335 research outputs found
Design of generalized fractional order gradient descent method
This paper focuses on the convergence problem of the emerging fractional
order gradient descent method, and proposes three solutions to overcome the
problem. In fact, the general fractional gradient method cannot converge to the
real extreme point of the target function, which critically hampers the
application of this method. Because of the long memory characteristics of
fractional derivative, fixed memory principle is a prior choice. Apart from the
truncation of memory length, two new methods are developed to reach the
convergence. The one is the truncation of the infinite series, and the other is
the modification of the constant fractional order. Finally, six illustrative
examples are performed to illustrate the effectiveness and practicability of
proposed methods.Comment: 8 pages, 16 figure
Time-domain response of nabla discrete fractional order systems
This paper investigates the time--domain response of nabla discrete
fractional order systems by exploring several useful properties of the nabla
discrete Laplace transform and the discrete Mittag--Leffler function. In
particular, we establish two fundamental properties of a nabla discrete
fractional order system with nonzero initial instant: i) the existence and
uniqueness of the system time--domain response; and ii) the dynamic behavior of
the zero input response. Finally, one numerical example is provided to show the
validity of the theoretical results.Comment: 13 pages, 6 figure
Description and Realization for a Class of Irrational Transfer Functions
This paper proposes an exact description scheme which is an extension to the
well-established frequency distributed model method for a class of irrational
transfer functions. The method relaxes the constraints on the zero initial
instant by introducing the generalized Laplace transform, which provides a wide
range of applicability. With the discretization of continuous frequency band,
the infinite dimensional equivalent model is approximated by a finite
dimensional one. Finally, a fair comparison to the well-known Charef method is
presented, demonstrating its added value with respect to the state of art.Comment: 9 pages, 9 figure
Some fundamental properties on the sampling free nabla Laplace transform
Discrete fractional order systems have attracted more and more attention in
recent years. Nabla Laplace transform is an important tool to deal with the
problem of nabla discrete fractional order systems, but there is still much
room for its development. In this paper, 14 lemmas are listed to conclude the
existing properties and 14 theorems are developed to describe the innovative
features. On one hand, these properties make the N-transform more effective and
efficient. On the other hand, they enrich the discrete fractional order system
theor
On the Leibniz rule and Laplace transform for fractional derivatives
Taylor series is a useful mathematical tool when describing and constructing
a function. With the series representation, some properties of fractional
calculus can be revealed clearly. This paper investigates two typical
applications: Lebiniz rule and Laplace transform. It is analytically shown that
the commonly used Leibniz rule cannot be applied for Caputo derivative.
Similarly, the well-known Laplace transform of Riemann-Liouville derivative is
doubtful for n-th continuously differentiable function. By the aid of this
series representation, the exact formula of Caputo Leibniz rule and the
explanation of Riemann-Liouville Laplace transform are presented. Finally,
three illustrative examples are revisited to confirm the obtained results
Non-Hermitian skin effect in a single trapped ion
Non-Hermitian skin effect (NHSE) describes the exponential localization of
all eigenstates toward boundaries in non-Hermitian systems, and has attracted
intense research interest of late. Here we theoretically propose a scheme in
which the NHSE significantly impacts the external motion of a single trapped
ion through complex spin-motion dynamics. On the one hand, we show the
competition between the NHSE and the coherent Bloch dynamics. On the other
hand, since the NHSE manifests as a non-reciprocal flow in occupied phonon
modes, we demonstrate that such dynamics can have potential applications in
cooling and sensing. Our proposal can be readily implemented using existing
experimental techniques, and offers a scalable (in terms of the available ions
and phonon modes) simulation platform for relevant non-Hermitian physics.Comment: 9 pages, 8 figure
Analytical calculation of the inverse nabla Laplace transform
The inversion of nabla Laplace transform, corresponding to a causal sequence,
is considered. Two classical methods, i.e., residual calculation method and
partial fraction method are developed to perform the inverse nabla Laplace
transform. For the first method, two alternative formulae are proposed when
adopting the poles inside or outside of the contour, respectively. For the
second method, a table on the transform pairs of those popular functions is
carefully established. Besides illustrating the effectiveness of the developed
methods with two illustrative examples, the applicability are further discussed
in the fractional order case
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