12,622 research outputs found
On the fractional damped oscillators and fractional forced oscillators
In this paper, we use the fractional calculus to discuss the fractional
mechanics, where the time derivative is replaced with the fractional derivative
of order . We deal with the motion of a body in a resisting medium where
the retarding force is assumed to be proportional to the fractional velocity
which is obtained by acting the fractional derivative on the position. The
fractional harmonic oscillator problem, fractional damped oscillator problem
and fractional forced oscillator problem are also studied
Center Manifolds of Differential Equations in Banach Spaces
The center manifold is useful for describing the long-term behavior of a
system of differential equations. In this work, we consider an autonomous
differential equation in a Banach space that has the exponential trichotomy
property in the linear terms and Lipschitz continuity in the nonlinear terms.
Using the spectral gap condition we prove the existence and uniqueness of the
center manifold. Moreover, we prove the regularity of the manifold with a few
additional assumptions on the nonlinear term. We approach the problem using the
well-known Lyapunov-Perron method, which relies on the Banach fixed-point
theorem. The proofs can be generalized to a non-autonomous system
Tight Bounds for Hashing Block Sources
It is known that if a 2-universal hash function is applied to elements of
a {\em block source} , where each item has enough
min-entropy conditioned on the previous items, then the output distribution
will be ``close'' to the uniform distribution. We
provide improved bounds on how much min-entropy per item is required for this
to hold, both when we ask that the output be close to uniform in statistical
distance and when we only ask that it be statistically close to a distribution
with small collision probability. In both cases, we reduce the dependence of
the min-entropy on the number of items from in previous work to
, which we show to be optimal. This leads to corresponding improvements
to the recent results of Mitzenmacher and Vadhan (SODA `08) on the analysis of
hashing-based algorithms and data structures when the data items come from a
block source.Comment: An extended abstract of this paper will appear in RANDOM0
Joint Channel Estimation and Data Detection for Multihop OFDM Relaying System under Unknown Channel Orders and Doppler Frequencies
In this paper, channel estimation and data detection for multihop relaying
orthogonal frequency division multiplexing (OFDM) system is investigated under
time-varying channel. Different from previous works, which highly depend on the
statistical information of the doubly-selective channel (DSC) and noise to
deliver accurate channel estimation and data detection results, we focus on
more practical scenarios with unknown channel orders and Doppler frequencies.
Firstly, we integrate the multilink, multihop channel matrices into one
composite channel matrix. Then, we formulate the unknown channel using
generalized complex exponential basis expansion model (GCE-BEM) with a large
oversampling factor to introduce channel sparsity on delay-Doppler domain. To
enable the identification of nonzero entries, sparsity enhancing Gaussian
distributions with Gamma hyperpriors are adopted. An iterative algorithm is
developed under variational inference (VI) framework. The proposed algorithm
iteratively estimate the channel, recover the unknown data using Viterbi
algorithm and learn the channel and noise statistical information, using only
limited number of pilot subcarrier in one OFDM symbol. Simulation results show
that, without any statistical information, the performance of the proposed
algorithm is very close to that of the optimal channel estimation and data
detection algorithm, which requires specific information on system structure,
channel tap positions, channel lengths, Doppler shifts as well as noise powers
Study on the mechanical system related to Hahn's discrete time derivative
In this paper, we use the quantum variational calculus related to Hahn's
discrete time derivative construct the deformed version for the classical
mechanics related to the Hahn's calculus. We deal with the deformed dynamics
such as the motion with constant velocity and the motion with constant
acceleration. Moreover, we extend our work to the motion of a body in a
resisting medium by using the new identity for an infinite ()-series,
where the retarding force is assumed to be proportional to the deformed average
velocity
Persistence Curves: A canonical framework for summarizing persistence diagrams
Persistence diagrams are one of the main tools in the field of Topological
Data Analysis (TDA). They contain fruitful information about the shape of data.
The use of machine learning algorithms on the space of persistence diagrams
proves to be challenging as the space is complicated. For that reason,
transforming these diagrams in a way that is compatible with machine learning
is an important topic currently researched in TDA. In this paper, our main
contribution consists of three components. First, we develop a general and
unifying framework of vectorizing diagrams that we call the Persistence Curves
(PCs), and show that several well-known summaries, such as Persistence
Landscapes, fall under the PC framework. Second, we propose several new
summaries based on PC framework and provide a theoretical foundation for their
stability analysis. Finally, we apply proposed PCs to two
applications---texture classification and determining the parameters of a
discrete dynamical system; their performances are competitive with other TDA
methods
A New Condition for Blow-up Solutions to Discrete Semilinear Heat Equations on Networks
The purpose of this paper is to introduce a new condition for some with
, where is
the first eigenvalue of discrete Laplacian , with which we
obtain blow-up solutions to discrete semilinear heat equations
\begin{equation*} \begin{cases}
u_{t}\left(x,t\right)=\Delta_{\omega}u\left(x,t\right)+f(u(x,t)), &
\left(x,t\right)\in S\times\left(0,+\infty\right),\\ u\left(x,t\right)=0, &
\left(x,t\right)\in\partial S\times\left[0,+\infty\right),\\
u\left(x,0\right)=u_{0}\geq0(nontrivial), & x\in\overline{S} \end{cases}
\end{equation*} on a discrete network . In fact, it will be seen that the
condition (C) improves the conditions known so far.Comment: 19 page
Sample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem
We generalize the PAC (probably approximately correct) learning model to the
quantum world by generalizing the concepts from classical functions to quantum
processes, defining the problem of \emph{PAC learning quantum process}, and
study its sample complexity. In the problem of PAC learning quantum process, we
want to learn an -approximate of an unknown quantum process
from a known finite concept class with probability using samples
, where are
computational basis states sampled from an unknown distribution and
are the (possibly mixed) quantum states outputted
by . The special case of PAC-learning quantum process under constant input
reduces to a natural problem which we named as approximate state
discrimination, where we are given copies of an unknown quantum state
from an known finite set , and we want to learn with probability
an -approximate of with as few copies of as possible. We
show that the problem of PAC learning quantum process can be solved with
samples when
the outputs are pure states and samples if the outputs can be mixed. Some
implications of our results are that we can PAC-learn a polynomial sized
quantum circuit in polynomial samples and approximate state discrimination can
be solved in polynomial samples even when concept class size is
exponential in the number of qubits, an exponentially improvement over a full
state tomography
Interactive Leakage Chain Rule for Quantum Min-entropy
The leakage chain rule for quantum min-entropy quantifies the change of
min-entropy when one party gets additional leakage about the information
source. Herein we provide an interactive version that quantifies the change of
min-entropy between two parties, who share an initial classical-quantum state
and are allowed to run a two-party protocol. As an application, we prove new
versions of lower bounds on the complexity of quantum communication of
classical information.Comment: A few terminology mistakes were corrected in this versio
Two supersolid phases in hard-core extended Bose-Hubbard model
The effect of the next-nearest-neighbor (nnn) tunneling on the hard-core
extended Bose-Hubbard model on square lattices is investigated. By means of the
cluster mean-field theory, the ground-state phase diagrams are determined. When
a modest nnn tunneling is introduced, depending on its sign, two distinct
supersolid states with checkerboard crystal structures are found away from
half-filing. The characters of various phase transitions out of these two
supersolid states are discussed. In particular, for the case with kinetic
frustration, the existence of a half supersolid phase possessing both solid and
unconventional superfluid orders is established. Our work hence sheds light on
the search of this interesting supersolid phase in real ultracold lattice gases
with frustrated tunnelings.Comment: 8 pages, 6 figures. Published versio
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