelectric substrate (Figure 1c). This
field contributes to the stiffness of the lattice and enhances the propagation
velocity of the acoustic wave [12]. A conductive material placed on top of the
substrate partially screens this electric field. When the conductivity
material changes as a function of gate voltage, magnetic field, or
temperature, the SAW velocity and attenuation coefficient also change [13].
These effects are plotted in Figure 1b and are described by the relation

$\frac{\delta
v_{s}}{v_{s}}-\frac{i\Gamma}{q}=\frac{K_{eff}^{2}}{2}\frac{1}{1+i\sigma^{\square}_{xx}(q,\omega)/\sigma_{m}},$
(1)

where $\sigma^{\square}_{xx}(q,\omega)$ is the longitudinal sheet conductance
(in the direction of the SAW propagation) of the nearby material at wave
vector $q$ and frequency $\omega=v_{s}q$, and $K_{\rm eff}^{2}$ is a
substrate-specific coupling coefficient. $\sigma_{m}$ is the characteristic
sheet conductance of the SAW and is given by
$\sigma_{m}=v_{s}(\epsilon+\epsilon_{0})$, where $\epsilon$ is the
permittivity of LiNbO3. $\sigma_{m}$ is of order $10^{-6}~{}\Omega^{-1}$ for
LiNbO3, and $K_{\rm eff}^{2}=0.053$ [14]. The response of the SAW to a
conducting sample is maximized when the sheet conductance of the sample
matches the characteristic conductance of the SAW, as shown in Figure 1.

There are two predominant methods for generating SAWs: delay lines and
resonant cavities. Most SAW experiments on GaAs/AlGaAs heterostructures use
the delay-line geometry, where a pulse of sound generated at one interdigital
transducer (IDT) is captured by another. Resonant cavities use the same pair
of IDTs but build up a standing wave between reflectors that constrain the
acoustic energy (see Figure 1a). The signal is then effectively amplified by
the quality factor ($Q$) of the cavity, which can easily be $10^{3}$ or
higher. This suggests that the resonant cavity geometry may be preferable to
the delay line geometry for small samples.

This argument can be quantified by examining how the measured phase changes
for the two geometries. For a delay line [15], a change in sound velocity
($\delta v/v$) translates to a change in the phase ($\delta\phi/360^{\circ}$)
of the SAW as

$\frac{\delta\phi}{360^{\circ}}=\frac{A_{s}}{\lambda\cdot W}\frac{\delta
v}{v},$ (2)

where $A_{s}$ is the sample area, $W$ is the width of the delay line
(perpendicular to the direction of propagation), and $\lambda$ is the
wavelength of the SAW. For a resonant cavity with quality factor $Q$ near its
resonant frequency, the phase shift is given by

$\frac{\delta\phi}{360^{\circ}}=\frac{Q}{\pi}\frac{A_{s}}{A_{c}}\frac{\delta
v}{v},$ (3)

where $A_{s}$ and $A_{c}$ are the sample and cavity areas, respectively.
Defining the ‘gain’ $G$ as the cavity phase shift divided by the delay line
phase shift, we obtain

$G=\frac{Q}{\pi}\frac{\lambda}{L},$ (4)

where $L$ is the length of the cavity or delay line.

Equation 4 suggests that, for a given $\delta v/v$ induced by a change in
sample conductivity, large phase shifts can be achieved with high quality
factor cavities. The factors $L$ and $\lambda$ are constrained by device
fabrication: we use $\lambda=20~{}\mu$m SAW cavities with an IDT separation of
$L=100~{}\mu$m. These length scales are easily achieved with standard
photolithography and leave enough space to fabricate a heterostructure in the
center of the cavity. These cavities regular achieve $Q$s between 1000 and
5000, producing a gain of up to 300, i.e. the phase shift is 300 times larger
when the devices is operated as a cavity instead of a delay line.

Experimental design.—Figure 1a shows a schematic of the two-port, Fabry-Perot
SAW resonators used in this study. These resonators consist of two IDTs for
excitation and detection of the SAW and two Bragg mirrors placed behind the
IDTs [16]. VdW heterostructures typically require gate and grounding
electrodes, as well as possible transport electrodes if desired, which we
fabricated from thinner metal than the Bragg mirrors to maintain a high cavity
$Q$ and to allow room for heterostructure fabrication.

We fabricated our resonators on commercially-available, 128∘ Y-Cut, SAW-grade,
black LiNbO3 wafers. The device has 15 nm gold electrodes and 300 nm,
electrically-isolated gold reflectors, where the mass of the gold loads the
resonator surface and confines the SAWs. A second resonator is describe in the
supplementary.

The monolayer graphene devices were mechanically exfoliated and encapsulated
with hBN on both sides to ensure high sample mobility. The device shown in
Figure 2a consists of a 37 nm bottom hBN layer, a graphene monolayer, a 64 nm
top hBN layer, and a 5 nm graphite top gate. The resonator had a $Q$ of 1500
after device fabrication.

Results.—Figure 2a shows the frequency and phase response of the device. IDT1
was driven at 10 mV with a Zurich Instrument UHF lockin amplifier with both
the graphene and the graphite gate grounded. The signal from IDT2 was filtered
with a ZX75BP-204 band pass filter from Mini-Circuits, amplified with a
AM-1571 amplifier from MITEQ, and measured with the lockin.

Two resonances near 190 MHz, labeled $f_{1}$ and $f_{2}$, are each accompanied
by a phase shift of approximately 180∘. Each resonance corresponds to a
different integer number of wavelengths in the cavity: $f_{1}-f_{2}=3$ MHz is
equal to surface wave velocity, 3980 $m/s$, divided by twice the effective
cavity length of 660 $\mu m$ (note that the Bragg mirrors make the effective
length much longer than the physical length of 240 $\mu m$). The quality
factor of the resonance at $f_{1}$ is 1500: we use this resonance for our
measurements.

Figure 2: Quantum transport in graphene using a surface acoustic wave
resonator. (a) Optical microscope images of the device. The lower image shows
a zoomed-out view of the full SAW resonator cavity. The image on the top left
shows a zoomed-in view of the center of the cavity with the detail of the pre-
patterned gold electrodes without the graphene sample (the 6 transport
electrodes were not used in this study). The image on the top right is a
false-color image showing the graphene device in detail. The monolayer
graphene is colored in light purple and the graphite top gate is colored in
light green. (b) Amplitude and phase of the transmitted voltage through SAW
resonant cavity as a function of frequency. Two resonances are indicated as
$f_{1}$ and $f_{2}$. (c) The change in the cavity phase at fixed frequency
$f_{1}$, as a function of magnetic field, at two different gate voltages, and
at 1.7 K. Quantum oscillations are clearly visible above 0.6 tesla, and
oscillations related to degeneracy-broken Landau levels are observed above 3
tesla. The data have been offset for clarity. (d) Landau fan diagram at 1.7 K,
from 1 tesla to 8 tesla. The color scale corresponds to the shift in SAW
velocity, proportional to the shift in cavity phase through Equation 3.

Figure 2c shows the change in cavity phase at the $f_{1}$ resonance as a
function of magnetic field. Traces at two different gate voltages are shown,
both taken at 1.7 K. Shubnikov-de Haas oscillations are clearly observed down
to approximately 0.6 tesla, illustrating both the sensitivity of the technique
and the quality of the sample. A standard Lifshitz-Kosevich analysis of the
oscillation data (details in the Supplementary Information) determines the
carrier densities to be $1.6\times 10^{12}$ cm-2 and $3.1\times 10^{12}$ cm-2
at -5 V and 0 V, respectively. We obtain the quantum lifetimes, $\tau_{q}$,
using a Dingle analysis: $\tau_{q}\approx 0.4$ ps at -5 V and $\tau_{q}\approx
0.26$ ps at 0 V, corresponding to quantum mobilities greater than $10^{4}$ cm2
V-1s-1. This high mobility demonstrates the compatibility of LiNbO3 as a
substrate material and the suitability of the cavity geometry and our
fabrication procedure with high-quality vdW devices.

Next, we measure as a function of gate voltage at fixed magnetic fields.
Figure 2d shows a Landau fan diagram over the entire available gate voltage
range and up to 8 tesla. Above 3 tesla, additional peaks emerge between the
main series of peaks at filling factors $\nu=2,6,10,14,$ etc. At high gate
voltage, there are clearly three intermediate peaks: these are the additional
Landau levels (LLs) that emerge when Zeeman and valley degeneracy are broken,
and these have been characterized extensively in high-quality transport
devices [17]. Note that we do not believe that any of the oscillatory features
we observe are due to the graphite gate [18]: the 4-fold LL degeneracy we
observe in Figure 2d is unique to graphene. The lack of oscillations from the
graphite gate is because the graphite conductance is too high: the gate
conductance is on the far right of Figure 1b and any modulation of that
conductance produces no significant velocity shift. A second device with
slightly lower mobility, but where we can gate to the Dirac point, shows
qualitatively similar features—this data is shown in the supplementary
information.

Discussion.—We have demonstrated that high-quality-factor, two-port, Fabry-
Perot SAW resonators fabricated on LiNbO3 can be used to measure quantum
transport in graphene heterostructures. We observe degeneracy-broken LLs at
less than 3 tesla and extract quantum lifetimes longer than 0.2 ps—both
indicative of high carrier mobility.

The measured changes in cavity phase shift and SAW amplitude can be converted
to changes in the conductivity of graphene at $q=2\pi/(20~{}{\rm\mu m})$ and
$\omega=2\pi\cdot 200~{}{\rm MHz}$. In GaAs/AlGaAs heterostructures, this
conversion is performed using Equation 1 with a simple relaxation-type model
and the results agree well with experiment [19].

To perform this conversion on our data, we perform finite-element simulations
of the full resonator and heterostructure assembly, including the graphite top
gate that which also contributes to the cavity phase shift (see Supplementary
Information). We used published conductivity data from the quantum Hall regime
of graphene to estimate the change in cavity phase shift between Landau levels
(see Supplementary Information) [20]. Our model predicts a phase shift of the
same order of magnitude that we observe in our data. The extraction of the
conductivity from the SAW resonator data could be made quantitative with two
improvements: 1) by greatly reducing the top-gate conductance by switching to
a low-conductivity material such as MoS2; and 2) by increasing the sample size
and/or reducing the total cavity area.

We used SAW resonators operating at 200 MHz in this study because the 20
$\mu$m feature size is easily achievable with cost-efficient, standard
photolithographic techniques. SAW resonators operating up into the GHz
frequency range, however, are a well-established and mature technology—a
typical smartphone contains more than a dozen SAW filters and delay lines.
Having demonstrated that LiNbO3 SAW resonators have the requisite sensitivity
for small heterostructures, it should now be possible to explore wave-vector
dependent effects down to hundreds-of-nanometer length scales. Immediately
accessible experiments based on existing theoretical proposals include
searching for the crossover from Dirac to Schrodinger-like behaviour in the
longitudinal conductivity of graphene [8], and measuring the SAW attenuation
of twisted bilayer graphene as a function of chemical potential to look for
non-Fermi-liquid signatures [9]. With further development, one could imagine
using SAW resonators to impose dynamic, periodic potentials on vdW
heterostrucures that are both tunable in period and switchable on nanosecond
timescales.

##  I Acknowledgments

The fabrication of vdW heterostructures on SAW devices was supported by the
Cornell Center for Materials Research with funding from the NSF MRSEC program
(DMR-1719875). B.J.R. and Y.F. acknowledge funding from the National Science
Foundation under grant no. DMR-1752784.

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# TranViT: An Integrated Vision Transformer Framework for Discrete Transit
Travel Time Range Prediction

Awad Abdelhalim Corresponding Author
Postdoctoral Research Associate
Department of Urban Studies and Planning
Massachusetts Institute of Technology  Cambridge  MA 02139  USA
Email<EMAIL_ADDRESS>
Jinhua Zhao
Associate Professor
Department of Urban Studies and Planning
Massachusetts Institute of Technology  Cambridge  MA 02139  USA
Email<EMAIL_ADDRESS>
##  1 Abstract

Accurate travel time estimation is paramount for providing transit users with
reliable schedules and dependable real-time information. This paper proposes
and evaluates a novel end-to-end framework for transit and roadside image data
acquisition, labeling, and model training to predict transit travel times
across a segment of interest. General Transit Feed Specification (GTFS) real-
time data is used as an activation mechanism for a roadside camera unit
monitoring a segment of Massachusetts Avenue in Cambridge, MA. Ground truth
labels are generated for the acquired images based on the observed travel time
percentiles across the monitored segment obtained from Automated Vehicle
Location (AVL) data. The generated labeled image dataset is then used to train
and evaluate a Vision Transformer (ViT) model to predict a discrete transit
travel time range (band). The results of this exploratory study illustrate
that the ViT model is able to learn image features and contents that best help
it deduce the expected travel time range with an average validation accuracy
ranging between 80%-85%. We also demonstrate how this discrete travel time
band prediction can subsequently be utilized to improve continuous transit
travel time estimation. The workflow and results presented in this study
provide an end-to-end, scalable, automated, and highly efficient approach for
integrating traditional transit data sources and roadside imagery to improve
the estimation of transit travel duration. This work also demonstrates the
value of incorporating real-time information from computer-vision sources,
which are becoming increasingly accessible and can have major implications for
improving operations and passenger real-time information.

Keywords: Travel time prediction, transit performance, vision transformers.

##  2 Introduction

Accurate and reliable travel time prediction plays a critical role in all
aspects of transportation planning. This is even more crucial when it comes to
public transit. Efficient operations, riders’ experience, and the general
perception of public transit are greatly shaped by the on-time performance of
the system. Travel and arrival time prediction, in general, and particularly
for public transit applications are fields of research that have been well
studied over the years. The recent advancements in machine learning and
artificial intelligence technologies have significantly improved the state of
practice. This is in part due to the wealth of data available for and from
day-to-day public transit operations, including high granularity data for
passenger count and movements (APC), automated fare collection (AFC),
automated vehicle locations (AVL), and the information from the General
Transit Feed Specification (GTFS). While these data sources offer an abundance
of scheduling, usage, and performance measures that can be used for tasks like
travel time prediction for transit, transit vehicles still - for the most
part- share road infrastructure with other roadway users. This necessitates
combining transit-specific data with more generalized data sources that can
offer information about the overall traffic state and roadway infrastructure,
which are often challenging to acquire.

The recent advancements in the fields of deep learning, machine perception,
and computer vision have made image data extremely useful. Tasks including
image classification, detection, and tracking of objects within images can be
accomplished with ease, with an ever-growing multitude of frameworks and
architectures to choose from. The biggest drawback when opting to utilize
computer vision architectures is that they remain extremely data-hungry,
requiring copious amounts of data that needs to be acquired and, more often
than not, manually labeled to train vision models. And while this is a
challenge facing the broader computer vision community, it is further
exacerbated in domain-specific applications like transit operations, where the
need to acquire external technology and talent for these tasks of data
acquisition and annotation is often cost-prohibitive. The authors of this
paper, see immense value in incorporating the domain knowledge of relevant
data sources to create a streamlined framework for image data acquisition,
labeling, and vision model training combining roadside imagery with transit
data sources. This study presents our exploratory framework for this concept
and experimental results from a pilot study conducted in a segment of
Massachusetts Avenue in Cambridge, MA, USA.

###  2.1 Objective and Contributions

In this study, we propose and evaluate TranViT, an end-to-end framework for
efficiently integrating real-time GTFS, AVL, roadside camera data, and a
Vision Transformer (ViT) architecture for predicting transit travel time
through a camera-monitored segment. We present a detailed breakdown of the
workflow to acquire, label, and successfully train the ViT model, resulting in
highly accurate and interpretable predictions. The following sections of this
paper are as follows: (a) a survey of related literature in transit travel
time prediction and applications of deep learning and computer vision, (b) a
detailed breakdown of the proposed TranViT framework, (c) results and analyses
of the case study, and, (d) discussion and conclusions.

###  2.2 Related Work

####  2.2.1 General Traffic and Transit Travel Time Prediction

There is a rich body of literature on travel time prediction, a topic that has
been well-studied for years. In the context of general traffic, this often
falls into larger-scale traffic state estimation, which along with travel time
and speed includes predicting traffic flow and density [wang2005real,
yang2005travel, work2008ensemble, yildirimoglu2013experienced]. Those works
among others have mostly relied on probe vehicle GPS and loop detector data,
which are often limited in size and temporal coverage, noisy, and required the
use of complementary techniques to overcome the data quality issues, including
particle and Kalman filtering, and machine learning [wang2008real,
chen2011real, jenelius2013travel] in combination with underlying traffic state
models and, in some cases, simulation modeling.

For transit, accurate travel time prediction is paramount for providing
transit users with reliable schedules and dependable real-time information
about their transit vehicles’ arrival times, and for the efficient
implementation of operation strategies such as transit signal priority
[zeng2014real, abdelhalim2018impact]. The methods used in general traffic
state estimation and travel prediction, particularly from probe vehicle
analyses, may not always translate directly into the context of transit.
Albeit sharing the same roadway infrastructure; the vehicle dynamics and
operations of transit (continuous start-stopping), among other factors like
passenger interactions with operators, influence the movement of transit
within traffic streams in a way that is not necessarily reflective of the
overall traffic. The need to monitor this level of complexity in size (fleets
as opposed to individual vehicles) and operation, however, has resulted in an
abundance of transit data sources being available to practitioners. Of the
aforementioned data sources, AVL data has particularly been at the center of
numerous research efforts. An early study by Cathey et al.
[cathey2003prescription] offered a generalized framework to utilize AVL data
for making transit arrival time predictions. The growing adoption of AVL
systems by transit agencies allowed researchers to develop real-time
applications [jeong2005prediction, shalaby2004prediction]. While these studies
have demonstrated the ability to obtain accurate travel time predictions from
AVL data, the lack of information regarding other influencing factors like
overall congestion state and the vehicles’ own dwell time were limiting
factors. Works by Yu et al. [yu2010hybrid, yu2011bus] have demonstrated that
incorporating additional external data sources (e.g. weather data) and using
stop-level travel time that captures this variation in dwell time more than
the route-level analysis results in improved predictions.

The past decade has witnessed substantial strides in data availability (higher
frequency AVL, GTFS-RT, APCs, etc) and real-time assessment and predictive
methodologies [park2020assessing, elliott2020modelling, aemmer2022measurement,
samal2017speedpro]. While state estimation methods based on Kalman filters,
and machine learning models based on boosting and ensemble remained popular
[zhang2015gradient, gaikwad2019performance], deep learning architectures based
on recurrent neural networks (RNNs) have also proven high competency in the
travel time prediction task. Studies by Zhou et al. [zhou2019learning] and
Pang et al. [pang2018learning] concluded that RNN-based models significantly
outperform other state-of-the-art methods. The proposed Long Short-Term Memory
(LSTM) RNN by Pang et al. achieved a minimum of 10% improvement in the mean
absolute error compared to other traditional models. Han et al. [han2020bus]
proposed a method based on position calibration and an LSTM model that
accurately predicts transit arrival time, with an error below two minutes for
the 8th downstream stop.

####  2.2.2 Computer Vision, Vision Transformers, and Related Works in
Transportation Applications

Computer vision (CV) is a field of artificial intelligence focused on the
derivation of useful information from images and image-based data (e.g.
videos). This includes tasks like image classification, object recognition,
and multi-object recognition and tracking. Albeit being proposed since the
early 1990s [lecun1998gradient], the rapid evolution of Convolutional Neural
Networks (CNNs) in the past decade helped establish their place as the
undisputed backbone for innumerable architectures that conduct the various
aspects of computer vision tasks. These CV-based methods have been widely
adopted for a variety of applications in the transportation field, including
traffic speed, turn count, and density estimation [buch2011review,
abdelhalim2021framework, gokasar2021real], safety [tageldin2014safety,
sayed2013automated, abdelhalim2021real], and autonomous driving
[janai2020computer]. The use of CV in transit applications remains extremely
rare, with few applications in railway intrusion detection [wang2021fast], and
a recent study by Sipetas et al. [sipetas2020estimation] who utilized a CV-
based video processing component as a part of a larger framework to estimate
left-behind subway passengers.

The Vision Transformer (ViT) architecture introduced by Dosovitskiy et al.
[dosovitskiy2020image] was inspired by the immense success of transformer
architectures in the field of natural language processing (NLP). The authors
of ViT state that they aimed to replicate the original transformer
architecture by Vaswani et al. [vaswani2017attention] as closely as possible.
Converse to CNNs, which have been the de-facto method for most computer vision
tasks, the vision transformer architecture doesn’t introduce the implicit bias
of the convolutional and pooling layers present in CNNs, allowing the trained
model to better extract global information from images. While this comes at
the cost of requiring more training data, the ViT-based models have shown to
outperform their CNN counterparts at their introduction in 2020. While there
has been a tug of war between ViT and CNN-based models since then, ViT-based
models have been the standout performer in tasks that are more involved than
simple everyday image classification, including but not limited to holography
[cuenat2021convolutional], classification of COVID-19 CT scan images
[gao2021covid], and identifying distracted driving [li2022distracted]. Those
studies provide an indication that the global-attention aspect of the ViT
architecture allows it to make more accurate predictions using information
that could be lost within convolution and pooling layers of a CNN. In
transportation-related applications, a recent study by Zheng and Jiang
[zheng2022evaluation] has concluded that CNN-based models outperform ViT for
the task of traffic sign classification. Albeit using a relatively small
dataset, this result does fit the hypothesis given that traffic sign images
contain locally-concentrated information (i.e. the sign itself, irrespective
of the rest of the image). Studies that evaluated vision transformers for
other transportation applications where the information needed to make
accurate predictions is expected to be sparse across the image show the
considerable benefits of using ViT. This includes the work of Liang et al.
[liang2022stargazer] demonstrating the ability of ViT in detecting driver
distraction, and a study by Abdelraouf et al. [abdelraouf2022using] who
demonstrated the ability of a ViT-based architecture to detect rain and
roadway surface conditions with an impressive F-1 score of up to 98%.

While computer vision methods and models have proven to add tremendous value
to different fields of science and practice, there currently remains a
significant gap between the abundance of transit data sources that are
utilized for travel time prediction and connecting these data sources with
methods and frameworks to extract complementary information through computer
vision, which can provide a significant boost to the state-of-the-art. The
authors believe that this gap is, by and large, due to the demanding process
of computer vision models’ data acquisition, labeling, and training. There is
an immense need for developing generalizable and transferable frameworks that
can seamlessly integrate these existing transit and urban data sources, and
streamlining the data acquisition, model training, and inference processes. We
propose TranViT as a pioneering example for this much-needed systems
integration task.

##  3 Methodology

###  3.1 Site of Study

The site of this exploratory study was near Central Square in Cambridge, MA,
USA, at the intersection of Massachusetts Avenue and Sidney Street. We
selected this site due to the availability of a public live camera streaming a
live view of this area, which is served by multiple Massachusetts Bay
Transportation Authority (MBTA) bus routes. The Google Earth view of the site
is illustrated in Figure 1. The north direction is indicated by the arrow at
the top-right of the figure. The top-left shows the position of the public
camera that was used to acquire images for this study, with the yellow lines
showing the field of view of the camera. The bus stops at the bottom-right
quarter of the image serve either direction of the MBTA Route 1 bus operating
between Nubian and Harvard Square. The inbound (to Boston) and outbound (to
Harvard Sq) directions of the route are respectively illustrated by the red
and blue arrows. Figure 2 shows the Google Maps view from the camera angle,
alongside the true camera perspective of the site.

Figure 1:  Site of study near Central Square in Cambridge, MA, USA.

(a) Google Maps view.

(b) Camera view.

Figure 2:  Camera perspective for the site of study.

###  3.2 Proposed Framework, Data Collection and Pre-processing, and ViT Model
Training

Figure 3:  TranViT data sources, modules, and workflow.

####  3.2.1 Data Sources and Acquisition

At the core of our proposed TranViT framework is the General Transit Feed
Specification real-time component (GTFS-RT). The GTFS-RT is available for the
MBTA vehicles through onboard GPS equipment which updates and publishes
vehicle location, heading, and occupancy data in real-time at high-frequency
intervals (up to every 1 second) [mbta_gtfs]. We utilize this high-frequency
of the GTFS-RT to trigger image acquisition _only_ as transit vehicles
approach the area of the study, which accomplishes the following:

  1. 1.

Enables linking the acquired image sequences and the travel times associated
with the trip ID that activated the image acquisition.

  2. 2.

Optimizes the image acquisition process, ensuring data is only collected as
needed.

Image acquisition from camera livestream at the site is activated once a
transit vehicle is approaching (within 500m). A total of 6 images are acquired
for each activation, with a 15-second wait time between the images to allow
for traffic movement across the intersection. If more than one transit vehicle
approach and activate the camera at the same time, a single acquisition stream
is initiated and separate images will be labeled by the trip ID and direction
of each transit vehicle that crossed the area during the given timeframe. The
outputs of the data acquisition process are a database containing the trip
IDs, directions, and the timestamp of the transit vehicles’ approach to the
site of study, and an image dataset with 6 images associated with each trip
ID. The data acquisition process for this pilot study is summarized by the
following:

  * •

Three separate rounds of data acquisition (to account for the seasonal
variations in daylight and roadside greenery).

    * –

Feb 23rd \- March 4th, 2022

    * –

March 28th \- April 6th, 2022

    * –

May 9th \- May 16th, 2022

  * •

Data was collected for each day in the time periods above between 6 AM - 9 PM.

  * •

Transit trip data is only recorded if the image acquisition process is
successful. Livestream buffering and connection issues can result in failed
acquisitions.

  * •

A total of 2,992 MBTA Route 1 bus trips were recorded.

  * •

A total of 17,905 images were acquired and associated with these trips.

Figure 3 shows an overview of the data sources and flow for each module sub-
component of our proposed framework, namely data acquisition, model training,
and real-time inference. The data attributes of data acquired from each source
are further described in Table 1.

Table 1: Data Sources and Attributes Used Source | Attribute | Type | Description
---|---|---|---
GTFS-RT | id | Integer | Unique trip identifier.
dir | Binary | Trip Direction (0 = outbound, 1 = inbound).
timestamp | Integer | Unix timestamp during image acquisition.
AVL | ts | Float | Total travel time across the segment (sec).
dwell | Float | Stop dwell time for a given trip (sec).
Camera | site image | 1280 x 720 x 3 Array | Image of the observed scene.

####  3.2.2 Generating Travel Time Bands Based on Effective Travel Time
Percentiles

The acquired trip-image dataset requires pre-processing to make it suitable
for the ViT image classification task. First, the overall travel time of
transit vehicles is acquired from the MBTA’s AVL database. The AVL database
also keeps a record of transit vehicles’ stop events, including the associated
dwell time with a stop event (if any). We associate the trip ID for each
transit vehicle in our dataset with the stop events at either of the two stops
in our site of study (in the inbound and outbound direction) and define the
effective travel time as the time spent by the vehicles on the 1-km segment
(which includes camera monitored area in addition to the 500m activation
buffer in either direction), minus any dwell time associated with that trip at
its respective stop on the site.

Figure 4:  Observed effective transit travel time for training dataset.

We use this effective travel time across the segment as an indicator of
traffic state during the time of image acquisition, with the assumption that
after accounting for stop events (if any) a transit vehicle’s behavior within
a traffic stream is representative of the overall traffic. This assumption,
however, doesn’t take into account the variation in vehicle dynamics, and the
impact of signal control and signal interruptions due to the unavailability of
real-time data sources for these factors at the time of this study. The
observed effective travel time for the dataset is illustrated in Figure 4. It
is worth noting that the dataset contains more outbound (1,536) than inbound
(1,456) trips.

The overall average effective travel time observed was 124 seconds, with a
minimum of 35 seconds and a maximum of 310 seconds. The descriptive statistics
are shown in Table 2. To normalize the labels for the training dataset, we
utilize a four-class labeling approach to discretize observed effective travel
time percentile and label them as follows:

  * •

Effective Travel Time $\leq$ 10% $\rightarrow$ Low.

  * •

10% $<$ Effective Travel Time $\leq$ 50% $\rightarrow$ Moderate.

  * •

50% $<$ Effective Travel Time $<$ 90% $\rightarrow$ Above Average.

  * •

Effective Travel Time $\geq$ 90% $\rightarrow$ High.

Figure 5 shows the distribution of these travel time bands per trip direction
and hour of the day.

Table 2: Descriptive Statistics for Effective Travel Time Statistic | Trip Direction
---|---
Overall | Inbound | Outbound
$\mu$ | 124 | 127 | 121
$\sigma$ | 38 | 41 | 35
10% | 79 | 79 | 79
50% | 121 | 124 | 118
90% | 160 | 166 | 156
Min | 35 | 35 | 35
Max | 310 | 309 | 310
Count | 2,992 | 1,456 | 1,536

(a) Inbound.

(b) Outbound.

Figure 5:  Travel time band distribution for acquired image dataset.

####  3.2.3 ViT Model Training and Fine-tuning

The ViT architecture approaches the image classification task with a similar
approach inspired by the transformer architecture introduced by Vaswani et al.
[vaswani2017attention], which has achieved incredible success in the field of
Natural Language Processing (NLP). The ViT model utilized as a part of our
framework is the one proposed by Dosovitskiy et al. [dosovitskiy2020image].
The gist of the model is that it simplifies the pixel-wise attention
calculation that would take place in a transformer’s encoder module by
splitting the image into $N$ fixed-size patches ($P$). Those fixed-size
patches are linearly inserted into the transformer encoder alongside their
positional embeddings, which simply tell the encoder where each patch belongs
in an image. Meaning that an input image x $\in\mathbb{R}^{H\times W\times C}$
is reshaped into xp $\in\mathbb{R}^{N\times(P^{2\ .\ }C)}$, where H, W, C are
respectively the height (720 pixels), width (1280 pixels), and the number of
color channels (3) in the source image. A constant latent vector ($D$) is used
across all layers, and learnable positional embedding parameters (Elin, Epos)
are utilized to extract an embedding token (zo). Flattened embeddings pass
through a series of encoders ($L$) each with a multi-head self-attention (MSA)
layer, and a feed-forward Multi-Layer Perceptron (MLP), both preceded by
normalization layers (LN). This process is mathematically described by the
following:

$\displaystyle\textbf{z}_{o}$ $\displaystyle=[\textbf{x}_{class};\
\textbf{x}_{p}^{1}\textbf{E};\ \textbf{x}_{p}^{2}\textbf{E};\ ...\ ;\
\textbf{x}_{p}^{N}\textbf{E}]+\textbf{E}_{pos},$
$\displaystyle\textbf{E}\in\mathbb{R}^{(P^{2\ .\ }C)\times D},\
\textbf{E}_{pos}\in\mathbb{R}^{(N+1)\times D}$ (1)
$\displaystyle\textbf{z}_{l}^{{}^{\prime}}$
$\displaystyle=\text{MSA}(\text{LN}(\textbf{z}_{l-1}))+\textbf{z}_{l-1},$
$\displaystyle l=1\ ...\ L$ (2) $\displaystyle\textbf{z}_{l}$
$\displaystyle=\text{MLP}(\text{LN}(\textbf{z}_{l}^{{}^{\prime}}))+\textbf{z}_{l}^{{}^{\prime}},$
$\displaystyle l=1\ ...\ L$ (3) y
$\displaystyle=\text{LN}(\textbf{z}_{L}^{0})$ (4)

Where y is the final image representation out of the transformer encoder, then
passed into a classification MLP outputting the most likely class for the
image, which in our case is the travel time band prediction based on the
effective travel time percentile. We start with the base ViT model pre-trained
on the ImageNet-21K dataset and open-sourced by Google Research
[dosovitskiy2020image]. To accommodate for the data-hungry nature of ViT,
prior to re-training on our data we increase the size of the dataset by
creating an image augmentation pipeline; randomly cropping, tilting, and
slightly adjusting the brightness and contrast of images acquired in our
dataset as shown in Table 3, while maintaining the original images’ class
label based on effective travel time rank. Every image is passed through the
pipeline 6 times, and the probability of an augmentation action is the
probability that it is performed on the image during an iteration in a random
magnitude within action bounds. This results in a final dataset with 78,385
images.

Table 3: Image Augmentation Pipeline Parameters Action | Augmentation Bounds | Probability
---|---|---
Lower | Upper
Crop | 560 $\times$ 560 | 1280 $\times$ 720 | 0.33
Rotate | -30 o | +30 o | 0.33
Brightness | -20% | +20% | 0.50
Contrast | -20% | +20% | 0.50

We initially train multiple instances of the ViT model while performing a grid
search to optimize the model’s hyper-parameters to minimize the overall F-1
Score. The bounds and optimal values of this fine-tuning process are shown in
Table 4. After finding optimal hyper-parameters, the vision transformer was
trained using a 5-fold cross-validation, with an 80-20 stratified training-
testing split for each fold. We also evaluate the model in terms of precision,
recall, and accuracy.

Table 4: ViT Parameter Fine-Tuning Parameter | Parameter Limits | Optimal
---|---|---
Lower | Upper
Hidden Layers | 2 | 12 | 12
Attention Heads | 2 | 12 | 12
Batch Size | 8 | 256 | 32
Learning Rate | 1$e^{-6}$ | 1$e^{-2}$ | 2$e^{-5}$
Dropout Probability | 0 | 0.25 | 0.10

##  4 Results and Discussion

###  4.1 ViT Performance on Effective Travel Time Band Prediction

As a reminder, the inbound and outbound directions discussed hereinafter are
illustrated by respectively the red and blue arrows in Figure 2(a). After
training on 80% of the images in the k-fold augmented dataset, the averaged
5-fold performance of the optimized ViT model on the test sets for either
direction is shown in the confusion matrices in Figure 6 below, illustrating
the true and predicted travel time band for test images. Figure 7 illustrates
the results normalized to the total number of true travel time band images in
the test set (sum over columns = 1, diagonal cells represent accuracy for each
travel time band).

(a) Inbound.

(b) Outbound.

Figure 6:  Confusion matrices for test subsets.

(a) Inbound.

(b) Outbound.

Figure 7:  Normalized confusion matrices for test subsets.

The results indicate that the model is able to differentiate the discrete
expected travel time ranges based on the observed effective transit travel
time used for labeling the dataset. While a higher number of
misclassifications happen between the intermediate states (moderate and above-
average conditions), the misclassification rate between the extreme conditions
(low and high) is very low. This result is of utmost importance, indicating
that when using image data in complementing the travel time prediction
process, the misclassifications will rarely lead to an extreme-end under or
overestimation. Detailed results on the ViT model’s performance are provided
in Table 5, showing the 5-folds average for each metric $\scriptstyle\pm$ the
range of variation for each metric across the 5-folds.

Table 5: 5-Fold Cross-Validation Test Sets Performance Metrics Class | Inbound | Outbound
---|---|---
Prc $\scriptstyle(\pm)$ | Recall $\scriptstyle(\pm)$ | F-1 $\scriptstyle(\pm)$ | Prc $\scriptstyle(\pm)$ | Recall $\scriptstyle(\pm)$ | F-1 $\scriptstyle(\pm)$
Low | 0.74 $\scriptstyle(0.05)$ | 0.81 $\scriptstyle(0.02)$ | 0.78 $\scriptstyle(0.01)$ | 0.76 $\scriptstyle(0.02)$ | 0.80 $\scriptstyle(0.02)$ | 0.79 $\scriptstyle(0.03)$
Normal | 0.84 $\scriptstyle(0.01)$ | 0.83 $\scriptstyle(0.02)$ | 0.83 $\scriptstyle(0.01)$ | 0.82 $\scriptstyle(0.01)$ | 0.82 $\scriptstyle(0.01)$ | 0.82 $\scriptstyle(0.01)$
Above Average | 0.81 $\scriptstyle(0.01)$ | 0.78 $\scriptstyle(0.01)$ | 0.80 $\scriptstyle(0.00)$ | 0.81 $\scriptstyle(0.01)$ | 0.79 $\scriptstyle(0.02)$ | 0.79 $\scriptstyle(0.01)$
High | 0.77 $\scriptstyle(0.02)$ | 0.81 $\scriptstyle(0.02)$ | 0.79 $\scriptstyle(0.01)$ | 0.74 $\scriptstyle(0.02)$ | 0.79 $\scriptstyle(0.03)$ | 0.75 $\scriptstyle(0.03)$
Accuracy | 0.81 $\scriptstyle(0.01)$ | 0.80 $\scriptstyle(0.01)$

Given that the ViT training and inference are based on images that provide a
constrained view of the area of study, variation in the number of vehicles and
other factors between images taken from the same sequence can occur (6 images
at 15-second intervals, as detailed in the Methodology). To account for this
variation, we run inference on image sequences for a given trip ID instead of
single images. Images from the test set of the best-performing fold were
grouped by their trip ID before making travel time band predictions based on
the average of all predictions for the image sequence. The results are
illustrated in Figures 8 and 9. Classification metrics are detailed in Table
6, where support is the number of actual occurrences of the class in the test
subset.

(a) Inbound.

(b) Outbound.

Figure 8:  Confusion matrices using averaged image sequence score.

(a) Inbound.

(b) Outbound.

Figure 9:  Normalized confusion matrices using averaged image sequence score. Table 6: Performance Metrics Using Averaged Image Sequence Score Class | Inbound | Outbound
---|---|---
Prc | Recall | F-1 | Support | Prc | Recall | F-1 | Support
Low | 0.93 | 0.77 | 0.84 | 201 | 0.94 | 0.76 | 0.84 | 213
Moderate | 0.83 | 0.91 | 0.87 | 539 | 0.83 | 0.89 | 0.86 | 557
Above Average | 0.81 | 0.84 | 0.82 | 396 | 0.78 | 0.87 | 0.82 | 433
High | 0.94 | 0.77 | 0.85 | 193 | 0.95 | 0.73 | 0.83 | 199
Accuracy | 0.85 | 0.84

The results in Table 6 demonstrate significant improvements in model
performance when using image sequences, particularly in nearly eliminating all
misclassifications between non-consecutive travel time ranges. While the
accuracy (the number of travel time band labels that were correctly classified
divided by the total occurrences of the class label in the training dataset)
slightly drops for the under-represented classes (low and high travel time
bands), the precision for those classes increases significantly. Given that
precision is an indicator of the quality of the prediction (the number of true
positives divided by the total number of class predictions made by the model)
a precision of over 93% for those extreme classes is of utmost importance when
those labels are to be used to enhance travel time predictions. The precision,
recall, F-1 score, and accuracy all increase for the moderate and above-
average travel time band labels.

Of the 2,992 trip-image sequences in the dataset acquired for this study,
2,731 transit trip IDs had two or more images in the test set of the best-
performing training fold. The effect of the number of images in a sequence is
illustrated in Figure 10. The line plots show the average classification
accuracy for each travel time band, while the envelopes illustrate the 95th
confidence interval for a given number of images in sequence. While accuracy
and confidence increase as the number of images used for prediction increases,
particularly for under-represented classes (the low and high travel time
bands), a decline is observed after the number of images in a sequence exceeds
6. This is attributable to the original number of images for each trip-image
sequence acquisition being 6 images, hence exceeding that number indicates the
presence of augmented images in a test sequence that could make it more
challenging to classify.

Figure 10:  Accuracy versus image count in a test sequence.

Figure 11 illustrates the variation in travel time band prediction accuracy
for both directions across different times of the day. Consistent with
previously discussed results, the model’s performance for the inbound
direction outperforms the outbound. The model was found to learn and make
better predictions during the AM and PM peak hours for both directions
compared to the off-peak hours. This may be attributed to the apparent
variation in traffic volumes observed during peak hours which is a good
indicator of the expected travel time across the monitored segment during
these times. Higher travel times can occur during off-peak hours without the
presence of high traffic due to traffic control, driving behavior, or vehicles
impeding access to bus stops. Such factors can not be identified from the
image data used in this study.

(a) Inbound.

(b) Outbound.

Figure 11:  Variation in classification accuracy by direction and time of day.

Next, we assess the interpretability of the results obtained by the ViT. This
is accomplished by mapping the averaged attention scores (values between 0 to
1) of the model’s 12 attention heads. Attention illustrates the parts of the
image from which the model learns to gain the most information in making the
predictions for travel time bands. Higher scores are illustrated by brighter
pixels. Figures 12 and 13 illustrate the attention maps for different
scenarios for the inbound and outbound directions.

(a) Normal.

(b) Rain.

(c) Snow.

(d) Night.

Figure 12:  Inbound direction attention maps across different conditions.

We observe the ViT model learning to make predictions in a very logical and
sensible manner, where a lot of attention is directed to the pixels containing
information about vehicles traveling in the direction of interest. For the
inbound direction illustrated in Figure 12, we see the majority of attention
being directed to the right side of the image, mostly to the vehicle in the
queue in the inbound direction (when present), but also towards the
downstream. The ViT learns to look in the opposing direction with lower
attention while learning to ignore the buildings that provide little to no
information (although attention to buildings’ indoor lights was observed, the
model might use it as an indicator for the time of day). Other sensible
actions illustrated by the ViT include paying attention to the snow when there
is a snowfall, which impacts the travel speeds across the segment. In Figure
12(d), it can be seen that at night the transformer learns to include
information from the inbound direction traffic light which becomes visible
from the camera’s vantage point in the evening.

We see the ViT model exhibiting similar behavior in the outbound direction in
Figure 13. The camera view for the outbound direction, however, is a lot more
limited compared to that of the inbound, as can be seen with the building on
the left of the image blocking the upstream, and a very limited view of the
downstream. This explains the better predictive performance for the model on
the inbound direction presented in the earlier evaluation tables and confusion
matrices. Asides from learning to look at vehicles and their locations, the
model was observed to pay more attention to the sky and buildings’ lights
during the night. This behavior seems to be an attempt to compensate for the
lack of information induced by the restricted camera view for the outbound
direction by making some form of time-of-day inference.

(a) Normal.

(b) Rain.

(c) Snow.

(d) Night.

Figure 13:  Outbound direction attention maps across different conditions.

###  4.2 Implications for Travel Time Prediction

We conclude this assessment with a brief proof-of-concept looking into what
this use of real-time, computer-vision-based travel time class predictions
could mean to the broader task of travel and arrival time estimation. We fit a
linear regression model to predict the effective segment travel time of the
2,731 vehicles in the image sequence test set based on the recorded
information for these vehicles which was obtained from GTFS real-time
component. The linear regression model (Ordinary Least Squares, OLS) makes
baseline travel time predictions based on the time of day, the direction of
travel, and the occupancy of the transit vehicle as it approaches the segment
Occupancy is a continuous variable showing the percentage of passengers
estimated from on-bus automatic passenger counters to total seating capacity,
ranges from 0 - 150%, with $\mu=34.6$ and $\sigma=26.67$. Hours of the day
were encoded as binary variables. We then run the same model, with the
addition of a predicted travel time band label (OLS+) obtained from the
inference of the images associated with the transit vehicle’s approach to the
segment. The predicted travel time bands are denoted by the TTB variables. The
results of actual versus predicted travel times are illustrated in Figure 14.

(a) AVL-based linear regression.

(b) With image-based band prediction.

Figure 14:  Exploring a linear regression fit for predicting transit travel
time through a segment.

Table 7 shows the coefficient estimates for the different models. Hours of the
day that were not found significant for any model were dropped from the table
for brevity. The model-agnostic addition of the travel time band labels
predicted from images works to create bounds that enhance the continuous
travel time estimation. This demonstratory evaluation shows significant
improvement achieved by the linear regression model, both in terms of the
predictions’ mean absolute error and r-squared. As the data and image sequence
acquisition for this study was initiated when a transit vehicle is within 500
meters of the camera, we utilized state predictions from the previous transit
vehicle traveling in the same direction and obtained comparable results,
indicating that the traffic state does not change drastically between
consecutive vehicles (10-minute scheduled headway for MBTA Route 1). Future
work will investigate the extent to which this look-ahead horizon can provide
reliable predictions, and assess the extent of performance improvements to
more sophisticated state-of-the-practice models.

Table 7: Linear Regression Model Estimates Variable | Type | Inbound | Outbound
---|---|---|---
OLS | OLS+ | OLS | OLS+
Occupancy | Continous | 0.23∗∗ | 0.04 | 0.02 | 0.03
Hour_6 | Binary | -25.11∗∗ | -0.04 | -15.63∗∗ | 1.59
Hour_7 | Binary | -12.92∗∗ | 2.59 | -15.79∗∗ | 2.5
Hour_9 | Binary | 8.53* | 4.71 | 11.54∗∗ | 5.42∗∗
Hour_12 | Binary | 6.12 | 0.08∗ | 11.10∗∗ | 6.81∗∗
Hour_13 | Binary | 1.49 | 2.29 | 2.49 | 5.06∗
Hour_14 | Binary | -2.47 | 3.56 | 7.42∗∗ | 7.58∗∗
Hour_16 | Binary | 21.79∗∗ | 11.49∗∗ | 7.04∗∗ | 3.35
Hour_17 | Binary | 24.85∗∗ | 6.25∗ | 22.47∗∗ | 7.70∗∗
Hour_18 | Binary | 25.10∗∗ | 11.37∗∗ | 37.19∗∗ | 15.54∗∗
Hour_19 | Binary | 3.11 | 4.36∗ | 18.18∗∗ | 9.31∗∗
Hour_20 | Binary | 6.14 | 9.95∗∗ | 11.82∗∗ | 5.71∗∗
Hour_21 | Binary | -8.35 | 1.04 | 9.87∗∗ | 8.38∗∗
TTB_Low | Binary | - | -30.14∗∗ | - | -24.62∗∗
TTB_Mod | Binary | - | 3.37∗∗ | - | 6.05∗∗
TTB_Aav | Binary | - | 40.46∗∗ | - | 35.85∗∗
TTB_High | Binary | - | 87.05∗∗ | - | 76.86∗∗
Intercept | 116.55 | 101.11 | 112.58 | 94.15
R-Square | 0.111 | 0.693 | 0.128 | 0.687

Coefficients denoted with "∗∗" are significant at $p<0.05$ level; "∗" are
significant at $p<0.10$.

##  5 Conclusions

This study provided a detailed implementation and assessment for TranViT, an
integrated framework utilizing a combination of traditional transit data
sources with roadside computer vision to improve real-time transit travel time
prediction. The exploratory assessment of our proposed framework was conducted
for a segment of Massachusetts Avenue in Cambridge, MA, USA, with the results
providing evidence for the potency of this framework. First, we introduce a
workflow for automated data acquisition and labeling using traditional transit
data sources. We then train the ViT component of the framework which was able
to successfully learn image features and contents that best help it deduce the
expected travel time range across the segment of interest, with a validation
accuracy ranging between 80%-85%. Finally, we demonstrate how this prediction
of the travel time range can subsequently be utilized to improve continuous
travel time prediction. We believe this end-to-end, scalable, automated, and
highly efficient approach for integrating transit data sources and roadside
imagery to extract real-time information addresses a fundamental gap impeding
the utilization of computer vision in transit operations, which can have major
implications for improving arrival time predictions and passenger real-time
information.

Unlike existing studies utilizing computer vision for transportation
applications, our framework doesn’t require having certain data in place. On
the contrary, the major contribution of this work is creating a generalized
workflow for acquiring and labeling the data for the computer vision tasks,
which could be extended to other use cases (e.g. focusing on areas of images
containing bus stops for anticipating the dwell time). Utilizing the GTFS
real-time component to initiate transit data and corresponding image sequence
acquisitions results in an extremely computationally-efficient workflow,
running on a single CPU and with no more than 128 MBs of RAM. Based on our
observations from this study, we expect a more demanding data acquisition and
training task for models to be deployed in areas where the scenery changes
noticeably between seasons (like our case in Cambridge, MA). The models we
trained in between rounds of data acquisition did not perform well without
retraining. The subsequent model re-training, however, is faster. This is
attributed to the gigantic number of parameters (86 Million in the base model)
that a ViT employs, which requires substantial training to ensure
generalizability. The output travel time range labels (based on the
percentiles used to create these ranks) can complement any existing travel
time prediction algorithm by adding a complementary real-time attribute
describing the currently observed state of traffic at an area of interest,
supplementing what models traditionally learn from historic AVL observations.
We concluded this study with a proof of concept demonstrating the potential
impact of this additional vision-based input for improving transit travel time
predictions. In practice, we anticipate that a data-driven deployment of
roadside cameras only in locations where high transit delays are observed
would be sufficient in providing reliable travel time predictions without the
need to monitor large segments of the transit network.

One of the limitations of this study is the need to generate augmented images
to satisfy the data-hungry nature of vision-transformed training. Future works
will look into acquiring a larger quantity of image data, alongside
integrating additional data sources that provide information on signal timing
(if the work is conducted in an intersection setting) or adapting the model in
ways that account for the impact of traffic control variations on segment
travel times. Another potential improvement is utilizing the observed speed
and/or acceleration profiles of the transit vehicles as opposed to their
travel time. This could either be accomplished by logging the transit
vehicles’ location coordinates from GTFS-RT during the data acquisition phase,
or by utilizing computer-vision-based trajectory tracking for all vehicles,
which would come at the expense of higher computational load but will allow
for the estimation of the true overall traffic speed and state more
accurately, leading to a better quality labeling for the training data.

##  6 Acknowledgements

The authors would like to thank the MBTA for providing data access which
enabled this study, and the MIT SuperCloud and Lincoln Laboratory
Supercomputing Center for providing high-performance computing resources that
have contributed to the research results reported in this paper.

##  7 Author Contribution Statement

The authors confirm their contribution to the paper as follows: study
conception and design: A.A., J.Z.; data collection: A.A.; analysis and
interpretation of results: A.A.; draft manuscript preparation: A.A., J.Z. Both
authors reviewed the results and approved the final version of the manuscript.
The authors do not have any conflicts of interest to declare.


#  Spectrum structure for eigenvalue problems involving mean curvature
operators in Euclidean and Minkowski spaces ∗

Ruyun Maa Hongliang Gaob Tianlan Chenc
a,b,cDepartment of Mathematics, Northwest Normal University, Lanzhou 730070, P
R China

###### Abstract

In this paper, we are concerned with quasilinear Dirichlet problem

$\left\\{\begin{aligned}
&-\Big{(}\frac{u^{\prime}(x)}{\sqrt{1+\kappa(u^{\prime}(x))^{2}}}\Big{)}^{\prime}=\lambda
u(x),\ \ \ \ \ 0<x<1,\\\ &u(0)=u(1)=0,\\\ \end{aligned}\right.$ $None$

where $\kappa\in(-\infty,0)\cup(0,\infty)$ is a constant. We show that any
nontrivial solution $u$ of (P) has only finite many of simple zeros in
$[0,1]$, all of humps of $u$ are same, and the first hump is symmetric around
the middle point of its domain. We also describe the global structure of the
set of nontrivial solutions of (P).

000E-mail address<EMAIL_ADDRESS>(R. Ma<EMAIL_ADDRESS>(H.Gao),
<EMAIL_ADDRESS>(T. Chen). 000∗Supported by the NSFC (No.11361054,
No.11201378), SRFDP(No.20126203110004), Gansu provincial National Science
Foundation of China (No.1208RJZA258).

Keywords. One-dimensional Euclidean-curvature operator, one-dimensional
Minkowski-curvature operator, eigenvalue, sign-changing solutions.

MR(2000) 35K59, 35K93

##  1 Introduction

This paper studies the spectrum structure of the quasilinear Dirichlet problem

$\left\\{\begin{aligned}
&-\Big{(}\frac{u^{\prime}(x)}{\sqrt{1+\kappa(u^{\prime}(x))^{2}}}\Big{)}^{\prime}=\lambda
u(x),\ \ \ \ \ 0<x<1,\\\ &u(0)=u(1)=0,\\\ \end{aligned}\right.$ $None$

where

${}^{\prime}=D:=\frac{d}{dx},$

and $\kappa\neq 0$ is a constant. This problem establishes a quasilinear
continuum deformation between the linear eigenvalue problem

$\left\\{\begin{aligned} &-u^{\prime\prime}(x)=\lambda u(x),\ \ \ \ \
0<x<1,\\\ &u(0)=u(1)=0,\\\ \end{aligned}\right.$ $None$

and (1.1), which is a quasilinear problem associated to the one-dimensional
Euclidean-curvature operator as $\kappa>0$ and to the one-dimensional
Minkowski-curvature operator as $\kappa<0$, respectively.

The quasilinear problem (1.1) can be equivalently written as

$\left\\{\begin{aligned}
&-\frac{u^{\prime\prime}(x)}{[1+\kappa(u^{\prime}(x))^{2}]^{3/2}}=\lambda
u(x),\ \ \ \ \ 0<x<1,\\\ &u(0)=u(1)=0,\\\ \end{aligned}\right.$ $None$

or

$\left\\{\begin{aligned} &-u^{\prime\prime}(x)=\lambda
u[1+\kappa(u^{\prime}(x))^{2}]^{3/2},\ \ \ \ \ 0<x<1,\\\ &u(0)=u(1)=0.\\\
\end{aligned}\right.$ $None$

Cano-Casanova, López-Gómez and Takimoto [1] used the standard techniques from
bifurcation theory [2-5] to study the existence of positive solutions for
(1.4) with $\kappa>0$. Their main results can be summarized in the following
list:

$\bullet$ Problem (1.4) has a positive solution if and only if

$8B^{2}<\lambda<\pi^{2},\ \ \ \ \ \
B=\int^{1}_{0}\frac{d\theta}{\sqrt{\theta^{-4}-1}}.$ $None$

$\bullet$ The positive solution of (1.4) is unique if it exists. Subsequently,
we denote it by $u_{\lambda}$.

$\bullet$ $u_{\lambda}$ is symmetric around $1/2$ for all $\lambda$ satisfying
(1.5).

$\bullet$ $u_{\lambda}^{\prime}(0)$ is strictly decreasing in
$(8B^{2},\pi^{2})$.

$\bullet$ $u_{\lambda}$ satisfies

$\lim_{\lambda\downarrow 8B^{2}}u^{\prime}_{\lambda}(0)=\infty,\ \ \ \ \ \
\lim_{\lambda\downarrow
8B^{2}}||u_{\lambda}||_{\infty}=\frac{1}{2B\sqrt{\kappa}},$ $None$

and

$\lim_{\lambda\uparrow\pi^{2}}||u_{\lambda}||_{\infty}=\lim_{\lambda\uparrow\pi^{2}}u_{\lambda}(1/2)=0.$
$None$

$\bullet$ The point-wise limit $u_{8B^{2}}:=\lim_{\lambda\downarrow
8B^{2}}u_{\lambda}$ satisfies

$u_{8B^{2}}^{\prime}(0)=-u_{8B^{2}}^{\prime}(1)=\infty.$ $None$

Their results are motivated by the pioneering results in Nakao [6] and have
been extended to the more general case

$\left\\{\begin{aligned}
&-\Big{(}\frac{u^{\prime}(x)}{\sqrt{1+\kappa(u^{\prime}(x))^{2}}}\Big{)}^{\prime}=\lambda
V(x)u(x),\ \ \ \ \ 0<x<1,\\\ &u(0)=u(1)=0,\\\ \end{aligned}\right.$

by Cano-Casanova, López-Gómez and Takimoto [7], where the the weight function
$V(x)\geq 0$ and $V(x)\not\equiv 0$ in $[0,1]$.

It is well-known that (1.2) has a sequence of eigenvalues

$\lambda_{n}=n^{2}\pi^{2},\ \ \ \ n\in\\{1,\,2,\,\cdots\\}.$

For each $n\in\mathbb{N}$, $\lambda_{n}$ is simple, and its eigenfunction
$\varphi_{n}=\sin n\pi x$ has exactly $n-1$ simple zeros in $(0,1)$.

Of course, the natural question is what would happen if we consider the sign-
changing solutions of (1.1) under $\kappa\neq 0$? It is the purpose of this
paper to study the higher eigenvalue case for (1.1). More precisely, we shall
show the following:

Theorem 1.1 Assume that $\kappa>0$. Let

$S^{\nu}_{n}:=\\{u\in C^{1}_{0}[0,1]\,|\,u\,\text{has
exactly}\;n-1\;\text{simple zeros in}\ (0,1),\ \text{and}\ \nu u\ \text{is
positive near}\ 0\\}.$

for $n\in N$ and $\nu\in\\{+,-\\}$. Then

1\. Problem (1.4) has a $S^{\nu}_{n}$-solution if and only if

$8n^{2}B^{2}<\lambda<n^{2}\pi^{2},\ \ \ \ \ \
B=\int^{1}_{0}\frac{d\theta}{\sqrt{\theta^{-4}-1}}.$ $None$

2\. The $S^{\nu}_{n}$-solution of (1.4) is unique if it exists. Subsequently,
we denote it by $u_{\lambda}$.

3\. All of the humps of $u_{\lambda}$ are same, and the first hump is
symmetric around $\frac{1}{2n}$ for all $\lambda$ satisfying (1.9).

4\. $u_{\lambda}^{\prime}(0)$ is strictly decreasing in
$(8n^{2}B^{2},n^{2}\pi^{2})$.

5\. $u_{\lambda}$ satisfies

$\lim_{\lambda\downarrow
8n^{2}B^{2}}||u_{\lambda}||_{\infty}=\frac{1}{2nB\sqrt{\kappa}},$

and

$\lim_{\lambda\uparrow
n^{2}\pi^{2}}||u_{\lambda}||_{\infty}=\lim_{\lambda\uparrow
n^{2}\pi^{2}}u_{\lambda}(\frac{1}{2n})=0.$

6\. The point-wise limit

$u_{8n^{2}B^{2}}:=\lim_{\lambda\downarrow 8n^{2}B^{2}}u_{\lambda}$

provides us with a solution of (1.4) at $\lambda=8n^{2}B^{2}$ which is
singular at $t=\frac{j}{n}$, $j=0,1,\cdots n$, in the sense that

$(-1)^{j}u_{8B^{2}}^{\prime}(\frac{j}{n})=\infty,\ \ \ \ \ j=0,1,\cdots n.$
$None$

Theorem 1.2 Assume that $\kappa<0$. Then

(1) For $n\in N$ and $\nu\in\\{+,-\\}$, (1.4) has a $S^{\nu}_{n}$-solution if
and only if

$n^{2}\pi^{2}<\lambda<\infty.$ $None$

(2) The $S^{\nu}_{n}$-solution of (1.4) is unique if it exists. Subsequently,
we denote it by $u_{\lambda}$.

(3) All of the humps of $u_{\lambda}$ are same, and the first hump is
symmetric around $\frac{1}{2n}$ for all $\lambda$ satisfying (1.11).

(4) $u_{\lambda}^{\prime}(0)$ is strictly increasing in
$(n^{2}\pi^{2},\infty)$.

(5) $u_{\lambda}$ satisfies

$\lim_{\lambda\uparrow\infty}||u_{\lambda}||_{\infty}\leq\frac{1}{2n\sqrt{-\kappa}},$

and

$\lim_{\lambda\downarrow
n^{2}\pi^{2}}||u_{\lambda}||_{\infty}=\lim_{\lambda\downarrow
n^{2}\pi^{2}}u_{\lambda}(\frac{1}{2n})=0.$

(6) The point-wise limit

$u_{\infty}:=\lim_{\lambda\uparrow\infty}u_{\lambda}$

satisfies

$u^{\prime}_{\infty}(\frac{j}{n}):=(-1)^{j}\frac{1}{\sqrt{-\kappa}},\ \ \ \
j\in\\{0,\,1,\,\cdots,n\\}.$

Throughout this paper we use the following notations and conventions. Given a
function $u\in C[a,b]$ it is said that $u\succ 0$ if $u\geq 0$ but
$u\not\equiv 0$. For any $V\in C[a,b]$, we shall denote by
$\sigma[-D^{2}+V;(a,b)]$ the principal eigenvalue of $-D^{2}+V$ under
homogeneous Dirichlet boundary conditions in the interval $(a,b)$. In
particular, $\sigma[-D^{2},(0,1)]=\pi^{2}$.

Let $E:=C^{1}_{0}[0,1]$ with the normal

$||u||_{1}:=\max\\{||u||_{\infty},||u^{\prime}||_{\infty}\\}.$

Then $S^{\nu}_{n}$ is open in $E$ for each $n\in N$ and $\nu\in\\{+,-\\}$.

For the related results on the quasilinear problem (1.1) and its more general
case (including the higher dimensional case), see D. Gilbarg and N. S.
Trudinger [8], S.-Y. Cheng and S.-T. Yau [9], R. Bartnik and L. Simon [10], C.
Bereanu, P. Jebelean and J. Mawhin [11], C. Bereanu, P. Jebelean, P. J. Torres
[12-13], M. F. Bidaut-Véron and A. Ratto [14], I. Coelho, C. Corsato, F.
Obersnel and P. Omari [15], R. López [16], J. Mawhin [17], A. E. Treibergs
[18], A. Azzollini [19], H. Pan and R. Xing [20] and references therein.

The contains of this paper have been distributed as follows. In Section 2, we
shows that any nontrivial solutions of (1.4) belongs to $S^{\nu}_{n}$ for some
$n\in\mathbb{N}$ and $\nu\in\\{+,-\\}$, the $S^{\nu}_{n}$-solutions has same
humps and the first hump is symmetric around $\frac{1}{2n}$, and $u$ is a
$S^{\nu}_{n}$ solution of (1.4) with $\kappa>0$ implies
$\lambda\in(0,n^{2}\pi^{2})$. Section 3 is devoted to show that (1.4) with
$\kappa>0$ has a $S^{\nu}_{n}$-solution if and only if
$8n^{2}B^{2}<\lambda<n^{2}\pi^{2}$, and complete the proof of Theorem 1.1. In
section 4, we state and prove some properties for the nontrivial solutions of
(1.4) with $\kappa<0$. Section 5 is devoted to show that (1.4) with $\kappa<0$
has a $S^{\nu}_{n}$-solution if and only if $n^{2}\pi^{2}<\lambda<\infty$, and
complete the proof of Theorem 1.2.

##  2 The properties of nontrivial solutions of (1.4) with $\kappa>0$

As the linearization of (1.4) at $(\lambda,u)=(\lambda,0)$ is given by (1.2),
by the local bifurcation theorem of Crandall and Rabinowitz [2,3], for each
$n\in\mathbb{N}$, (1.1) possesses a curve of $S^{\nu}_{n}$-solutions emanating
from $(\lambda,u)=(\lambda,0)$ at $\lambda=n^{2}\pi^{2}$. Actually,
$n^{2}\pi^{2}$ is the unique bifurcation value from $u=0$ to
$S^{\nu}_{n}$-solutions of (1.4).

The next result shows that any nontrivial solution $u$ of (1.4) cannot have a
degenerate zero.

Lemma 2.1 Let $u$ be a nontrivial solution of (1.4) with $\kappa\neq 0$ for
some $\lambda\in(0,\infty)$. Then, $u\in S^{\nu}_{n}$ for some
$\nu\in\\{+,-\\}$ and $n\in\mathbb{N}$.

Proof. Suppose on the contrary that

$u(\tau)=u^{\prime}(\tau)=0$

for some $\tau\in[0,1]$. Then $u$ is a solution of the initial value problem

$\left\\{\begin{aligned} &-u^{\prime\prime}(x)=\lambda
u[1+\kappa(u^{\prime}(x))^{2}]^{3/2},\ \ \ \ \ 0<x<1,\\\
&u(\tau)=u^{\prime}(\tau)=0\\\ \end{aligned}\right.$ $None$

which implies that $u(x)\equiv 0$ for $x\in[0,1]$. This is a contradiction.
$\Box$

The next result shows that this bifurcation is sub-critical.

Lemma 2.2 Let us assume $\kappa>0$. Let $u$ be a $S^{\nu}_{n}$-solution of
(1.1) for some $\lambda\in(0,\infty)$. Then $0<\lambda<n^{2}\pi^{2}$.
Moreover,

(1) All the positive bumps of $u$ have the same shape, and any such bump $B$
satisfies

(i) $B$ is symmetric about its mid-point $m_{B}$;

(ii) $u^{\prime}$ is strictly decreasing on $B$, so $B$ contains exactly one
zero of $u^{\prime}$ at $m_{B}$;

(2) All the negative bumps of $u$ have the same shape, and any such bump $D$
satisfies

(i) $D$ is symmetric about its mid-point $m_{D}$;

(ii) $u^{\prime}$ is strictly increasing on $D$, so $m_{D}$ contains exactly
one zero of $u^{\prime}$ at $m_{D}$;

(3) The positive and negative bumps have the same shape.

Proof. Without loss of generality, we assume $u\in S^{+}_{n}$. Let

$b:=u^{\prime}(0).$

Then $b>0$. Notice that $u$ is also a nontrivial solution of the initial value
problem

$\left\\{\begin{aligned}
&-\Big{(}\frac{u^{\prime}(x)}{\sqrt{1+\kappa(u^{\prime}(x))^{2}}}\Big{)}^{\prime}=\lambda
u(x),\ \ \ \ \ 0<x<1,\\\ &u(0)=0,\ \ \ u^{\prime}(0)=b.\\\
\end{aligned}\right.$ $None$

Denote

$v=u^{\prime}.$ $None$

Then (1.3) can be written in the form

$\frac{v^{\prime}}{(1+\kappa v^{2})^{3/2}}=\lambda u$

and, hence

$\frac{vv^{\prime}}{(1+\kappa v^{2})^{3/2}}=\lambda uu^{\prime}$

or, equivalently,

$\frac{d}{dx}\Big{(}-\frac{1}{\kappa}(1+\kappa
v^{2})^{-1/2}+\lambda\frac{u^{2}}{2}\Big{)}=0.$

Therefore, for every $x\in[0,1]$, we have that

$\lambda\frac{\kappa}{2}u^{2}(x)-\frac{1}{\sqrt{1+\kappa
v^{2}(x)}}=-\frac{1}{\sqrt{1+\kappa b^{2}}}.$ $None$

It now follows from (2.4) that if $b=0$ then $u\equiv 0$ on $[0,1]$. Hence, if
$u\not\equiv 0$ then any zeros of $u$ are simple, and $u\neq 0$ at any zero of
$u^{\prime}$.

Let $\tau\in[0,1)$ be a zero of $u$. Suppose that $u>0$ on a maximal interval
$P$ with $\tau$ is as a left end (similar arguments apply to maximal intervals
$N$ on which $u<0$). Then from the equation in (2.2),

$\frac{u^{\prime}(x)}{\sqrt{1+\kappa(u^{\prime}(x))^{2}}}=\frac{u^{\prime}(\tau)}{\sqrt{1+\kappa(u^{\prime}(\tau))^{2}}}-\lambda\int^{x}_{\tau}u(s)ds,\
\ \ \ \ x\in P.\\\ $ $None$

Since the function

$\psi(s):=\frac{s}{\sqrt{1+\kappa s^{2}}}$

is strictly increasing on $(-\infty,\infty)$, it follows from (2.5) that
$u^{\prime}$ is strictly decreasing on $P$. Hence, $P$ contains at most one
zero of $u^{\prime}$.

Now denote that $P=(x_{l},x_{r})$, $m:=(x_{r}+x_{l})/2$. Then, by (2.4),

$u(x_{l})=u(x_{r})=0,\ \ \ \ u^{\prime}(x_{l})=-u^{\prime}(x_{r})>0.$ $None$

By (2.6), it is easy to check that both $u(x)$ and $u(x_{r}+x_{l}-x)$ are
solutions of initial value problem

$\left\\{\begin{aligned}
&-\Big{(}\frac{w^{\prime}(x)}{\sqrt{1+\kappa(w^{\prime}(x))^{2}}}\Big{)}^{\prime}=\lambda
w(x),\ \ \ \ \ x_{l}<x<x_{r},\\\ &w(x_{l})=0,\ \ \
w^{\prime}(x_{l})=u^{\prime}(x_{l}).\\\ \end{aligned}\right.$

Hence, by the above uniqueness result, the solution curves on the intervals
$[x_{l},m]$, $[m,x_{r}]$ are symmetric, and $u^{\prime}(m)=0$. These results
show that any positive (and negative) bumps have the properties (i) and (ii)
described in the theorem.

To prove (3), we take $n\geq 2$. Let

$0=\tau_{0}<\tau_{1}<\cdots<\tau_{n}=1$

be the zeros of $u$ in $[0,1]$. Let us consider the initial value problem

$\left\\{\begin{aligned} &-v^{\prime\prime}(x)=\lambda
u[1+\kappa(v^{\prime}(x))^{2}]^{3/2},\ \ \ \ \ 0<x<1,\\\ &v(0)=0,\ \ \
v^{\prime}(0)=u^{\prime}(0).\\\ \end{aligned}\right.$ $None$

Since the equation in (1.3) is autonomous, it is easy to check that the
function

$v(x):=-u(x-\tau_{1}),\ \ \ \ x\in[\tau_{1},\tau_{2}]$

is also a solution of the equation in (2.7). Now by the uniqueness results for
the initial value problem (2.7), $u\equiv v$ in $[\tau_{0},\tau_{1}]$, and
accordingly $u\equiv v$ in $[0,1]$, which implies the first positive bump and
the first negative bump have the same shape.

To show that $0<\lambda<n^{2}\pi^{2}$, multiplying the equation in (2.1) by
$\sin(n\pi x)$ and integrating in $(0,\frac{1}{n})$, we are lead to

$\displaystyle n^{2}\pi^{2}\int^{1/n}_{0}\sin(n\pi x)u(x)dx$
$\displaystyle=-\int^{1/n}_{0}\sin(n\pi x)u^{\prime\prime}(x)dx$
$\displaystyle=\lambda\int^{1/n}_{0}\sin(n\pi x)u(x)V(x)dx$
$\displaystyle>\lambda\int^{1/n}_{0}\sin(n\pi x)u(x)dx,$

where $V:=[1+(u^{\prime})^{2}]^{3/2}$. Because $V\succ 1$, for as $u=0$ if
$V=1$. Therefore, $\lambda<n^{2}\pi^{2}$. The proof is complete. $\Box$

##  3 Interval of $\lambda$ in which (1.4) with $\kappa>0$ has
$S^{\nu}_{n}$-solutions

Recall that

$B:=\int^{1}_{0}\frac{d\theta}{\sqrt{\theta^{-4}-1}}.$

Lemma 3.1 [1, Theorem 3.2.] The set

$\mathcal{S}_{1}:=\\{(\lambda,||u_{\lambda}||_{\infty})\,|\,(\lambda,u_{\lambda})\
\text{is a symmetric positive solutions of (1.4)}\\}$

is a differentiable monotone curve, which connects
$(8B^{2},\frac{1}{2B\sqrt{\kappa}})$ with $(\pi^{2},0)$. For each
$\lambda\in(8B^{2},\pi^{2})$, the problem (1.4) admits a unique symmetric
positive solution. Moreover, if $u_{\lambda}$ stands for the unique symmetric
positive solution of (1.4), then $u^{\prime}_{\lambda}(0)$ is strictly
decreasing in $(8B^{2},\pi^{2})$ and

$u^{\prime}_{\lambda}(0)=\infty.$

Furthermore, the point-wise limit

$u_{8B^{2}}:=\underset{\lambda\downarrow 8B^{2}}{\lim}\;u_{\lambda}$

provides us with a solution of (1.4) at $8B^{2}$ which is singular at $0$ and
$1$ in the sense that

$u^{\prime}_{8B^{2}}(0)=-u^{\prime}_{8B^{2}}(1)=\infty.$

In this section, we shall extend the above result to the case of nodal
solutions.

Theorem 3.2  For each $n\in N$ and $\nu\in\\{+,-\\}$, the set

$\mathfrak{S}^{\nu}_{n}:=\\{(\lambda,||u_{\lambda}||_{\infty})\,|\,(\lambda,u_{\lambda})\
\text{is a}\ S^{\nu}_{n}\text{-solutions of}\ (1.4)\\}$

is a differentiable monotone curve, which connects
$(8n^{2}B^{2},\frac{1}{2B\sqrt{\kappa}})$ with $(\pi^{2}n^{2},0)$. For each
$\lambda\in(8n^{2}B^{2},n^{2}\pi^{2})$, the problem (1.4) admits a unique
${S}^{\nu}_{n}$-solution. Moreover, if $u_{\lambda}$ stands for the unique
${S}^{\nu}_{n}$-solution of (1.4), then $u^{\prime}_{\lambda}(0)$ is strictly
decreasing in $(8n^{2}B^{2},n^{2}\pi^{2})$ and

$\underset{\lambda\downarrow
8n^{2}B^{2}}{\lim}\,u^{\prime}_{\lambda}(0)=\infty.$

Furthermore, the point-wise limit

$u_{8n^{2}B^{2}}:=\underset{\lambda\downarrow 8n^{2}B^{2}}{\lim}\;u_{\lambda}$

provides us with a solution of (1.4) at $8n^{2}B^{2}$ which is singular at $0$
and $1$ in the sense that

$(-1)^{j}u^{\prime}_{8n^{2}B^{2}}(\frac{j}{n})=\infty,\ \ \ \ \
j\in\\{0,1,\cdots,n\\}.$

Proof. By Lemma 2.2, to study the nodal solutions of (1.1) in
$\mathfrak{S}^{\nu}_{n}$, it is enough to study the positive solutions of

$\left\\{\begin{aligned}
&-\Big{(}\frac{u^{\prime}(x)}{\sqrt{1+\kappa(u^{\prime}(x))^{2}}}\Big{)}^{\prime}=\lambda
u(x),\ \ \ \ \ 0<x<\frac{1}{n},\\\ &u(0)=u(\frac{1}{n})=0.\\\
\end{aligned}\right.$ $None$

The change of variable

$u(x)=v(y),\ \ \ \ y=xn,\ \ \ 0\leq x\leq\frac{1}{n}$ $None$

transforms (3.1) into

$\left\\{\begin{aligned}
&-\Big{(}\frac{v^{\prime}(y)}{\sqrt{1+\tilde{\kappa}(v^{\prime}(y))^{2}}}\Big{)}^{\prime}=\tilde{\lambda}v(y),\
\ \ \ \ 0<y<1,\\\ &v(0)=v(1)=0,\\\ \end{aligned}\right.$ $None$

where we are denoting

$\tilde{\kappa}:=\kappa\,n^{2},\ \ \ \ \ \
\tilde{\lambda}:=\frac{\lambda}{n^{2}}.$ $None$

As (3.3) is of the same type as (1.4), by the analysis already done in Section
2, it becomes apparent that (3.1) possesses a positive symmetric solution
around in $\frac{1}{2n}$ if and only if

$8B^{2}<\tilde{\lambda}<\pi^{2}\ \Leftrightarrow\
8n^{2}B^{2}<\lambda<n^{2}\pi^{2}.$ $None$

Furthermore, we may use Lemma 3.1 to get the desired results. $\Box$

##  4 The properties of nontrivial solutions of (1.4) with $\kappa<0$

As the linearization of (1.4) at $(\lambda,u)=(\lambda,0)$ is given by (1.2),
by the local bifurcation theorem of Crandall and Rabinowitz [2,3], for each
$n\in\mathbb{N}$ and $\nu\in\\{+,-\\}$, (1.4) possesses a curve of
$S^{\nu}_{n}$-solutions emanating from $(\lambda,u)=(\lambda,0)$ at
$\lambda=n^{2}\pi^{2}$. Actually, $n^{2}\pi^{2}$ is the unique bifurcation
value from $u=0$ to $S^{\nu}_{n}$-solutions of (1.4).

The next result shows that this bifurcation is sup-critical in the case
$\kappa<0$.

Lemma 4.1 Let $\kappa<0$. Let $u$ be a $S^{\nu}_{n}$-solution of (1.4) for
some $\lambda\in(0,\infty)$. Then $n^{2}\pi^{2}<\lambda<\infty$. Moreover,

(1) All the positive bumps of $u$ have the same shape, and any such bump $B$
satisfies

(i) $B$ is symmetric about its mid-point $m_{B}$;

(ii) $u^{\prime}$ is strictly decreasing on $B$, so $B$ contains exactly one
zero of $u^{\prime}$ at $m_{B}$;

(2) All the negative bumps of $u$ have the same shape, and any such bump $D$
satisfies

(i) $D$ is symmetric about its mid-point $m_{D}$;

(ii) $u^{\prime}$ is strictly increasing on $D$, so $m_{D}$ contains exactly
one zero of $u^{\prime}$ at $m_{D}$;

(3) The positive and negative bumps have the same shape.

Proof. Using the same method to prove Lemma 2.2, with obvious changes, we may
get the that the humps of $u$ have the same shape.

To show that $n^{2}\pi^{2}<\lambda<\infty$, multiplying the equation in (2.1)
by $\sin(n\pi x)$ and integrating in $(0,\frac{1}{n})$, we are lead to

$\displaystyle n^{2}\pi^{2}\int^{1/n}_{0}\sin(n\pi x)u(x)dx$
$\displaystyle=-\int^{1/n}_{0}\sin(n\pi x)u^{\prime\prime}(x)dx$
$\displaystyle=\lambda\int^{1/n}_{0}\sin(n\pi x)u(x)V(x)dx$
$\displaystyle<\lambda\int^{1/n}_{0}\sin(n\pi x)u(x)dx,$

where $V:=[1+\kappa(u^{\prime})^{2}]^{3/2}$. Because $1\succ V$, for as $u=0$
if $V\equiv 1$. Therefore, $\lambda>n^{2}\pi^{2}$. The proof is complete.
$\Box$

##  5 Interval of $\lambda$ in which (1.4) with $\kappa<0$ has
$S^{\nu}_{n}$-solutions

We only determine the maximum interval of $\lambda$ in which (1.4) with
$\kappa<0$ has $S^{+}_{n}$-solutions since the other case can be treated
similarly.

From Lemma 4.1, $u\in S^{+}_{n}$, the humps of $u$ are same, and the first
hump is symmetric around $\frac{1}{2n}$, and

$u^{\prime}(x)>0\ \ \text{if}\ x\in[0,\frac{1}{2n}),\ \ \ \
u^{\prime}(\frac{1}{2n})=0,\ \ \ \ u^{\prime}(x)<0\ \text{if}\
x\in(\frac{1}{2n},\frac{1}{n}].$ $None$

So we only need to study the maximum interval of $\lambda$ in which (1.4) with
$\kappa<0$ has symmetric positive solutions in $(0,\frac{1}{n})$.

First, we consider the special case that $u\in S^{+}_{1}$ and establish the
following

Theorem 5.1 The set

$\mathfrak{S}_{1}^{+}:=\\{(\lambda,||u_{\lambda}||_{\infty})\,|\,(\lambda,u_{\lambda})\
\text{is a}\ S^{+}_{1}\text{-solutions of}\ (1.4)\\}$

consists of a differentiable monotone curve in
$[\pi^{2},\infty)\times[0,\frac{1}{2\sqrt{-\kappa}})$. For each
$\lambda\in(\pi^{2},\infty)$, the problem (1.4) admits a unique
$S^{+}_{1}$-solution. Moreover, if $u_{\lambda}$ stands for the unique
$S^{+}_{1}$-solution of (1.4), then $u^{\prime}_{\lambda}(0)$ is strictly
increasing in $(\pi^{2},\infty)$. Furthermore, the point-wise limit

$u_{\infty}(t):=\underset{\lambda\uparrow\infty}{\lim}\;u_{\lambda}(t),$

which satisfies

$u^{\prime}_{\infty}(0)=-u^{\prime}_{\infty}(1)=\frac{1}{\sqrt{-\kappa}}.$

To prove Theorem 5.1, we need some preliminaries.

Denote

$u_{0}=u(\frac{1}{2}),\ \ \ \ \ v_{0}=u^{\prime}(0),\ \ \ \ v=u^{\prime}.$

Thanks to (5.1) with $n=1$, (1.5) can be written in the form

$-\frac{v^{\prime}}{(1+\kappa v^{2})^{3/2}}=\lambda u.$

and hence

$-\frac{vv^{\prime}}{(1+\kappa v^{2})^{3/2}}=\lambda uu^{\prime},$

or, equivalently,

$\frac{d}{dx}\Big{(}-\frac{1}{\kappa}(1+\kappa
v^{2})^{-1/2}+\lambda\frac{u^{2}}{2}\Big{)}=0.$

Therefore, for every $x\in[0,1]$, we have that

$\lambda\frac{\kappa}{2}u^{2}(x)-\frac{1}{\sqrt{1+\kappa
v^{2}(x)}}=-\frac{1}{\sqrt{1+\kappa
v_{0}^{2}}}=\lambda\frac{\kappa}{2}u_{0}^{2}-1,$ $None$
$v(x)=u^{\prime}(x)=\sqrt{\frac{1}{-\kappa}}\sqrt{1-\big{\\{}1-\lambda\frac{\kappa}{2}(u_{0}^{2}-u^{2}(x))\big{\\}}^{-2}}\
\ \ \text{for all}\ x\in[0,\frac{1}{2}],$ $None$

and

$v(x)=u^{\prime}(x)=-\sqrt{\frac{1}{-\kappa}}\sqrt{1-\big{\\{}1-\lambda\frac{\kappa}{2}(u_{0}^{2}-u^{2}(x))\big{\\}}^{-2}}\
\ \ \text{for all}\ x\in[\frac{1}{2},1].$ $None$

Thus, necessarily

$\displaystyle\frac{1}{2}$
$\displaystyle=\sqrt{-\kappa}\int^{1/2}_{0}\frac{u^{\prime}(x)}{\sqrt{1-\big{\\{}1-\lambda\frac{\kappa}{2}(u_{0}^{2}-u^{2}(x))\big{\\}}^{-2}}}dx$
$\displaystyle=\sqrt{-\kappa}\int^{u_{0}}_{0}\frac{1}{\sqrt{1-\big{\\{}1-\lambda\frac{\kappa}{2}(u_{0}^{2}-u^{2})\big{\\}}^{-2}}}du$
$\displaystyle=\sqrt{-\kappa}\int^{1}_{0}\frac{u_{0}}{\sqrt{1-\big{\\{}1-\lambda\frac{\kappa}{2}u_{0}^{2}(1-\theta^{2})\big{\\}}^{-2}}}d\theta$

where $\theta:=\frac{u}{u_{0}}.$

Equivalently,

$\frac{1}{2}=J(\lambda,u_{0}),$ $None$

where $J:D(J)\to\mathbb{R}_{+}:=(0,\infty)$ is the function defined by

$J(\lambda,\xi)=\sqrt{-\kappa}\int^{1}_{0}\frac{\xi}{\sqrt{1-\big{\\{}1-\lambda\frac{\kappa}{2}\xi^{2}(1-\theta^{2})\big{\\}}^{-2}}}d\theta$
$None$

for all $(\lambda,\xi)\in D(J)$, where

$D(J)=\Big{\\{}(\lambda,\xi)\,|\,(\lambda,\xi)\in\mathbb{R}_{+}\times\mathbb{R}_{+}\Big{\\}}.$

Owing to (5.3), it can be easily seen that the symmetric positive solutions of
(1.4) are in one-to-one correspondence with the pairs $(\lambda,u_{0})\in
D(J)$ satisfying (5.5).

The next result provides us with some important monotonicity properties of the
function $J(\lambda,\xi)$.

Proposition 5.2 (i) For every $\xi>0$, the function $\lambda\mapsto
J(\lambda,\xi)$ is decreasing and

$\lim_{\lambda\downarrow 0}J(\lambda,\xi)=\infty.$ $None$

(ii) For every $\lambda>0$, the function $\varphi(\xi):=J(\lambda,\xi)$
satisfies $\varphi^{\prime}(\xi)>0$, and, hence, it is increasing.

Proof. The proof of (i) is straightforward and, hence, we omit its details.
The proof of (ii) is more delicate. From (5.6), differentiating with respect
to $\xi$ and rearranging terms, we find that

$\displaystyle\varphi^{\prime}(\xi)$
$\displaystyle=\sqrt{-k}\int_{0}^{1}\\{1-[1-\lambda\frac{k}{2}\xi^{2}(1-\theta^{2})]^{-2}\\}^{-\frac{3}{2}}[1-\lambda\frac{k}{2}\xi^{2}(1-\theta^{2})]^{-3}\lambda
k\xi^{2}(1-\theta^{2})d\theta$
$\displaystyle+\sqrt{-k}\int_{0}^{1}\\{1-[1-\lambda\frac{k}{2}\xi^{2}(1-\theta^{2})]^{-2}\\}^{-\frac{1}{2}}d\theta$
$\displaystyle=\sqrt{-k}\int_{0}^{1}\\{1-[1-\lambda\frac{k}{2}\xi^{2}(1-\theta^{2})]^{-2}\\}^{-\frac{3}{2}}\\{[1-\lambda\frac{k}{2}\xi^{2}(1-\theta^{2})]^{-3}\lambda
k\xi^{2}(1-\theta^{2})$ $\displaystyle\ \ \ \
+1-[1-\lambda\frac{k}{2}\xi^{2}(1-\theta^{2})]^{-2}\\}d\theta.$

Notice that $k<0$ and set

$\alpha(\theta):=[1-\lambda\frac{k}{2}\xi^{2}(1-\theta^{2})]^{-1},~{}~{}0\leq\theta\leq
1.$ $None$

Then

$0<\alpha(\theta)\leq 1~{}~{}~{}\text{and}~{}~{}~{}1-\alpha^{2}(\theta)\geq
0~{}~{}~{}\forall\theta\in[0,1]$

and

$\varphi^{\prime}(\xi)=\sqrt{-k}\int_{0}^{1}(1-\alpha^{2})^{\frac{3}{2}}[\alpha^{3}\lambda
k\xi^{2}(1-\theta^{2})+(1-\alpha^{2})]d\theta.$

On the other hand, according to (5.8), it becomes apparent that

$\alpha\lambda k\xi^{2}(1-\theta^{2})=2(\alpha-1)$

and, hence,

$\displaystyle\varphi^{\prime}(\xi)$
$\displaystyle=\sqrt{-k}\int_{0}^{1}(1-\alpha^{2})^{\frac{3}{2}}[2\alpha^{2}(\alpha-1)+(1-\alpha^{2})]d\theta$
$\displaystyle=2\sqrt{-k}\int_{0}^{1}(1-\alpha^{2})^{\frac{3}{2}}(1-\alpha)^{2}(\alpha+\frac{1}{2})d\theta>0.$

The proof is complete. $\Box$

Proof of Theorem 5.1. By the fact

$-\kappa v_{0}^{2}<1.$

This together with the symmetric property of $u$ imply that

$\xi<\frac{1}{2\sqrt{-\kappa}}.$

From Proposition 5.2, there exists an interval
$(0,\rho)\subseteq\big{(}0,\frac{1}{2\sqrt{-\kappa}}\big{)}$, such that for
every $\xi\in(0,\rho)$, there exists a unique $\lambda(\xi)>0$ satisfying

$J\big{(}\lambda(\xi),\xi\big{)}=\frac{1}{2},$

i.e.

$\frac{1}{2}=\frac{1}{\sqrt{\lambda(\xi)}}\int^{1}_{0}\frac{1-\lambda\frac{\kappa}{2}\xi^{2}(1-\theta^{2})}{\sqrt{1-\theta^{2}-\lambda\frac{\kappa}{4}\xi^{2}(1-\theta^{2})^{2}}}d\theta$

and, therefore, since $\pi^{2}<\lambda(\xi)$, letting $\xi\to 0$, we get

$\frac{1}{2}=\frac{1}{\sqrt{\lambda(0)}}\int^{1}_{0}\frac{d\theta}{\sqrt{1-\theta^{2}}}=\frac{1}{\sqrt{\lambda(0)}}\frac{\pi}{2}.$

Hence, $\lambda(0):=\lim\limits_{\xi\rightarrow 0}\lambda(\xi)=\pi^{2}.$

By the monotonicity and continuity of $J$, $\xi\to\lambda(\xi)$ must be
continuous and strictly increasing. The differentiability of
$\mathfrak{S}^{+}_{1}$ is a direct consequence from Proposition 5.2(ii) and
the implicit function theorem.

The function $u^{\prime}_{\lambda}(0)$ is strictly increasing for
$\lambda\in(\pi^{2},\infty)$ is an immediate consequence of the fact
$\xi\to\lambda(\xi)$ is strictly increasing in $(0,\rho)$ and the relation

$\lambda\frac{\kappa}{2}u^{2}(x)-\frac{1}{\sqrt{1+\kappa
v^{2}(x)}}=-\frac{1}{\sqrt{1+\kappa
v_{0}^{2}}}=\lambda\frac{\kappa}{2}\xi^{2}-1.$

On the other hand, let

$\underset{\lambda\uparrow\infty}{\lim}\;u_{\lambda}(\frac{1}{2})=u_{\infty}(\frac{1}{2})=:\xi^{*}.$

Then $\xi^{*}>0$. Combining this with the fact $-\frac{1}{\sqrt{1+\kappa
v_{0}^{2}}}=\lambda\frac{\kappa}{2}\xi^{2}-1$ and letting $\lambda\to\infty$,
we get

$\underset{\lambda\uparrow\infty}{\lim}u^{\prime}_{\lambda}(0)=u^{\prime}_{\infty}(0)=\frac{1}{\sqrt{-\kappa}},$

and this completes the proof. $\Box$

Now, we are in the position to prove Theorem 1.2.

Proof of Theorem 1.2. By Lemma 4.1, to study the set of
$S^{\nu}_{n}$-solutions of (1.4), it is enough to study the set of positive
solutions of

$\left\\{\begin{aligned}
&-\Big{(}\frac{u^{\prime}(x)}{\sqrt{1+\kappa(u^{\prime}(x))^{2}}}\Big{)}^{\prime}=\lambda
u(x),\ \ \ \ \ 0<x<\frac{1}{n},\\\ &u(0)=u(\frac{1}{n})=0.\\\
\end{aligned}\right.$ $None$

The change of variable

$u(x)=v(y),\ \ \ \ y=xn,\ \ \ 0\leq x\leq\frac{1}{n},$ $None$

transforms (5.9) into

$\left\\{\begin{aligned}
&-\Big{(}\frac{v^{\prime}(y)}{\sqrt{1+\tilde{\kappa}(v^{\prime}(y))^{2}}}\Big{)}^{\prime}=\tilde{\lambda}v(y),\
\ \ \ \ 0<y<1,\\\ &v(0)=v(1)=0,\\\ \end{aligned}\right.$ $None$

where we are denoting

$\tilde{\kappa}:=\kappa\,n^{2},\ \ \ \ \ \ \tilde{\lambda}:=\lambda/n^{2}.$
$None$

Notice that all of the conclusions of Theorem 5.1 for (5.11) are still valid
for (5.9) via the transformation (5.10). Thus, Theorem 1.2 can be deduced from
(5.12) and Theorem 5.1. $\Box$

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