32,052 research outputs found
A useful variant of the Davis--Kahan theorem for statisticians
The Davis--Kahan theorem is used in the analysis of many statistical
procedures to bound the distance between subspaces spanned by population
eigenvectors and their sample versions. It relies on an eigenvalue separation
condition between certain relevant population and sample eigenvalues. We
present a variant of this result that depends only on a population eigenvalue
separation condition, making it more natural and convenient for direct
application in statistical contexts, and improving the bounds in some cases. We
also provide an extension to situations where the matrices under study may be
asymmetric or even non-square, and where interest is in the distance between
subspaces spanned by corresponding singular vectors.Comment: 12 page
Data-Driven Multi-step Demand Prediction for Ride-Hailing Services Using Convolutional Neural Network
Ride-hailing services are growing rapidly and becoming one of the most
disruptive technologies in the transportation realm. Accurate prediction of
ride-hailing trip demand not only enables cities to better understand people's
activity patterns, but also helps ride-hailing companies and drivers make
informed decisions to reduce deadheading vehicle miles traveled, traffic
congestion, and energy consumption. In this study, a convolutional neural
network (CNN)-based deep learning model is proposed for multi-step ride-hailing
demand prediction using the trip request data in Chengdu, China, offered by
DiDi Chuxing. The CNN model is capable of accurately predicting the
ride-hailing pick-up demand at each 1-km by 1-km zone in the city of Chengdu
for every 10 minutes. Compared with another deep learning model based on long
short-term memory, the CNN model is 30% faster for the training and predicting
process. The proposed model can also be easily extended to make multi-step
predictions, which would benefit the on-demand shared autonomous vehicles
applications and fleet operators in terms of supply-demand rebalancing. The
prediction error attenuation analysis shows that the accuracy stays acceptable
as the model predicts more steps
Dissipation Effects in Hybrid Systems
The dissipation effect in a hybrid system is studied in this Letter. The
hybrid system is a compound of a classical magnetic particle and a quantum
single spin. Two cases are considered. In the first case, we investigate the
effect of the dissipative quantum subsystem on the motion of its classical
partner. Whereas in the second case we show how the dynamics of the quantum
single spin are affected by the dissipation of the classical particle.
Extension to general dissipative hybrid systems is discussed.Comment: 4+ pages, 4 figure
Metallic Icosahedron Phase of Sodium at Terapascal Pressures
Alkali metals exhibit unexpected structures and electronic behavior at high
pressures. Compression of metallic sodium (Na) to 200 GPa leads to the
stability of a wide-band-gap insulator with the double hexagonal hP4 structure.
Post-hP4 structures remain unexplored, but they are important for addressing
the question of the pressure at which Na reverts to a metal. Here we report the
reentrant metallicity of Na at the very high pressure of 15.5 terapascal (TPa),
predicted using first-principles structure searching simulations. Na is
therefore insulating over the large pressure range of 0.2-15.5 TPa. Unusually,
Na adopts an oP8 structure at pressures of 117-125 GPa, and the same oP8
structure at 1.75-15.5 TPa. Metallization of Na occurs on formation of a stable
and striking body-centered cubic cI24 electride structure consisting of Na12
icosahedra, each housing at its center about one electron which is not
associated with any Na ions.Comment: 5 pages, 4 figures, PRL (2015
Geometric phase in dephasing systems
Beyond the quantum Markov approximation, we calculate the geometric phase of
a two-level system driven by a quantized magnetic field subject to phase
dephasing. The phase reduces to the standard geometric phase in the weak
coupling limit and it involves the phase information of the environment in
general. In contrast with the geometric phase in dissipative systems, the
geometric phase acquired by the system can be observed on a long time scale. We
also show that with the system decohering to its pointer states, the geometric
phase factor tends to a sum over the phase factors pertaining to the pointer
states.Comment: 4 page
Geometric phases induced in auxiliary qubits by many-body systems near its critical points
The geometric phase induced in an auxiliary qubit by a many-body system is
calculated and discussed. Two kinds of coupling between the auxiliary qubit and
the many-body system are considered, which lead to dephasing and dissipation in
the qubit, respectively. As an example, we consider the XY spin-chain
dephasingly couple to a qubit, the geometric phase induced in the qubit is
presented and discussed. The results show that the geometric phase might be
used to signal the critical points of the many-body system, and it tends to
zero with the parameters of the many-body system going away from the critical
points
Impact of intrinsic biophysical diversity on the activity of spiking neurons
We study the effect of intrinsic heterogeneity on the activity of a
population of leaky integrate-and-fire neurons. By rescaling the dynamical
equation, we derive mathematical relations between multiple neuronal parameters
and a fluctuating input noise. To this end, common input to heterogeneous
neurons is conceived as an identical noise with neuron-specific mean and
variance. As a consequence, the neuronal output rates can differ considerably,
and their relative spike timing becomes desynchronized. This theory can
quantitatively explain some recent experimental findings.Comment: 4 pages, 5 figure
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